43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Physics [16 ed.] 8194767733, 9788194767732

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EBD_8341

EBD_8341

EBD_8341

EBD_8341

P-1

Physical World, Units and Measurements

1

Topic-1 : Unit of Physical Quantities 1 ҈


MCQs with One Correct Answer


 矄�泐
ᨨ

鼀�
�
和ᔼ
盐 �
�矄
҈


├�
鼀

ᔼ

ᆠ�
盐 �矄
�
犨 �ᆠ
 
盐 �
ᨨ
 �
�矄

�
ᆠ鼀
   
盐 �
ᨨ
鼀矄矄
�矄
�

�

 
盐鼀�ᆠ
盐
ᨨ
 矄�泐 ᨨ
 
鼀�
�矄
᲼㝐�
҈





҈黨
ᔼ㡰桌 㡰







㡰

ᆠ㡰




㡰 ҈


㡰
҈


꾀
鼀
矄鼀
ᆠ泐� �

ᆠ盐 
ᨨ �
㜠 �矄
犨�
 矄�泐 
�矄
矄盐ᆠ
ᨨ

鼀� �ᆠ
�

ᨨ �
㜠 �矄㡰

 ᆠ�ᆠ
�
쟼 泐
ᨨ
ᨨ �
Ӹ㜠 �矄
�矄
�ᆠ泐 ᨨᨨ�ᨨ 
ᨨ

矄
犨
矄



ᆠᨨ 矄 
ᨨ ᨨᨨ�ᨨ �泐
矄
和ᔼ
盐 �

ᔴ㡰

㡰

ᆠ㡰 ᨨ盐⏌鼀㡰
7



q
B




F

ᨨᆠ


GM e


G

盐 �矄 

꾀㡰 GMeMs , G
 盐 �矄
���ᨨ  ᆠᨨ 矄 


�


㡰 ᨨ 鼀㜠矄ᆠᨨ 

ᔴᨨ盐鼀
ᔼ 꾀㡰 ᔴᆠ� ᆠ 㡰 ᔼ 盐ᆠ ᆠ ᔴ㡰 㝐� �ᆠ
ᔼ 盐ᆠ�ᨨ

Me

鼀矄矄
ᨨ
 

Ms

鼀矄矄
ᨨ
 
和盐





ᔴᨨ盐鼀
ᔼᔼ

矄㡰  㡰
ᨨ㡰

├�㡰҈

�㡰 ��㡰 ���㡰 �㡰 㡰

ᔴᨨ盐鼀
ᔼᔼ ᨨ 鼀㜠矄ᆠᨨ  ᆠᨨ盐ᨨ鼀
ᨨ盐҈ ᆠᨨ盐ᨨ鼀
ᨨ㡰҈

犨ᨨ
鼀㜠鼀㡰҈ ᨨ㜠矄ᆠᨨ 
鼀㡰҈ ᲼҈黨黨

㜠

㝐├矄桌

10 Subjective Problems

㡰 ᨨ㡰
ᆠᨨ盐ᨨ鼀㡰鼀㡰 ᲼


䰘桌

㡰 鼀㡰

矄ᆠᨨ 㡰


Re

�盐矄
ᨨ
 


Match the Following ᔴᨨ盐鼀
ᔼ

矄ᆠᨨ 㡰


Me

鼀矄矄
ᨨ
 



㡰 


F


⼄㡰 ├�ᨨ�鼀㡰
鼀㡰

q

ᆠ �
B

鼀� �ᆠ
�

᲼㝐�
졠 � 
꾀�

�



҈桌 As m m
m
㡰 ᆠ㡰 㡰 As As m A 
SI
盐 �
ᨨ
� 盐ᆠ ᆠ

 
 泐
ᆠ

犨�
矄 ᲼҈黨黨

㜠


㝐├矄桌 㡰 犨⏌鼀 㡰 ᨨ㜠矄ᆠ⏌鼀

RT

R

盐 �矄 M �矄
ᆠᨨ 矄 
T

矄ᨨ盐
鼀盐
M

鼀ᨨ
鼀矄矄





෌�
 
㝐⧤和
盐 �矄
ᨨ
ᆠ
ᨨ
 
ᨨᨨ犨� �

⼄盐 ��矄
꾀㡰 Ӹᨨ盐 �矄
鼀ᨨ盐盐矄

᲼҈黨

桌

㡰 㝐� �ᆠ
ᔼ 盐ᆠ�ᨨ

ᔴ㡰 ᨨ犨
ᨨ

 矄

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EBD_8341 P-2

Physics

Topic-2 : Dimensions of Physical Quantities 1

MCQs with One Correct Answer 黨


҈













�






䰘


㡰



҈

E ҈
 
z =
 B CR m
e
 � 
犨 
C㜠ᆠᆠ� ᆠ

R㜠 矄�矄 ᆠ

l㜠 �
E㜠쟼ᆠ�ᆠ
�

B㜠鼀� �ᆠ
�
 
e

m

㜠

矄ᆠ 鼀���泐
 
鼀��泐
矄ᆠ�泐



᲼㝐�

和


�






ᔼᔼ㡰桌 㡰 x, y
 
z

 
矄鼀
�鼀 矄�ᨨ
㡰 졠 泐
x
 
z

 
矄鼀
�鼀 矄�ᨨ
ᆠ㡰 졠 泐
x  
y

 
矄鼀
�鼀 矄�ᨨ
㡰 졠 泐
y
 
z

 
矄鼀
�鼀 矄�ᨨ
�鼀 矄�ᨨ 
ᨨ鼀盐
ᨨ
 鼀
ᆠᨨ 盐ᆠ��泐
�矄
 
K  ᨨ矄
 
鼀盐


᲼㝐�
和










ᔼ㡰桌 㡰 㝐 

⧤ 㡰 㝐 

⧤
ᆠ㡰 㝐 
⧤ 㡰 㝐 
⧤҈ 꾀
⼄盐 �泐
x
�矄
��
泐
IFv
⏌WL
㡰
�
鼀矄
ᨨ
鼀ᨨ鼀 
ᨨ � �
I

ᨨᆠ
F

ᨨᆠ�泐
v

犨ᨨ├
W
 
 �
L

 �鼀 矄�ᨨ 
ᨨ鼀盐
ᨨ
x
�矄
矄鼀
矄
 
ᨨ

᲼㝐�
和










ᔼᔼ㡰桌 㡰  ᆠ├矄
ᆠᨨ 矄  㡰 ᨨᆠ
ᆠᨨ 矄  ᆠ㡰  �泐
 矄�泐 㡰 ᆠᨨ �ᆠ� 
ᨨ
�矄ᆠᨨ矄�泐 꾀鼀ᨨ盐 
ᨨ
矄ᨨ
 �泐
ᆠ�
ᨨ
 
 Ἐ矄
矄盐 ᆠ 
盐 �


盐 �
�鼀
�矄
 � 

矄ᨨ
ᆠᨨ 矄 
�鼀 矄�ᨨ
ᨨ
矄ᨨ
ᆠᨨ 矄 
�矄

᲼㝐�
和









ᔼᔼ㡰桌 㡰 㝐

㡰 㝐
 ᆠ㡰 㝐

҈ 㡰 㝐 
ᔼ
矄



꾀
 
ᨨᆠ


ᆠ ᨨ矄
矄
盐 鼀  盐 �矄

 
 
�鼀 矄�ᨨ
ᨨ
Ӹᨨ盐 �Ἐ矄
鼀ᨨ盐盐矄
犨�


᲼㝐�
和










ᔼ㡰桌

҈ ᆠ㡰 꾀

㡰 꾀҈
㡰 꾀  㡰 꾀  ᔼ
鼀ᨨ鼀 盐鼀
㡰


꾀㡰
 
�鼀
 㡰

├
ᨨ

  盐 鼀 
⼄盐 ��矄
 
 
�鼀 矄�ᨨ 
ᨨ鼀盐
ᨨ  �泐
�矄










᲼㝐�
和










ᔼᔼ㡰桌 㡰 ᲼
꾀 
桌 㡰 ᲼꾀҈ 
桌 ᆠ㡰 ᲼꾀҈⏌
-҈ 桌 㡰 ᲼҈⏌
꾀 -҈桌 ␔ �ᆠ
ᨨ
 
ᨨᨨ犨� �
ᆠᨨ鼀� �ᨨ 矄
矄
 
�鼀 矄�ᨨ

ᨨ
ᆠ�ᆠ
矄�矄 ᆠ
Î0
�矄
 
鼀���泐
ᨨ
ᆠ盐盐鼀  
m

�矄
 
鼀��泐
ᨨ
ᆠ盐盐鼀㡰䁸 ᲼㝐�
҈

꾀�


҈黨
ᔼ㡰桌


⼄盐 ��矄
x =

m
m



㡰


e
e



y=

ᆠ㡰

e
m


e
㡰 m


ᔼ
 
ᨨ鼀盐

㦨
�Ӹ



 


�鼀 矄�ᨨ 矄
ᨨ ᆠᆠ� ᆠ
 
鼀� �ᆠ
�

矄ᆠ�泐

␔ 
  
�鼀 矄�ᨨ 矄
ᨨ
Ӹ
�
和ᔼ
盐 �矄
䁸 ᲼㝐�
҈

꾀�


҈黨
ᔼᔼ㡰桌 㡰 ᲼㝐҈




꾀
桌 㡰 ᲼㝐




꾀
桌 ᆠ㡰 ᲼㝐





꾀
桌 㡰 ᲼㝐





꾀桌

ᔼ
和ᔼ
盐 �矄

 
�鼀 矄�ᨨ 矄
ᨨ


Î

�矄
m


᲼㝐�


꾀�


҈黨
ᔼ㡰桌 㡰 꾀

㝐҈҈ 㡰 꾀 㝐  ⏌
ᆠ㡰 꾀 㝐 㡰 꾀

㝐҈
҈


l

r

c
 
v
矄 
� 盐ᆠ ᆠ

矄�矄 ᆠ
ᆠᆠ� ᆠ
 
ᨨ�

矄ᆠ�泐


�鼀 矄�ᨨ
ᨨ 

҈

l
�
和ᔼ
盐 �矄
犨�




᲼㝐�
҈





҈黨
ᔼᔼ㡰桌 rcv 㡰 ᲼꾀


桌 㡰 ᲼꾀҈ 桌 ᆠ㡰 ᲼ 꾀桌 㡰 ᲼
桌 
ᨨᆠ
ᨨ
� ᆠ�ᨨ
犨
犨ᨨ
ᨨ鼀矄
�矄
��
泐 ҈҈
æ x
ö F = ab   ç ç akT ÷÷ 錐
犨 
x
�矄
 
�矄 ᆠ

├
�矄
  è ø ᨨ鼀


ᆠᨨ 矄 
 

�矄
鼀盐
 
a
 
b
 犨ᨨ
ᆠᨨ 矄 矄


�鼀 矄�ᨨ 矄
ᨨ
b
�矄
᲼㝐�
҈҈




҈黨
ᔼ㡰桌 㡰 㝐


㡰 㝐
 
ᆠ㡰 㝐 
㡰 㝐


҈

ᔼ
矄
㡰

ᆠᆠ�ᨨ
꾀㡰
 
ᨨᆠ
 㡰

ᆠᨨ 矄� 矄
盐 鼀 
盐 �矄

 
�鼀 矄�ᨨ
ᨨ
Ӹᨨ盐 �矄
鼀ᨨ盐盐矄 犨�




᲼㝐�
҈҈




҈黨
ᔼᔼ㡰桌 㡰 
꾀

㡰 
꾀

ᆠ㡰 
꾀
㡰 
꾀
hc�
犨 
ᆠ
�矄
矄
ᨨ G �� 

෌
盐 �矄
���ᨨ 
ᆠᨨ 矄 
 
h
�矄
 
 ᆠ├矄 ᆠᨨ 矄 

�鼀 矄�ᨨ
ᨨ
f
�矄
 
ᨨ



᲼㝐�
黨




҈黨
ᔼ㡰桌 㡰  㡰  �泐 ᆠ㡰 鼀ᨨ鼀 盐鼀 㡰 ᨨ盐鼀 ҈

쟼 矄矄�ᨨ
ᨨ
�鼀
�
鼀矄
ᨨ
෌
盐 �矄
���ᨨ  ᆠᨨ 矄 㡰


 ᆠ├Ἐ矄
ᆠᨨ 矄 㡰
 
ᆠ
矄
ᨨ
�� 㡰
�矄 ᨨᨨ�ᨨ 
ᨨ


᲼㝐�
黨




҈黨
ᔼᔼ㡰桌

҈
꾀
⼄盐 �泐
f
�矄
��
泐
f
㦨


㡰

ᆠ� ෌

㡰

ᆠ ෌

ᆠ㡰

෌

㡰

෌

ᆠ ᆠ ҈�

�鼀 矄�ᨨ 矄
ᨨ
矄ᨨ� �
ᨨ �
V

�
ᨨᨨᆠ�ᆠ  ᆠ
�
盐 �矄
ᨨ
 ᆠ├s
ᆠᨨ 矄 
�h

矄
ᨨ
�� 
�c  
෌��ᨨ 
ᆠᨨ 矄 
�G
 
鼀
A
�矄
᲼㝐�






҈黨
ᔼ㡰桌 㡰 h
⏌c�⏌G҈⏌A
҈ 㡰 h⏌G
⏌c⏌
A
҈ ᆠ㡰 h
⏌
e҈⏌
G
⏌
A҈ 㡰 h
G⏌
C҈⏌
A҈ ҈



�

B


犨 
B
�矄
鼀� �ᆠ
�
 
m0
m
�矄
 
鼀� �ᆠ
鼀��泐
ᨨ
ᆠ盐盐鼀

�矄
᲼㝐�






҈黨
ᔼᔼ㡰桌 㡰 㝐 
㡰 㝐
҈ ᆠ㡰 㝐

㡰㝐҈ 

�鼀 矄�ᨨ 矄
ᨨ


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P-3

Physical World, Units and Measurements

҈䰘



ᆠ ᆠ�矄�ᆠ
�矄 ᆠ

犨 �ᆠ
⼄盐 盐鼀
���ᨨ   ᆠ矄

矄�� � �ᆠ 

 
 ᆠ├
 �

ᆠ

鼀�  ᨨ鼀

矄盐�
ᆠᨨ鼀� �ᨨ
ᨨ
 
盐 鼀 
 泐矄�ᆠ ᆠᨨ 矄 矄
෌


 
ᆠ

␔ �ᆠ
ᨨ
 
ᨨᨨ犨� �
ᆠᨨᆠ泐 ��矄
 
 ᆠ├
 � 䁸 ᲼㝐�
졠 � 
꾀�
҈�



҈
桌 ҈

҈



҈黨








҈


҈ æ ෌ ö

 㡰 ෌
ᆠ 㡰 ç  ÷ ᆠ㡰

ᆠ 㡰 ෌ ᆠ ෌ èᆠ ø

�鼀
 㡰

ᨨᆠ�泐
ᔴ㡰
 
 �盐
鼀ᨨ鼀 盐鼀
 㡰
 ᆠ ᨨ矄
矄
盐 鼀 
⼄盐 ��矄
� 矄
ᨨ
鼀矄矄

 �  
�鼀

ᔼ
鼀矄
ᨨ
 矄

 
�鼀 矄�ᨨ 矄
ᨨ
鼀矄矄
犨ᨨ盐 


᲼㝐�
졠 � 
꾀�





҈䰘桌 㡰 ᲼
㝐
桌㦨᲼
҈
ᔴ


桌 㡰 ᲼
㝐
桌㦨᲼
҈
ᔴ


桌 ᆠ㡰 ᲼
㝐
桌㦨᲼
҈
ᔴ

҈
桌 㡰 ᲼
㝐
桌㦨᲼

ᔴ


桌 ᔼ
 
ᨨᨨ犨� �
ἘᔼἘ
 矄
ᨨ
ᆠ盐 
 
ᨨ 
矄泐鼀ᨨ矄 
 �
盐矄盐
鼀 � �

ᔴ ᨨᨨ矄
 
ᨨ�ᨨ
  ᆠᨨ矄ᨨ 矄
ᨨ
 
�鼀 矄�ᨨ 矄
ᨨ
ᆠ�ᆠ
ᆠᨨ 盐ᆠ��泐





᲼㝐�
졠 � 
꾀�
黨



҈
桌 㡰 㝐҈   ᔼ 㡰 㝐҈   ᔼ
҈   ᆠ㡰 㝐  ᔼ 㡰 㝐  ᔼ
ᔼ
ᆠᨨ �ᆠ
ᆠ �


ᆠᨨ
鼀矄矄
鼀

矄
ᨨ
�� 
�

ᆠ盐盐鼀
ᆠ
 
 ᆠ├矄
ᆠᨨ 矄 


├
矄
盐 鼀  ⼄盐 ��矄

 
鼀��泐
ᨨ
ᆠ盐盐鼀
m

ᆠ

 矄矄 �
盐 �矄
ᨨ



᲼㝐�
졠 � 
꾀�
҈҈



҈�桌

æ 鼀ᆠ
ö æ ᆠ ö æ ö æ ö 㡰 ç
÷ 㡰 ç
÷ ᆠ㡰 ç
÷ 㡰 çç
÷÷ è 鼀 ø è ᆠ ø è  ø è 鼀 ø ᔼ
 
ᆠᆠ� ᆠ
ᨨ

 ᨨᆠᆠ�ᨨ
�矄
鼀矄盐
�
鼀矄 ᨨ

盐 �
�u
鼀
泐
ᆠᨨ鼀� � �
 
ᆠ�ᆠ
ᆠ �
�e
ᨨ 
�盐矄
�


 ᆠ├矄
ᆠᨨ 矄 
� 
 
矄
ᨨ
��  �c
 
᲼㝐�
졠 � 
꾀�
҈




҈�桌 ᆠ 


ᆠ 
㡰 盐 =
ᆠ㡰 盐 = 㡰 盐 = ᆠ 

 
ᨨ鼀
 
ᨨᨨ犨� �
ᆠᨨ鼀� �ᨨ 矄
ᨨ
 泐矄�ᆠ
ᆠᨨ 矄 矄  矄矄
 ᨨ盐�
 �
盐矄盐
矄泐鼀ᨨ矄㡰
 
ᨨ 泐 ᆠᨨ鼀� �ᨨ

 
犨ᨨ盐

 
矄鼀
盐
�
�   矄泐矄鼀矄
ᨨ
盐 �矄

�矄
᲼㝐�
졠 � 
꾀�
҈




҈
桌





᲼ Î
桌
 ᨨ
 
�鼀 矄�ᨨ 
ᨨ鼀盐
ᨨ
 
鼀���泐 ᨨ
ᆠ盐盐鼀

ᔼ
㝐
㦨
鼀矄矄


㦨
 �


㦨
�鼀
  꾀
㦨
ᆠ�ᆠ
ᆠ盐 

 




᲼㝐�


҈桌 㡰 Î

㦨
᲼㝐҈





꾀桌 㡰 Î

㦨
᲼㝐҈





꾀
桌 ᆠ㡰 Î
㦨
᲼㝐҈




҈
꾀
桌 㡰 Î

㦨
᲼㝐҈




҈
꾀桌
�

�鼀 矄�ᨨ 矄
ᨨ
 �盐
鼀ᨨ鼀 盐鼀

 

  ᆠᆠ� ᆠ


矄ᆠ�泐
᲼㝐�
졠 � 
꾀�






҈桌 㡰 㝐
҈꾀


-

㝐 -҈-

㡰 㝐
-




㝐 -҈-

꾀
ᆠ㡰 㝐
-҈

-

㝐
꾀






䰘







㡰 盐 =




㡰

㡰

ᆠ㡰

2


peᨨ






pe ᨨ ෌鼀



鼀
㦨
鼀矄矄
ᨨ
ᆠᨨ 㡰

m ᨨ eᨨ ෌

ᆠ




p m ᨨ eᨨ ෌ ᆠ
ᔼ
鼀矄
ᨨ
矄�矄 ᆠ

 
�鼀


 
�鼀 矄�ᨨ 矄
ᨨ
�ᨨ m
ᨨ
 
鼀��泐
m
 
鼀���泐
e
�矄
e ᲼㝐�
졠 � 
꾀�
҈҈



҈
桌 㡰 ᲼ 
桌 㡰 ᲼
҈桌 ᆠ㡰 ᲼
桌 㡰 ᲼

桌 㡰





ᆠ


黨


㡰 㝐
-҈

-

㝐 -҈-

꾀
␔ �ᆠ
ᨨ
 
ᨨᨨ犨� �
矄

�  
�鼀 矄�ᨨ 矄䁸 㡰 矄矄盐

Ӹᨨ盐 �矄
鼀ᨨ盐盐矄


和矄矄 ᲼


�和桌 㡰 쟼㝐

ᨨ �
�  ᆠ

쟼ᆠ�ᆠ
ᨨ � ᆠ㡰 

␔ᨨ├
ᨨ 

쟼 �泐 㡰 �ᨨ
鼀ᨨ鼀 

쟼ᆠ�ᆠ
盐

쟼ᆠ�ᆠ
� a az 矄矄盐
 矄
ᨨ
�矄 ᆠ
矄

P =   æç - ö÷
è kq ø b 犨 
a

b

ᆠᨨ 矄 矄

z
�矄
�矄 ᆠ

k
�矄
ᨨ鼀 矄 ᆠᨨ 矄 
 
q
�矄
鼀盐


�鼀 矄�ᨨ
ᨨ
b
 ᲼



和桌 㡰 M
L
T
㡰 M҈L҈T ҈ ᆠ㡰 M
L
T
㡰 M҈L҈T
DV 꾀
⼄盐 �泐
X �矄
��
泐
e
L
犨 
Î

�矄
  Dt 鼀���泐
ᨨ
 

矄ᆠ

L �矄

 �

DV
�矄

ᨨ � �  ᆠ
 
Dt
�矄

�鼀
� 


�鼀 矄�ᨨ 
ᨨ鼀盐 ᨨ
X �矄
 
矄鼀
矄
 
ᨨ ᲼


҈和桌 㡰 矄�矄 ᆠ 㡰 ᆠ � ᆠ㡰 ᨨ� 㡰 ᆠ盐  æ ҈ö


�鼀 矄�ᨨ
ᨨ
ç ÷ e
E

 e


鼀���泐
ᨨ
 è
ø 矄ᆠ

E

ᆠ�ᆠ
�㡰 ᲼



和桌 㡰 MLT҈ 㡰 ML
T
ᆠ㡰 ML҈T
㡰 ML
T҈ Integer Value Answer



ᨨ
� 
 
�矄 ᆠ
d ᨨ
犨 �ᆠ

矄�� 
ᆠ

矄

ᆠ泐
�
ᨨ��泐
ᆠᨨ ��ᨨ 矄


�犨泐矄㜠 �� 
盐矄矄 �鼀 矄�ᨨ 矄
 
矄矄盐鼀矄
 
 
�矄 ᆠ
 矄
ᨨ

 
鼀矄矄
 矄�泐
r
ᨨ
 
ᨨ�

�  矄�泐
ᨨ犨⏌㡰
S
ᨨ  
�� 
ᨨ鼀
 
矄�� 
 
�矄
⼄盐 ᆠ泐
f


 ��  � 矄
 
d
�矄
ᨨᨨ�ᨨ 
ᨨ
S҈⏌n


盐
ᨨ
n
�矄 ᲼꾀



҈
桌 4

Fill in the Blanks

҈

�鼀 矄�ᨨ
ᨨ
ᆠ�ᆠ
ᆠᨨ 盐ᆠ��泐
�矄
힘힘힘힘힘힘힘힘힘
᲼҈黨黨䰘
㜠
҈
㝐├桌 


⼄盐�ᨨ
ᨨ
矄
ᨨ

�矄
�矄
��
泐 æ a ö çè P +
÷ø V - b㡰 = RT


�鼀 矄�ᨨ 矄
ᨨ
 
ᆠᨨ 矄  V a
�矄
힘힘힘힘힘힘힘힘힘

᲼҈黨黨䰘
㜠


㝐├矄桌

WWW.JEEBOOKS.IN

EBD_8341 P-4








Physics

ᔼ
 
ᨨ鼀盐
X
㦨

YZ


X
 
Z

�鼀 矄�ᨨ 矄
ᨨ ᆠᆠ� ᆠ
 
鼀� �ᆠ
� 盐ᆠ�ᨨ
矄ᆠ�泐

 �鼀 矄�ᨨ 矄
ᨨ
Y
�
MKSQ
矄泐鼀

힘힘힘힘힘힘힘힘힘
힘힘힘힘힘힘힘힘
᲼҈黨


㜠


㝐├矄桌  ᆠ├矄
ᆠᨨ 矄 
矄
�鼀 矄�ᨨ
힘힘힘힘힘힘힘힘
᲼҈黨
�
㜠


㝐├矄桌 6

�


MCQs with One or More Than One Correct

和ᨨ鼀�鼀矄
�
�矄
ᆠᨨ  � 
ᨨ
ᆠᨨ 矄盐ᆠ

矄泐矄鼀
ᨨ
盐 �矄 矄ᨨ
 

⼄盐 ��矄
ᆠ

 矄矄
�
鼀矄
ᨨ
ᨨ 泐
ᨨ   泐矄�ᆠ
⼄盐 �泐

ᔼ
ᨨ 
矄盐ᆠ
矄泐矄鼀

�鼀 矄�ᨨ 矄
ᨨ �  
⼄盐 ��矄

��
�
鼀矄
ᨨ

⼄盐 �泐
X
矄 ᨨᨨ犨矄
᲼ᨨ矄��ᨨ 桌
㦨
᲼Xa桌錐
᲼矄桌
㦨
᲼Xb桌錐
᲼ᆠᆠ�ᨨ 桌
㦨᲼Xp桌錐 ᲼� 
鼀ᨨ鼀 盐鼀桌
㦨
᲼Xq桌錐
᲼ ᨨᆠ桌
㦨
᲼Xr桌




᲼꾀





桌 㡰 a + p =
b





䰘






黨


ᆠ㡰 p - q + r = a 㡰 p + q + r = b 
盐矄
ᆠᨨ 矄�

矄泐矄鼀
ᨨ
盐 �矄
�
犨 �ᆠ
鼀矄矄
   �盐
鼀ᨨ鼀 盐鼀

�鼀 矄�ᨨ 矄矄

ᔼ
 �
矄 �鼀 矄�ᨨ 矄
ᨨ


犨 �ᆠ
ᨨ
 
ᨨᨨ犨� �
矄鼀 矄㡰
�矄⏌ 
ᆠᨨᆠ䁸


᲼꾀



҈黨桌 
㡰 
�鼀 矄�ᨨ
ᨨ
ᨨᆠ
�矄
L 㡰 
�鼀 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
�矄
L҈ ᆠ㡰 
�鼀 矄�ᨨ
ᨨ
 �泐
�矄
L
㡰 
�鼀 矄�ᨨ
ᨨ
ᨨ犨
�矄
L� 꾀
 � 矄ᆠ
l㡰
 矄
ᨨ
 
鼀���泐
e㡰
ᨨ
 �ᆠ�ᆠ
鼀�

ᨨ鼀


ᆠᨨ 矄 
├㡰

 
矄ᨨ盐 鼀盐
 㡰

 
盐鼀

盐 �
ᨨ盐鼀
 㡰
ᨨ
ᆠ�

ᆠ �
�ᆠ矄

 
 
ᆠ �
⼄㡰
ᆠ�
泐
ᆠ
ᨨ  
�ᆠ矄

␔ �ᆠ
ᨨ
 
ᨨᨨ犨� �
 矄矄�ᨨ 矄㡰
ᨨ
I �矄㡰
�鼀 矄�ᨨ 泐
ᆠᨨᆠ䁸
᲼꾀


҈
桌

æ ⼄
ö l = ç ÷ 㡰 ç e├  ÷ è ø

æ e├  ö ÷ 㡰 l = çç
÷ è ⼄ ø

æ ö ⼄
÷ ᆠ㡰 l = çç
⏌ ÷ è e ├  ø

æ ö ⼄
÷ 㡰 l = çç ҈⏌ ÷ è e ├  ø

ᔼ
鼀矄
ᨨ
ᨨ �
�  ᆠ
V

ᆠ�ᆠ
ᆠ盐 
I
鼀���泐
e


鼀��泐
m

 
矄
ᨨ
�� 
c

  �鼀 矄�ᨨ 泐
ᆠᨨᆠ
⼄盐�ᨨ 矄㡰
�矄㡰
᲼꾀



҈�桌 㡰 m
ᔼ

㦨
e

㡰 m
ᔼ
㦨
m
 ᆠ㡰 ᔼ
㦨
e
ᆠ 㡰 m
ᆠᔼ
㦨
e
  ᆠ├矄
ᆠᨨ 矄 
h

矄
ᨨ
�� 
c
 
���ᨨ  ᆠᨨ 矄 
G

盐矄
ᨨ
ᨨ鼀

盐 �
ᨨ
 �
L
 

盐 �
ᨨ 鼀矄矄
M


 
ᆠᨨᆠ
ᨨ�ᨨ 矄㡰
�矄㡰 ᲼꾀



҈�桌 㡰 M
µ
c 㡰 M
µ
G ᆠ㡰 L
µ
h






㡰 p + q - r = b

㡰 L
µ
G


[e 0 ]
 ᨨ
 
�鼀 矄�ᨨ 
ᨨ鼀盐
ᨨ
 
鼀���泐 ᨨ
 

ᆠ盐盐鼀

 
[m 0 ]
 
ᨨ
 
鼀��泐
ᨨ
  ᆠ盐盐鼀

ᔼ
M
㦨
鼀矄矄

L
㦨
 �


T
㦨
�鼀
 
I
㦨
ᆠ�ᆠ ᆠ盐 
᲼҈黨黨

㜠


㝐├矄桌

㡰 ᲼e
桌 = M -҈ L- T
Ι 㡰 ᲼e
桌 = M -҈ L- T
Ι
ᆠ㡰 ᲼m
桌 = M LT -
Ι -
㡰 ᲼m
桌 = M L
T -҈ Ι
҈

�矄
ᨨ
 泐矄�ᆠ
⼄盐 ��矄
 

 
矄鼀 �鼀 矄�ᨨ 矄
�矄
㡰

᲼҈黨黨�和桌 㡰 泐 ᨨ矄
盐鼀
 
ᆠᨨ �ᆠ� 
ᨨ
�ᆠ�ᨨ

㡰 ᔴ盐�
 
⼄盐 ᆠ泐
ᨨ

�� 
犨 ᆠ㡰  

 
���ᨨ 
ᨨ � 㡰  ᆠ├矄
ᆠᨨ 矄 
 
ᨨ⼄盐



�鼀 矄�ᨨ 矄
ᨨ
 
⼄盐 ��矄
�
ᨨ 
ᨨ
鼀ᨨ㡰
ᨨ
  ᨨᨨ犨� �
�矄

 
矄鼀

ᔼ � 泐
 
�
矄㡰 㡰 ᨨ⼄盐
 
␔ᨨ├ ᲼҈黨


㜠


㝐├矄桌 㡰 꾀 �盐
鼀ᨨ鼀 盐鼀
 
␔ᨨ├ ᆠ㡰 쟼 �泐
 
Ӹᨨ盐 �矄
鼀ᨨ盐盐矄 㡰 �� 
泐
 
␔ � 7

Match the Following



㝐ᆠ
�矄
ᔼ
犨�
�矄
ᔼᔼ
 
矄ᆠ
 
ᆠᨨᆠ
 矄犨 盐矄� �
 
ᆠᨨ矄
��
ᨨ犨
 
�矄矄
᲼꾀



҈桌 �矄
ᔼ �矄
ᔼᔼ 
ᨨ鼀


ᆠᨨ 矄  ҈
᲼㝐
㜠҈桌 턼
ᔴᨨ �ᆠ� 
ᨨ
�矄ᆠᨨ矄�泐

᲼㝐҈ ҈桌
 ᆠ├
ᆠᨨ 矄  
᲼㝐 ⧤҈桌 和
鼀
ᆠᨨ 盐ᆠ��泐

᲼㝐

⧤҈桌 ᔴᨨ矄
 턼 和 㡰  ҈

㡰 
҈
ᆠ㡰

҈  㡰
҈



㝐ᆠ
 
 泐矄�ᆠ
⼄盐 ��矄
��
�
ᆠᨨ盐鼀
ᔼ
犨� �鼀 矄�ᨨ 矄
 矄矄
�
鼀矄
ᨨ
鼀矄矄
M㡰

 �
L㡰
�鼀
T㡰

 
ᆠ �
Q㡰
��
�
ᆠᨨ盐鼀
ᔼᔼ
 
犨�
  ᆠᨨᆠ
 矄犨
�� 矄
 
鼀ᆠ 
⼄盐 �泐
�

盐 ᨨ鼀
�
泐ᨨ盐
 矄犨
ᨨᨨ├
᲼҈黨

㜠


㝐├矄桌 ᔴᨨ盐鼀
ᔼ ᔴᨨ盐鼀
ᔼᔼ 꾀 �盐
鼀ᨨ鼀 盐鼀 ML
T
 
 ML
Q


ᨨ⼄盐 ML
T҈ ᔴᆠ� ᆠ MLT҈Q
ᔼ 盐ᆠ ᆠ M ҈ L 
T
Q
矄�矄��泐 L
T 
8

Comprehension/Passage Based Questions

矄矄� ᔼ
ᆠᨨ鼀� �ᆠ
 ᨨ泐

 
ᆠ�ᆠ
 
鼀� �ᆠ
  ᨨ鼀  

ᨨ
ᆠ
ᨨ 

 ᨨ

 
�鼀 矄�ᨨ 矄
ᨨ
ᆠ�ᆠ  
鼀� �ᆠ
⼄盐 ��矄
鼀盐矄
矄ᨨ


ᨨ
ᆠ
ᨨ 

ᔼ

 
⼄盐矄�ᨨ 矄
ᨨ犨

᲼E桌
 
᲼B桌
矄 
ᨨ
�鼀 矄�ᨨ 矄
ᨨ
ᆠ�ᆠ  
鼀� �ᆠ
�矄
矄ᆠ�泐

犨 �
᲼e
桌
 
᲼m
桌
矄 
ᨨ �鼀 矄�ᨨ 矄
ᨨ
 
鼀���泐
 
鼀��泐
ᨨ

矄ᆠ 矄ᆠ�泐

᲼L桌
 
᲼T桌

�鼀 矄�ᨨ 矄
ᨨ
 �
 
�鼀 矄ᆠ�泐

꾀
 
⼄盐 ��矄

��
�
和ᔼ
盐 �矄
᲼꾀



҈
桌
�

�ᨨ
犨
᲼E桌
 
᲼B桌
�矄 㡰 ᲼E桌
㦨
᲼B桌
᲼L桌҈
᲼T桌 㡰 ᲼E桌
㦨
᲼B桌᲼L桌᲼T桌 ᆠ㡰 ᲼E桌
㦨
᲼B桌
᲼L桌
᲼T桌҈ 㡰 ᲼E桌㦨᲼B桌
᲼L桌҈
᲼T桌҈

WWW.JEEBOOKS.IN

P-5

Physical World, Units and Measurements







�ᨨ
犨
᲼e
桌
 
᲼m
桌
�矄 㡰 ᲼m
桌
㦨
᲼e
桌
᲼L桌

᲼T桌
㡰 ᲼m
桌
㦨
᲼e
桌
᲼L桌

᲼T桌
҈

ᆠ㡰 ᲼m
桌
㦨
᲼e
桌
᲼L桌
᲼T桌 㡰 ᲼m
桌
㦨
᲼e
桌҈
᲼L桌

᲼T桌







10 Subjective Problems
䰘


␔�
 
�鼀 矄�ᨨ 矄
ᨨ
 
ᨨᨨ犨� �
�
鼀矄
ᨨ
鼀矄矄
�鼀

 �
 
ᆠ � ᲼҈黨


㜠


㝐├矄桌

�㡰 鼀� �ᆠ
盐 ��㡰 ����泐
鼀ᨨ盐盐矄 꾀
�矄
盐

ᨨ鼀

 ᨨ矄�ᨨ
盐 
犨

ᨨ矄ᆠ�矄 犨�

�ᨨ
T
ᨨᨨ�ᨨ 
ᨨ
p a d b E c

␔ 
�P
�矄
  矄�ᆠ
矄矄盐

�d
�矄
 
 矄�泐
ᨨ
犨
 
�E
�矄
 
ᨨ  �泐
ᨨ
 
 ᨨ矄�ᨨ

� 
 
盐矄
ᨨ
a

b
 
c
᲼҈黨
҈㜠

㝐├矄桌

Topic-3 : Errors in Measurements & Experimental Physics 1 ҈









MCQs with One Correct Answer

꾀
矄ᆠ犨
�盐�
矄
�

��矄�ᨨ 矄
ᨨ
�矄
ᆠ�ᆠ盐
矄ᆠ

 ᆠ�ᆠ盐
矄ᆠ
�矄


盐 �矄
 
ᨨ
 
�ᆠ
矄ᆠ
鼀├� �
�ᨨ
ᨨ
盐矄

ᨨ
ᨨ 
ᆠᨨ鼀
ᨨ�ᨨ
ᨨ
 
ᆠ�ᆠ盐 矄ᆠ


�矄ᆠ鼀 
ᨨ


�
鼀鼀
�矄
ᨨ�ᆠ
ᨨ
 
�ᆠ 矄ᆠ


盐
ᨨ
ᨨ
ᨨ
� ᨨ

 
 
矄 ᆠᨨ盐 
ᨨ
 
矄ᆠ犨
�盐�


矄ᆠ�泐

᲼㝐�
和










ᔼ㡰桌 㡰 ��



m鼀 㡰 ᨨ矄��

҈

m鼀 ᆠ㡰 ᨨ矄��



҈
鼀鼀 㡰 ᨨ矄��



҈
m鼀 꾀
矄盐 
鼀矄盐� �
 
�鼀
ᨨ

 ᆠ�
ᨨ
ᆠ�ᆠ盐 ᆠᨨ矄矄㜠矄ᆠ�ᨨ
犨�
 

ᨨ

 �
矄ᆠ
ᆠᨨ矄
  ᨨᨨ犨� �
ᨨ盐
� �矄
�
�

鼀鼀

�
��
鼀鼀

�

�
鼀鼀
�

�
鼀鼀


�
ᨨ
 矄
ᨨ盐
� �
�矄
�
�䰘�
鼀鼀  
 
矄 
��ᨨ
ᨨ
 

�矄



䰘黨�
鼀鼀

 �
�鼀
ᨨ
 
 ᆠ�
矄 ᨨ盐
  ᨨ

ᆠᨨ ᲼㝐�
和










ᔼᔼ㡰桌 矄

㡰 �
�䰘�
±



䰘黨㡰
鼀鼀 㡰 �
�䰘�
±



䰘

㡰
鼀鼀 ᆠ㡰 �
�

±



䰘
㡰
鼀鼀 㡰 �
�

±



䰘㡰
鼀鼀
꾀
 泐矄�ᆠ
⼄盐 �泐
z
 矄
ᨨ
ᨨ盐
ᨨ矄矄
a

b

c  
d

矄
z
㦨






�




 a b




ᆠ �矄
ᨨ
ᨨ
�
 
鼀㜠 cd  矄盐鼀 
ᨨ
a

b

c
 
d


奠

҈
�奠


奠
 


�奠
㜠 矄ᆠ�泐


ᆠ �
ᨨ
ᨨ
�
z
�矄

᲼㝐�
和


�






ᔼ㡰桌 㡰 ҈


�奠 㡰 ҈

�奠 ᆠ㡰 ҈
�奠 㡰 ҈

�奠 矄� �
矄ᆠ犨
�盐�
ᨨ
�ᆠ


҈
ᆠ鼀
 
�

��矄�ᨨ 矄
ᨨ
�矄 ᆠ�ᆠ盐
矄ᆠ

 
 �ᆠ├ 矄矄
ᨨ

ᨨᆠ
�矄
鼀矄盐

ᔼ 矄 ᨨ盐
ᆠᨨᆠ泐

ᆠᨨ
矄
[㝐�
Sep. 03, 2020 (I)] 㡰

҈
҈
ᆠ鼀㡰

҈


ᆠ鼀 ᆠ㡰

҈
�
ᆠ鼀 㡰

҈

ᆠ鼀


矄
ᆠᨨ盐 
ᨨ
 
鼀�
矄ᆠ
ᨨ

 �
ᆠ�矄
�矄 ҈
鼀鼀

ᔼ矄
 �
矄ᆠ
�矄
��
� ᨨ
҈

��矄�ᨨ 矄
  ᆠᨨ� ᆠ�
犨�
黨
��矄�ᨨ 矄
ᨨ
 
鼀�
矄ᆠ

␔ 
犨矄
 ᨨ盐ᆠ � �
ᆠ
ᨨ 

 
䰘
��矄�ᨨ
ᨨ
 �
矄ᆠ ᆠᨨ� ᆠ�矄
犨�

��矄�ᨨ
ᨨ
鼀�
矄ᆠ
 
 
ᨨ
ᨨ  �
矄ᆠ
�矄
泐� �
�� 
矄�
ᨨ
 
ᨨ
ᨨ
鼀�
矄ᆠ
␔ 
 �矄
 �
�矄
盐矄
ᨨ
鼀矄盐
 �
ᨨ

ᆠ泐�   
ᨨ
ᨨ
 
 �
矄ᆠ
犨

҈
ᆠ鼀
 



ᆠ鼀
 

和
ᆠᨨ� ᆠ�矄
犨�

鼀�
矄ᆠ
��矄�ᨨ


 � ᨨ
 
ᆠ泐� 
�矄


和
�矄
 �
矄ᆠ
��矄�ᨨ 㡰 [㝐�
Sep.
02, 2020 (I)]

㡰 


ᆠ鼀 㡰 

҈
ᆠ鼀 ᆠ㡰 

䰘
ᆠ鼀 㡰

黨黨
ᆠ鼀 ᔼ
 
矄ᆠ犨
ᨨ

矄ᆠ犨㜠�盐�
�矄
��
矄�
ᨨ�ᨨ 矄

�
鼀ᨨ矄 泐

鼀鼀
ᨨ
 
鼀�
矄ᆠ

ᔼ
 

�

��矄�ᨨ 矄
ᨨ
  ᆠ�ᆠ盐
矄ᆠ
 
矄
ᆠᨨ盐 
ᨨ
 
矄ᆠ犨
�盐�
�矄
᲼㝐�
黨







ᔼ㡰桌 㡰



҈
ᆠ鼀㡰




鼀鼀
ᆠ㡰


҈
ᆠ鼀 㡰



҈
鼀鼀 䰘
ᨨ
 
ᨨ盐
矄矄
ᨨ
 
鼀矄盐
 泐矄�ᆠ
⼄盐 ��矄
矄 ��
ᨨ犨

␔ �ᆠ
ᨨ
 
ᨨᨨ犨� �
ᨨ�ᨨ 矄
�矄
ᆠᨨᆠ䁸 ᲼㝐�
黨







ᔼᔼ㡰桌 꾀㡰 꾀҈
㦨






҈
㦨



䰘



ᔴ҈
㦨

�


㡰 꾀

㦨









㦨
҈






ᔴ

㦨





ᔴ㡰 꾀
㦨

�




㦨
҈黨


҈


ᔴ
㦨



҈
 㡰 꾀

㦨

�



㦨



҈黨҈

ᔴ

㦨
҈黨
� 㡰 꾀

졔


졔
ᔴ



꾀҈
졔
҈
졔
ᔴ҈


꾀
졔

졔
ᔴ


꾀

졔



졔
ᔴ
㡰 꾀҈
졔
҈
졔
ᔴ҈
㦨
꾀

졔


졔
ᔴ

㦨
꾀
졔

졔
ᔴ
㦨
꾀

졔



졔
ᔴ
ᆠ㡰 꾀

졔


졔
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EBD_8341 P-6

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p

WWW.JEEBOOKS.IN

P-7

Physical World, Units and Measurements






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EBD_8341 P-8





�






Physics

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 
犨�
ᨨ





鼀鼀
犨�

盐 ᆠ� 泐 ᨨ
±



�
鼀鼀


ᨨ
ᨨ
 ᆠ泐
҈


├�


矄盐 
矄ᨨ 鼀矄盐矄
 
�鼀
ᨨ
 
犨�
ᨨ





鼀鼀
犨�


盐 ᆠ� 泐
ᨨ
±



҈
鼀鼀

├
g
㦨
黨


鼀⏌矄

 ᆠ㡰

 Ӹᨨ盐 �Ἐ矄
鼀ᨨ盐盐矄
ᨨ� 
ᨨ鼀
 
� �
�矄 ᲼


䰘桌 㡰 



±


㡰
맔
҈
҈҈
⏌鼀
㡰 



±



㡰
맔
҈
҈҈
⏌鼀
ᆠ㡰 



±


҈㡰
맔
҈
҈҈
⏌鼀
㡰 



±



�㡰
맔
҈
҈҈
⏌鼀
꾀
矄盐 
 ᨨ鼀矄

 �鼀 
ᨨ
鼀� �ᨨ
ᨨ æ
p
l ö g ç =
÷


ᨨ
�
 �
l
�矄
Dl
 
�
�鼀
T
�矄
DT è T ø  
n
�矄
盐鼀
ᨨ
�鼀矄
 
� �
�矄
├

 鼀矄盐鼀 
ᨨ
g
�矄
鼀ᨨ矄
ᆠᆠ盐
ᨨ ᲼




㜠
㝐

҈桌 Dl DT n 㡰 �
鼀鼀



矄ᆠ ҈
㡰 �
鼀鼀



矄ᆠ

ᆠ㡰 �
鼀鼀

҈
矄ᆠ
҈
㡰 ҈
鼀鼀

҈
矄ᆠ �
ᔼ

矄ᆠ犨
�盐�

 
ᨨ
ᨨ
鼀� 矄ᆠ
ᆠᨨ� ᆠ�矄
犨� � 
��矄�ᨨ
ᨨ
ᆠ�ᆠ盐
矄ᆠ
�
��盐
�㡰


ᆠ�ᆠ盐 ��矄�ᨨ
ᨨ
矄ᆠ犨
�盐�

�


ᔼ
鼀ᨨ矄


�
鼀鼀
ᨨ
鼀�

矄ᆠ
�
ᨨ 
ᨨ�ᨨ


�鼀
ᨨ
 

�
��盐
��㡰 �矄 ᲼




㜠
㝐

҈桌


҈
�


��盐
�㡰




�



��盐
��㡰

䰘






㡰


�
鼀鼀 㡰




鼀鼀 ᆠ㡰 ҈



鼀鼀 㡰 ҈

�
鼀鼀 꾀
犨�
ᨨ
 �
l
㦨


±





ᆠ鼀
 
�盐矄
r
㦨


�
±




�
ᆠ鼀  
鼀矄矄
m
㦨



±





�鼀

㝐 �鼀盐鼀
ᆠ �
ᨨ
�

 矄�泐
�矄 ᲼



和桌 㡰
㡰
ᆠ㡰 ҈ 㡰


꾀
ᆠ盐
矄

矄�
ᨨ
 �
҈


맔
҈

鼀

ᔴᆠ盐
�矄
ᨨ盐鼀
㡰 ҈
䰘
맔
҈


鼀 㡰 ҈
䰘
맔
҈


鼀 ᲼


和桌 

 ᆠ㡰 ҈
䰘

맔
҈
鼀 㡰 ҈
䰘

맔
҈


鼀 2

黨


Integer Value Answer


 �泐
ᨨ

矄泐矄鼀
矄

盐 ᆠ�ᨨ
ᨨ
�鼀

�矄
��
矄 Et㡰
㦨
A

 a
㡰
犨 
a
㦨




矄҈


鼀矄盐鼀 
ᨨ A
矄

ᨨ
ᨨ
҈

�奠

ᔼ
 
ᨨ
�
 
鼀矄盐鼀 
ᨨ �鼀
�矄
҈
�
奠

 
ᆠ �
ᨨ
�
 
盐
ᨨ
Et㡰
 t
㦨
�
矄
�矄 ᲼꾀



҈�桌




盐� �
和矄
 �鼀 

ᨨ
ᨨ
 
 �
矄ᆠ
�矄 犨




맔
҈


鼀
 


�
맔
҈


鼀
ᨨ
 
鼀�
矄ᆠ






��矄�ᨨ
ᨨ
 
 �
矄ᆠ
 ᆠ泐
ᆠᨨ� ᆠ�矄 犨�
ᨨ 
ᨨ
 
鼀�
矄ᆠ
��矄�ᨨ 矄

␔ 

��ᨨ  ᨨ
ᨨ


├�
�矄
�
ᨨ
 
犨�

 
ᨨ
ᨨ
 
 � 矄ᆠ
矄�
�矄
犨




맔
҈


鼀
 


�
맔
҈


鼀
ᨨ  
鼀�
矄ᆠ
盐
ᨨ犨
 

�
��矄�ᨨ
ᨨ
 �
矄ᆠ ᆠᨨ� ᆠ�矄
犨�
ᨨ 
ᨨ
 
鼀�
矄ᆠ
��矄�ᨨ 矄


 � ᨨ
 
 �
鼀�ᆠ
犨�
�矄


鼀
 
�矄
ᆠᨨ矄矄㜠矄ᆠ�ᨨ 
 �矄


맔
҈
䰘
鼀



矄
ᆠᨨ盐 
ᨨ
 
 �
矄ᆠ
�矄
҈


맔 ҈
�
鼀


鼀 �鼀盐鼀
ᆠ �
ᨨ
�
 
Ӹᨨ盐 �矄 鼀ᨨ盐盐矄
ᨨ
 
犨�
�矄 ᲼꾀



҈
桌 3

Numeric Answer


҈

 矄�泐
ᨨ

矄ᨨ�
鼀
矄 
�矄
鼀� 
泐 鼀矄盐� �
�矄
鼀矄矄
 
�矄
�鼀


鼀 �鼀盐鼀
ᨨ
�

æ x ö  
 矄�泐
ᨨ
 
矄 
�矄
çè ÷ 奠

ᔼ
 
�
ᨨ矄 ҈

ø �
鼀矄盐� �
 
鼀矄矄
 
 
�鼀




奠
 
҈
�奠 矄ᆠ�泐

 
盐
ᨨ
x
�矄
힘힘힘힘힘힘
᲼㝐�
和










ᔼ㡰桌


犨ᨨ
ᆠᆠ�ᨨ矄
犨�
ᆠᆠ� ᆠ
盐矄
C҈
㦨





᠀
҈

  
C

㦨




᠀
҈�


ᆠᨨ

ᆠ
�
矄�矄


ᨨ� �
ᆠᨨ矄矄
 �矄
ᆠᨨ鼀� �ᨨ
�矄
V
㦨
�



᠀







 ᆠ �
ᨨ
�
 
ᆠᆠ盐�ᨨ
ᨨ
 
 �泐
矄ᨨ
�

 �矄
ᆠᨨ鼀� �ᨨ
ᨨ
ᆠᆠ�ᨨ矄
�矄
힘힘힘힘힘힘힘



᲼꾀





桌

꾀
ᨨ�ᆠ
 ᆠ
矄
҈
�
鼀
ᨨ �
矄ᆠ
� �
ᨨ盐
⼄盐 ��矄�ᨨ 矄
�
ᆠ
ᆠ鼀

␔ �
鼀矄盐� �
 
ᨨᆠ
 �
ᨨ 
ᆠᨨ 
 矄

 
 矄
�矄
├

䰘�
ᆠ鼀
鼀├
ᨨ
 
矄ᆠ  
 
ᨨᆠ
�
�矄
├


�
ᆠ鼀
鼀├


�鼀�
ᨨ
  ᨨᆠ
�
ᨨ
 
ᨨ 
矄�
ᨨ
 
 矄
ᨨ矄
犨� �鼀�
�
 
�矄
├

҈�
ᆠ鼀
鼀├

ᔼ
 �矄
 �鼀 
 
ᆠ �
ᨨ
�
 
鼀矄盐鼀 
ᨨ
 
ᨨᆠ
 � ᨨ
 
 矄
�矄
힘힘힘





᲼꾀



҈黨桌


꾀
矄
犨�
ᨨ
�鼀


�
mm
 
Ӹᨨ盐 �矄
鼀ᨨ盐盐矄

맔
҈
҈҈
N m

ᆠ�矄

ᨨ
ᨨ
鼀矄矄
M


 �
ᨨ
  犨�
犨�
 
ᨨ
�矄
҈


m

꾀
 �
矄ᆠ
犨�
҈

��矄�ᨨ 矄 �矄
ᆠ 
ᨨ
 
 
ᨨ
 �矄
犨�

 
ᨨ
 
矄
犨�
�矄 
  ᆠ
犨�
ᨨ
犨 �ᆠ

鼀�
矄ᆠ

ᨨ
矄
ᆠᨨ盐 
҈

mm

�矄
ᆠ 


҈

��矄�ᨨ 矄
ᨨ
 
 �
矄ᆠ ᆠᨨ矄ᨨ 
ᨨ
黨
��矄�ᨨ 矄
ᨨ
 
鼀�
矄ᆠ

ᔼ ��泐

  ᨨ
ᨨ
 �
矄ᆠ
ᆠᨨ� ᆠ�矄
犨�
 
ᨨ
ᨨ
鼀�
矄ᆠ
ᔼ
 
ᨨ
ᨨ
 
矄
犨�
�矄
� ᆠ矄
泐
҈


kg

   �
矄ᆠ
��矄�ᨨ
犨 �ᆠ
ᆠᨨ� ᆠ�矄
犨�

鼀�
矄ᆠ ��矄�ᨨ
�矄
힘힘힘힘힘힘힘힘힘힘

├
�
㦨҈

m s


 
p 㦨




᲼꾀



҈
桌
�

矄�
ᨨ

ᆠ盐
�矄
鼀矄盐
泐
 �
ᆠ�矄
҈
��矄�ᨨ 矄
ᨨ

 �
矄ᆠ
ᆠᨨ� ᆠ�
犨�
黨
��矄�ᨨ 矄
ᨨ 鼀�
矄ᆠ

犨 
҈
��矄�ᨨ
ᨨ
鼀�
矄ᆠ
�矄
҈
鼀鼀㡰

 鼀�
矄ᆠ
矄
҈

鼀鼀
 
�矄
��矄�ᨨ
ᨨ
 �
矄ᆠ ᆠᨨ� ᆠ�矄
犨�
 
鼀�
矄ᆠ

㝐矄矄
ᨨ
 
ᆠ盐
�矄


䰘

�
� 
 
 矄�泐
ᨨ
 
ᆠ盐
�
ᨨ�
矄�� � �ᆠ  ᲼


�
㜠


㝐├矄桌 ��盐矄



ᔼ
和矄
 �鼀 

犨 �ᆠ
�矄
盐矄
ᨨ
� 
Ӹᨨ盐 �矄 㝐ᨨ盐盐矄
ᨨ
矄�ᆠ�泐

 
�鼀
ᨨ
 �鼀 
犨�
�矄 D
㦨



�
ᆠ鼀
鼀矄盐
泐

矄ᆠ
ᨨ
矄
ᆠᨨ盐 




҈
ᆠ鼀㡰  
 �
�矄
L
㦨
҈҈

ᆠ鼀
鼀矄盐
泐

矄ᆠ
ᨨ
矄

WWW.JEEBOOKS.IN

P-9

Physical World, Units and Measurements


䰘


ᆠᨨ盐 


҈
ᆠ鼀㡰

꾀
犨�� 
ᨨ
�

N
ᆠ盐矄矄

  矄�ᨨ
ᨨ

X 㦨


҈
�
ᆠ鼀
鼀矄盐
泐

鼀�ᆠᨨ鼀
ᨨ
矄
ᆠᨨ盐 



҈ᆠ鼀㡰

� 
鼀 �鼀盐鼀
ᨨ矄矄�
ᨨ
�
 
盐矄
ᨨ Ӹᨨ盐 �矄
鼀ᨨ盐盐矄

和ᆠ犨
�盐�
 
鼀
矄ᆠ

 ᨨ鼀
ᨨ
᲼




㜠


㝐├矄桌 꾀
矄ᆠ犨
�盐�
� �
҈


⼄盐
��矄�ᨨ 矄
 

�ᆠ
ᨨ  �
҈
鼀鼀
�矄
盐矄
ᨨ
鼀矄盐
 
�鼀
ᨨ

犨�
ᨨ  �
�


ᆠ鼀


鼀�
矄ᆠ
� �
�矄
҈
鼀鼀
 

䰘 ᆠ�ᆠ盐
��矄�ᨨ
ᆠᨨ� ᆠ�矄
犨�
 
鼀�
矄ᆠ

� 
  ᆠ盐
矄盐 ᆠ

ᨨ
犨�
�
ᆠ鼀

ᨨ
ᨨ�
矄�� � �ᆠ  ��盐
盐矄 p = 6








㡰
䰘

MCQs with One or More than One Correct Answer

ᔼ

 �鼀 
ᨨ
鼀� 
 
ᆠᆠ�ᨨ
盐
ᨨ
�㜠 �泐
�

 
ᨨ鼀盐
盐矄
ᨨ
 
�鼀
�ᨨ
ᨨ

�ᨨ�ᆠ 鼀ᨨ�ᨨ
�矄
=
p


黨


�



᲼




㜠


㝐├矄桌

䰘 - 㡰


盐矄
ᨨ

 

 ��

鼀矄盐
ᨨ
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矄



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矄
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矄鼀 矄㡰
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q
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q
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黨


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l
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├ ᨨ犨
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q
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ᨨ
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q

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q
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㡰 
矄ᨨ盐
ᨨ
�
d
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ᆠᨨ 矄  㡰 
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d
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d
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�
d
ᆠ矄矄 �҈
꾀
矄盐 
盐矄矄
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矄�鼀
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g

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盐
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矄ᆠ
ᨨ
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 
ᆠᨨ矄


矄ᆠᨨ 矄
ᨨ



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ᨨ
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ᨨ矄�ᨨ

犨 �ᆠ ᨨ
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ᨨᨨ犨� �
矄鼀 矄㡰
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㡰
盐䁸 ᲼

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桌 㡰 쟼ᨨ
DT
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鼀矄盐� �
T

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�ᨨ

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矄ᆠᨨ 矄 㡰 쟼ᨨ
DT
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T

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矄ᆠᨨ  ᆠ㡰 ᆠ �
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g
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�奠 8

Comprehension/Passage Based Questions

矄矄� ᔼ
 
鼀矄盐鼀 
ᨨ矄
�

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�   
⼄盐 ��矄
 ├ ᨨ犨

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�
�矄
ᨨ矄矄�
ᨨ
鼀� 
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ᨨ
�
 泐   
⼄盐 �泐

�矄
�矄
ᨨ 
泐
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盐矄
ᨨ
矄�矄
  矄�ᨨ

 
盐 ᆠ� �
 
  矄�ᨨ

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�矄
ᨨ犨
ᨨ
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ᨨ

ᨨ  鼀

ᆠᨨ 矄�
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�ᨨ
z
㦨
x⏌y

ᔼ
 
ᨨ矄
�
x

y
 
z 
Dx

Dy
 
Dz

矄ᆠ�泐

 

҈

x ± Dx x æ Dx ö æ Dy ö z
± Dz
㦨

㦨
ç҈ ± ÷
÷ ç҈ ± y ± Dy yè x øè y ø ҈

æ Dy ö


矄�矄
  矄�ᨨ
ᨨ
ç ҈ ± ÷

ᨨ
�矄
ᨨ犨
�
Dy⏌y

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m
Dy⏌y㡰

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

和ᨨ
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ᨨ
�
z
犨�
 æ Dx Dy ö + Dz
㦨
z ç ÷
y ø è x

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鼀├矄
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矄矄盐鼀�ᨨ
 
Dx⏌x =
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Dy⏌y
=
҈

 ᨨ

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⼄盐 ��矄
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᲼꾀



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桌 ҈  a 㡰 �

ᔴᨨ 矄�
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r =
ᨨ
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泐 ҈ + a 㡰 鼀矄盐� �

�鼀 矄�ᨨ 矄矄
⼄盐 �泐
a

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�
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ᨨ
a
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Da Da⏌a
=
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�
鼀� � �
r䁸 Da
D a
D a
a D a 㡰
㡰
ᆠ㡰
㡰 ҈ + a㡰 ҈ + a 㡰 ҈  a 㡰 ҈  a
㡰 �
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ᨨ
x
=
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x㡰
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x

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Dl

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l

�

s҈

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㡰


 ᆠ㡰



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҈

10 Subjective Problems �

ᔼ
n
��矄�ᨨ
ᨨ
鼀�
矄ᆠ
ᆠᨨ� ᆠ�矄
犨�
n졔҈㡰
��矄�ᨨ 矄 ᨨ
 �
矄ᆠ

෌�
ᨨ 
鼀�
矄ᆠ
��矄�ᨨ
�矄
⼄盐
ᨨ �a
盐 �矄

� 
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矄
ᆠᨨ盐 
ᨨ
 
 �
᲼



㜠


㝐├矄桌

WWW.JEEBOOKS.IN

EBD_8341 P-10

Physics

Topic-4 : Miscellaneous (Mixed Concepts) Problems 8

Comprehension/Passage Based Questions

矄矄� 꾀
 矄
ᆠᨨᆠ�ᨨ
ᨨ
⼄盐
盐鼀
ᨨ
ᆠᨨ 矄
 
ᨨ矄�� �ᨨ 矄
�矄
ᆠ
盐
矄鼀

ᔴ�
矄ᨨ�矄
ᆠᨨ � � �
�  ᨨ矄��
�ᨨ 矄
矄盐ᨨ盐 
泐

ᆠᨨ 矄
ᆠ
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
矄

盐
矄鼀


�

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盐鼀
 矄�泐
ᨨ

ᆠᨨ 矄
ᆠ
ᨨ
鼀矄矄
�鼀

␔ 
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ᆠᨨ 矄

矄盐ᆠ
ᨨ

ᆠ�ᆠ �

 泐

�矄ᆠ
�泐
犨泐
ᨨ鼀
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泐
ᨨ矄�� �ᨨ 矄

ᔼ
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ᆠ�ᆠ
�
ᆠᨨ鼀矄
ᨨ

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��
ᨨ ᨨ矄ᆠ�
ᨨ盐
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�ᨨ 矄
犨�

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 �盐
⼄盐 ᆠ泐 �w
犨 �ᆠ
�矄
ᆠ
 
矄鼀
⼄盐 ᆠ泐

ᨨ
矄盐矄�
  ᨨ矄ᆠ��ᨨ 矄


�鼀
泐� �
ᆠ�ᆠ
�
矄
ᨨ

�
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
 �盐
⼄盐 ᆠ泐
w

犨 


ᨨ
 
 �泐
�矄 矄ᨨ
 


ᨨ
�
�矄
 ᆠ

꾀矄
w ᨨᆠ 矄
w

 


ᆠᨨ 矄

矄
ᨨ
矄ᨨ  ᆠ
ᨨ� 
 

 
 �泐
�矄  ᆠ

�矄
�矄
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  �ᨨ
ᨨ
��
 ᆠ��泐
ᨨ
鼀矄
᲼

҈҈桌 ҈


├� �
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ᆠᨨ �ᆠ
ᆠ �
矄
�
 
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鼀���泐
矄 �e


矄
�鼀 矄�ᨨ 
 泐矄�矄
ᨨ
鼀� 
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ᆠᨨᆠ  矄矄�ᨨ
ᨨ
w
㡰



Ne 㡰 me


me
Ne

ᆠ㡰

Ne
me


㡰

Ne
me


쟼矄�鼀
 
犨 �

犨 �ᆠ
矄鼀
 ᆠ�ᨨ
犨� ᨨᆠᆠ盐
ᨨ

鼀
� �
 
 矄�泐
ᨨ
ᆠᨨ 矄 
»

´ ҈

䰘
鼀

├� �
e

㦨
҈
҈҈
 
鼀矄矄
鼀
» ҈


犨 
 矄
⼄盐 ��矄

�
ᨨ
和ᔼ
盐 �矄
㡰



鼀 㡰



鼀 ᆠ㡰 


鼀 㡰



鼀

Answer Key Topic-1 : Unit of Physical Quantities ҈
㡰



㡰






ᆠ

㡰 Topic-2 : Dimensions of Physical Quantities

҈
҈҈

҈
҈
䰘


㡰

㡰 㡰 ҈

㡰 㡰


㡰 ҈  ᲼㝐   꾀
桌 

㡰 



ᆠ㡰


҈




黨


ᆠ㡰

㡰 㡰 ҈

ᆠ㡰 ᆠ㡰


㡰 ҈  ᲼㝐   꾀
桌 

ᆠ

㡰




ᆠ㡰

�
҈�

�


҈


㡰

ᆠ㡰 ᨨ 㡰҈

㡰 㡰


㡰  
᲼㝐 
턼
桌 



ᆠ㡰

䰘
҈䰘

䰘








㡰

㡰 ҈

ᆠ㡰



҈ ᲼㝐 桌 

㡰



㡰 㡰 㡰 ᆠ㡰

黨
҈黨

黨
�

�


㡰 ҈

㡰


ᆠ㡰 



㡰 

ᆠ㡰




㡰 ᆠ㡰 㡰 

ᆠ㡰 㡰

Topic-3 : Errors in Measurements & Experimental Physics ҈
҈











㡰 㡰 㡰 㡰 
㡰



҈҈

҈
҈

҈


㡰 㡰 ᆠ㡰 ᆠ㡰 ҈
�
㡰


䰘




ᆠ鼀
㡰


҈











㡰 㡰 㡰 㡰 ҈

㡰



҈







㡰 ᆠ㡰 㡰 㡰 ҈
黨㡰

�
҈











ᆠ㡰 ᆠ㡰 㡰 㡰 


㡰








㡰
黨


ᆠ㡰 �

㡰



҈�

�
�

�


㡰 䰘
ᨨ 盐矄㡰

㡰 ҈

㡰 ҈䰘
㡰


㡰
䰘
㡰 

ᆠ㡰 䰘





�
ᆠ鼀㡰




�҈


ᆠ㡰 �

㡰

ᆠ㡰 黨
ᨨ 盐矄㡰 㡰 ҈

ᆠ㡰 ҈黨
㡰 㡰


㡰
黨
㡰 㡰 

㡰 黨

㡰 ҈

黨
맔
҈
҈

⏌鼀
㡰 æ  ö �
ᆠ㡰 �

ç ÷ è +҈ø

Topic-4 : Miscellaneous (Mixed Concepts) Problems ҈
ᆠ㡰



㡰

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P-11

Motion in a Straight Line

Motion in a Straight Line

2

Topic-1 : Distance, Displacement & Uniform Motion 1 ҈






꾀
�ᆠ
�矄
鼀ᨨ� �
犨�
矄
v = b x
ᨨ �
ᨨ矄�� x㜠 �矄

ᔴᆠ盐
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矄
ᨨ
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
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t
㦨 t 矄矄盐鼀  
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ᨨ���


㦨

㡰





᲼㝐�
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꾀



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ᔼᔼ桌 㡰

b
t


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ᆠ㡰

b t

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t


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t

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ᆠ㡰
҈
⏌
鼀⏌矄

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ᨨ矄��ᨨ

㡰


ᔼ
҈


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A
ᨨ
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B

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矄鼀�ᆠ�ᆠ
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ᨨᆠ�泐 A ᲼҈黨黨黨和
㜠


㝐├矄桌

҈ m ⏌ s
ᨨ犨矄
ᨨ


10 Subjective Problems



㡰 
҈

鼀⏌矄

҈ m ⏌ s
ᨨ犨矄
ᨨ 㜠犨矄


�鼀

㡰 


MCQs with One or More than One Correct Answer

㡰
ᨨ

�矄 ᆠ

ᨨ矄��ᨨ

꾀
�ᆠ
鼀ᨨ矄
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ᆠ�ᆠ
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R

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�矄 ᆠ ᆠᨨ
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�  
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矄鼀 鼀ᨨ�ᨨ

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ᨨ矄
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�
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桌 ᨨᆠ�泐

Fill in the Blanks

4

MCQs with One Correct Answer

꾀 矄犨
 
ᨨᨨ犨� �
��� � 矄ᨨ 矄
�
�































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鼀⏌矄

1.0m

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��ᨨ
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矄 ᨨ犨
�
 
��盐

ᆠ㡰 ҈


鼀⏌矄

ᨨ矄
�
盐䁸



᲼҈黨䰘黨桌 㡰 ᨨ B

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EBD_8341 P-12

Physics

Topic-2 : Non-uniform Motion

҈


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MCQs with One Correct Answer


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v㡰
 
�鼀
t㡰
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矄��  � 
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矄 ᨨ犨
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S
�矄
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ᨨ
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泐
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t
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C

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鼀 
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盐
ᨨ
鼀矄矄


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矄
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犨
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
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b
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c
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


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c

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b
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x
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쟼







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


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矄�� 
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矄


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�鼀

�鼀

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P-13

Motion in a Straight Line

ᆠ㡰

㡰

2 ᨨᆠ�泐

ᨨᆠ�泐

�矄 ᆠ

�矄 ᆠ

黨


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t
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t


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a

t㡰
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졠 � 
㝐y
҈




҈
桌

Integer Value Answer

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鼀ᨨ� �
�
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ᨨ


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졔
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A
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x �ᆠ�ᨨ
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
鼀⏌矄 a

a

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鼀

㡰 3 t

t

a ᆠ㡰

a

Numeric Answer

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矄
矄矄
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t
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t
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矄
犨�
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WWW.JEEBOOKS.IN

EBD_8341 P-14

Physics

Topic-3 : Relative Velocity in One Dimension 1 ҈













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L

M

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
㝐├矄桌 9 Assertion and Reason Type Questions 和T꾀TE㝐ENT㜠҈


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和T꾀TE㝐ENT㜠



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㝐├矄桌

Topic-4 : Motion Under Gravity 1 ҈


MCQs with One Correct Answer

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P-15

Motion in a Straight Line

(a) t =

2.

2 æhö ç ÷ 3 ègø

(The figures are schematic and not drawn to scale)

h g

(b) t = 1.8

240

æhö 2h (c) t = 3.4 ç ÷ (d) t = 3g ègø A Tennis ball is released from a height h and after freely falling on a wooden floor it rebounds and reaches height

(a)

h . The velocity versus height of the ball during its motion 2

(c)

may be represented graphically by : (graph are drawn schematically and on not to scale) [Main 4 Sep. 2020 (I)] h/2

h(v)

(b)

(c)

h h/2

h(v)

(d)

h h/2

h(v)

A body is thrown vertically upwards. Which one of the following graphs correctly represent the velocity vs time? [Main 2017]

3.

(b)

(a)

240

8 (y2 – y1) m

12

t(s)

(d) t® 8

6.

v

v

t(s)

12

(y2 – y1 ) m

t(s)

12

12

t(s)

From a tower of height H, a particle is thrown vertically upwards with a speed u. The time taken by the particle, to hit the ground, is n times that taken by it to reach the highest point of its path. The relation between H, u and n is: [Main 2014] (a) 2gH = n2u2 (b) gH = (n – 2)2 u2 d (c) 2gH = nu2 (n – 2) (d) gH = (n – 2)u2 A ball is dropped vertically from a height d above the ground. It hits the ground and bounces up vertically to a height d/2. Neglecting subsequent motion and air resistance, its velocity v varies with the height h above the ground as [2000S]

5.

h(v)

h

h

(y2 – y1 ) m

(b)

240

h/2

(a)

240

8

v

v

(y2 – y1) m

(a)

(b)

(c)

(d)

3 Numeric Answer A ball is dropped from the top of a 100 m high tower on a

7.

1 s before hitting the ground, it covers a 2 distance of 19 m. Acceleration due to gravity (in ms–2) near the surface on that planet is _______. [Main 8 Jan. 2020 (II)]

planet. In the last

(c)

(d)

Two stones are thrown up simultaneously from the edge of a cliff 240 m high with initial speed of 10 m/s and 40 m/s respectively. Which of the following graph best represents the time variation of relative position of the second stone with respect to the first ? (Assume stones do not rebound after hitting the ground and neglect air resistance, take g = 10 m/ s2) [Main 2015]

4.

5 True / False Two balls of different masses are thrown vertically upwards with the same speed. They pass through the point of projection in their downward motion with the same speed (Neglect air resistance). [1983 - 2 Marks]

8.

Answer Key Topic-1 : Distance, Displacement & Uniform Motion 4. (2R, pR) 5. (b) Topic-2 : Non-uniform Motion 3. (b) 4. (d) 5. (c) 6. (c) 7. (a) 13. (20) 14. (3) 15. (a, c, d) Topic-3 : Relative Velocity in one Dimension

1.

(b)

2. (b)

3. (b)

1. 11.

(a) (b)

2. (b) 12. (8)

1.

(a)

2. (d)

3. (c)

1.

(c)

2. (c)

3. (a)

8. (c)

9. (b)

10. (a)

ædö ç v ÷ 7. (False) 8. (b) è ø Topic-4 : Motion Under Gravity 4. (b) 5. (c) 6. (a) 7. (8.00) 8. (True) 4. (c)

5. (5)

6.

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EBD_8341 P-16

Physics

3

Motion in a Plane

Topic-1 : Vectors MCQs with One Correct Answer uur uuur uur uuur 
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Numeric Answer ®





)

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P-17

Motion in a Plane

Topic-2 : Motion in a Plane with Constant Acceleration 1 ҈


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EBD_8341 P-18

Physics

Topic-3 : Projectile Motion 1 1.

A particle of mass m is projected with a speed u from the p ground at an angle q = w.r.t. horizontal (x-axis). When 3 it has reached its maximum height, it collides completely inelastically with another particle of the same mass and velocity uiˆ. The horizontal distance covered by the combined mass before reaching the ground is: [Main 9 Jan. 2020 (II)] (a)

2.

3 3 u2 8 g

(b)

3 2 u2 4 g

u2 5 u2 2 2 (c) (d) 8 g g The trajectory of a projectile near the surface of the earth is given as y = 2x – 9x2. If it were launched at an angle q0 with speed v0 then (g = 10 ms–2): [Main 12 April 2019 (I)] 5 1 (a) q0 = sin –1 and v0 = ms–1 3 5 3 æ 2 ö ms–1 (b) q0 = cos–1 ç ÷ and v0 = 5 è 5ø

3.

4.

5.

9 æ 1 ö (c) q0 = cos–1 ç ÷ and v0 = ms–1 3 è 5ø 3 æ 2 ö (d) q0 = sin –1 ç ÷ and v0 = ms–1 5 è 5ø A shell is fired from a fixed artillery gun with an initial speed u such that it hits the target on the ground at a distance R from it. If t1 and t2 are the values of the time taken by it to hit the target in two possible ways, the product t1t2 is : [Main 12 April 2019 (I)] (a) R/4g (b) R/g (c) R/2g (d) 2R/g Two particles are projected from the same point with the same speed u such that they have the same range R, but different maximum heights, h1 and h2. Which of the following is correct ? [Main 12 April 2019 (II)] (a) R2 = 4 h1h2 (b) R2 =16 h1h2 (c) R2 = 2 h1h2 (d) R2 = h1h2 A plane is inclined at an angle a = 30o with respect to the horizontal. A particle is projected with a speed u = 2 ms–1, from the base of the plane, as shown in figure. The distance from the base, at which the particle hits the plane is close to: (Take g=10 ms–2) [Main 10 April 2019 (II)]

(a) 20 cm (b) 18 cm

(c) 26 cm

A body is projected at t = 0 with a velocity 10 ms–1 at an angle of 60° with the horizontal. The radius of curvature of its trajectory at t = 1s is R. Neglecting air resistance and taking acceleration due to gravity g = 10 ms–2, the value of R is: [Main 11 Jan. 2019 (I)] (a) 10.3 m (b) 2.8 m (c) 2.5 m (d) 5.1 m 7. Two guns A and B can fire bullets at speeds 1 km/s and 2 km/s respectively. From a point on a horizontal ground, they are fired in all possible directions. The ratio of maximum areas covered by the bullets fired by the two guns, on the ground is: [Main 10 Jan. 2019 (I)] (a) 1 : 16 (b) 1 : 2 (c) 1 : 4 (d) 1 : 8 8. The position of a projectile launched from the origin at r t = 0 is given by r = 40iˆ + 50 ˆj m at t = 2s. If the projectile was launched at an angle q from the horizontal, then q is (take g = 10 ms–2) [Main Online April 9, 2014] 2 -1 -1 4 -1 3 -1 7 (a) tan (c) tan (d) tan (b) tan 3 5 2 4 9. A projectile is given an initial velocity of (iˆ + 2 ˆj ) m/s, where iˆ is along the ground and ˆj is along the vertical. If g = 10 m/s2 , the equation of its trajectory is [Main 2013] (a) y = x - 5 x 2 (b) y = 2 x - 5 x 2 (c) 4 y = 2 x - 5 x 2 (d) 4 y = 2 x - 25 x 2 10. A ball projected from ground at an angle of 45° just clears a wall in front. If point of projection is 4 m from the foot of wall and ball strikes the ground at a distance of 6 m on the other side of the wall, the height of the wall is : [Main Online April 22, 2013] (a) 4.4 m (b) 2.4 m (c) 3.6 m (d) 1.6 m 6.

MCQs with One Correct Answer

(d) 14 cm

(

2

)

Integer Value Answer

11. A ball is thrown from ground at angle q with horizontal and with an initial speed u0. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is V1. After hitting the ground, the ball rebounds at the same angle q but with a reduced speed of u0/a. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is 0.8 V1, the value of a is ____ [Adv. 2019]

12. A train is moving along a straight line with a constant acceleration ‘a’. A boy standing in the train throws a ball forward with a speed of 10 m/s, at an angle of 60° to the horizontal. The boy has to move forward by 1.15 m inside the train to catch the ball back at the initial height. The acceleration of the train, in m/s2, is [Adv. 2011]

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P-19

Motion in a Plane

5 ҈


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Topic 4 : Relative Velocity in Two Dimensions & Uniform Circular Motion 1 ҈


MCQs with One Correct Answer

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EBD_8341 P-20

Physics

When a car sit at rest, its driver sees raindrops falling on it vertically. When driving the car with speed v, he sees that raindrops are coming at an angle 60º from the horizontal. On furter increasing the speed of the car to (1 + b)v, this angle changes to 45º. The value of b is close to: [Main Sep. 06, 2020 (II)] (a) 0.50 (b) 0.41 (c) 0.37 (d) 0.73 The stream of a river is flowing with a speed of 2 km/h. A swimmer can swim at a speed of 4 km/h. What should be the direction of the swimmer with respect to the flow of the river to cross the river straight? [Main 9 April 2019 (I)] (a) 90° (b) 150° (c) 120° (d) 60° Ship A is sailing towards north-east with velocity km/hr where points east and , north. Ship B is at a distance of 80 km east and 150 km north of Ship A and is sailing towards west at 10 km/hr. A will be at minimum distance from B in: [Main 8 April 2019 (I)] (a) 4.2 hrs.(b) 2.6 hrs. (c) 3.2 hrs. (d)2.2 hrs. Two particles A, B are moving on two concentric circles of radii R1 and R2 with equal angular speed w. At t = 0, their positions and direction of motion are shown in the

2.

3.

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5.

p ö æ figure : ç t = 2w ÷ø è

6.

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a

(c)

(d)

r r A boat which has a speed of 5 km/hr in still water crosses a river of width 1 km along the shortest possible path in 15 minutes. The velocity of the river water in km/hr is [1988 - 1 Mark]

(a) 1

X

9.

B

® ® The relative velocity vA - vB is given by: (a) w(R1 + R2) iˆ (b) –w(R1 + R2) iˆ (c) w(R2 – R1) iˆ (d) w(R1 – R2) iˆ

r

a

A

R2

(b) r

8.

R1

a

a

(a)

[Main 12 Jan. 2019 (II)] Y

A particle is moving along a circular path with a constant speed of 10 ms–1. What is the magnitude of the change in velocity of the particle, when it moves through an angle of 60° around the centre of the circle? [Main Online April 10, 2015] (a) 10 3m/s (b) zero (d) 10 m/s (c) 10 2m/s If a body moving in circular path maintains constant speed of 10 ms–1, then which of the following correctly describes relation between acceleration and radius? [Main Online April 10, 2015]

(b) 3

(c) 4

(d) 41 A river is flowing from west to east at a speed of 5 metres per minute. A man on the south bank of the river, capable of swimming at 10 metres per minute in still water, wants to swim across the river in the shortest time. He should swim in a direction [1983 - 1 Mark] (a) due north (b) 30° east of north (c) 30° west of north (d) 60° east of north

Answer Key Topic-1 : Vectors 1.

(d)

2. (c)

3. (a)

4. (a)

5. (a)

6. (195)

7. (90)

8. (2.00)

Topic-2 : Motion in a Plane with Constant Acceleration 1. 10.

(d) 2. (a) (a, b, c)

3. (c)

4. (d)

5. (d)

6. (b)

7. (c)

8. (580)

9. (a, b, c, d)

7. (a)

8. (c)

9. (b)

Topic-3 : Projectile Motion 1. 11.

(a) 2. (c) (4.00) 12. (5)

3. (d) 13. (True)

4. (b)

5. (a)

6. (b)

10. (b)

Topic-4 : Relative Velocity in two Dimensions & Uniform Circular Motion 1.

(a)

2. (d)

3. (c)

4. (b)

5. (c)

6. (d)

7. (c)

8. (b)

9. (a)

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P-21

Laws of Motion

Laws of Motion

4

Topic-1 : Ist, IInd & IIIrd Laws of Motion MCQs with One Correct Answer

1 ҈






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ᨨᆠ
F = k  v y �$ + v x $ 㡰

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vx
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x r r  
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v

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a
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v × a
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矄ᆠ矄 �
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矄犨矄
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�  泐
盐矄






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ᨨ
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


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g �矄
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g ö ҈ g ö ҈ 
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÷÷ � ø g� è
g � è ø 꾀
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ᨨ鼀
  �ᨨ盐 


ᆠᨨᆠ
鼀ᨨ鼀 盐鼀㜠 �� 
㜠 㡰
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�矄


㡰

�


҈

᲼㝐�
黨
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҈黨
ᔼ桌

dM t 㡰 = bv
t 㡰
dt 犨 
v t㡰
�矄
�矄
� 矄  ᨨ盐矄
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
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矄�
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᲼㝐�
和


�






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꾀矄

矄盐

�矄
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� ᆠ矄矄





㡰 -bv t 㡰

㡰 -

bv M t 㡰


bv M t 㡰

㡰 -

bv 
M t 㡰

ᆠ㡰 


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矄鼀

ᨨ
鼀矄矄
m
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

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ᨨᆠ
mkv
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v
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鼀 �鼀盐鼀
�� 
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泐
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和










ᔼᔼ㡰桌





㡰

㡰

ᆠ㡰

㡰

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�ᆠ
ᨨ
鼀矄矄
鼀
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
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� 
犨� 鼀ᨨ鼀 盐鼀


和� �
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�鼀

㦨




ᨨᆠ

㦨
k
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  矄鼀
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矄ᨨ
 

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鼀ᨨ鼀 盐鼀
ᆠ  �矄
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ᨨ


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k
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
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ᨨ

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᲼㝐�
҈҈




҈黨
ᔼᔼ桌

㡰

ku
҈  -҈ g
k

㡰

҈ æ ku
ö  ҈ +
g ÷ø k çè

㡰

ᆠ㡰

ku
҈  -҈ k
g

㡰

҈ æ ku
ö  ҈ +
k çè g ÷ø

ᆠ㡰




k 

㡰


k 

㡰




 k
p ├

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EBD_8341 P-22

䰘


Physics

AB

BC  
CD

矄�� 
� 
矄�鼀 矄


鼀� �盐 ᨨ
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ᨨ
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ᨨ
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ᨨᆠ
ᨨ
 
ᨨ泐
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t
㦨


m矄 ᨨ
t
㦨
҈
m矄
�矄















矄
᲼҈黨黨

㜠


㝐├矄桌 ᔴ




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ᨨ
鼀矄矄
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盐ᨨ
泐

ᨨᆠ

��
泐





ᨨᆠ
㡰

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犨
=

㡰

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矄
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҈




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ᨨ
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>>
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矄


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ᨨ �
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᲼㝐�
졠 � 
꾀�
҈҈



҈�桌




5











쟼





҈


�鼀
m 矄㡰

҈


҈
҈


True / False

҈

꾀
ᨨᆠ├
鼀ᨨ矄
ᨨ犨
泐
盐矄 � �
 
矄盐ᨨ盐 � �
� ᆠ├犨矄
᲼҈黨

桌 6

MCQs with One or More than One Correct Answer

҈
꾀
  ᆠ
鼀
ᆠ 
ᨨ
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 ᲼҈黨


㜠


㝐├矄桌 㡰 �矄

� �
鼀
泐
 � ��ᨨ
㡰 ᆠ

ᨨ


� �
鼀
ᆠ盐矄
 

�矄 ᨨ� �
ᨨ盐 
 
矄盐
ᆠ㡰 �矄

� �
鼀
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�
 �矄
鼀
㡰 ᆠ

ᨨ


� �
鼀
ᆠ盐矄
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
�矄 ᨨ� �
ᨨ盐
�矄
ᨨ犨
 �矄




㡰

mv 2 2(L - nr)

㡰

mv 2 L - 2nr

9

mv 2 㡰 ᨨ L - nr 꾀
ᨨ泐
ᨨ
鼀矄矄
�
├�
盐 
 
ᆠ�ᨨ
ᨨ
ᆠᨨ 矄 
ᨨᆠ r r = ˆ� + ˆ
矄
ᨨᆠ�泐


㦨


矄
矄
 =
�ˆ -
ˆ 鼀⏌矄

ᆠ㡰 黨




(

泐

)

r r  


㦨
҈
矄
矄
 = +
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
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졠 � 
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҈҈



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桌

҈



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( -ˆ +
ˆ) 

㡰

æ  ˆ
ˆö ç - � + ÷  è � � ø

ᆠ㡰

( �ˆ -
ˆ) 

㡰

æ ˆ
ˆö ç � - ÷  è� � ø

꾀
�ᆠ
鼀ᨨ矄
�
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X㜠Y
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盐 
 
� 盐 ᆠ
ᨨ
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矄盐ᆠ
 
�矄
� 
鼀ᨨ鼀 盐鼀
�矄 r pt 㡰 = A ᲼�ˆ ᆠᨨ矄kt 㡰 - ˆ 矄� kt 㡰桌

犨 
A
 
k
 ᆠᨨ 矄 矄


 �
犨
 
ᨨᆠ
 
 
鼀ᨨ鼀 盐鼀 �矄 ᲼


䰘桌 㡰
㡰 
ᆠ㡰
� 㡰 黨
4

҈҈


Fill in the Blanks

Assertion and Reason Type Questions

҈

和鼀 㜠҈


ᔼ
�矄
矄�
ᨨ
盐
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 
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�
ᨨ


�ᨨ盐 
  和鼀 㜠



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ᨨ
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犨ᨨ
矄盐 ᆠ矄
�
ᆠᨨ ᆠ
᲼



桌 㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
盐錐
和鼀 㜠

�矄

ᆠᨨᆠ
  �ᨨ
ᨨ
和鼀 㜠҈ 㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

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盐錐
和鼀 㜠

�矄
졠

ᆠᨨᆠ
  �ᨨ
ᨨ

和鼀 㜠҈ ᆠ㡰 和鼀 
㜠҈
�矄
盐

和鼀 㜠

�矄
矄 㡰 和鼀 
㜠҈
�矄
矄

和鼀 㜠

�矄
盐 ҈�
和T꾀TE㝐ENT㜠҈




꾀
ᆠᨨ
ᆠᨨ矄



和ᨨ鼀
�矄 矄 
├
ᨨ
�

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ᆠᨨ
ᆠ

盐
ᨨ盐
犨� ᨨ盐 �矄ᨨ�� �
 
�矄 矄
ᨨ鼀
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
和T꾀TE㝐ENT㜠



ᨨ
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 
�矄

⼄盐
  ᨨᨨ矄�
ᆠ�ᨨ
᲼


䰘桌 㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
盐錐
和㜠 鼀 㜠

�矄

ᆠᨨᆠ
  �ᨨ
ᨨ
和鼀 㜠҈ 㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
盐錐
和㜠 鼀 㜠

�矄
졠

ᆠᨨᆠ
  �ᨨ
ᨨ
和鼀 㜠҈ ᆠ㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
矄 㡰 和鼀 㜠҈
�矄
矄

和鼀 㜠

�矄
盐



鼀� �盐
ᨨ
 
ᨨᆠ
�
犨ᨨ 矄㡰
ᆠ� �
ᨨ

ᨨ泐 �矄
犨�
�鼀
t
�
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P-23

Laws of Motion

Topic-2 : Motion of Connected Bodies, Pulley & Equilibrium of Forces





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EBD_8341 P-24

Physics


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P-25

Laws of Motion

Topic-3 : Friction

An insect is at the bottom of a hemispherical ditch of radius 1 m. It crawls up the ditch but starts slipping after it is at height h from the bottom. If the coefficient of friction between the ground and the insect is 0.75, then h is : (g = 10 ms–2) [Main Sep. 06, 2020 (I)]

Two masses m1 = 5 kg and m2 = 10 kg, connected by an inextensible string over a frictionless pulley, are moving as shown in the figure. The coefficient of friction of horizontal surface is 0.15. The minimum weight m that should be put on top of m2 to stop the motion is: [Main 2018] T m (a) 18.3 kg m2 m 2

(a) 0.20 m

(b) 27.3 kg

1 1.

2.

(b) 0.45 m

(c) 0.60 m (d) 0.80 m A block of mass 5 kg is (i) pushed in case (A) and (ii) pulled in case (B), by a force F=20 N, making an angle of 30 o with the horizontal, as shown in the figures. The coefficient of friction between the block and floor is m = 0.2. The difference between the accelerations of the block, in case (B) and case (A) will be : (g =10 ms–2) [Main 12 April 2019 (II)]

–2

3.

5.

MCQs with One Correct Answer

m1

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8. (a) 8 N (b) 16 N (c) 40 N (d) 12 N A block of mass 10 kg is kept on a rough inclined plane as shown in the figure. A force of 3 N is applied on the block. The coefficient of static friction between the plane and the block is 0.6. What should be the minimum value of force P, such that the block doesnot move downward? (take g = 10 ms–2) [Main 9 Jan. 2019 (I)]

1

(d)

2 - n2

1 1 - n2

A rocket is fired vertically from the earth with an acceleration of 2g, where g is the gravitational acceleration. On an inclined plane inside the rocket, making an angle q with the horizontal, a point object of mass m is kept. The minimum coefficient of friction mmin between the mass and the inclined surface such that the mass does not move is : [Main Online April 9, 2016] (a) tan2q (b) tanq (c) 3 tanq (d) 2 tan q Given in the figure are two blocks A and B of weight 20 N and 100 N, respectively. These are being pressed against a wall by a force F as shown. If the coefficient of friction between the blocks is 0.1 and between block B and the wall is 0.15, the frictional force applied by the wall on block B is: [Main 2015] F

A

B

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A given object takes n times more time to slide down a 45° rough inclined plane as it takes to slide down a perfectly smooth 45° incline. The coefficient of kinetic friction between the object and the incline is : [Main Online April 15, 2018] 1 1 1- 2 (b) 1 - 2 (a) n n (c)

–2

(a) 0.4 ms (b) 3.2 ms (c) 0.8 ms–2 (d) 0 ms–2 Two blocks A and B masses mA=1 kg and mB = 3 kg are kept on the table as shown in figure. The coefficient of friction between A and B is 0.2 and between B and the surface of the table is also 0.2. The maximum force F that can be applied on B horizontally, so that the block A does not slide over the block B is : [Take g = 10 m/s2] [Main 10 April 2019 (II)]

T

(c) 43.3 kg

(a) 32 N

45°

(b) 18 N

(c) 23 N

(d)

25 N

(a) 120 N (c) 100 N

(b) 150 N (d) 80 N

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EBD_8341 P-26

Physics

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P-27

Laws of Motion


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ᔼᔼ㡰桌 5






Integer Value Answer

MCQs with One or More than One Correct Answer

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꾀
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P
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矄�� �
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A
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t
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 �矄
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7

턼

A 

B ᔴ

Match the Following


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 ᲼꾀



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g
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턼㜠


㜠

和㜠 10 Subjective Problems


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㝐桌 ᨨ
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�
鼀⏌矄

q

WWW.JEEBOOKS.IN

EBD_8341 P-28


黨


Physics

犨ᨨ
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B
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P-29

Laws of Motion

Topic-4 : Circular Motion & Banking of Road 1 1.

2.

5.

MCQs with One Correct Answer

A disc rotates about its axis of symmetry in a hoizontal plane at a steady rate of 3.5 revolutions per second. A coin placed at a distance of 1.25cm from the axis of rotation remains at rest on the disc. The coefficient of friction between the coin and the disc is (g = 10m/s2) [Main Online April 15, 2018] (a) 0.5 (b) 0.7 (c) 0.3 (d) 0.6 A conical pendulum of length 1 m makes an angle q = 45° w.r.t. Z-axis and moves in a circle in the XY plane.The radius of the circle is 0.4 m and its centre is vertically below O. The speed of the pendulum, in its circular path, will be : [Main Online April 9, 2017] (Take g = 10 ms–2) Z (a) 0.4 m/s

(a)

(c) 5 6.

O q

(b) 4 m/s (c) 0.2 m/s 3.

7.

X Y

q f O

4.

(b)

1 sin f = cos q 2 3 sin f = cos q 4

2 (c) sin f = cos q (d) 3 A ball of mass (m) 0.5 kg is attached to the end of a string having length (L) 0.5 m. The ball is rotated on a horizontal circular path about vertical axis. The maximum tension that the string can bear is 324 N. The maximum possible value of anguar velocity of ball (in radian/s) m is [2011] (a) 9 (b) 18 (c) 27

L

8.

v

v

(b)

(d)

v

v

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A simple pendulum with a bob of mass m swings with an angular amplitude of 40°. When its angular displacement is 20°, the tension in the string is greater than mg cos 20°. [1984 - 2 Marks] 6

C

(d) 2 m/s A particle is released on a vertical smooth semicircular track from point X so that OX makes angle q from the vertical (see figure). The normal reaction of the track on the particle vanishes at point Y where OY makes angle f with the horizontal. Then: [Main Online April 19, 2014]

(a) sin f = cos f

A small block is shot into each of the four tracks as shown below. Each of the tracks rises to the same height. The speed with which the block enters the track is the same in all cases. At the highest point of the track, the normal reaction is maximum in [2001S]

MCQs with One or More than One Correct Answer

A wire, which passes through the hole in a small bead, is bent in the form of quarter of a A circle. The wire is fixed vertically on ground as shown in the figure. The bead is released from near the top of the wire and it slides along the wire 90° without friction. As the bead B moves from A to B, the force it applies on the wire is [Adv. 2014] (a) always radially outwards (b) always radially inwards (c) radially outwards initially and radially inwards later (d) radially inwards initially and radially outwards later A simple pendulum of length L and mass (bob) M is oscillating in a plane about a vertical line between angular limit – f and + f . For an angular displacement q ( | q | < f ), the tension in the string and the velocity of the bob are T and V respectively. The following relations hold good under the above conditions : [1986 - 2 Marks] (a) T cos q = Mg. MV 2 L (c) The magnitude of the tangenial acceleration of the bob | aT | = g sin q (d) T = Mg cos q

(b) T – Mg cos q =

(d) 36

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EBD_8341 P-30

Physics

10 Subject Problems 꾀
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P-31

Work, Energy and Power

5

Work, Energy and Power

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EBD_8341 P-32

7.

8.

Physics

The work done on a particle of mass m by a force, é ù x y ˆj ú Kê iˆ + 2 2 32 2 2 32 (x + y ) êë ( x + y ) úû (K being a constant of appropriate dimensions), when the particle is taken from the point (a, 0) to the point (0, a) along a circular path of radius a about the origin in the x – y plane is [Adv. 2013] Kp Kp 2K p (a) (b) (c) (d) 0 a a 2a If W1, W2 and W3 represent the work done in moving a particle from A to B along three different paths 1,2 and 3 respectively (as shown) in the gravitational field of a point mass m, find the correct relation between W1, W2 and W3 [2003S] B

(a) W1> W2 > W3 (b) W1= W2 = W3

9.

2

1

Integer Value Answer

11. A light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses 0.36 kg and 0.72 kg. Taking g = 10 m/s2, find the work done (in joules) by the string on the block of mass 0.36 kg during the first second after the system is released from rest. [2009] 3

m

Numeric Answer

12. A particle is moved along a path AB-BC-CD-DE-EF-FA, as ur shown in figure, in presence of a forced F = (ayi$ + 2ax $j ) N , where x and y are in meter and a = –1Nm–1. The work done on the particle by this force ur [Adv. 2019] F will be ___ Joule.

2 3

(c) W1< W2 < W3 (d) W2> W1 > W3

10. A uniform chain of length L and mass M is lying on a smooth table and one third of its length is hanging vertically down over the edge of the table. If g is acceleration due to gravity, the work required to pull the hanging part on to the table is [1985 - 2 Marks] (a) MgL (b) MgL/3 (c) MgL/9 (d) MgL/18

A

y

A force F = - K ( yiˆ + xjˆ) (where K is a positive constant) acts on a particle moving in the xy plane. Starting from the origin, the particle is taken along the positive x axis to the point (a, 0), and then parallel to the y axis to the point (a, a), The total work done by the force F on the particle is [1998S - 2 Marks] (a) – 2Ka2 (b) 2Ka2 (c) – Ka2 (d) Ka 2

A

1.0

B

D

.5 F

C

E 0.5

0

x

1.0

Topic-2 : Energy 1 1.

MCQs with One Correct Answer

A particle moves in one dimension from rest under the influence of a force that varies with the distance travelled by the particle as shown in the figure. The kinetic energy of the particle after it has travelled 3 m is : [Main 7 Jan. 2020 II]

3.

4.

1

1 (d) n 2 n n A body of mass 1 kg falls freely from a height of 100m, on a platform of mass 3 kg which is mounted on a spring having spring constant k = 1.25 × 106 N/m. The body sticks to the platform and the spring’s maximum compression is found to be x. Given that g = 10 ms–2, the value of x will be close to : [Main 11 April 2019 (I)] (a) 40 cm (b) 4 cm (c) 80 cm (d) 8 cm A uniform cable of mass ‘M’ and length ‘L’ is placed on a

(a) n

(b)

2

(c)

th

2.

(a) 4 J (b) 2.5 J (c) 6.5 J (d) 5 J A spring whose unstretched length is l has a force constant k. The spring is cut into two pieces of unstretched lengths 11 and l2 where, l1 = nl2 and n is an integer. The ratio k1/k2 of the corresponding force constants, k1 and k2 will be: [Main 12 April 2019 (II)]

æ 1ö horizontal surface such that its ç ÷ part is hanging è nø below the edge of the surface. To lift the hanging part of the cable upto the surface, the work done should be: [Main 9 April 2019 (I)] 2 MgL MgL MgL (a) (b) (c) (d) nMgL 2 2 2n n n2

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P-33

Work, Energy and Power

5.

A particle moves in one dimension from rest under the influence of a force that varies with the distance travelled by the particle as shown in the figure. The kinetic energy of the particle after it has travelled 3 m is : [Main 8 April 2019 (I)]

11. A point particle of mass m, moves long the uniformly rough track PQR as shown in the figure. The coefficient of friction, between the particle and the rough track equals m. The particle is released, from rest from the point P and it comes to rest at a point R. The energies, lost by the ball, over the parts, PQ and QR, of the track, are equal to each other, and no energy is lost when particle changes direction from PQ to QR. The value of the coefficient of friction m and the distance x (= QR), are, respectively close to : [Main 2016] P

6.

(a) 4 J (b) 2.5 J (c) 6.5 J (d) 5 J A block of mass m, lying on a smooth horizontal surface, is attached to a spring (of negligible mass) of spring constant k. The other end of the spring is fixed, as shown in the figure. The block is initally at rest in its equilibrium position. If now the block is pulled with a constant force F, the maximum speed of the block is:[Main 9 Jan. 2019 (I)] m

(a) 7.

2F mk

(b)

F p mk

(c)

F

pF mk

(a) -

k 4a

(c) zero

2

(b)

10.

mk

k 2r 2

. Its total energy is: [Main 2018]

k

2a 2 3 k (d) 2 a2

A body of mass m = 10–2 kg is moving in a medium and experiences a frictional force F = –kv2. Its intial speed is v0 = 10 ms–1. If, after 10 s, its energy is

9.

F

A particle is moving in a circular path of radius a under the action of an attractive potential U = -

8.

(d)

1 2 mv0 , the value of k 8

will be: [Main 2017] (a) 10–4 kg m–1 (b) 10–1 kg m–1 s–1 (c) 10–3 kg m–1 (d) 10–3 kg s–1 An object is dropped from a height h from the ground. Every time it hits the ground it looses 50奠 of its kinetic energy. The total distance covered as t ® ¥ is [Main Online April 8, 2017] 8 5 h (a) 3h (b) ¥ (c) (d) h 3 3 A time dependent force F = 6t acts on a particle of mass 1 kg. If the particle starts from rest, the work done by the [Main 2017] force during the first 1 second will be (a) 9 J (b) 18 J (c) 4.5 J (d) 22 J

h=2m

30° Horizontal Surface

R Q

(a) 0.29 and 3.5 m (b) 0.29 and 6.5 m (c) 0.2 and 6.5 m (d) 0.2 and 3.5 m 12. A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up? Fat supplies 3.8 × 107 J of energy per kg which is converted to mechanical energy with a 20奠 efficiency rate. Take g = 9.8 ms–2 : [Main 2016] –3 –3 (a) 9.89 × 10 kg (b) 12.89 × 10 kg (c) 2.45 × 10–3 kg (d) 6.45 × 10–3 kg 13. A particle is moving in a circle of radius r under the action of a force F = ar2 which is directed towards centre of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy = 0 for r = 0) : [Main Online April 11, 2015] 5 3 4 3 1 3 αr ar (b) ar (c) (d) ar3 6 3 2 14. A block of mass m = 0.1 kg is connected to a spring of unknown spring constant k. It is compressed to a distance x from its equilibrium position and released from rest. After approaching half the distance æç x ö÷ from equilibrium è 2ø position, it hits another block and comes to rest momentarily, while the other block moves with a velocity 3 ms–1. The total initial energy of the spring is : [Main Online April 10, 2015] (a) 0.3 J (b) 0.6 J (c) 0.8 J (d) 1.5 J

(a)

th

æ1ö 15. A bullet looses ç ÷ of its velocity passing through ènø one plank. The number of such planks that are required to stop the bullet can be: [Main Online April 19, 2014] (a)

n2 2n 2 (b) 2n - 1 n -1

(c) infinite

(d) n

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EBD_8341 P-34

҈



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Physics

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
ᆠᨨ鼀� �ᨨ
�矄
ᆠᨨ鼀矄矄
泐


䰘�
ᆠ鼀

 E E �ᨨ
ᨨ

 �泐
矄ᨨ
�
꾀
 

�矄
A


A
�矄 EB EB ⼄盐 ᲼㝐�
졠 � 
꾀�
黨



҈桌 ᨨ


҈
黨  㡰 㡰 ᆠ㡰 㡰  黨 ҈

꾀
ᨨᆠ├
ᨨ
鼀矄矄


├�
�矄

ᨨ
鼀ᨨ
ᨨ �
 
㜠 �矄

ᔼ
�矄 
矄
 
ᨨ鼀

㦨


ᨨ 犨矄
�
�矄
矄盐ᆠ
ᨨ

�鼀㜠   
ᨨᆠ
Ft㡰
�
 
x
�ᆠ�ᨨ


ᨨᆠ
Ft㡰
�矄 犨�
t
矄
矄 ᨨ犨
�
 
��盐


├� �ᆠ
 �泐
ᨨ
  ᨨᆠ├
 


�
矄ᆠᨨ 矄
�矄


᲼

҈
桌 Ft㡰




꾀
�
矄� �
犨�
矄� �㜠ᆠᨨ 矄 
k
�矄
盐 �
ᨨ鼀
  ᆠ�� �
 

ᨨᆠ├
ᨨ
鼀矄矄
M
�矄
ᆠ 
ᨨ
�矄
ᨨ犨
 



鼀矄矄
�矄
矄
犨�
 
矄� �
� ��泐
盐 矄ᆠ 



 
鼀 �鼀盐鼀
  矄�ᨨ
�
 
矄� �
�矄 ᲼



和桌 Mg Mg
Mg
Mg 㡰 ᆠ㡰 㡰 k k k
k
҈
꾀
�ᆠ

犨 �ᆠ
�矄
ᆠᨨ 矄� 
ᨨ
鼀ᨨ
ᨨ �
 
x㜠 �矄
�矄
矄盐ᆠ
ᨨ

ᨨᆠ
�
 
矄鼀
�ᆠ�ᨨ
犨 �ᆠ �矄
犨�
 
�矄 ᆠ
x
ᨨ
 
�ᆠ
ᨨ鼀
 
ᨨ���
矄 Fx㡰
㦨
kx
졔
ax



k
 
a

ᨨ矄��
ᆠᨨ 矄 矄

ᨨ x ³


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盐 ᆠ�ᨨ 
ᨨ鼀
ᨨ
 
ᨨ �
 �泐
Ux㡰
ᨨ  
�ᆠ
�矄 ᲼



和桌

㡰

U(x)

U(x)

㡰

㡰

�



X

㡰



X

㡰

X


N

㡰 䰘
�



�s

ᆠ㡰 �





s

O

t U(x)

҈



㡰 ҈




 꾀
ᨨᆠ├
B㡰
�矄
ᆠ 
ᨨ
犨ᨨ
盐 矄ᆠ 
矄� �矄
S҈
  S

犨�
矄� �
ᆠᨨ 矄 矄
k
 

k

矄ᆠ�泐
矄
��

ᔼ㡰



ᨨ 
 矄

ᆠ 
ᨨ
� �ᆠ
矄盐ᨨ矄
M҈
  M

ᨨ
ᆠ 
ᨨ
 
犨矄


矄� �矄
 
矄盐ᨨ矄 
����
鼀矄矄


�矄
ᨨ
�ᆠ�ᨨ
 泐犨 

 ᨨᆠ├
B
�矄
�矄ᆠ
ᨨ犨矄
犨
҈
泐

矄鼀
�矄 ᆠ
x  ��盐
ᔼᔼ㡰
 
矄


ᨨᆠ├
盐 矄
 
鼀ᨨ矄
 鼀 �鼀盐鼀
�矄 ᆠ
y
ᨨ犨矄
犨



�矄ᆠ鼀 矄
x
 
y 
鼀矄盐
犨�
矄ᆠ
ᨨ
 
⼄盐���盐鼀
ᨨ矄��ᨨ
ᨨ  
ᨨᆠ├
B


�ᨨ
y⏌x
�矄


᲼



桌
҈ M҈ M
S S҈
ᔼ B x


M


҈黨


S


҈ S҈

B

M҈

ᔼᔼ

x 㡰
㡰
ᆠ㡰 ҈⏌
㡰 ҈⏌
꾀
�ᆠ
�矄
ᆠ
泐

ᨨᆠ
F
㦨
kx

犨 
k
�矄

졔 ᆠᨨ 矄 

ᔼ矄
ᨨ �
 �泐

x
㦨


�矄
ᨨ

␔ �ᆠ
ᆠ盐 ᆠᨨᆠ泐
矄 矄
 
��ᨨ
ᨨ
ᨨ �
 �泐
ᨨ  
ᨨᆠ├
犨�
矄ᆠ
ᨨ
x ᲼



和桌 U

U(x)

ᆠ㡰

X




꾀
矄� �
ᨨ
ᨨᆠ㜠ᆠᨨ 矄 
k
�矄
ᆠ盐
� ᨨ
犨ᨨ
�ᆠ矄
矄盐ᆠ  
ᨨ 
�ᆠ
�矄
ᨨ盐
 
 �
ᨨ
 
ᨨ 


  ᨨ �
�ᆠ
犨�


ᨨᆠ㜠ᆠᨨ 矄 
ᨨ ᲼҈黨黨黨和
㜠


㝐├矄桌 㡰 
⏌㡰k 㡰 ⏌
㡰k ᆠ㡰 
k 㡰

k

犨ᨨ
鼀矄矄矄
ᨨ
҈
�鼀
 


�鼀

鼀ᨨ� �
犨�
⼄盐
├� �ᆠ  ��矄


�ᨨ
ᨨ
 
鼀� �盐矄
ᨨ
 �
�  鼀ᨨ鼀 
�矄 ᲼҈黨

桌 㡰



҈ 㡰 㡰 ҈


҈


҈ ᆠ㡰 ҈



2

Integer Value Answer




ᔴᨨ 矄�

��ᆠ
矄 
�
PQ �
 
�ᆠ
  犨�
OP
㦨

鼀
  OQ
㦨


鼀

꾀
ᨨᆠ├
ᨨ
鼀矄矄
҈
├�
�矄
盐 ᨨ �
 
�
ᨨ鼀
P
ᨨ
Q
犨�

ᨨᆠ
ᨨ
҈



犨 �ᆠ
�矄 犨泐矄

ᨨ
� 
PQ
矄
 
��盐
�� 㡰

꾀矄矄盐鼀� �

ᨨ
�ᆠ�ᨨ 矄矄
ᨨ矄矄矄

 
├� �ᆠ
 �泐
ᨨ
 
ᨨᆠ├
犨 

�
ᆠ 矄
Q
�矄
n 맔
҈
㡰
ᨨ盐矄


盐
ᨨ
n �矄
├ ᆠᆠ�ᨨ
盐
ᨨ
��泐
㦨
҈

鼀矄
㡰



᲼꾀



҈
桌 Q

U



鼀

㡰

x

㡰

x

黨
U

ᆠ㡰

O

U x

㡰

x


鼀

P


�

꾀
ᨨᆠ├
ᨨ
鼀矄矄


҈

├�
�矄
ᆠ 
ᨨ

矄� �
ᨨ
ᨨᆠ㜠 ᆠᨨ 矄 


⏌鼀


ᆠᨨ �ᆠ� 
ᨨ
�ᆠ�ᨨ
犨
  ᨨᆠ├
 
 
ᨨᨨ
�矄


҈

ᔼ ��泐
 
ᨨᆠ├
�矄

矄
   
矄� �
�矄
盐 㜠矄ᆠ 

꾀
�鼀盐矄
�矄
��
ᨨ
 
ᨨᆠ├

WWW.JEEBOOKS.IN

P-35

Work, Energy and Power

矄
矄 ᨨ犨
�
 
��盐


ᨨᆠ├
矄�矄

�矄 ᆠ
ᨨ




鼀
 
ᆠᨨ鼀矄
ᨨ
矄
ᨨ
 
�矄
�鼀


� ��
ᨨᆠ�泐
ᨨ  
ᨨᆠ├
�
鼀⏌矄
�矄

㦨
⏌҈




�矄 ᲼

҈҈桌

3





䰘


Numeric Answer

꾀
ᆠ�ᆠ├

ᨨ
鼀矄矄


҈�
├�
�矄
 ᨨ犨
�ᆠ泐
盐
泐
 ᨨ犨� �
鼀ᆠ � 
矄ᨨ
 
�
�矄矄
ᨨ

鼀 �鼀盐鼀
�� 
ᨨ


鼀
 
� �
 
鼀ᆠ � 

ᔼ
 

盐矄 � �
 
 �矄

ᆠᨨ 矄 
ᨨᆠ
F
ᨨ
 

 
鼀ᨨ矄 ᨨ�ᨨ 泐

�矄 ᆠ
ᨨ




鼀
犨 �
盐 ᆠ � �
 

 
盐
ᨨ
F
�
㡰
�矄
g
㦨
҈

鼀矄
㡰
힘힘힘힘힘힘힘힘힘힘
[Main 3 Sep. 2020 (I)] 꾀
�ᆠ
m 㦨

├�㡰
矄�矄
ᨨ犨

�ᆠ�ᨨ 矄矄
ᆠ├
꾀졠ᔴ㡰 矄� �
ᨨ鼀
矄


ᨨ� 
A
 �� 


鼀㡰

꾀 
ᆠ � � C

 
�ᆠ
ᆠᨨ � 盐矄
ᨨ
鼀ᨨ
泐
�
�
矄

ᨨᆠ�
␔ 
�
ᆠ � �
�矄
�� 矄
ᨨ� 
P
 �� 
҈
鼀㡰

  ├� �ᆠ
 �泐
ᨨ
 
�ᆠ
�
J㡰
�矄

 ��盐
犨
�矄 矄ᆠ 鼀�ᆠ
 
ᨨ
ᨨ
矄ᆠ錐
├
g 㦨
҈

鼀矄
㡰
¾¾¾ . ᲼Main 䰘







ᔼ㡰桌


㡰 v
-
gh =

҈ gR


 gR 㡰 v

-
gh =
ᆠ㡰  
ᆠ �
ᨨᆠ
⼄盐�

ᨨ� 矄
x
 
z
�矄
ᨨ 㡰  
ᆠ �
ᨨᆠ
⼄盐�
�矄
鼀 �鼀盐鼀

ᨨ� 矄
x  
z
黨
꾀
�ᆠ
ᨨ
鼀矄矄
m
�矄
� ��泐

矄

 
ᨨ���

ᔼ
�矄 矄盐ᆠ
ᨨ

ᨨᆠ
 
矄矄
鼀ᨨ� �
ᨨ �
 
x㜠 �矄

ᔼ矄 ├� �ᆠ
 �泐
K
ᆠ  �矄
犨�
�鼀
矄
⧤⏌
㦨
g

犨 
g �矄

ᨨ矄��
ᆠᨨ 矄 
ᨨ
ᨨ�
�鼀 矄�ᨨ 矄

␔ �ᆠ ᨨ
 
ᨨᨨ犨� �
矄鼀 矄
�矄
㡰
盐䁸 ᲼꾀



҈
桌 㡰 
ᨨᆠ
�
ᨨ
 
�ᆠ
�矄
ᆠᨨ 矄  㡰 
矄
ᨨ
 
�ᆠ
�矄
ᨨᨨ�ᨨ 
ᨨ
�鼀 ᆠ㡰 
�矄 ᆠ
ᨨ
 
�ᆠ
ᨨ鼀
 
ᨨ���
� ᆠ矄矄 � 泐
犨�
�鼀 㡰 
ᨨᆠ
�矄
ᆠᨨ 矄� 

꾀
矄鼀

矄矄
鼀ᨨ� �
ᨨ鼀
A
ᨨ

� 
ᆠ├
矄
矄 ᨨ犨

�
 
��盐

和盐 ᆠ
AB
矄
�ᆠ�ᨨ

ᨨ鼀
A
ᨨ
B
 
 ᨨ矄
犨� ᨨ盐
矄�� �

和盐 ᆠ
BC
�矄
�ᆠ�ᨨ 矄矄

KA

KB
  KC

├� �ᆠ
 ��矄
ᨨ
 


A

B
 
C

矄ᆠ�泐




᲼




㜠
�㝐

҈桌 C

�� 

A

A

P C



鼀

hC

hA

B

O

6




㡰

hA > hC 錐 K B > KC

㡰

hA > hC 錐 KC > K A

ᆠ㡰

hA = hC 錐 K B = KC

㡰

hA < hC 錐 K B > KC

MCQs with One or More than One Correct Answer

꾀
矄盐 
矄├矄
盐

鼀
 
鼀├矄

 �


犨�
  ᨨ�ᨨ 

⏌矄 
矄矄
矄
矄 ᨨ犨
�
 
��盐㡰

  ᨨᨨ鼀
ᨨ
 
鼀
犨�
矄
v

 
犨 矄
ᨨ
盐
ᨨ盐  ᨨ

矄鼀�ᆠ�ᆠ盐

xyz ᨨ
�盐矄
R
盐� �
犨 �ᆠ
⏌ 矄 
ᆠ 矄

鼀 �鼀盐鼀
�� 
h

ᨨ� 
y㡰
ᨨ鼀
 
�ᨨ盐  矄
矄 ᨨ犨
�
 
��盐

꾀矄矄盐鼀
 
 
 �泐
ᨨ矄矄
�矄

����
 
 
ᨨᆠ
⼄盐�
ᨨ
 �矄
盐

 
�� 矄 ᨨ� 
�矄
ᨨ�
泐
�矄⏌ 
犨�� 
ᨨ 泐


g
�矄
  ᆠᆠ�ᨨ
盐
ᨨ
��泐㡰 ᲼꾀





桌 z R

y

x h 


҈
꾀
矄ᨨ 
�
ᨨ

矄� �
ᨨ
 �

�矄
犨 �
�

�ᆠ ᆠ�ᆠ
犨�
 
ᨨ 
 
ᨨ
 
矄� �

 
ᆠ 

꾀
 ᆠ�
� 矄 
ᨨ
�鼀

 
矄ᨨ 
�矄

�矄
ᨨ犨矄
ᨨ矄��ᨨ
 
矄

矄
u


鼀� �盐
ᨨ

 
ᆠ  �
�
�矄 ᨨᆠ�泐
矄
�
ᆠ 矄

ᨨ矄��ᨨ
犨 
 
矄� �
�矄 ᨨ�ᨨ 
�矄 ᲼҈黨黨
和
㜠


㝐├矄桌 㡰

u
-
gL

㡰


gL

ᆠ㡰

u
- gL

㡰


u
- gL㡰



꾀
�ᆠ
�矄
ᆠ
盐ᨨ
泐

ᨨᆠ
ᨨ
ᆠᨨ 矄 
鼀� �盐 犨 �ᆠ
�矄
犨泐矄
 �ᆠ盐
ᨨ
 
ᨨᆠ�泐
ᨨ
  �ᆠ


鼀ᨨ�ᨨ
ᨨ
 
�ᆠ
├矄
ᆠ
�

 

ᔼ ᨨᨨ犨矄
 

᲼҈黨
䰘
㜠


㝐├矄桌 㡰 �矄
ᨨᆠ�泐
�矄
ᆠᨨ 矄  㡰 �矄
ᆠᆠ�ᨨ
�矄
ᆠᨨ 矄  ᆠ㡰 �矄
├� �ᆠ
 �泐
�矄
ᆠᨨ 矄 
㡰 �
鼀ᨨ矄
�

ᆠ�ᆠ盐



WWW.JEEBOOKS.IN

EBD_8341 P-36

Physics

Match the Following

7


꾀
�ᆠ
ᨨ
盐 �
鼀矄矄
�矄
鼀ᨨ� �
ᨨ �
 
x㜠 �矄
盐 
 
� 盐 ᆠ
ᨨ

ᨨᆠ
 
�矄
ᨨ
 �泐
�矄
ᆠᨨ 矄

ᨨ盐
ᨨ矄矄� ᨨ鼀矄
ᨨ
 
ᨨ �
 �泐
ᨨ
 
�ᆠ

��
�
ᆠᨨ盐鼀
ᔼ
a
 
U

ᆠᨨ 矄 矄㡰

㝐ᆠ
 
ᨨ �
 ��矄
�
ᆠᨨ盐鼀

ᔼ
ᨨ
 
ᆠᨨ矄ᨨ � �
矄鼀 矄㡰
�
ᆠᨨ盐鼀
ᔼᔼ
ᔴᨨ盐鼀
ᔼ U
꾀㡰 U҈x㡰
㦨



㡰

U
x㡰
㦨


ᔴᨨ盐鼀
ᔼᔼ é æ x ö
ù ê҈ - ç ÷ ú êë è a ø úû




U
æ x ö
ç ÷
èaø


ù
é æ x ö   ê- æ x ö ú ç ÷ ᔴ㡰 ç ÷
êë è a ø úû èaø ù é 
ê x  ҈ æ x ö ú ç ÷ 㡰 U
x㡰
㦨


êë a  è a ø úû

U
Ux㡰
㦨



㡰


ᨨᆠ
ᆠ� �
ᨨ
 
�ᆠ
�矄
ᨨ

x
㦨
a

⼄㡰


ᨨᆠ
ᆠ� �
ᨨ
 
�ᆠ
�矄
ᨨ

x
㦨



㡰


ᨨᆠ
ᆠ� �
ᨨ
 
�ᆠ
�矄
ᨨ

x
㦨

a

矄㡰


�ᆠ
 � ᆠ矄

ᆠ�
ᨨᆠ
ᨨ犨矄
x
㦨


�
 
��ᨨ
x




㡰


�ᆠ
犨�
ᨨ
 �泐


Comprehension/Passage Based Questions

8

꾀和和꾀෌E 꾀
矄鼀
ᨨᆠ├
ᨨ
鼀矄矄
҈
├�
�矄
矄
ᨨ鼀
矄

 
ᨨ
ᨨ
 ᨨ盐�
ᆠ├


ᆠ├
�矄

ᆠ�ᆠ盐
ᆠ
ᨨ
�盐矄



鼀


ᨨᆠ├ 矄�矄
ᨨ �
 
ᆠ├
犨� ᨨ盐
ᨨ� �
 

�ᆠ�ᨨ 
ᨨᆠ ᆠ矄
ᨨ
�
�
 
�ᆠ�ᨨ
ᨨᨨ矄�
ᨨ
 
� 矄  ᨨ盐矄
ᨨᆠ�泐



犨ᨨ├
ᨨ 
�
ᨨᆠᨨ鼀� �
 
�ᆠ�ᨨ
盐
ᨨ
 
ᨨ� 
턼

矄 矄 ᨨ犨
�
 
��盐
ᨨ犨

�矄
҈�


 ├
 
ᆠᆠ�ᨨ
盐
ᨨ
��泐

�
㦨
҈

鼀矄
㡰 泐

᲼꾀



҈桌

 




턼

졠




鼀� �盐
ᨨ
 
ᨨ鼀
ᆠ�ᨨ
 
ᆠ矄
ᨨ
  ᨨᆠ├

 
ᨨ� 
턼
�矄 㡰 䰘
�


㡰





ᆠ㡰 ҈҈
�


㡰


�


�

矄
ᨨ
 
ᨨᆠ├
犨 
�
ᆠ 矄
 
ᨨ� 
턼
�矄 㡰 �
鼀矄҈ ᆠ㡰

҈
 鼀矄 -҈

㡰 ҈

鼀矄҈ 㡰


鼀矄҈

9

U

ᆠ
ᨨ矄ᆠ�
ᨨ盐
 
ᨨ� 
x
㦨




Assertion and Reason Type Questions



和T꾀TE㝐ENT҈


꾀
ᨨᆠ├
ᨨ
鼀矄矄
m
矄矄
鼀ᨨ� �
ᨨ
 ᨨ盐�
ᨨ�ᨨ 
矄盐 ᆠ
犨�

ᨨᆠ�泐
v

ᔼ
矄ᨨ矄
盐
ᨨ �ᆠ�ᨨ
犨
 
ᨨᆠ├
 
 
矄盐 ᆠ
 
鼀ᨨ� �  ᨨ盐�

ᆠ�
�矄 ᆠ


矄盐 ᆠ
�矄
ᨨ犨
�
ᨨ


 �
ᨨ


犨�
 
ᨨ�ᨨ 
 
 
矄鼀
ᨨᆠ├
�矄 鼀
ᨨ
�ᨨ
盐
ᨨ
 
矄盐 ᆠ
犨�
 
矄鼀
� ��
ᨨᆠ�泐 v


ᆠ矄
�
 
鼀ᆠ  �ᆠ
 �泐
�
 
矄ᆠᨨ  矄�盐�ᨨ
�矄
矄鼀
 
 
�
 
�矄
矄�盐�ᨨ
和T꾀TE㝐ENT




ᆠᨨ �ᆠ� 
ᨨ
�ᆠ�ᨨ
犨
  ᨨᆠ├
 
 
矄盐 ᆠ
ᆠ矄矄
犨�
 
� ᆠ矄
�
   �
ᨨ
� ᆠ� �ᨨ
᲼


䰘桌 㡰 和鼀 ҈
�矄
盐

和鼀 

�矄
盐錐
和鼀 

�矄

ᆠᨨᆠ
  �ᨨ
ᨨ
和鼀 ҈ 㡰 和鼀 ҈
�矄
盐

和鼀 

�矄
盐錐
和鼀 

�矄
졠

ᆠᨨᆠ
  �ᨨ
ᨨ
和鼀 ҈ ᆠ㡰 和鼀 ҈
�矄
盐

和鼀 

�矄
矄 㡰 和鼀 ҈
�矄
矄

和鼀 

�矄
盐 10 Subjective Problems 䰘
꾀
矄� �

犨�
ᨨ 
 
� 
ᨨ

���
犨

矄矄� �
ᨨ 
� 
�ᆠ�ᨨ 矄矄
盐泐


�矄 ᆠ
ᨨ

鼀
ᨨ鼀
 
犨
矄

ᨨ� 
鼀矄矄

M
㦨

├�
ᆠ 
ᨨ
�


�矄 ᆠ
ᨨ ҈鼀
ᨨ鼀
 
犨

꾀
鼀矄矄
m 㦨


�
├�
ᆠ 

 
 M  
�矄


矄
矄ᨨ
 
  矄� �
�矄
ᨨ�ᨨ 
犨
 
犨  
 
盐泐
 
�ᆠ
泐ᨨ   
盐泐

␔ 
犨�

 
矄 m 犨�
犨 �ᆠ
 
鼀矄矄
M
犨�
�
  犨
犨 
 
鼀矄矄
m
�矄
矄
䁸 ᲼҈黨
�
㜠


㝐├矄桌

WWW.JEEBOOKS.IN

P-37

Work, Energy and Power





꾀


�
├�
ᨨᆠ├
矄�矄
ᨨ鼀
 
ᨨ� 
A
矄
��㡰

ᨨ
 ᨨ�ᨨ 
ᆠ├
犨�

� ��
矄
ᨨ

鼀⏌矄
ᨨ犨矄
 犨�� 矄矄
ᨨ�ᨨ 
矄� �
ᨨ
 �
҈
鼀
 
ᨨᆠ ᆠᨨ 矄 



犨ᨨ ⏌鼀



AB
ᨨ
 

ᆠ├

�矄 �ᆠ�ᨨ 矄矄
 
 


BC
矄
 
ᆠᨨ �ᆠ� 矄
ᨨ
矄�ᆠ  
├� �ᆠ
�ᆠ�ᨨ
矄





 




矄ᆠ�泐

ᔼ
  �矄 ᆠ矄
AB

 
BD



鼀
 


҈


鼀
矄ᆠ�泐

�   
ᨨ
�矄 ᆠ

 ᨨ盐�
犨 �ᆠ
 
ᨨᆠ├
鼀ᨨ矄
 ᨨ �
ᆠᨨ鼀矄
ᨨ
矄
ᆠᨨ鼀泐
 ├
�
㦨
҈
鼀
⏌
矄
㡰 ᲼҈黨

㜠
䰘

㝐├矄桌

A

黨


B

C

D

犨ᨨ
ᨨᆠ├矄
A
 
B
 ᆠᨨ

ᆠ
ᨨ
ᆠ
ᨨ 
泐
 B 矄� �
 

矄� �
錐
 
矄� � 矄矄矄
ᨨ

�ᆠ�ᨨ 矄矄 C 盐泐
矄
矄 ᨨ犨
�
 
��盐
ᨨᆠ├
B A 矄�矄
ᨨ
 
ᨨ�ᨨ 
ᨨ



矄盐 ᆠ
ᨨ

矄�ᨨ 泐
ᨨᆠ├ C
 
 
ᨨᆠ├
A
矄�矄
ᨨ �  
�ᆠ
矄�
ᨨ
C

ᨨ 犨�
 
矄鼀
盐 � ᨨ鼀
矄



ᆠᨨ �ᆠ� 
ᨨ
�ᆠ�ᨨ
犨
 
矄盐 ᆠ矄
ᨨ
ᨨᆠ├矄 �矄





ᨨᆠ
ᆠᨨ 矄 
ᨨ
 
矄� �
�矄
҈黨


犨ᨨ 矄⏌鼀

ᔼ

鼀矄矄
ᨨ
ᨨᆠ├
A
�矄



⧤�


ᆠᆠ盐
 
鼀矄矄
ᨨ
ᨨᆠ├
B
   
 �泐
矄ᨨ
�
 
矄� �
᲼҈黨


㜠
�
㝐├矄桌 D A




G

F E C 㡰 㡰 ᔼ
 
��盐矄
㡰
 
㡰
AC

DG
 
GF

� 
� ᆠ�   矄

BC
㦨
EF
㦨
x
 
AB
㦨
DE
㦨
y

꾀
矄鼀
ᨨᆠ├
ᨨ
鼀矄矄 M
�矄
矄
ᨨ鼀
 
ᨨ� 
A

ᔼ
矄�矄
ᨨ犨
AC
  ᆠ 矄
C
犨�

矄
Vc


矄鼀
ᨨᆠ├
�矄
矄 ᨨ鼀
矄
ᨨ鼀
 
ᨨ� 
D

ᔼ
矄�矄
ᨨ犨
DGF
 
ᆠ 矄  
ᨨ� 
F
犨�
矄
VF


ᆠᨨ �ᆠ� 矄
ᨨ
├� �ᆠ �ᆠ�ᨨ 矄
犨
 
ᨨᆠ├
 
ᨨ
 
矄盐 ᆠ
AC
  DGF

m
᲼҈黨

桌 ᔴᆠ盐
VC
 
VF

҈
␔ 


�矄
 ᨨ犨
盐

 
鼀� �盐
ᨨ
�矄
鼀ᨨ鼀 盐鼀 ᆠ矄矄
 
 
� ᆠ矄矄

ᨨ矄
 �矄
�ᨨ
  ᆠᨨ 矄�ᨨ
ᨨ
鼀ᨨ鼀 盐鼀
� ᆠ�䁸 ᲼҈黨䰘黨桌


꾀
矄� �
ᨨ
ᨨᆠ
ᆠᨨ 矄 
k
�矄
ᆠ盐
� ᨨ
 
⼄盐
矄
␔ 
�矄
ᨨᆠ
ᆠᨨ 矄 
ᨨ
ᆠ
䁸 ᲼҈黨䰘
桌

꾀
盐
�矄
�
ᨨ鼀

� 

ᔼ
 
� 
ᆠᨨ�矄
泐

鼀�  犨  
 
├� �ᆠ
 �泐
ᨨ
 
� 
�矄
�
 

⼄盐 ᨨ
矄矄
 
 
ᨨ
 
盐
᲼҈黨䰘
桌 B

Topic-3 : Power

҈


S

ᆠᨨ矄
ᨨ

҈

㦨
䰘


␔

�
㦨
҈

鼀矄
㡰᲼㝐�
䰘







ᔼ桌 㡰 ҈
䰘
鼀矄҈ 㡰 ҈
黨
鼀矄҈ ҈ ᆠ㡰 ҈
�
鼀矄 㡰



鼀矄҈ 꾀
�ᆠ
ᨨ
鼀矄矄
㝐
�矄
鼀ᨨ� �
�

ᆠ�ᆠ
ᨨ
� 
�盐矄
�
矄盐ᆠ

犨泐
 
�矄
ᆠ �
ᆠᆠ�ᨨ

�鼀

�矄 ��
泐




犨 

�矄

ᆠᨨ 矄 


ᨨ犨
� ᨨ
 
�ᆠ
泐
 
ᨨᆠ
ᆠ� �
ᨨ
�

�矄

᲼㝐�
졠 � 
꾀�
҈




҈
桌

S

҈
㝐




㡰 㝐



ᆠ㡰 㝐



㡰 㝐


 꾀
ᆠ
ᨨ
犨�� 
␔
�矄
ᨨ

� ᆠ� 
ᨨ
 
�矄矄
泐
҈

鼀
ᨨ

�矄 ᆠ
ᨨ
҈
⧤鼀
 
�矄

ᆠᨨ 矄 
�ᆠ�ᨨ 

MCQs with One Correct Answer

1

꾀
�ᆠ
�矄
鼀ᨨ� �
盐 ��ᆠ�ᨨ 泐
ᨨ

ᨨ�ᨨ 
  盐 
 
ᆠ�ᨨ
ᨨ

ᆠᨨ 矄 
ᨨ犨
矄盐泐� �
 �泐 矄ᨨ盐ᆠ


�矄ᆠ鼀 
s㡰
㜠
�鼀
t㡰
�
 
矄ᆠ�矄  
鼀ᨨ�ᨨ
ᨨ
 
�ᆠ
�矄
� 矄

犨
矄ᆠ 鼀�ᆠ泐  

ᨨ
ᨨ
矄ᆠ㡰

᲼㝐�


和






ᔼᔼ㡰桌 S




㡰

㡰

㡰 t

t

S





␔
ᨨ
 
ᆠ

␔ �
鼀ᨨ� �
盐 �
ᨨ
 
ᨨ



矄
ᨨ
҈

鼀矄҈

 
ᆠ
矄
ᨨ犨
. ᔼ
�
矄
ᨨ犨

ᨨᆠ

ᆠ㡰

㡰 t





t

꾀




ᆠ�ᆠ
鼀ᨨᨨ
� 矄

ᨨ
� �

鼀 �鼀盐鼀 ᨨ
ᨨ
ᆠᆠ�泐
ᨨ





├�

ᔼ
 
�ᆠ�ᨨ 
ᨨᆠ
ᨨ
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�矄







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矄
ᨨ
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ᨨ

盐
ᨨ
�矄


犨 �
鼀ᨨ� �
ᨨ犨 �

矄
v
 
盐
ᨨ
v �矄

᲼㝐�
졠 � 
꾀�
黨



҈
桌 㡰


鼀矄҈ 㡰 �
鼀矄҈ ᆠ㡰 ҈�
鼀矄҈ 㡰 ҈

鼀矄҈

WWW.JEEBOOKS.IN

EBD_8341 P-38

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䰘






Physics

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犨� 㜠ᨨ犨
� ᨨ
ᆠᨨ 矄
犨� 
 �泐
� ᨨ
ᆠ�ᆠ  �泐

꾀矄矄盐鼀
 
 
� ᨨ
ᆠᨨ 矄

� 
ᆠ�ᨨ

ᨨ
 
犨� 
 �泐
� ᆠ
泐
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矄
� ᨨ
ᆠ�ᆠ  �泐

ᨨ
犨� 
矄
n

 
ᆠ�ᆠ
ᨨ犨
ᨨ盐盐
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 鼀ᨨ矄
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ᨨᨨ�ᨨ 
ᨨ
᲼㝐�
졠 � 
꾀�

�



҈桌 㡰 n
㡰 n
ᆠ㡰 n 㡰 n 꾀
䰘

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ᨨ鼀

ᆠᨨ盐ᆠ � � ᨨ矄��ᨨ

ᨨ
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鼀
盐矄 矄
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ᨨᆠ
F
ᨨ
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
ᆠ 
ᨨ
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泐


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鼀
 ᨨ

矄

꾀 
 

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ᆠ
�

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泐
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鼀盐矄ᆠ矄 �矄


 ├

�
㦨
҈

鼀矄
㡰 ᲼㝐�
졠 � 
꾀�





҈桌 㡰




맔
҈

␔矄

 
矄 㡰




맔
҈

␔矄

├
ᨨ ᆠ㡰




맔
҈


␔矄

 
矄 㡰




맔
҈


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 �泐
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 �泐

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 
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泐
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 �泐

ᨨ
犨� 
矄
v

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ᆠ�ᆠ
ᨨ犨 ᨨ盐盐
犨�

ᨨᨨ�ᨨ 
ᨨ ᲼



和桌 㡰 v 㡰 v
ᆠ㡰 v  㡰 v
꾀
�ᆠ
ᨨ
鼀矄矄
鼀
�矄
鼀ᨨ� �
�

ᆠ�ᆠ盐

ᨨ
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r
矄盐ᆠ
 
�矄
ᆠ �
ᆠᆠ�ᨨ
ac
�矄

泐� � 犨�
�鼀
t
矄
ac
㦨
k
rt

犨 
k
�矄


ᆠᨨ 矄 



ᨨ犨 �
ᨨ
 
�ᆠ矄
泐
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ᨨᆠ
ᆠ� �
ᨨ
�
�矄
᲼҈黨黨

㜠
҈
㝐├桌

㡰 mk
r
t
p mk
r
t

� mk r t 㡰 㡰 ᨨ ᆠ㡰  꾀
ᨨ泐
�矄
鼀ᨨ
ᨨ �

矄�� 
� 
泐

鼀ᆠ �  �� �
ᆠᨨ 矄 
ᨨ犨


�矄 ᆠ
鼀ᨨ
泐
  ᨨ泐
�
�鼀
t
�矄
ᨨᨨ�ᨨ 
ᨨ ᲼҈黨

㜠


㝐├矄桌 㡰 t ҈⏌
㡰 t ⏌
ᆠ㡰 t ⏌
㡰 t
ᔼ

鼀ᆠ � 
�矄
盐�ᆠ
犨�
ᨨ� ᲼҈黨

桌 㡰  
鼀ᆠ  �ᆠ
 �
ᨨ
 
鼀ᆠ � 
� ᆠ矄矄
㡰  
鼀ᆠ  �ᆠ
 �ᆠ� ᆠ泐
ᨨ
 
鼀ᆠ � 
� ᆠ矄矄
ᆠ㡰 ᨨ
�矄
鼀ᆠ  �ᆠ
 �
 
 �ᆠ� ᆠ泐 � ᆠ矄
㡰 �矄
 �ᆠ� ᆠ泐
� ᆠ矄矄

盐
�矄
鼀ᆠ  �ᆠ
 � ᆠ矄矄


㡰

黨


҈



2

Integer Value Answer

҈҈
꾀
�ᆠ
ᨨ
鼀矄矄




├�
�矄
鼀ᨨ� �
�
ᨨ 
�鼀 矄�ᨨ
盐  
ᨨᆠ
 
�矄

ᆠᨨ 矄 
ᨨ犨


�
␔
ᨨ
 
�ᆠ
ᔼ
 
� ��
矄
�
鼀矄҈㡰
ᨨ
 
�ᆠ
�矄
ᨨ

  矄
�
鼀矄҈㡰
 
�
矄
�矄


᲼꾀



҈桌 3

Numeric Answer

҈

꾀
ᨨ泐
ᨨ
鼀矄矄


├�
�矄
�
泐

 �� 
�� �
 ᆠᨨ 矄 
ᨨ犨
ᨨ
҈
⏌矄


ᨨ泐
矄矄
ᨨ鼀
矄
  鼀ᨨ矄
�

矄�� 
� 

꾀 
黨
矄ᆠᨨ 矄

 
ᨨ泐
矄 鼀ᨨ

�矄 ᆠ
�
鼀㡰
힘힘힘힘힘힘
᲼㝐�

�
和






ᔼᔼ㡰桌

Topic-4 : Collisions 1 ҈





MCQs with One Correct Answer

�ᆠ
꾀
ᨨ
鼀矄矄
鼀҈
鼀ᨨ� �
犨�
ᨨᆠ�泐
 �$ + $ 㡰 鼀矄-҈ ᆠᨨ�矄
犨�
 ᨨ 
�ᆠ

ᨨ
鼀矄矄
鼀

犨 �ᆠ
�矄

矄 r r � ��泐


V҈
 
V


 
ᨨᆠ��矄
ᨨ
�ᆠ矄
꾀
  
 
ᆠᨨ�矄�ᨨ
矄ᆠ�泐

ᔼ
鼀҈
㦨

鼀

 
 
ᆠᨨ�矄�ᨨ

r r r V҈ = $� +  $ 㡰 鼀矄 -҈

 
 �
犨
V҈
 
V

�矄







᲼㝐�


和






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º ᆠ㡰 

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ᨨ
鼀矄矄矄
m


m


m
 

m

 �
�

� 
ᨨ


�ᆠ�ᨨ 矄矄
ᨨᨨ

꾀 ᨨ 
ᨨᆠ├
ᨨ
鼀矄矄
m

鼀ᨨ� �
犨� 矄
v
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矄鼀
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

꾀
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ᆠᨨ�矄�ᨨ 矄 
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� 矄�ᆠ

泐
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m
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ᨨ

[4 Sep.
2020 (I)] v

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m

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ᨨ
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҈
├�
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犨�
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矄�ᆠ├矄
ᨨ
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ᔼ
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ᨨ
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矄�├矄
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ᨨᨨ
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g
㦨
҈

鼀⏌矄


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ᨨ
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 
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ᨨ
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桌 ᲼㝐�

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ᔼᔼ㡰桌 㡰


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҈
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
 꾀
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ᨨ
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m
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鼀矄矄

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v ˆ
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v
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2 Sep. 2020 (I)] u
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ᨨ
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P-39

Work, Energy and Power

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WWW.JEEBOOKS.IN

EBD_8341 P-40

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ᨨ� 
A䁸



᲼


黨桌 v

A


v

Integer Value Answer





ᨨᆠ矄
A, B
 
C

├
�

矄�� 
� 
ᨨ
 �ᆠ�ᨨ 矄矄
ᨨ�ᨨ 
矄盐 ᆠ

矄

鼀矄矄矄
m, 2m  
m, 矄ᆠ�泐


ᨨᆠ
A
鼀ᨨ矄
ᨨ犨矄
B
犨�
 矄
黨
鼀⏌矄
 
鼀├矄

矄�ᆠ
ᆠᨨ�矄�ᨨ
犨�
�

  

B
鼀├矄
ᆠᨨ鼀泐
� 矄�ᆠ
ᆠᨨ�矄�ᨨ
犨�
C.
꾀 鼀ᨨ�ᨨ 矄
ᨨᆠᆠ盐
ᨨ
 
矄鼀
矄�� 
� 

� 
 
�  矄
�
鼀⏌矄㡰
ᨨ
 
ᨨᆠ
ᔴ
᲼


黨桌 m


m

m

꾀



ᔴ



꾀
ᨨ
ᨨ
鼀矄矄
m, 矄盐矄 
泐

矄� �
ᨨ
 �
l1, �矄 ��

鼀� �鼀盐鼀
ᨨᆠ�泐
⼄盐�
ᨨ
ᆠᨨ鼀

盐
ᆠ�ᆠ
�

 
�ᆠ
 

꾀
 
�� 矄
ᨨ� 

�
ᆠᨨ�矄
矄�ᆠ泐 犨�
 ᨨ 
ᨨ
ᨨ
鼀矄矄
m 矄盐矄 
泐

矄� �
ᨨ
 � l2, 犨 �ᆠ
�矄
� ��泐

矄

ᨨ
 
矄� �矄

鼀矄矄㜠矄矄  
�   矄�

ᔼ
 
矄ᆠᨨ 
ᨨ

 
ᆠᨨ�矄�ᨨ
ᆠ⼄盐�矄  
鼀� �鼀盐鼀
矄
⼄盐�
ᨨ
ᆠᨨ鼀

盐
ᆠ�ᆠ
�
  l �ᆠ
 

 
�ᨨ
҈
�矄 ᲼꾀



҈桌 l
3

Numeric Answer




犨ᨨ
ᨨ�矄
ᨨ
 
矄鼀
鼀矄矄

鼀ᨨ� �
犨�
 
矄鼀 矄

盐
�
�  
�ᆠ�ᨨ 矄
�

 

泐

 ᆠᨨ鼀泐
� 矄�ᆠ
ᆠᨨ�矄�ᨨ
 
鼀ᨨ
ᨨ� 
   犨�

� 
矄
犨 �ᆠ
�矄

ᨨ
 �
� ��
矄

  �
犨
 
� ��
ᨨᆠ��矄
ᨨ
 
犨ᨨ
ᨨ�矄
�

᲼㝐�


和






ᔼ㡰桌 �㡰
�矄
힘힘힘힘힘힘

�
꾀
�ᆠ
ᨨ
鼀矄矄
m
�矄
鼀ᨨ� �
ᨨ �
 
x㜠 �矄
犨�
� �� ᨨᆠ�泐
u�ˆ

ᔼ
ᆠᨨ�矄
矄�ᆠ泐
犨�

�ᆠ
ᨨ
鼀矄矄 ҈

鼀

矄
 
 
鼀ᨨ矄
犨�

�矄
� ��
├� �ᆠ  �泐
矄
��盐㡰

ᔼ
矄� q҈ = n 矄� q


 
盐
ᨨ
n �矄
힘힘힘힘힘힘힘힘힘힘힘
[㝐� 2 Sep. 2020 (II)] 鼀 q҈ 鼀 ҈

鼀 q
u�$ ҈

鼀


꾀
ᨨ泐
A

ᨨ
鼀矄矄
m
㦨


҈
├�
矄

� ��
ᨨᆠ�泐
ᨨ
 �ˆ 鼀矄҈
ᔼ
ᆠᨨ�矄
矄�ᆠ泐
犨�
 ᨨ 
ᨨ泐

B
ᨨ
 
矄鼀 鼀矄矄
犨 �ᆠ
矄

� ��
ᨨᆠ�泐
ᨨ
� ˆ 鼀矄҈

꾀  r ᆠᨨ�矄�ᨨ

A
鼀ᨨ矄
犨�

ᨨᆠ�泐
v =
�ˆ + ˆ


 �泐 x ᨨ
B
 
ᆠᨨ�矄�ᨨ
�矄
犨�
矄
J


盐
ᨨ
x
�矄 ҈
힘힘힘힘힘힘힘
᲼㝐�









ᔼ桌

(


҈


㡰
㡰
 ᆠ㡰
㡰 ҈

犨ᨨ
�ᆠ矄
ᨨ
鼀矄矄矄
m҈
 
m

�
ᨨᆠ�
鼀ᨨ�ᨨ

r r 
ᨨᆠ��矄
v҈  
v

矄ᆠ�泐

�鼀
t
㦨



泐 r r ᆠᨨ�

�鼀 tᨨ

�
ᨨᆠ��矄
ᆠᨨ鼀
v҈ Ἐ
 
v
Ἐ
 �鼀

t ᨨ
犨 �
矄�
鼀ᨨ� �
�
�


盐
ᨨ r r r r ᲼


҈和桌 ( m҈v҈ Ἐ+ m
v
Ἐ) - ( m҈v҈ + m
v
)
�矄 㡰 ᨨ 㡰 ( m҈ + m
) gtᨨ ҈ 㡰
m҈
졔
m
㡰gtᨨ ᆠ㡰 ( m҈ + m
) gtᨨ


)


䰘
꾀

�矄
ᨨᆠ
ᨨ鼀
 
�ᨨ盐 


 �
ᨨ

�ᨨ
犨�  
ᨨ�ᨨ 
矄盐 ᆠ

ᔼ
ᆠ 矄

鼀 �鼀盐鼀
�� 
ᨨ
҈

m  
盐 矄
ᨨ
 
�ᨨ盐 

ᨨ
�� �
 
�ᨨ盐 
ᨨ  
�矄
�鼀

�
ᨨ矄矄

ᨨ
�矄
├� �ᆠ
 �泐

ᔼ鼀鼀�泐  
 
ᨨ盐 ᆠ

 
ᨨᆠ�泐
ᨨ
 

鼀├矄

 � ᨨ

ᨨ
犨�
 
ᨨ�ᨨ 
矄盐 ᆠ


鼀 �鼀盐鼀
�� 
� ᆠ 矄
 
 
ᨨ盐 ᆠ

�
metres

�矄
힘힘힘힘힘힘힘힘힘힘힘
᲼꾀



҈
桌

WWW.JEEBOOKS.IN

P-41

Work, Energy and Power

6




MCQs with One or More than One Correct Answer

꾀
矄鼀
�ᆠ
ᨨ
鼀矄矄
m
鼀ᨨ� �
� 矄�

泐

ᨨᨨ犨  
矄�� 
盐
ᨨ �
 
盐
 �矄
盐 �ᨨ矄
矄�ᆠ ᆠᨨ�矄�ᨨ

犨ᨨ
 矄


盐
矄
ᨨ
�ᆠ�ᨨ
 
�
�矄 ᆠᨨ矄

ᨨ 
 
泐


矄盐 ᆠ
犨 �
 
ᨨ 
 
�矄 �
犨�

泐
鼀ᨨ

�矄ᨨ
矄
矄 ᨨ犨
�
��盐
␔ 
 
�矄 ᆠ
ᨨ
 
�矄ᨨ
ᨨ鼀
ᆠᨨ矄
 
�矄
L㦨
L
 
�ᆠ
矄
�矄
v
㦨
v




�矄ᨨ
�矄
鼀ᨨ
� 犨
 dL v


犨 
dL
�矄
  
泐
ᨨ犨
矄
V 矄盐ᆠ
 
V >
mw
㡰
 
� ᆠ  �
�
 
 �
ᨨ
 
矄� �
�矄
矄
��
泐
  ᨨ�ᨨ
᲼㝐�
黨







ᔼᔼ桌

WWW.JEEBOOKS.IN

P-49

System of Particles and Rotational Motion

㡰


æ mw
ö ç ÷  çè k ÷ø


mw
㡰 k

mw
mw
㡰 k k 꾀
�ᆠ
ᨨ
鼀矄矄
m
�矄
� 
ᨨ
ᨨ 
 
ᨨ

�� 
矄� � � �
ᨨᆠ
ᆠᨨ 矄 
k
 
盐 矄ᆠ 
 �
l

 ᨨ 
 
�矄
� 


矄泐矄鼀
�矄
��

 �盐
矄
w ᨨ盐
 
� 
 
ᨨ
 
矄� �
矄盐ᆠ
 
�
ᨨ矄
�
 ᆠ�ᆠ
�
��泐

矄ᆠ


 
矄ᆠ
�
 
矄� � �矄

᲼㝐�









ᔼ桌

ᆠ㡰





㡰

ml w

㡰




k - mw










ml w


ml w k + mw k + mw 꾀
矄鼀ᨨᨨ
犨�
ᨨ
 �

p
�矄
 
� ᨨ

ᆠ�ᆠ
 
├
�


�ᆠ
 

꾀

ᆠ
矄�
矄鼀ᨨᨨ 泐
ᨨ
 
犨�
␔ 
 
ᆠ�ᆠ
�矄
ᨨ� �
犨�
 �盐
矄
w
ᨨ盐
  �ᆠ
�鼀
꾀

矄
矄 ᨨ犨
�
��盐

 

�矄

矄 犨�
矄ᆠ
ᨨ
 
ᆠ�ᆠ盐
� �

ᨨ矄��ᨨ

矄
矄 ᨨ犨



 
盐
ᨨ
w

�矄
⼄盐
ᨨ


᲼㝐�
҈

꾀



҈黨
ᔼᔼ桌

ᆠ㡰 �


ml w
k - wm

㡰

g
g ᆠᨨ矄 q ᆠᨨ矄 q 㡰
l l g
g 矄� q ᆠ㡰 㡰

矄� q
l
l ᔴᨨ ᆠ
鼀� 盐
�矄
鼀
泐
鼀� � �
ᆠ鼀 

矄ᨨ 
  矄 
�

ᨨ� �
ᆠ泐� �ᆠ
盐鼀

ᔼ
 
盐鼀
ᨨ矄
ᨨᨨ 矄

 
� �� 矄
鼀�
矄盐ᆠ├
ᨨ
 
犨
ᨨ
 
盐鼀  
ᨨ
鼀� � �
ᨨ
� �� 矄
ᨨ矄
ᨨ
├
ᆠ

 鼀 �鼀盐鼀
ᨨ�ᨨ 
矄
ᨨ
 
盐鼀
�
ᨨ盐�ᨨ 矄
 鼀� 盐
鼀㡰
ᨨ
 矄盐
ᨨ
鼀� � �
�矄
ᆠᨨ矄
ᨨ

 ├
 
�盐矄
ᨨ
 
盐鼀
ᨨ

҈

�
鼀
 
�矄
 
ᨨ
 ᨨ�ᨨ 㡰
᲼㝐�
졠 � 
꾀�
҈




҈
桌 㡰
䰘

㡰


ᆠ㡰 ҈
 㡰




犨ᨨ
� �ᆠ
�矄ᆠ矄
ᨨ
矄鼀
�盐矄
R

ᨨ� �
ᨨ盐  �
 矄
�
ᨨᨨ矄�
�ᆠ�ᨨ 矄
犨�
 
矄鼀
ᆠᨨ 矄   �盐
矄
w


�矄ᆠ矄

�
 
矄鼀
ᨨ�ᨨ 
 
꾀
�鼀
t
㦨



 
ᨨ� 矄
P
 
Q

ᆠ� �
ᆠ
ᨨ 
矄 矄 ᨨ犨
�
 
��盐


�
矄
犨
 
犨ᨨ ᨨ� 矄
P
 
Q
�矄
vr

ᔼ
ᨨ 
�鼀
�ᨨ
T㡰
ᨨ
ᨨ�ᨨ
ᨨ  
�矄ᆠ矄

vr 矄

盐 ᆠ�ᨨ
ᨨ
�鼀
�矄
矄
矄 
泐 ᲼

҈
桌

㡰

黨


㡰

㡰



g 㡰
g ⏌ r 㡰
r 㡰
�⏌r ᆠ㡰  g 㡰 ⏌  꾀
�ᆠ
�矄
鼀ᨨ� �
犨�

盐 � ᨨ鼀
矄
�

ᆠ�ᆠ盐
ᨨ� ᨨ
�盐矄

�

ᆠ 
ᨨᆠ
� 矄泐
ᨨᨨ�ᨨ 
ᨨ
  n
ᨨ犨
ᨨ



ᔼ
 
�ᨨ
ᨨ
ᨨ�ᨨ
ᨨ
 
�ᆠ
�矄

 








᲼㝐�


҈
桌 ⏌
㡰 µ

ᨨ
 泐

㡰 µ n ⏌
+҈

㡰





䰘


ᆠ㡰 µ  +҈㡰⏌
㡰 µ ⏌
꾀
矄 
盐 � ᨨ鼀
ᨨ
ᨨ
鼀矄矄
㝐
 
 �
l
�矄
�ᨨ 
ᨨ 
 
矄ᨨ
 
�
ᆠ
ᨨ
�

�ᆠ
 
矄 ��盐㡰


�矄
����
�ᆠ�ᨨ

 
�ᨨ


  
�矄

�ᆠ泐
ᨨ
 
�ᨨ
 
 
矄



 �盐
ᆠᆠ�ᨨ
ᨨ
 
ᨨ
犨 
�
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EBD_8341 P-50

Physics

4 ҈҈


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angle q
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P-51

System of Particles and Rotational Motion

4.

Shown in the figure is rigid and uniform one meter long rod AB held in horizontal position by two strings tied to its ends and attached to the ceiling. The rod is of mass 'm' and has another weight of mass 2 m hung at a distance of 75 cm from A. The tension in the string at A is : [Main Sep. 02, 2020 (I)] (a) 0.5 mg (b) 2 mg (c) 0.75 mg (d) 1 mg A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a < R) by applying a force F at its centre 'O' perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is : [Main Sep. 02, 2020 (I)] F

O

æ R-aö (a) Mg 1 - ç ÷ è R ø

2

R

a 2

æ R ö (b) Mg ç ÷ -1 è R-aø

a a2 (d) Mg 1 - 2 R R Consider a uniform rod of mass M = 4m and length l pivoted about its centre. A mass m moving with velocity v making p angle q = to the rod’s long axis collides with one end of 4 the rod and sticks to it. The angular speed of the rod-mass system just after the collision is: [Main 8 Jan. 2020 I]

(c) Mg

5.

6.

7.

3 v (b) 7 2l

4v 3 2v (d) 7l 7 l A particle of mass m is moving along a trajectory given by x = x0 + a cos w1t y = y0 + b cos w2 t The torque, acting on the particle about the origin, at t = 0 is: [Main 10 Apr. 2019 I] 2$ 2$ (a) m ( - x0 b + y0 a ) w1 k (b) +my0 aw1 k 2 2 $ (c) zero (d) -m( x0 bw2 - y0 aw1 )k The time dependence of the position of a particle of mass r m = 2 is given by r (t ) = 2 t i$ - 3t 2 $j . Its angular momentum, with respect to the origin, at time t = 2 is : [Main 10 Apr. 2019 II] $ $ (a) 48 i + j (b) 36k$

(a)

(

(

)

(c) -34 k$ - i$ 8.

)

3v 7l

9.

(a) 4.0×10–6 Nm (b) 1.6×10–5 Nm (c) 7.9×10–6 Nm (d) 2.0×10–5 Nm A rectangular solid box of length 0.3 m is held horizontally, with one of its sides on the edge of a platform of height 5m. When released, it slips off the table in a very short time t = 0.01 s, remaining essentially horizontal. The angle by which it would rotate when it hits the ground will be (in radians) close to : [Main 8 Apr. 2019 II]

(a) 0.5 (b) 0.3 (c) 0.02 (d) 0.28 10. A particle of mass 20 g is released with an initial velocity 5 m/s along the curve from the point A, as shown in the figure. The point A is at height h from point B. The particle slides along the frictionless surface. When the particle reaches point B, its angular momentum about O will be : [Main 12 Jan. 2019 II] (Take g = 10 m/s2) O

(c)

(d) -48k$ A metal coin of mass 5 g and radius 1 cm is fixed to a thin stick AB of negligible mass as shown in the figure The system is initially at rest. The constant torque, that will make the system rotate about AB at 25 rotations per second in 5s, is close to : [Main 10 Apr. 2019 II]

a = 10 m A h = 10 m

B (a) 2 kg-m2/s

(b) 8 kg-m2/s

(c) 6 kg-m2/s

(d) 3 kg-m2/s uur uur 11. A slab is subjected to two forces F1 and F2 of same uur magnitude F as shown in the figure. Force F2 is in XYplane while force F1 acts along z-axis at the point r r 2i + 3 j . The moment of these forces about point O will

(

)

be :

[Main 11 Jan. 2019 I] Z F1 O

F2

y

30º 4m 6m x

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EBD_8341 P-52

Physics

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WWW.JEEBOOKS.IN

P-53

System of Particles and Rotational Motion

20.

21.

22.

23.

24.

25.

of the particle is 12 rad s–1, the magnitude of its angular momentum about a point on the ground right under the centre of the circle is : [Main Online April 11, 2015] (a) 14.4 kg m2s–1 (b) 8.64 kg m2s–1 2 –1 (c) 20.16 kg m s (d) 11.52 kg m2s–1 A bob of mass m attached to an inextensible string of length l is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed w rad/s about the vertical. About the point of suspension: [Main 2014] (a) angular momentum is conserved. (b) angular momentum changes in magnitude but not in direction. (c) angular momentum changes in direction but not in magnitude. (d) angular momentum changes both in direction and magnitude. A ball of mass 160 g is th rown up at an an gle of 60° to the horizontal at a speed of 10 ms–1. The angular momentum of the ball at the highest point of the trajectory with respect to the point from which the ball is thrown is nearly (g = 10 ms–2) [Main Online April 19, 2014] (a) 1.73 kg m2/s (b) 3.0 kg m2/s 2 (c) 3.46 kg m /s (d) 6.0 kg m2/s A particle of mass 2 kg is moving such that at time t, its r position, in meter, is given by r (t ) = 5iˆ - 2t 2 ˆj . The angular momentum of the particle at t = 2s about the origin in kg m–2 s–1 is : [Main Online April 23, 2013] (a) - 80 k$ (b) (10iˆ - 16 ˆj ) (c) -40k$ (d) 40k$ A bullet of mass 10 g and speed 500 m/s is fired into a door and gets embedded exactly at the centre of the door. The door is 1.0 m wide and weighs 12 kg. It is hinged at one end and rotates about a vertical axis practically without friction. The angular speed of the door just after the bullet embeds [Main Online April 9, 2013] into it will be : (a) 6.25 rad/sec (b) 0.625 rad/sec (c) 3.35 rad/sec (d) 0.335 rad/sec A small mass m is attached to a z massless string whose other end is fixed at P as shown in the figure. The P mass is undergoing circular motion in the x-y plane with centre at O and constant angular speed w. If the m O angular momentum of the system, calculated about O and P are denoted w r r by LO and LP respectively, then r r (a) LO and LP do not vary with time r r (b) LO varies with time while LP remains constant r r (c) LO remains constant while LP varies with time r r (d) LO and LP both vary with time A particle is confined to rotate in a circular path decreasing linear speed, then which of the following is correct? [2005S]

r L (angular momentum) is conserved about the centre r (b) only direction of angular momentum L is conserved (c) It spirals towards the centre (d) its acceleration is towards the centre. 26. A block of mass m is at rest under the action of force F against a wall as shown in figure. Which of the following statement is incorrect? [2005S] a (a) f = mg [f friction force] (a)

a

(b) F = N [N normal force]

F

(c) F will not produce torque (d) N will not produce torque 27. A horizontal circular plate is rotating about a vertical axis passing through its centre with an angular velocity wo. A man sitting at the centre having two blocks in his hands stretches out his hands so that the moment of inertia of the system doubles. If the kinetic energy of the system is K initially, its final kinetic energy will be [2004S] (a) 2 K (b) K/ 2 (c) K (d) K/ 4 28. A particle undergoes uniform circular motion. About which point on the plane of the circle, will the angular momentum of the particle remain conserved? [2003S] (a) centre of the circle (b) on the circumference of the circle. (c) inside the circle (d) outside the circle. 29. Consider a body, shown in figure, consisting of two identical balls, each of mass M connected by a light rigid rod. If an impulse J=MV is imparted to the body at one of its ends, what would be its angular velocity? [2003S] L

(a) V/L (b) 2V/L M (c) V/3L

M

J = MV (d) V/4L 30. A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its centre. A tortoise is sitting at the edge of the platform. Now, the platform is given an angular velocity w0. When the tortoise move along a chord of the platform with a constant velocity (with respect to the platform), the angular velocity of the platform w(t) will vary with time t as [2002S] w(t )

w(t )

(a)

w0

(b)

w0

t

t

w( t )

w( t )

(c) w0

(d) w0 t

t

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EBD_8341 P-54

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 �盐
ᨨᆠ�泐 ᲼꾀



҈桌 �

矄҈㡰
ᨨ
 
矄泐矄鼀
�矄 䰘
꾀
� 泐
矄
ᆠᨨ 矄�矄矄
ᨨ
犨ᨨ
矄矄
꾀
鼀矄矄



M矄㡰
 
 鼀矄矄
҈҈M矄㡰

犨 
Ms
�矄
 
鼀矄矄
ᨨ
 
矄盐

泐
 矄
泐
�矄 ᆠ

 

ᨨ� �
ᨨ盐
 �
ᆠ  ᨨ
鼀矄矄

犨 �ᆠ
�矄
矄�ᨨ 泐


�ᨨ
ᨨ
 
ᨨ
 �盐 鼀ᨨ鼀 盐鼀
ᨨ
 
� 泐
矄
ᨨ
 
 �盐
鼀ᨨ鼀 盐鼀
ᨨ 矄

ᨨ盐
 
ᆠ 
ᨨ
鼀矄矄
�矄 ᲼

҈
桌 3

Numeric Answer



꾀
 �
ᨨ
ᨨ
鼀矄矄


黨
├�
 
 �
҈
鼀
�矄
矄盐矄 

 矄

ᨨ鼀
ᨨ 
 
矄ᨨ
 
�
ᆠ
泐
ᨨ矄ᆠ�
�
  �ᆠ
 

꾀
�ᆠ
ᨨ
鼀ᨨ


҈
├�
鼀ᨨ� �
�

矄��  � 
犨�
ᨨᆠ�泐



鼀⏌矄
�矄
 
ᨨ

�矄
ᨨᨨ鼀
鼀ᨨ矄 ᨨ� 
 
矄�ᆠ├矄
ᨨ
�
矄
��盐㡰


 �盐
矄 �
⏌矄㡰
ᨨ
 
ᨨ
�鼀鼀�泐
 
 
ᆠᨨ�矄�ᨨ
犨�
 힘힘힘힘힘힘힘힘힘힘힘힘힘힘
᲼㝐�
和


�






ᔼᔼ㡰桌 黨
꾀
矄ᨨ
ᨨ



├�
鼀矄矄
�矄
矄 � �
ᨨ
 
�鼀
ᨨ

ᆠ�ᆠ盐  ᨨ鼀
ᨨ
鼀矄矄




├�
ᨨ� �
ᨨ盐
�矄
 �矄

� ᨨ盐�ᨨ 矄

鼀� 盐
鼀㡰


矄ᨨ
ᨨ犨
矄矄 鼀ᨨ� �
ᨨ犨矄
 
ᆠ 
ᨨ
 
 ᨨ鼀

␔ 
犨�
  
ᨨ�ᨨ 
矄
�
鼀㡰
ᨨ
 
 ᨨ鼀
犨 
  矄ᨨ
ᆠ 矄
�矄
ᆠ 
힘힘힘힘힘힘힘힘힘힘
[Main Sep. 03, 2020 (I)]



盐

盐 � ᨨ鼀
鼀
矄ᆠ
ᨨ�ᨨ 泐
ᨨ
泐ᨨ盐
   � 
� �矄
犨�
 
 
ᨨ 






ᆠ鼀
 
 
�� 
ᨨ  
黨




ᆠ鼀

␔ 
泐ᨨ盐
鼀
ᨨ
鼀ᨨ
ᨨ
 
� �矄 矄ᨨ犨泐
ᨨ犨矄
 
ᆠ 

� ��泐
ᨨ 泐
 
 
� �
矄�矄 犨�
矄ᆠ
ᨨ
 
矄ᆠ
 
 
�� 
� �
ᨨ矄
ᨨ
꾀 
矄ᨨ鼀
�矄 ᆠ

 
 
� �
矄ᨨ矄
 
 
�� 
ᨨ  矄矄
矄�� �


 
�� 
� �
矄ᨨ矄


�矄 ᆠ
xR ᨨ鼀
 
ᆠ 
�




ᆠ鼀㡰
ᨨ
 
矄ᆠ
 
 
 
ᨨ  矄矄
矄�� �
��

�矄
 矄
ᆠ盐矄
ᨨ
  �  ᆠ
�
 
�ᆠ�ᨨ 
ᨨᆠ矄
ᨨ
 
犨ᨨ
� �矄

ᔼ
  ᆠᨨ �ᆠ� 矄
ᨨ
矄�ᆠ
 
泐 鼀�ᆠ
�ᆠ�ᨨ
犨
  � �矄
 
 
矄ᆠ






 





矄ᆠ�泐

  盐
ᨨ
xR
�
ᆠ鼀㡰
�矄
힘힘힘힘힘힘
᲼꾀





桌 4

Fill in the Blanks


҈
꾀
矄ᨨ 
ᨨ
鼀矄矄
m

�
ᨨ
 
 
ᨨ

矄� �

�矄
犨 � ᨨ盐 
�

ᨨ�ᨨ 
ᆠ�ᆠ

�ᆠ
 
ᨨᆠ
盐
ᨨ ��泐㡰


 �
ᨨ
 
矄� �
�矄
盐ᆠ
�盐泐 ├� �
 
 �盐
鼀ᨨ鼀 盐鼀
ᨨ
 
矄ᨨ 
ᨨ盐
  ᆠ 
ᨨ
 
ᆠ�ᆠ
ᆠᨨ 矄 



 
 矄�ᨨ
�
 
矄� � �矄
��
泐
T
㦨
Arn
犨 
A
�矄

ᆠᨨ 矄 

r
�矄
  � 矄  ᨨ盐矄
�盐矄
ᨨ
 
ᆠ�ᆠ
 
n
㦨





᲼҈黨黨
㜠
҈
㝐├桌

WWW.JEEBOOKS.IN

P-55

System of Particles and Rotational Motion










꾀
ᆠ泐� 
ᨨ
鼀矄矄
M
 
�盐矄
R
�矄
矄� �
ᨨ

ᨨ�ᨨ   ᨨ鼀
犨 �ᆠ
�矄

ᨨ
 
xy
 㡰
犨�
�矄
 �矄 � 
ᨨ �
 
y㜠 �矄
 

ᨨ
ᨨ
ᨨ盐
�矄
 �矄

  ᨨ鼀
�矄
��

鼀ᨨ�ᨨ
�
 
x㜠�ᆠ�ᨨ
��
泐
x
㦨
A ᆠᨨ矄
 w t㡰


�矄
ᨨ
矄�� �
犨
 
ᆠ泐� 
   ᨨ鼀


鼀 �鼀盐鼀
ᨨ⼄盐
ᆠ� �
ᨨ
 
ᆠ泐� 
盐� � �矄
鼀ᨨ�ᨨ
�矄


















᲼҈黨


㜠


㝐├矄桌 꾀
矄鼀ᨨᨨ
盐 � ᨨ鼀
ᨨ
ᨨ
 �
L
 
鼀矄矄
M
矄
犨ᨨ � �ᆠ
矄
ᨨ
����
矄�

ᆠ
ᨨ
鼀矄矄
m

犨 �ᆠ ᆠ
矄�
泐
ᨨ �
 
ᨨ

ᔼ ��泐
 
犨ᨨ
矄

  
ᆠ 
ᨨ
 
ᨨ
 
 
矄泐矄鼀
�矄
ᨨ� �
犨�


 �盐
ᨨᆠ�泐
w

ᨨ盐

 �矄
 盐ᆠ盐
ᨨ
  ᨨ
 
矄矄� �
 ᨨ盐�
 
鼀�ᨨ� 
ᨨ
 
ᨨ
矄 ��盐㡰



ᨨ
  
ᨨᆠ矄

␔ 
 
矄
ᆠ  
 矄
ᨨ
 
ᨨ

 
 �盐
ᨨᆠ�泐
ᨨ
 
矄泐矄鼀
�矄








᲼҈黨


㜠


㝐├矄桌






 w





ᔴᨨ 矄�

ᨨ泐
ᨨ
鼀矄矄
҈


├�

矄

 
ᨨ���

�鼀 r t
㦨



꾀
ᨨᆠ
F = at �ˆ + bˆ 㡰
�矄
�
ᨨ
 
ᨨ泐

犨  a
㦨
҈


矄҈
 
b
㦨
҈


N


ᨨ⼄盐
ᆠ� �
ᨨ
 
ᨨ泐 r ᨨ盐
 
ᨨ���

�鼀
t
㦨
҈


s
�矄
t

␔ �ᆠ
ᨨ
  ᨨᨨ犨� �
矄鼀 矄
�矄
㡰
盐䁸 ᲼꾀



҈
桌 r ҈ 㡰  t = Nm  r 㡰 
ᨨ⼄盐
t
�矄
�
 
�ᆠ�ᨨ
ᨨ
 
盐 �
ᆠᨨ
졔 kˆ r ҈ ᆠ㡰 
ᨨᆠ�泐
ᨨ
 
ᨨ泐

t
㦨
҈
s
�矄
v = �ˆ +
ˆ 㡰ms ҈
҈ 㡰 
鼀� �盐
ᨨ
�矄ᆠ鼀 
ᨨ
 
ᨨ泐

t
㦨
҈s
�矄
m

䰘

ᨨ �
 �泐
ᨨ

�ᆠ
ᨨ
鼀矄矄
m


�矄 ᆠ r
ᨨ鼀

� 
ᨨ� 
O
�矄
��
泐
Vr㡰
㦨
kr
⏌


犨 
k
�矄
 ᨨ矄��
ᆠᨨ 矄 
ᨨ
ᨨ�
�鼀 矄�ᨨ 矄

�矄
�ᆠ �矄
鼀ᨨ� �
�

ᆠ�ᆠ盐
ᨨ�
ᨨ
�盐矄
R
ᨨ盐
 
ᨨ� 
O
ᔼ
v
�矄
 
矄
ᨨ
 
�ᆠ
 
L
�矄
 
鼀� �盐
ᨨ �矄
 �盐
鼀ᨨ鼀 盐鼀
ᨨ盐
O

犨 �ᆠ
ᨨ
 
ᨨᨨ犨� � 矄鼀 矄
�矄
㡰
盐䁸 ᲼꾀



҈
桌 k R
m

㡰 v
㦨





㡰 v
㦨


mk
R



꾀
犨 
ᨨ
�盐矄

 
鼀矄矄
㝐
�矄
ᆠ

 
ᨨᨨ鼀
ᨨ 
� 
矄
ᨨ
�� 

矄
矄 ᨨ犨
�
 
��盐

꾀
ᆠᨨ 矄  ᨨᆠ
�矄
ᆠᨨ � 盐ᨨ盐矄泐
�
ᨨ
 
矄盐 ᆠ
ᨨ
 
犨  矄ᨨ
 
�
盐矄
ᆠ�鼀矄
 
矄
犨� ᨨ盐
矄�� �

ᔴᨨ 矄�  
ᨨ⼄盐
t
ᨨ盐

 �矄
ᨨ鼀
ᨨ
 
 
ᨨ
 
 矄矄� �
 ᨨ盐�
 
ᨨ� 
턼

␔ �ᆠ
ᨨ
 
ᨨᨨ犨� � ᲼꾀



҈䰘桌 ᨨ�ᨨ 矄
�矄⏌
ᆠᨨᆠ䁸

ᆠ㡰 L
㦨
mk R







5 True / False 꾀
 �
盐 � ᨨ鼀
ᆠ�ᆠ盐
�矄ᆠ
ᨨ
鼀矄矄
M
 
�盐矄
R
�矄 ᨨ� �
�

ᨨ�ᨨ 
 
ᨨ盐

 �矄
矄矄� �
 ᨨ盐� �矄
ᆠ 
 
 �ᆠ盐
ᨨ
�矄
 
犨�

 �盐 ᨨᆠ�泐
w

꾀 ᨨ 
�矄ᆠ
ᨨ
 
矄鼀
�鼀 矄�ᨨ 矄
盐
ᨨ 鼀矄矄
M⏌

�矄
ᆠ
� 泐
ᨨ
 
�矄
�矄ᆠ
ᆠᨨ �泐

  �盐
ᨨᆠ�泐
ᨨ
 
矄泐矄鼀
ᨨ犨
�矄

w ⏌ �
᲼҈黨


㜠

㝐├矄桌 6


�


k R m

㡰 L
㦨


和

MCQs with One or More than One Correct Answer

꾀
ᨨ
ᨨ
鼀矄矄
m
 
 �
L

�ᨨ

ᨨ 
ᨨ
�矄
 矄

�矄  �� �
�ᆠ泐

꾀
盐
ᨨ
 
矄鼀
鼀矄矄
鼀ᨨ� �
 矄
v
矄�├矄
 
ᨨ
ᨨ�ᨨ 泐


�矄 ᆠ
x
ᨨ鼀
�矄 �ᨨ
 
 
�矄
鼀
�
�


ᆠᨨ鼀�  矄泐矄鼀
ᨨ犨
ᨨ矄
犨�
 �盐
矄
w
ᨨ盐
 
�ᨨ



鼀 �鼀盐鼀
 �盐
矄
wM
�矄
ᆠ �
ᨨ
x = xM.



᲼꾀





桌

x

L

v

vx 㡰 w =
L + x
L ᆠ㡰 xM = 

҈
vx 㡰 w =
L + ҈
x
v  㡰 wM =
L

턼

 

㡰 ᔼ
 
ᨨᆠ
�矄
�

ᨨ� 

 � �泐
 

ᆠ矄矄
ᆠᨨ � 盐ᨨ盐矄泐
矄
 
犨 
ᆠ�鼀矄 㡰 ᔼ
 
ᨨᆠ
�矄
�
ᨨ鼀
ᨨ
 
ᆠ�ᆠ盐鼀  ᆠ
 ᨨ� 

 
t
�矄
ᆠᨨ 矄  ᆠ㡰 ᔼ
 
ᨨᆠ
�矄
�
ᨨ鼀
ᨨ
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WWW.JEEBOOKS.IN

EBD_8341 P-56

Physics

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mg
mg =
 
N҈  q =
mg ¹
 
N
= ҈ + m m ҈


㡰

m҈ =

m
¹

 
N҈  q =

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P-57

System of Particles and Rotational Motion

��


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ᔼᔼ㡰桌

MCQs with One Correct Answer

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O' O




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MR
M R
+ H
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MH MR ᆠ㡰 㡰  

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ᔼᔼ㡰桌

5 䰘
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EBD_8341 P-58





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P-59

System of Particles and Rotational Motion

part is converted into a uniform solid sphere. Let I1 be the moment of inertia of the new sphere about its axis. The ratio I1/I2 is given by : [Main 10 Apr. 2019 II] (a) 185 (b) 140 (c) 285 (d) 65 17. A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of q, where q is the angle by which it has rotated, is given as kqn. If its moment of inertia is I then the angular acceleration of the disc is: [Main 9 April 2019 I] k k k 2k q q (c) 2 I q q (a) (b) (d) 4I I I 18. A thin smooth rod of length L and mass M is rotating freely with angular speed w0 about an axis perpendicular to the rod and passing through its center. Two beads of mass m and negligible size are at the center of the rod initially. The beads are free to slide along the rod. The angular speed of the system, when the beads reach the opposite ends of the rod, will be: [Main 9 Apr. 2019 II] M w0 M w0 (a) (b) M +m M + 3m M w0 M w0 (c) (d) M + 6m M + 2m 19. A thin circular plate of mass M and radius R has its density varying as r(r) = r0 r with r0 as constant and r is the distance from its center. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is I = a MR2. The value of the coefficient a is: [Main 8 April 2019 I] 3 (a) 1 2 (b) 3 5 (c) 8 5 (d) 2 20. Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm), about its axis be 1. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is: [Main 12 Jan. 2019 I] (a) 12 cm (b) 16 cm (c) 14 cm (d) 18 cm 21. The moment of inertia of a solid sphere, about an axis parallel to its diameter and at a distance of x from it, is ‘I(x)’. Which one of the graphs represents the variation of I(x) with x correctly? [Main 12 Jan. 2019 II]

I(x)

I(x)

(a)

(b)

O

x

O

I(x)

I(x)

(c)

22.

x

triangle DEF is removed from ABC, the moment of inertia of the remaining figure about the same axis is I. Then : [Main 11 Jan. 2019 I] A

E

D

G B

23.

C

F

15 3 (b) I = I0 (a) I = I0 16 4 9 I (c) I = I0 (d) I = 0 16 4 a string is wound around a hollow cylinder of mass 5 kg and radius 0.5 m. If the string is now pulled with a horizontal force of 40 N, and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect the mass and thickness of the string) [Main 11 Jan. 2019 II] 40 N

(a) 20 rad/s2 (b) 16 rad/s2 2 (c) 12 rad/s (d) 10 rad/s2 24. A circular disc D1 of mass M and radius R has two identical discs D2 and D3 of the same mass M and radius R attached rigidly at its opposite ends (see figure). The moment of inertia of the system about the axis OO’, passing through the centre of D1, as shown in the figure, will be : [Main 11 Jan. 2019 II] O'

D2

O

D3

D1

(a) MR2

(b) 3MR2

(c)

4 MR 2 (d) 5

2 MR 2 3

m are connected at the two ends of a 2 massless rigid rod of length l. The rod is suspended by a thin wire of torsional constant k at the centre of mass of the rod-mass system (see figure). Because of torsional constant k, the restoring toruque is t= kq for angular displacement q. If the rod is rotated by q0 and released, the tension in it when it passes through its mean position will be: [Main 9 Jan. 2019 I]

25. Two masses m and

(d)

x x O O An equilateral triangle ABC is cut from a thin solid sheet of wood. (See figure) D, E and F are the mid-points of its sides as shown and G is the centre of the triangle. The moment of inertia of the triangle about an axis passing through G and perpendicular to the plane of the triangle is I0. If the smaller

(a)

3k q0 2 l

(b)

2k q0 2 l

(c)

k q0 2 l

(d)

k q0 2 2l

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EBD_8341 P-60






Physics

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WWW.JEEBOOKS.IN

P-61

System of Particles and Rotational Motion

36. From a circular disc of radius R and mass 9M, a small disc of radius R/3 is removed from the disc. The m oment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through O is [2005S] (a) 4MR2 R 3

40 MR 2 (b) 9

R

(c) 10 MR 2 (d) 37.

2R/3

37 MR 2 9

IO and IP respectively. Both these axes are perpendicular to the plane of the lamina. The ratio IP / IO to the nearest integer is [2012] 42. Four solid spheres each of diameter 5 cm and mass 0.5 kg are placed with their centers at the corners of a square of side 4 cm. The moment of inertia of the system about the diagonal of the square is N ´ 10–4 kg– m2, then N is. [2011] 3

One quarter sector is cut from a uniform circular disc of radius R. This sector has mass M. It is made to rotate about a line perpendicular to its plane and passing through the center of the original disc. Its moment of inertia about the axis of rotation is [2001S]

Numeric Answer

43. ABC is a plane lamina of the shape of an equilateral triangle. D, E are mid points of AB, AC and G is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through G and perpendicular to the plane ABC is I0. If part ADE is removed, the moment of inertia of the NI 0 where N is an 16 integer. Value of N is _______. [Main Sep. 04, 2020 (I)]

remaining part about the same axis is (a)

38.

1 MR 2 (b) 2

A

1 MR 2 (d) (c) 2 MR 2 8 A thin wire of length L and uniform linear mass density r is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX' is [2000S]

(a)

rL3 8p

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2 3

(c)

39.

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16p

X

2

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3

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8p2 16p Let I be the moment of inertia of a uniform square plate about an axis AB that passes through its centre and is parallel to two of its sides. CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle q with AB. The moment of inertia of the plate about the axis CD is then equal to [1998S - 2 Marks] (a) I (b) I sin2 q 2 (c) I cos q (d) I cos2 (q/2) 2

2

rL3

Integer Value Answer

40. The densities of two solid spheres A and B of the same r radii R vary with radial distance r as rA(r) = k æç ö÷ and R è ø 5 ærö rB(r) = k ç ÷ , respectively, where k is a constant. The è Rø moments of inertia of the individual spheres about axes passing through their centres are IA and IB, respectively. If IB n = , the value of n is [Adv. 2015] I A 10 41.

A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing though O and P is

E

D

G B

C

44. A circular disc of mass M and radius R is rotating about its axis with angular speed w1. If another stationary disc R having radius and same mass M is dropped co-axially 2 on to the rotating disc. Gradually both discs attain constant angular speed w2. The energy lost in the process is p奠 of the initial energy. Value of p is ___________. [Main Sep. 04, 2020 (I)] 45. An massless equilateral triangle EFG of side 'a' (As shown in figure) has three particles of mass m situated at its vertices. The moment of inertia of the system about the N ma 2 line EX perpendicular to EG in the plane of EFG is 20 where N is an integer. The value of N is _________. [Main Sep. 03, 2020 (II)] X

E

F

a

G

46. One end of a straight uniform 1 m long bar is pivoted on horizontal table. It is released from rest when it makes an angle 30° from the horizontal (see figure). Its angular speed when it hits the table is given as ns -1 , where n is an integer. The value of n is ________ . [Main 9 Jan. 2020 I]

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EBD_8341 P-62


䰘


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P-63

System of Particles and Rotational Motion

sufficiently high so that the block does not slide before toppling, the minimum force required to topple the block is [2000S] (a) infinitesimal P F (b) mg/4 M L (c) mg/2

joining AB and Cd (see figure). It is given a light push so that it starts rolling with its centre O moving parallel to CD in the direction shown. As it moves, the roller will tend to: [Main 2016] B

D

(d)

O

2 A

7.

C

(a) go straight. (b) turn left and right alternately. (c) turn left. (d) turn right. A loop of radius r and mass m rotating with an angular velocity w0 is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero.What will be the velocity of the centre of the hoop when it ceases to slip ? [Masin 2013]

mg(1 - m) Integer Value Answer

12. Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds v1 and v2, respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and v1 = 3 m/s then v2 in m/s is [Adv. 2015] (g = 10 m/s2) A

v1 = 3m/s 30m

rw0 rw0 rw0 (b) (c) (d) rw0 3 4 2 A small object of uniform density rolls up a curved surface with an initial velocity v. It reaches up to a maximum height 3v 2 of with respect to the initial position. The object is 4g [2007]

B

(a) 8.

v

9.

10.

11.

(a) ring (b) solid sphere (c) hollow sphere (d) disc A disc is rolling without slipping with angular velocity w. P and Q are two points equidistant from the centre C. The order of magnitude of velocity is [2004S] (a) vQ > vC > vP (b) vP > vC > vQ C P (c) vP = vC , vQ = vC/2 Q (d) vP < vC > vQ A cylinder rolls up an inclined plane, reaches some height, and then rolls down (without slipping throughout these motions). The directions of the frictional force acting on the cylinder are [2002S] (a) up the incline while ascending and down the incline descending (b) up the incline while ascending as well as descending (c) down the incline while ascending and up the incline while descending (d) down the incline while ascending as well as descending. A cubical block of side L rests on a rough horizontal surface with coefficient of friction m . A horizontal force F is applied on the block as shown. If the coefficient of friction is

C

v2 27m D

13. A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2N on the ring and rolls it without slipping with an acceleration of 0.3 m/s2. The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is (P/10). The value of P is [2011] 3

Numeric Answer

14. Consider a uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force F at a point b above its centre of mass (see figure). If the coefficient of friction is m = 0.4, the maximum possible value b of 100 × for box not to topple before moving is a ________. [Main 7 Jan. 2020 II]

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EBD_8341 P-64

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P-65

System of Particles and Rotational Motion






m


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㜠


㝐├矄桌

WWW.JEEBOOKS.IN

EBD_8341 P-66

Physics

㡰 ᆠ㡰 2 黨


M w
L 㡰
M w
L 㡰
Integer Value Answer

ᆠ㡰

M w
L

犨�







Mw L


F

졠 
矄ᆠᨨ 
 
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ᨨᆠ矄

 
 �盐
矄
ᨨ  
�矄ᆠ
�

矄҈
�矄 ᲼꾀



҈
桌

҈



꾀
盐 � ᨨ鼀
ᆠ盐
ᨨ
矄�
a
 
鼀矄矄
m
矄矄
ᨨ

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

꾀
ᨨ�ᨨ 
ᨨᆠ
F
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ᨨ
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

�� 
a⏌

ᨨ
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
鼀� �鼀盐鼀 盐
ᨨ
F
ᨨ
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�� 矄
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꾀矄矄盐鼀
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ᆠ盐
ᨨ矄
ᨨ
矄�㡰
᲼҈黨


㜠


㝐├矄桌 5

҈҈


True / False

꾀
� �盐

ᨨ
盐 � ᨨ鼀  �ᆠ├ 矄矄
 
 矄�泐
�矄
鼀 ᨨ
ᨨ
ᨨ盐

 �矄  �ᆠ盐
ᨨ
 
 
ᨨ  

 
㡰
矄矄� � A B C  ᨨ盐�
A

㡰
矄矄� �
 ᨨ盐� B

泐
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�ᆠ�ᨨ
ᨨ F  
矄鼀
ᨨᆠ

F


C
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ᨨ
AB㡰
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矄 ᨨ犨
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矄鼀
᲼҈黨
�
㜠

㝐├矄桌 6

҈



Fill in the Blanks

MCQs with One or More than One Correct Answer

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ᨨᨨ
犨� ᨨ盐
矄�� �

␔ �ᆠ
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矄鼀 矄㡰
�矄⏌
ᆠᨨᆠ

犨 
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ᨨ
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犨�
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Mg ҈



g L ҈
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矄ᨨ�
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P
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Q
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矄鼀
鼀矄矄
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矄鼀
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ᨨ� �
ᨨ犨

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ᨨ鼀
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P
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犨 �
Q
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␔ �ᆠ
矄鼀 矄㡰
�矄㡰 ᆠᨨᆠ䁸 ᲼

҈
桌

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F
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4

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P
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Q
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
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矄鼀 �鼀
㡰 ᔴ泐� 矄
P
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ᆠ㡰 ᨨ
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Q
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鼀
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҈
鼀⏌矄

꾀
矄鼀

ᨨ 鼀矄矄


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├�

鼀ᨨ� �
犨�
ᨨᆠ�泐



鼀⏌矄
�
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ᨨᨨ矄� �ᆠ�ᨨ
�矄
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

�� 
ᨨ


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盐
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҈

鼀⏌矄

ᔼ鼀鼀�泐
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  ᆠᨨ�矄�ᨨ


᲼

҈҈桌

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盐
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ᔴ㝐
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矄ᨨ
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m
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矄鼀ᨨᨨ ᨨ�ᨨ 
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m
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m
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v
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ᨨ
 
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ᆠ  ᨨ
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ᨨ
 �泐
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ᨨ
鼀矄矄
ᨨᆠ�泐
泐
w

E  
vc
矄ᆠ�泐

犨

 
ᆠᨨ�矄�ᨨ

᲼҈黨黨҈
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㝐├桌 㡰 vc
㦨



m v 㡰 w
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a v �a ᔴ v ᆠ㡰 w = �a 㡰

mv
E= �


a


a


v 鼀m

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P-67

System of Particles and Rotational Motion

7

Match the Following




҈

ᔼ
 
ᔴᨨ盐鼀 㜠ᔼ
ᨨ犨

ᨨ盐
�  
 矄
ᨨ

�ᆠ 
��
矄
盐 ᆠ�ᨨ 矄
ᨨ
�鼀

ᔼ
 矄
盐 ᆠ�ᨨ 矄

a
 
b 
ᨨ矄��
ᆠᨨ 矄 矄
ᨨ
ᨨ�
�鼀 矄�ᨨ 矄
 
a ¹ b
ᔼ
ᆠ
ᆠ矄

 
ᨨᆠ
ᆠ� �
ᨨ
 
�ᆠ
�矄
�  ᨨ
ᨨ
ᆠᨨ 矄�

ᔼ
ᔴᨨ盐鼀 㜠ᔼᔼ

�
 泐矄�ᆠ
⼄盐 ��矄 r r ᨨ
 
�ᆠ

鼀 �ᨨ   �矄
 
� 
鼀ᨨ鼀 盐鼀
 �矄  
 �盐
鼀ᨨ鼀 盐鼀
ᨨ盐
 
ᨨ���

⧤
�矄
 
├� �ᆠ  �泐



�矄
 
ᨨ �
 �泐
 
쟼
�矄
 
ᨨ
 �泐
㝐ᆠ
ᆠ

�
�矄㜠ᔼ
犨�
 ᨨ矄
⼄盐 ��矄
�
�矄㜠ᔼᔼ
犨 �ᆠ

ᆠᨨ 矄
ᨨ
 




















᲼꾀



҈
桌

턼

和
㡰 㡰 ᆠ㡰 㡰

ᔴᨨ盐鼀 ᔼ r 㡰 = a  �ˆ+ b  ˆ r 㡰 = a ᆠᨨ矄 w �ˆ+ b 矄� w ˆ r 㡰 = aᆠᨨ矄 w �ˆ+ 矄� w ˆ㡰 r b 㡰 = a �ˆ+ 
ˆ





ᔴᨨ盐鼀 ᔼᔼ r  r  ⧤







҈




�
쟼 
®
҈









�錐
턼
®



�錐

®








�錐
和
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� 
®
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




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턼
®


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




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和
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®






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턼
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®
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錐
和
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� 
®
҈






�錐
턼
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®








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和
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҈䰘
ᔴᨨ盐鼀 㜠ᔼᔼ
矄 ᨨ犨矄
�
矄泐矄鼀矄
�
犨 �ᆠ
犨ᨨ
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

矄

 
Ӹ

꾀矄ᨨ
�
ᆠ
ᆠ矄

ᨨ� 

�矄
矄 ᨨ犨

ᔴᨨ盐鼀 㜠ᔼ ��矄
矄ᨨ鼀
矄鼀 矄
ᨨ盐

 ⏌ᨨ
Ӹ

㝐ᆠ
 矄
矄鼀 矄
ᨨ
 
ᨨ�
矄泐矄鼀矄㡰
ᨨ鼀
ᔴᨨ盐鼀
ᔼᔼ








᲼


黨桌 ᔴᨨ盐鼀 㜠ᔼ ᔴᨨ盐鼀
ᔼᔼ 꾀㡰 
ᨨᆠ
 
泐

ᨨ
Ӹ
矄
 㡰 Ӹ  鼀� �盐
Mg. 

ᨨᆠ├
Ӹ
ᨨ

鼀矄矄
M  
ᨨ

� 

� ᆠ� 
 


矄�矄
ᨨ
�
犨�
 ᆠᨨ 矄 
ᨨᆠ�泐
㡰 
���ᨨ 
ᨨ �
 �泐
ᨨ 
�矄
ᆠᨨ � 盐ᨨ盐矄泐
� ᆠ矄� �


⼄㡰




 Ӹ 

犨ᨨ
� �
鼀� 矄
Ӹ
 


ᆠ
ᨨ
鼀矄矄
M, 
├
�
�ᆠ�ᨨ 矄矄
�ᆠ 矄�ᆠ
矄 
矄ᨨ
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 泐

ᆠ
ᨨ 

Ӹ
矄矄
ᨨ
 
矄

 
  �矄
�
�
�
⼄盐���盐鼀


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 
ᨨ鼀ᨨ矄
ᨨ� 
ᨨ
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矄 
ᨨ
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ᨨ
 
犨ᨨ
� �矄


犨 ᨨ
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�

� 
 
�矄
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
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ᨨᆠ�泐
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 �泐
ᨨ
 
矄泐矄鼀 
졔Ӹ
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ᆠᨨ � 盐ᨨ盐矄泐
ᆠ矄� �


㡰 
ᨨ⼄盐
ᨨ
 
犨�� 
ᨨ
Ӹ
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�矄
ᨨ


㡰 꾀
盐泐
Ӹ
ᨨ
鼀矄矄
m0 �矄
� 
ᨨ


 ᨨ盐�
 ᆠ鼀


꾀
ᨨᆠ├
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鼀矄矄
㝐
 �矄
ᨨ鼀

矄� �
  �ᨨ矄
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 
盐泐
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ᨨ
  


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�

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 
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犨�

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ᨨᆠ�泐




Ӹ 

矄㡰 Ӹ

 

꾀
矄 
Ӹ
ᨨ
鼀矄矄
M
�矄
盐
�

ᨨ 㜠�矄ᆠᨨ盐矄
�⼄盐�

├
�

ᆠᨨ � 
 矄


矄 
�矄
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�
鼀ᨨ矄
ᨨ犨
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 
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㡰 꾀
矄 
Ӹ
ᨨ
鼀矄矄
M
�矄
� �
犨�
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鼀�  ᨨᆠ�泐
�

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�⼄盐�

├
�
 ᆠᨨ � 


Ӹ

 

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EBD_8341 P-68

Physics

8

Comprehension Based Questions

Passage 1 One twirls a circular ring (of mass M and radius R) near the tip of one's finger as shown in Figure 1. In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is r. The finger rotates with an angular velocity w0. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is m and the acceleration due to gravity is g. [Adv. 2017]

Now consider two similar systems as shown in the figure: Case (a) the disc with its face vertical and parallel to x-z plane; Case (b) the disc with its face making an angle of 45° with x-y plane and its horizontal diameter parallel to x-axis. In both the cases, the disc is welded at point P, and the systems are rotated with constant angular speed w about the z-axis. z

Q

w

y

P x

Case (a)

z

w

Q

45°

R x

r R Figure 1

Figure 2

18.

19.

The total kinetic energy of the ring is 1 Mw02 (R - r) 2 (a) (b) Mw02 R 2 2 3 Mw02 (R - r)2 (c) (d) Mw02 (R - r) 2 2 The minimum value of w0 below which the ring will drop down is (a)

g m(R - r)

(c)

3g 2m(R - r)

y

P

(b)

2g m(R - r)

(d)

g 2m(R - r)

Passage 2 The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass. These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless, stick, as shown in the figure. When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed w, the motion at any instant can be taken as a combination of (i) a rotation of the centre of mass of the disc about the z-axis and (ii) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points P and Q). Both these motions have the same angular speed w in this case [2012]

Case (b)

20. Which of the following statements about the instantaneous axis (passing through the centre of mass) is correct? (a) It is vertical for both the cases (a) and (b) (b) It is vertical for case (a); and is at 45° to the x-z plane and lies in the plane of the disc for case (b). (c) It is horizontal for case (a); and is at 45° to the x-z plane and is normal to the plane of the disc for case (b). (d) It is vertical for case (a); and is 45° to the x-z plane and is normal to the plane of the disc for case (b). 21. Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?

2w for both the cases

(a) It is

(b) It is w for case (a); and w/ 2 for case (b) (c) It is w for case (a); and 2w for case (b) (d) It is w for both the cases. Passage 3 A uniform thin cylindrical disk of mass M and radius R is attached to y two identical massless springs of spring constant k which are fixed 2d to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance d from its centre. The axle is massless and both the d d springs and the axle are in R V0 horizontal plane. x The unstretched length of each spring is L. The disk is initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall. The disk rolls without slipping with r velocity V = V ˆi . The coefficient of friction is µ. [2008] 0

0

22. The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is (a) – kx (b) – 2kx (c) – 2kx/3 (d) – 4kx/3

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P-69

System of Particles and Rotational Motion





졠


ᆠ 
ᨨ
鼀矄矄
ᨨ
 
�矄├
盐 �ᨨ矄
矄�鼀
鼀ᨨ �ᆠ 鼀ᨨ�ᨨ
犨�
 �盐
⼄盐 ᆠ泐
w
⼄盐
ᨨ
 ├
├
├
├ 㡰 ᆠ㡰 㡰 㝐 㝐 㝐 㝐


鼀 �鼀盐鼀
盐
ᨨ


ᨨ
犨 �ᆠ
 
�矄├
犨�
ᨨ
犨� ᨨ盐 矄�� �
�矄


㝐

㡰




㡰

m�

㝐 ├

㡰

m�

㝐 㝐 �㝐
ᆠ㡰 m�
㡰
m�
├ ├
├

矄矄�



犨ᨨ
�矄ᆠ矄
A
 
B

鼀ᨨ盐 
ᆠᨨ �泐
ᨨ

�ᆠ
 

 �矄ᆠ矄

鼀ᨨ鼀 矄
ᨨ
� �
I
 


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  ᆠᨨ鼀鼀ᨨ
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A
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
� ��
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w 盐矄� �

 
 �

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
矄� �

ᆠᨨ鼀矄矄
泐
 �矄 ᆠ
x

�矄ᆠ
B
�矄
�鼀

 �盐
ᨨᆠ�泐
w
泐
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矄� � � �
 
矄鼀
矄� �
ᆠᨨ 矄 
 
ᆠᨨ鼀矄矄
泐

�矄 ᆠ x


ᨨ
 
�矄ᆠ矄
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�
 
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
ᨨ矄矄
ᨨ

├� �ᆠ
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�
 
ᨨ
ᨨᆠ矄矄
�矄

᲼


䰘桌 I w
I w
I w
I w
㡰 ᆠ㡰 㡰


 ␔ 
�矄ᆠ
B �矄
ᨨ盐� 
�
ᆠᨨ ᆠ
犨�
�矄ᆠ
A

 泐
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 �盐
ᨨᆠ�泐
�
�鼀
t


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ᨨ
ᨨ 
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泐
 
ᨨ 
盐� �
 �矄
�ᨨ
�矄 ᲼


䰘桌  Iw I w 黨
I w 黨I w 㡰









㡰



ᆠ㡰




㡰


t t
t
t


�ᨨ
x҈⏌x


�矄 ᲼


䰘桌 ҈ ҈ 㡰
㡰 ᆠ㡰 㡰









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҈

�ᆠ矄
A

B  
C

ᆠ
ᨨ
鼀矄矄
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ᆠ
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x㜠y
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A
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w
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꾀

w

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䰘


10 Subjective Problems





黨






꾀
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
ᨨ  鼀矄矄
M
 
�鼀 矄�ᨨ

a
맔
b
�矄

�
ᨨ�ᨨ  ᨨ矄��ᨨ
泐
矄�├� �


 矄鼀
矄
ᆠ
ᨨ
鼀矄矄 m㡰

盐 �

 矄ᆠᨨ 



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矄�├� �
�
 
矄 

��ᨨ
ᨨ
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


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ᨨ
 
矄
犨�
 

�矄
矄�ᆠ

␔ 
�矄
v䁸 ᲼




㜠

㝐桌 ෌�
n
㦨
҈



M 㦨

├�

m 㦨



҈
├�錐
b
㦨


m錐
a
㦨
҈
鼀錐 g
㦨
҈

鼀⏌矄
㡰


犨ᨨ
� �ᆠ
矄

ᆠ
ᨨ
鼀矄矄 M
 
 �
L

矄� �
ᨨ
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WWW.JEEBOOKS.IN

EBD_8341 P-70








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P-71

System of Particles and Rotational Motion

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EBD_8341 P-72

Physics

Answer Key Topic-1 : Centre of Mass, Centre of Gravity & Principle of Moments ҈


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Topic-3 : Torque, Couple and Angular Momentum ҈
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Topic-5 : Rolling Motion ҈
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Topic-6 : Miscellaneous (Mixed Concepts) Problems ҈
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P-73

Gravitation

Gravitation

7

Topic-1 : Kepler's Laws of Planetary Motion 1 ҈


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R⏌䰘 10 Subjective Problems

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

㡰 TA
㦨
TB ᔼ
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�矄 ᆠ
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矄盐
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盐鼀
ᨨ
泐矄
�

泐
犨ᨨ盐
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᲼҈黨黨

㜠


㝐├矄桌 㡰


� 㡰 ҈
黨 ᆠ㡰 ҈


� 㡰 䰘
6

矄盐
和
矄盐ᆠ
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
ᨨ
� �
ᆠ矄
�矄


æ TA ö æ RA ö çè T ÷ø = çè R ÷ø B B



㡰 ҈
㦨


㡰 ҈
㦨



꾀
� 泐
矄
矄泐矄鼀
ᆠᨨ 矄�矄矄
ᨨ
犨ᨨ
矄矄
A
 
B
犨 �ᆠ 
�鼀
�ᨨ
TA
 
TB

�盐矄
RA  
RB
 
鼀矄矄 MA
 
MB



᲼




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矄�矄
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矄鼀
矄 矄

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�ᨨ矄
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
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ᨨ盐
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ᨨ� ᨨ
S҈
�矄
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├鼀

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S

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S҈

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泐
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矄ᨨ 盐
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S҈
᲼҈黨


㜠


㝐├矄桌

WWW.JEEBOOKS.IN

EBD_8341 P-74

Physics

Topic-2 : Newton's Universal Law of Gravitation 1 ҈


MCQs with One Correct Answer




꾀
矄�� 
ᨨ
ᨨ
 �

  矄
ᨨ鼀
x 㦨 a
ᨨ
x 㦨

졔




���ᨨ 
ᨨᆠ
�
 矄
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ᨨ� 
鼀矄矄
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x 㦨


�
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鼀矄矄
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盐 �
 �
ᨨ
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ᨨ
�矄
꾀
졔
x


�矄
��

᲼㝐�
҈





҈黨
ᔼ桌 泐
㡰

㡰

é æ ҈ ù ҈ö ෌鼀 ê 꾀 ç - ÷ -  ú a + L a ø ë è û

㡰

é æ҈ ù ҈ ö ෌鼀 ê 꾀 ç ÷ -  ú a a + L ø ë è û

ᆠ㡰

é æ ҈ ù ҈ö - ÷ +  ú ෌鼀 ê 꾀 ç a + L a è ø ë û

෌㝐





é æ҈ ù ҈ ö ෌鼀 ê 꾀 ç ÷ +  ú ë èa a+Lø û

├
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矄盐
ᨨ鼀
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ᨨ





맔
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├鼀
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맔
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├鼀
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�
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├�
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
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D ҈
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(

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)

(

)



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҈+

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矄 
ᨨ
鼀矄矄
㝐
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�盐矄



矄鼀
矄 
ᨨ �盐矄

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桌

ᆠ㡰

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鼀矄矄
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⼄盐��矄 
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





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Topic-3 : Acceleration due to Gravity 1 ҈


矄� �
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ᔼ
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犨�� 矄
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和






ᔼᔼ㡰桌

MCQs with One Correct Answer

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盐
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R
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h
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ç ÷
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[Main 2 Sep. 2020 (II)]

㡰

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R - R


WWW.JEEBOOKS.IN

P-75

Gravitation





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꾀
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

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



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҈


鼀⏌矄
҈

ᔼ
 
�盐矄
ᨨ
 

犨
ᨨ
矄 � ├
泐
ᨨ 
ᆠ 

�矄 鼀矄矄
鼀� � �
 
矄鼀

 
ᆠᆠ�ᨨ
盐
ᨨ
��泐 ᨨ
 
 矄
矄盐 ᆠ
犨ᨨ盐
᲼҈黨
҈
㜠


㝐├矄桌 㡰 ᆠ矄 㡰
鼀�
盐 ᆠ  � ᆠ㡰 � ᆠ矄 㡰 
ᨨ 4

㡰


졠

R
ᨨ
�
 
ᨨ犨

犨�
ᨨ
 
矄鼀
 �
 

�  � 鼀矄矄
 矄�泐
҈

├�
鼀҈
� ᨨ
�

ᔼ
 
犨�
�矄
ᨨ
ᨨ盐ᆠ � �  泐犨 

 
ᨨᆠ
�

 
ᨨ
ᨨ
 
犨�
泐

矄ᨨ

ᨨ� �
�
�
ᆠ
�矄
├
 
�盐矄
ᨨ
쟼
㦨


맔
҈


鼀
   
ᆠᆠ�ᨨ
盐
ᨨ
��泐
ᨨ
쟼
�矄
҈

鼀矄
㡰 ᲼꾀



҈
桌 㡰 黨

 㡰 ҈


 ᆠ㡰 ҈


 㡰 ҈�

 ҈҈

ᆠ  �
�
 
盐
ᨨ
ᆠᆠ�ᨨ
ᨨ

ᨨ犨矄 矄盐

犨 
 
鼀ᨨᨨ
ᆠᨨ鼀矄
ᨨ鼀
 
ᨨ矄��ᨨ
ᨨ
矄ᨨ ᆠ�矄
ᨨ
 
ᨨ矄��ᨨ
ᨨ
 
ᨨ 
矄�
ᨨ

�
�  犨�
矄盐
�矄


Fill in the Blanks

҈

盐鼀�ᆠ
盐

ᨨ
 
 �盐
ᨨᆠ�泐
ᨨ
ᨨ�ᨨ
ᨨ  

矄 ᨨ盐













⏌矄
�
ᨨ
ᨨ
鼀├
   ᆠ�
ᆠᆠ�ᨨ
盐
ᨨ
��泐
⼄盐
ᨨ
ᨨ
᲼҈黨


㜠


㝐├矄桌

Topic-4 : Gravitational Field, Potential and Potential Energy 1 ҈


MCQs with One Correct Answer

犨ᨨ
 矄

鼀矄矄矄
M
 
҈

M
 
 �
��

a  

a

矄ᆠ�泐


矄�ᨨ
犨
 
ᆠ 矄 ᨨ
 
 矄
�矄
҈
a

꾀
ᨨ泐
ᨨ
鼀矄矄
鼀
�矄
�
ᨨ鼀
  矄盐 ᆠ
ᨨ
 
�
 
ᨨ犨矄
 
矄鼀
 
ᨨ �  
� 
ᨨ� � �
 �
ᆠ 矄

ᨨ
 
ᨨ泐
ᨨ


ᨨ

ᆠ
 
矄盐 ᆠ
ᨨ
矄鼀
 

 
鼀� �鼀盐鼀
�� � 矄

�矄

᲼㝐�


和






ᔼᔼ㡰桌 㡰
ᆠ㡰

GM a

GM
ma

㡰
㡰

GM a

 �GM a


WWW.JEEBOOKS.IN

EBD_8341 P-76





Physics

졠
 
x㜠 �矄
 


�矄 ᆠ
x
ᨨ鼀
 
ᨨ���

  ���ᨨ 
�
盐
ᨨ

鼀矄矄
�矄�盐�ᨨ
�矄
��
泐 Ax

 x + a
㡰⏌



A

 x
+ a
㡰

҈

 x
+ a
㡰











ᆠ㡰 A x
+ a
㡰
㡰 A x + a 㡰


鼀矄矄
 矄�泐
ᨨ

 
ᨨ
�盐矄
R
�矄
犨�
 








㡰 r =  R 㡰 r = R
҈ � R 㡰 r = ᆠ㡰 r = R  黨 ᔴᨨ 矄�
犨ᨨ
矄ᨨ�
矄 矄
ᨨ
��
R҈
㦨
҈m

R
㦨
m
  鼀矄矄矄
M҈
 
M


矄ᆠ�泐


���ᨨ 
� m҈ 盐
ᨨ
矄 
҈
 



矄 ᨨ犨


盐
ᨨ
m
�矄

᲼㝐�









ᔼ桌





෌��ᨨ 
� E

෌㝐 a

㡰 ҈

҈


෌㝐 a

꾀
矄
�ᆠ
�矄
鼀ᨨ� �
�
ᆠ�ᆠ盐
ᨨ�
�
 
���ᨨ 

ᆠ㡰 ⏌

�矄

ᆠᨨ 矄  㡰
�矄

ᆠᨨ 矄  꾀
ᨨ泐
ᨨ
鼀矄矄
鼀
�矄
鼀ᨨ� �
�

ᆠ�ᆠ盐
ᨨ�
ᨨ
�盐矄
R ᨨ盐

 
ᨨ
鼀矄矄
M

꾀
矄ᨨ鼀
� 矄 

�
矄�矄
� ᨨ 犨ᨨ
⼄盐
鼀矄矄矄


�矄
鼀矄矄
鼀ᨨ矄
�

ᆠ�ᆠ盐
ᨨ� R

 
 
ᨨ 
鼀矄矄

�

ᆠ�ᆠ盐
ᨨ�
ᨨ


R


�  ᆠ
犨
 
� 
 
� ��
ᨨ
 ��矄
�矄
᲼㝐�
졠 � 
꾀�
҈�



҈
桌 �盐矄


GMm GMm GMm GMm 㡰 + ᆠ㡰 㡰
R
R
R
R ᨨ鼀

矄ᨨ�
矄 
ᨨ
鼀矄矄
㝐
 
�盐矄



矄 �ᆠ ᨨ�ᨨ
ᨨ
�盐矄
⏌

�矄
鼀ᨨ

矄
矄 ᨨ犨
�
 
��盐


├� �
���ᨨ 
ᨨ �

㦨




㦨
¥

 
ᨨ �
  
ᆠ 
ᨨ
 
ᆠ�泐
 盐矄
ᨨ鼀
�矄

G = grav�tat�ᨨnal cᨨnstant㡰























᲼㝐�


҈�桌

㡰

黨








-

҈

҈

҈









� �盐矄 R


҈ ҈ ҈ 㡰 ᆠ㡰 㡰 

 꾀
矄ᨨ�
矄 
ᨨ
鼀矄矄
�㝐
 
�盐矄
�
�矄
矄盐ᨨ盐 
泐 
盐 � ᨨ鼀
ᆠᨨ ᆠ �ᆠ
矄 �ᆠ
矄 
ᨨ
 �ᆠ├ 矄矄


  鼀矄矄

㝐


���ᨨ 
�

�矄 ᆠ
�
ᨨ鼀
  ᆠ 
犨�






᲼㝐�
黨
꾀�


҈黨
ᔼ桌
෌㝐


㡰

෌㝐


ᆠ㡰

෌㝐


㡰


-
෌㝐 -
෌㝐 -෌㝐 -෌㝐 㡰 ᆠ㡰

㡰 
␔ �ᆠ
ᨨ
 
ᨨᨨ犨� �
鼀ᨨ矄
ᆠᨨ矄泐
�ᆠ矄
 
ᆠᨨᆠ ��ᨨ
ᨨ
 
���ᨨ 
ᨨ �
㡰
盐
ᨨ

�  
ᨨ
�盐矄

 
盐 � ᨨ鼀
鼀矄矄
 矄�泐
䁸
 ��盐矄 
ᨨ
犨
ᨨ
矄ᆠ㡰᲼㝐�
졠 � 
꾀�
҈҈



҈�桌

㡰

㡰





ᆠ㡰 ҈

҈


ᨨ
�盐矄







㡰

෌㝐 a

K


ᔼ � 泐
  r
ᆠᨨᆠ
�ᨨ
犨
 
�盐矄

ᨨ
 
�ᆠ矄
ᨨ�  
�矄
�ᨨ


᲼㝐�


꾀�


҈黨
ᔼᔼ桌
㡰 ⏌
�矄

ᆠᨨ 矄  㡰 ⏌ 
�矄

ᆠᨨ 矄 




�


㡰 ҈
҈



�
ᨨ盐ᆠ
泐

鼀矄矄
 矄�泐
r r 㡰 =

æ r
ö �矄 ᆠ
r
ᨨ鼀
�矄
ᆠ 
矄
rr 㡰 = r
çç҈ -
÷÷


  è R ø ���ᨨ 
�
�矄
鼀 �鼀盐鼀


᲼㝐�

和






ᔼᔼ㡰桌





䰘


A

㡰 ҈




෌㝐 a


�
 
x㜠�ᆠ�ᨨ


鼀� �盐
ᨨ

���ᨨ 
ᨨ �
ᨨ
 
x㜠 �矄


�矄 ᆠ
x

├� � �矄
盐
ᨨ

ᨨ

� � �泐

�矄

[Main 4 Sep. 2020 (I)] 㡰

㡰 ҈
�


҈




෌㝐

黨a a 黨a a
ᨨ盐
� �ᆠ
�ᆠ矄
ᨨ
鼀矄矄
㝐

ᨨᆠ

 
ᆠᨨ 矄 ᨨ

矄⼄盐
ᨨ
矄�
�a

␔ 
矄 ᨨ盐

 �
矄
�
ᆠ ᨨ
 鼀
ᨨ矄
盐 
 
� 盐 ᆠ
ᨨ
ᨨ 矄 ���ᨨ 
�
�

ᆠ�ᆠ盐
ᨨ�
ᆠ�ᆠ盐鼀矄ᆠ�� �
  矄⼄盐
䁸





᲼㝐�


꾀�


҈黨
ᔼ桌

㡰 㡰

㡰

㡰 ᆠ㡰

㡰



졠

졠



㡰



졠

㡰



졠

WWW.JEEBOOKS.IN

P-77

Gravitation





҈҈

���ᨨ 
�
�

��ᨨ
�矄
��
泐 ®

� = � ⏌ ├��ˆ + ҈
 ⏌ ├�ˆ


ᆠ  �
�
 
���ᨨ  ᨨ �
 �泐
ᨨ

�ᆠ
ᨨ
鼀矄矄
҈
├�
犨 
�
�矄
├

ᨨ鼀
 
ᨨ���
ᨨ

ᨨ� 
䰘
鼀



鼀㡰
�矄
᲼㝐�
졠 � 
꾀�
҈黨



҈
桌 㡰 䰘҈
 㡰 ҈ �
 ᆠ㡰 
䰘҈
 㡰 ҈
 鼀
鼀 ҈ ҈

҈ 


犨ᨨ
泐ᨨ �ᆠ
 矄
ᨨ
鼀矄矄矄
鼀҈
 
鼀



矄 犨 
 泐

� � �
�矄 ᆠ


ᆠ盐矄
ᨨ
  ���ᨨ 
ᨨᆠ
 泐
鼀ᨨ
ᨨ犨矄
ᆠ
ᨨ 
ᨨ �  
� 
ᨨ� � �
 �
ᆠ 矄

␔ 
�矄
 �
矄
犨 

 �
矄�ᨨ
�矄
�䁸






᲼㝐�
졠 � 
꾀�
҈




҈
桌 和
ᨨ
鼀҈
�矄
҈
 
 
ᨨ
鼀

�矄

㡰 㡰 ҈
㦨

҈ = 鼀



෌
෌ 
= 鼀҈  ( 鼀҈ + 鼀
)  ( 鼀҈ + 鼀
)

ᆠ㡰

҈ = 鼀҈


෌
෌ 
= 鼀
 ( 鼀҈ + 鼀
)  ( 鼀҈ + 鼀
)

㡰

҈ = 鼀



෌
෌ 
= 鼀

鼀҈ 鼀


㡰

㝐矄矄
ᨨ
ᆠᨨ鼀 矄 
㦨
㝐

MCQs with One or More than One Correct Answer

҈䰘

鼀� �盐矄
ᨨ
 
���ᨨ 
�

�矄 ᆠ
r҈  
r

ᨨ鼀
 
ᆠ 
ᨨ

盐 � ᨨ鼀
矄 
ᨨ
�盐矄
R
  鼀矄矄
m

F҈
 
F

矄ᆠ�泐



᲼҈黨黨

㜠


㝐├矄桌 F҈ r҈ 㡰 F = r �
r҈ < R   r
< R

㡰

F҈ r

= F
r҈
�
r҈ > R   r
> R

ᆠ㡰

F҈ r҈ = F
r
�
r҈ > R   r
> R

F҈ r҈
= F
r

�
r҈ < R   r
< R ҈

꾀
矄ᨨ�
矄 
ᨨ
盐 � ᨨ鼀
 矄�泐
 
�盐矄


盐 �矄
�矄 ᨨᆠ
犨�
�矄
ᆠ 

 
ᨨ���
O
ᨨ
ᆠᨨᨨ� 矄

犨ᨨ 矄 矄
ᨨ
⼄盐
��
҈
盐 �

犨�
 �
ᆠ 矄

A






㡰
 








㡰
矄ᆠ�泐


├
ᨨ盐
ᨨ
 
矄ᨨ� � �
 � 
矄 �ᆠ
ᆠ��矄
矄
矄 ᨨ犨
�
�� y



᲼҈黨黨㜠

㝐├矄桌 㡰





㡰


  ҈�
ᔼ
g
�矄
 
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EBD_8341 P-78

Physics

Topic-5 : Motion of Satellites, Escape Speed and Orbital Velocity 1.

2.

3.

A satellite is in an elliptical orbit around a planet P. It is observed that the velocity of the satellite when it is farthest from the planet is 6 times less than that when it is closest to the planet. The ratio of distances between the satellite and the planet at closest and farthest points is : [Main 6 Sep. 2020 (I)] (a) 1 : 6 (b) 1 : 3 (c) 1 : 2 (d) 3 : 4 A body is moving in a low circular orbit about a planet of mass M and radius R. The radius of the orbit can be taken to be R itself. Then the ratio of the speed of this body in the orbit to the escape velocity from the planet is : [Main 4 Sep. 2020 (II)] 1 (a) (b) 2 2 (c) 1 (d) 2 A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth’s radius Re. By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so

7.

vA n = . vB 4 The value of n is :

(a) 4 8.

(a) 5.

T 2 µ R (b) T 2 µ R3 (c)

T2 µ

R3

(d) T µ R

A body A of mass m is moving in a circular orbit of radius R about a planet. Another body B of mass

6.

1

m collides with 2

r æ vö A with a velocity which is half çè ÷ø the instantaneous 2 r velocity v or A. The collision is completely inelastic. Then, the combined body: [Main 9 Jan. 2020 I] (a) continues to move in a circular orbit (b) Escapes from the Planet’s Gravitational field (c) Falls vertically downwards towards the planet (d) starts moving in an elliptical orbit around the planet The energy required to take a satellite to a height 'h' above Earth surface (radius of Eareth = 6.4 × 10 3 km) is E1 and kinetic energy required for the satellite to be in

(c) 2

(d) 3

A satellite of mass m is launched vertically upwards with an initial speed u from the surface of the earth. After it reaches height R (R = radius of the earth), it m so that subsequently the 10 satellite moves in a circular orbit. The kinetic energy of the rocket is (G is the gravitational constant; M is the mass of the earth): [Main 7 Jan. 2020 I]

3 times larger. Due to this the farthest 2 distance from the centre of the earth that the satellite reaches is R. Value of R is : [Main 3 Sep. 2020 (I)] (a) 4Re (b) 2.5Re (c) 3Re (d) 2Re

4.

(b) 1

[Main 9 Jan. 2020 II]

ejects a rocket of mass

that it become

K The mass density of a spherical galaxy varies as over r a large distance 'r' from its centre. In that region, a small star is in a circular orbit of radius R. Then the period of revolution, T depends on R as : [Main 2 Sep. 2020 (I)]

a circular orbit at this height is E2. The value of h for which E1 and E2 are equal, is: [Main 9 Jan. 2019 II] (a) 1.6 × 103 km (b) 3.2 × 103 km (c) 6.4 × 103 km (d) 28 × 104 km Planet A has mass M and radius R. Planet B has half the mass and half the radius of Planet A. If the escape velocities from the Planets A and B are vA and vB, respectively, then

(a)

m æ 2 113 GM ö çu + ÷ 20 è 200 R ø

3m æ 5GM ö u+ (c) ç 8 è 6 R ÷ø

9.

2

æ 2 119 GM ö (b) 5m çè u ÷ 200 R ø mæ 2GM ö u(d) 20 çè 3 R ÷ø

2

A spaceship orbits around a planet at a height of 20 km from its surface. Assuming that only gravitational field of the planet acts on the spaceship, what will be the number of complete revolutions made by the spaceship in 24 hours around the planet ? [Given : Mass of Planet = 8×10 22 kg, Radius of planet = 2×10 6 m, Gravitational constant G = 6.67×10–11Nm2/kg2] [Main 10 April 2019 II] (a) 9

(b) 17

(c) 13

(d) 11

10. A rocket has to be launched from earth in such a way that it never returns. If E is the minimum energy delivered by the rocket launcher, what should be the minimum energy that the launcher should have if the same rocket is to be launched from the surface of the moon? Assume that the density of the earth and the moon are equal and that the earth’s volume is 64 times the volume of the moon. [Main 8 April 2019 II] (a)

E 64

(b)

E 32

(c)

E 4

(d)

E 16

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P-79

Gravitation

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EBD_8341 P-108

Physics

Topic-2 : Specific Heat Capacity and Thermodynamical Processes 1 1.

MCQs with One Correct Answer

Three different processes that can occur in an ideal monoatomic gas are shown in the P vs V diagram. The paths are lebelled as A ® B, A ® C and A ® D. The change in internal energies during these process are taken as EAB, EAC and EAD and the workdone as WAB, WAC and WAD. The correct relation between these parameters are : [Main 5 Sep. 2020 (I)]

D C

P A

2.

3.

4.

5.

6.

the gas molecule changes from t1 to t2. If

T2

T1

V (a) EAB = EAC < EAD, WAB > 0, WAC = 0, WAD < 0 (b) EAB = EAC = EAD, WAB > 0, WAC = 0, WAD > 0 (c) EAB < EAC < EAD, WAB > 0, WAC > WAD (d) EAB > EAC > EAD, WAB < WAC < WAD In an adiabatic process, the density of a diatomic gas becomes 32 times its initial value. The final pressure of the gas is found to be n times the initial pressure. The value of n is : [Main 5 Sep. 2020 (II)] 1 (a) 32 (b) 326 (c) 128 (d) 32 A balloon filled with helium (32°C and 1.7 atm.) bursts. Immediately afterwards the expansion of helium can be considered as : [3 Sep. 2020 (I)] (a) irreversible isothermal (b) irreversible adiabatic (c) reversible adiabatic (d) reversible isotherm7al A thermodynamic cycle xyzx is shown on a V-T diagram.

(a) 2 7.

8.

9.

The P-V diagram that best describes this cycle is: (Diagrams are schematic and not to scale) [Main 8 Jan. 2020 I]

(a)

(b)

Cp Cv

= g for this

t2 gas then a good estimate for t is given by: 1 [Main 7 Jan. 2020 I]

T1>T2 B

A litre of dry air at STP expands adiabatically to a volume of 3 litres. If g = 1.40, the work done by air is: (31.4 = 4.6555) [Take air to be an ideal gas] [Main 7 Jan. 2020 I] (a) 60.7 J (b) 90.5 J (c) 100.8 J (d) 48 J Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between

(b)

1 2

(c)

æ 1ö çè ÷ø 2

g

(d)

æ 1ö çè ÷ø 2

g +1 2

A sample of an ideal gas is taken through the cyclic process abca as shown in the figure. The change in the internal energy of the gas along the path ca is – 180 J, The gas absorbs 250 J of heat along the path ab and 60 J along the path bc. The work down by the gas along the path abc is: [Main 12 Apr. 2019 I]

(a) 120 J (b) 130 J (c) 100 J (d) 140 J A cylinder with fixed capacity of 67.2 lit contains helium gas at STP. The amount of heat needed to raise the temperature of the gas by 20oC is : [Given that R = 8.31 J mol – 1 K – 1] [Main 10 Apr. 2019 I] (a) 350 J (b) 374 J (c) 748 J (d) 700 J One mole of an ideal gas passes through a process where é 1 æ V ö2 ù pressure and volume obey the relation P = P0 ê1 - ç 0 ÷ ú . êë 2 è V ø úû Here Po and Vo are constants. Calculate the charge in the temperature of the gas if its volume changes from Vo to 2Vo. [Main 10 Apr. 2019 II]

(a)

1 Po Vo 2 R

(b)

5 Po Vo 4 R

3 Po Vo 1 Po Vo (d) 4 R 4 R 10. Following figure shows two processes A and B for a gas. If DQA and DQB are the amount of heat absorbed by the

(c)

(c)

(d)

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P-109

Thermodynamics

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P-111

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X
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᲼꾀





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P

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V
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T �矄
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V
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
� 
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矄鼀
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䰘


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P �矄






᲼҈黨黨

㜠


㝐├矄桌 ҈

5 A P, V

D

P,
V V

Integer Value Answer

True / False


҈

ᆠ盐矄
A
 
B
�
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��盐
矄 ᨨ犨
P㜠V
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

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ᨨ
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�矄

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泐
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A
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�
㜠

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҈
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ᆠᨨ鼀矄矄
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ᨨ

P

3

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꾀
 �� 
├矄
�
�
鼀ᨨ
ᨨ
�



ᔴ
 
҈
鼀

  ᆠᨨ鼀矄矄矄
�
��ᆠ泐
ᨨ
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ᨨ
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A B

⧤� 㡰
 
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� 
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�矄  T�
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盐
ᨨ
a
�矄 ᲼

҈
桌





Fill in the Blanks

6

V MCQs with One or More than One Correct Answer




꾀
鼀� 盐
ᨨ
�
�矄
ᆠᨨ � � �
�
鼀ᨨ矄
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 
҈
鼀ᨨ
ᨨ
���
�ᨨ鼀�ᆠ
�矄
�矄
� ��泐

矄矄盐

WWW.JEEBOOKS.IN

EBD_8341 P-112





Physics

P


ᨨ盐鼀
V


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T


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V
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෌�

҈


㦨


錐




黨

錐
R
�矄 �矄
ᆠᨨ 矄 㡰


᲼꾀



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␔
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
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V
V


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꾀 T


T 㡰 ᔼ  
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CDA
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
᲼


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A

B

D

҈


C ҈




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
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®
B
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B
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C
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A
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B
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C
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 �泐
ᨨ
 
�矄
 
 
犨ᨨ├ ᨨ 
泐
 
�矄

⼄盐
�
鼀� �盐
�

��ᆠ ᨨᆠ矄矄
ᆠ㡰  
�  
 �泐
ᨨ矄
ᨨ
ᆠ  �
�

�矄ᨨ 鼀 ᨨᆠ矄矄
㡰 ᨨ

�矄

ᨨ
鼀ᨨ
�

��ᆠ
ᨨᆠ矄矄
7

Match the Following


䰘
㝐ᆠ
 
 鼀ᨨ泐 鼀�ᆠ
ᨨᆠ矄矄矄
├� �
ᆠ
�
 矄泐矄鼀
犨�
 
ᆠᨨᆠ
ᆠᨨ ��ᨨ 矄

ᔼ
 



DQ
�矄
  
矄盐�

DW
�矄
 
犨ᨨ├
ᨨ 
 
DU
�矄
ᆠ  �
�

�  
 �泐
ᨨ
 
矄泐矄鼀
᲼㝐�


和






ᔼᔼ㡰桌 ᨨᆠ矄矄 ᔴᨨ ��ᨨ

ᔼ㡰 꾀��ᆠ 꾀㡰 DW
㦨

ᔼᔼ㡰 ᔼ矄ᨨ 鼀 㡰 DQ
㦨

ᔼᔼᔼ㡰 ᔼ矄ᨨᆠ ᨨ�ᆠ ᔴ㡰 DU ¹

DW ¹



DQ ¹
ᔼ㡰 ᔼ矄ᨨ�ᆠ 㡰 DU
㦨

㡰 ᔼ㡰㜠꾀㡰

ᔼᔼ㡰㜠㡰

ᔼᔼᔼ㡰㜠㡰

ᔼ㡰㜠㡰 㡰 ᔼ㡰㜠㡰

ᔼᔼ㡰㜠꾀㡰

ᔼᔼᔼ㡰㜠㡰

ᔼ㡰㜠ᔴ㡰 ᆠ㡰 ᔼ㡰㜠꾀㡰

ᔼᔼ㡰㜠꾀㡰

ᔼᔼᔼ㡰㜠㡰

ᔼ㡰㜠ᔴ㡰 㡰 ᔼ㡰㜠㡰

ᔼᔼ㡰㜠㡰

ᔼᔼᔼ㡰㜠꾀㡰

ᔼ㡰㜠ᔴ㡰


ᔼ

 鼀ᨨ泐 鼀�ᆠ
ᨨᆠ矄矄
ᨨ

�
鼀ᨨ ᨨ鼀�ᆠ
�矄
 
� � �矄�鼀

矄ᨨ
泐
 
�矄
�矄
��
泐

D

犨 
T
�矄
鼀盐
ᨨ
 
矄泐矄鼀
 
D
�矄
  � � �矄�鼀
ᆠ  �
�

 鼀ᨨ泐 鼀�ᆠ
⼄盐 �泐
X
ᨨ
  矄泐矄鼀

ᨨ

鼀ᨨ
ᨨ
鼀ᨨ ᨨ鼀�ᆠ
�
�矄 æT ö æV ö  X =
҈ ç ÷ +
҈ ç ÷



�矄
�矄
ᆠᨨ 矄 

V
T è Aø è VA ø �矄
ᨨ盐鼀
ᨨ
�矄
T꾀
 
V㜠

ᆠᨨ 矄 矄



�矄㜠ᔼ
ᨨ犨
��矄
矄ᨨ鼀
⼄盐 ��矄
� ᨨ
�
 ᨨᆠ矄矄
 
�矄㜠ᔼᔼ
��矄
矄ᨨ鼀
ᨨ矄矄�
盐矄
ᨨ
 矄 ⼄盐 ��矄



᲼꾀



҈黨桌

WWW.JEEBOOKS.IN

P-113

Thermodynamics

�矄㜠ᔼ ᔼ㡰

␔ᨨ├
ᨨ 
泐
 

矄泐矄鼀
�
ᨨᆠ矄矄 ҈®
® ᔼᔼ㡰 ᔴ  �
�
�    �泐
�
ᨨᆠ矄矄 ҈®
® ᔼᔼᔼ㡰 
矄ᨨ
泐  
矄泐矄鼀
�

ᨨᆠ矄矄
҈®
® ᔼ㡰 
矄ᨨ 泐
 
矄泐矄鼀 �
ᨨᆠ矄矄
҈®


�矄㜠ᔼᔼ ҈ RT
⏌ ln
㡰 

턼㡰

҈ RT


 㡰


和㡰








҈  㡰








졔
l

㡰  � 㡰







ᔼ
 
ᨨᆠ矄矄
ᆠ�
ᨨ盐
ᨨ
ᨨ 
鼀ᨨ
ᨨ
鼀ᨨ ᨨ鼀�ᆠ
� ҈ �矄
�矄
矄 ᨨ犨
�
 
��盐
㜠��鼀
犨�
P
V

㦨
RT

  
ᆠᨨᆠ
鼀ᆠ
�矄
P P



P


 ҈ V



黨




V


V

㡰 ᔼ®턼

ᔼᔼ®

ᔼᔼᔼ®和

ᔼ® 㡰 ᔼ®和

ᔼᔼ®

ᔼᔼᔼ®턼

ᔼ®

ᆠ㡰 ᔼ®턼

ᔼᔼ®

ᔼᔼᔼ®

ᔼ® 㡰 ᔼ®턼

ᔼᔼ®和

ᔼᔼᔼ®

ᔼ® ᔼ

 鼀ᨨ泐 鼀�ᆠ
ᨨᆠ矄矄
ᨨ

�
鼀ᨨ ᨨ鼀�ᆠ
�矄
 
� � �矄�鼀

矄ᨨ
泐
 
�矄
�矄
��
泐

D

犨 
T
�矄
鼀盐
ᨨ
 
矄泐矄鼀
 
D
�矄
  � � �矄�鼀
ᆠ  �
�

 鼀ᨨ泐 鼀�ᆠ
⼄盐 �泐
X
ᨨ
  矄泐矄鼀

ᨨ

鼀ᨨ
ᨨ
鼀ᨨ ᨨ鼀�ᆠ
�
�矄 æT ö æV ö  X =
҈ ç ÷ +
҈ ç ÷


R
�矄
�矄
ᆠᨨ 矄 

V
è TA ø è VA ø �矄
ᨨ盐鼀
ᨨ
�矄
T꾀
 
V꾀

ᆠᨨ 矄 矄



�矄㜠ᔼ
ᨨ犨
��矄
矄ᨨ鼀
⼄盐 ��矄
� ᨨ
�
 ᨨᆠ矄矄
 
�矄㜠ᔼᔼ
��矄
矄ᨨ鼀
ᨨ矄矄�
盐矄
ᨨ
 矄 ⼄盐 ��矄



᲼꾀



҈黨桌 �矄㜠ᔼ �矄㜠ᔼᔼ ҈ RT
ln
ᔼ㡰 ␔ᨨ├
ᨨ 
泐
  㡰  矄泐矄鼀
�
ᨨᆠ矄矄 ҈®
® ҈ RT
ᔼᔼ㡰 ᔴ  �
�
�   턼㡰   �泐
�
ᨨᆠ矄矄 ҈®
® ᔼᔼᔼ㡰 
矄ᨨ
泐  㡰


 
矄泐矄鼀
�
ᨨᆠ矄矄 ҈®
® ᔼ㡰 
矄ᨨ
泐 和㡰  
矄泐矄鼀
�
ᨨᆠ矄矄 ҈®









҈




졔
l

㡰  � 㡰



ᔼ
 
ᨨᆠ矄矄
ᆠ�
ᨨ盐
ᨨ
ᨨ 
鼀ᨨ
ᨨ
鼀ᨨ ᨨ鼀�ᆠ
�

 㡰

҈ �矄
�矄
矄
矄 ᨨ犨
�
 
TV㜠��鼀
犨�



㦨



  
ᆠᨨᆠ
鼀ᆠ
�矄


㡰 ᔼ®

ᔼᔼ®

ᔼᔼᔼ®턼

ᔼ®

㡰 ᔼ®和

ᔼᔼ®

ᔼᔼᔼ®턼

ᔼ® ᆠ㡰 ᔼ®

ᔼᔼ®

ᔼᔼᔼ®

ᔼ® 㡰 ᔼ®

ᔼᔼ®

ᔼᔼᔼ®

ᔼ®和 �

졠 
鼀ᨨ
ᨨ

鼀ᨨ ᨨ鼀�ᆠ
�
�矄
盐 �ᨨ矄
ᨨ盐  鼀ᨨ泐 鼀�ᆠ
ᨨᆠ矄矄矄
矄
矄 ᨨ犨
矄ᆠ 鼀�ᆠ泐
�
  㜠��鼀
ᨨ犨

꾀鼀ᨨ �
 矄
ᨨ盐
ᨨᆠ矄矄矄

ᨨ 
�矄 �矄ᨨ�ᆠ

ᨨ 
�矄
�矄ᨨᆠ ᨨ�ᆠ

ᨨ 
�矄
�矄ᨨ 鼀
 
ᨨ 
�矄 ��ᆠ

㝐ᆠ
 
ᨨᆠ矄矄矄
鼀 �ᨨ 
�
�矄㜠ᔼ
犨�  
ᆠᨨ矄ᨨ � �
矄鼀 矄
�
�矄㜠ᔼᔼ



᲼꾀



҈
桌  ᔼᔼ




ᔼᔼᔼ

ᔼ ᔼ




턼



和
㡰 㡰 ᆠ㡰 㡰

ᔼ和Tᔼ ᔼ
ᨨᆠ矄矄
ᔼ






ᔼ和Tᔼᔼ ␔ᨨ├
ᨨ 
泐
 
�矄 �矄
ᨨ ᔼ
ᨨᆠ矄矄
ᔼᔼ

鼀盐
ᨨ
 
�矄 鼀� 矄
盐 ᆠ  � ᔼ
ᨨᆠ矄矄
ᔼᔼᔼ 
ᨨ

�矄
 ᆠ  � 犨
 
�矄
 
�矄 矄盐ᨨ盐 � �矄 ᔼ
ᨨᆠ矄矄
ᔼ

␔ᨨ├
ᨨ 
泐
 
�矄
�矄 6P


®

錐
턼
®
錐

®
҈錐
和
®


®
҈錐
턼
®
錐

®

錐
和
®


®
錐
턼
®

錐

®
҈錐
和
®


®
錐
턼
®

錐

®

錐
和
®
҈ ҈


WWW.JEEBOOKS.IN

EBD_8341 P-114

Physics

ᔼ EᔴTᔼ졠N和
턼

Nᨨ

�҈

�

 
�



泐
ᨨ�泐
鼀ᆠ � �
 
� ᨨ鼀�ᨨ
��
�
 
 
ᆠᨨ盐鼀 矄
ᨨ
 
ᨨᨨ犨� �

꾀
�
�矄
�矄
盐 �ᨨ� �

ᆠ泐ᆠ�ᆠ
 鼀ᨨ泐 鼀�ᆠ
ᨨᆠ矄矄
�
�  
犨泐矄
矄
矄 ᨨ犨
�
 
ᆠᨨ矄ᨨ � �


��鼀矄
�

ᆠᨨ盐鼀

ᨨ
 


ᔴᨨ 矄�
ᨨ 泐
 

ᨨ鼀
矄
҈
ᨨ
矄



␔
 ᨨ矄
 
ᆠᨨ矄ᨨ � �
犨ᨨ├
ᨨ 
ᨨ
 
矄泐矄鼀

 ⼄盐�ᨨ 矄
 
ᨨ矄
�
 


矄 
ᨨ�ᨨ 矄
矄
盐矄
�
 鼀ᨨ泐 鼀�ᆠ
ᨨᆠ矄矄矄


Ӹ
�矄
 
�ᨨ
ᨨ

ᆠᆠ��矄 
ᆠᨨ 矄 
矄矄盐
 
ᆠᨨ 矄 
ᨨ盐鼀


盐鼀
ᨨ
鼀ᨨ矄
�
 
�矄
�矄















᲼꾀



҈䰘桌 ᔴᨨ盐鼀
҈ ᔼ㡰

␔҈®
=

ᔴᨨ盐鼀



҈ 

- ҈҈ 㡰 g -҈

ᔼᔼ㡰 ␔҈®
= -
+ ҈

�㡰

ᔴᨨ盐鼀


ᔼ矄ᨨ 鼀

��㡰 ᔼ矄ᨨᆠ ᨨ�ᆠ

㡰



턼㡰



҈




 ҈






ᔼᔼᔼ㡰 ␔҈®
=


���㡰 ᔼ矄ᨨ�ᆠ

҈




 㡰

  ҈ æ ö ᔼ㡰 ␔҈®
= -  ç
÷ è ҈ ø

�㡰

꾀��ᆠ

和㡰




 �҈


�



�


�



␔ �ᆠ
ᨨ 
ᨨ
 
ᨨᨨ犨� �
ᨨ�ᨨ 矄
ᆠᨨᆠ泐
矄 矄
  鼀ᨨ泐 鼀�ᆠ
ᨨᆠ矄矄
 
�矄
盐矄
矄

ᆠᨨᆠ�ᨨ
�
  鼀� �ᨨ
ᨨ
 
矄
ᨨ
矄ᨨ盐 
�

�
�矄䁸 㡰
ᔼ㡰
��㡰
턼㡰 㡰 ᔼ㡰
��㡰
 㡰 ᆠ㡰 ᔼᔼᔼ㡰
�㡰
 㡰 㡰 ᔼ㡰
�㡰
턼㡰 ␔ �ᆠ
ᨨ 
ᨨ
 
ᨨᨨ犨� �
ᨨ�ᨨ 矄
�矄
 
ᆠᨨᆠ ᆠᨨ鼀� �ᨨ 䁸 㡰 ᔼ㡰
��㡰
和㡰 㡰 ᔼᔼᔼ㡰
��㡰
和㡰 ᆠ㡰 ᔼᔼ㡰
�㡰
㡰 㡰 ᔼᔼ㡰
�㡰
 㡰 ␔ �ᆠ
ᨨ
 
ᨨᨨ犨� �
ᨨ�ᨨ 矄
�矄
 
ᨨ 泐
ᆠᨨᆠ 矄 �ᨨ
ᨨ

ᨨᆠ矄矄
�
犨 �ᆠ
D
㦨
D턼

D䁸 㡰 ᔼᔼ㡰
�㡰
 㡰 㡰 ᔼᔼᔼ㡰
���㡰
㡰 ᆠ㡰 ᔼᔼ㡰
���㡰
和㡰 㡰 ᔼᔼ㡰
���㡰
㡰 졠 
鼀ᨨ
ᨨ

鼀ᨨ ᨨ鼀�ᆠ
�
�矄
�矄
├
ᨨ �
犨ᨨ
ᆠ泐ᆠ�ᆠ ᨨᆠ矄矄矄
쟼
®

®
෌
®
쟼
 
쟼
®

®

®
쟼
矄
矄 ᨨ犨

�
 

��鼀


ᨨᆠ矄矄矄
� ᨨ

盐泐
�矄ᨨᆠ ᨨ�ᆠ
�矄ᨨ�ᆠ

�矄ᨨ 鼀
ᨨ
��ᆠ



᲼꾀



҈桌

 









쟼

 


෌ 

㝐ᆠ
 
 矄
�
�矄
ᔼ
犨�
 
鼀� �盐矄
ᨨ
 
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P-115

Thermodynamics

Codes: P (a) 4 (b) 4 (c) 3 (d) 1

Q 3 3 1 3

R 2 1 2 2

(a) Sketch the complete process on a p – V diagram. (b) Find the total work done by the gas, the total change in its internal energy and the final temperature of the gas. [Give your answer in terms of p1, V1, V2, Q and R] 59. One mole of an ideal monatomic gas is taken round the cyclic process ABCA as shown in Figure. Calculate [1998 - 8 Marks]

S 1 2 4 4

P

10 Subjective Problems 55.

56.

A cylinder of mass 1 kg is given heat of 20,000J at atmospheric pressure. If initially the temperature of cylinder is 20°C, find [2005 - 6 Marks] (a) final temperature of the cylinder. (b) work done by the cylinder. (c) change in internal energy of the cylinder (Given that specific heat of cylinder = 400 J kg–1 °C–1, coefficient of volume expansion = 9 × 10–5 °C–1, Atmospheric pressure = 105 N/m2 and Density of cylinder = 9000 kg/m3) A monoatomic ideal gas of two moles is taken through a cyclic process starting from A as shown in figure. The volume ratios are is 27oC.

V VB = 2 and D = 4. If the temperature TA at A VA VA

[2001-10 Marks]

V

VD

D

C

B

VB VA O

A TA

TB T

57.

Calculate, (a) the temperature of the gas at point B, (b) heat absorbed or released by the gas in each process, (c) the total work done by the gas during the complete cycle. Express your answer in terms of the gas constant R. Two moles of an ideal monatomic gas is taken through a cycle ABCA as shown in the P-T diagram. During the process AB, pressure and temperature of the gas vary such that PT = Constant. If T1 = 300 K, calculate [2000 - 10 Marks]

3P0

P0

B

A

C

V0

2V0

V

(a) the work done by the gas. (b) the heat rejected by the gas in the path CA and the heat absorbed by the gas in the path AB; (c) the net heat absorbed by the gas in the path BC; (d) the maximum temperature attained by the gas during the cycle. 60. One mole of a diatomic ideal gas (g = 1.4) is taken through a cyclic process starting from point A. The process A ® B is an adiabatic compression, B ® C is isobaric expansion, C ® D is an adiabatic expansion, and D ® A is isochoric. The volume ratios are VA / VB = 16 and VC / VB = 2 and the temperature at A is TA = 300 K.Calculate the temperature of the gas at the points B and D and find the efficiency of the cycle. [1997 - 5 Marks] 61. At 27°C two moles of an ideal monoatomic gas occupy a volume V. The gas expands adiabaticallly to a volume 2V. Calculate (i) the final temperature of the gas, (ii) change in its internal energy, and (iii) the work done by the gas during this process. [1996 - 5 Marks] 62. One mole of a mono atomic ideal gas is taken through the cycle shown in Fig: [1993 - 4+4+2 Marks] A ® B : adiabatic expansion

P

A B

B ® C : cooling at constant volume C ® D : adiabatic compression D

C

D ® A : heating at constant volume. V The pressure and temperature at A, B, etc. are denoted by

PA , TA , PB , TB etc., respectively. Given that TA = 1000 K,

58.

(a) the work done on the gas in the process AB and (b) the heat absorbed or released by the gas in each of the processes. Give answer in terms of the gas constant R. Two moles of an ideal monatomic gas, initially at pressure p1 and volume V1, undergo an adiabatic compression until its volume is V2. Then the gas is given heat Q at constant volume V2. [1999 - 10 Marks]

PB= (2/3)PA and PC = (1/ 3) PA , calculate the following quantities: (i) The work done by the gas in the process A ® B (ii) The heat lost by the gas in the process B ® C. (iii) Thetemperature TD. [Given : (2 / 3)2/5 = 0.85] 63. Two moles of helium gas undergo a cyclic process as shown in Fig. Assuming the gas to be ideal, calculate the following quantities in this process [1992 - 8 Marks]

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EBD_8341 P-116

Physics



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Topic-3 : Carnot Engine, Refrigerators and Second Law of Thermodynamics 1 ҈


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P-117

Thermodynamics

A solid body of constant heat capacity 1 J/°C is being heated by keeping it in contact with reservoirs in two ways: [Main 2015] (i) Sequentially keeping in contact with 2 reservoirs such that each reservoir supplies same amount of heat. (ii) Sequentially keeping in contact with 8 reservoirs such that each reservoir supplies same amount of heat. In both the cases body is brought from initial temperature 100°C to final temperature 200°C. Entropy change of the body in the two cases respectively is : (a) ln2, 2ln2 (b) 2ln2, 8ln2 (c) ln2, 4ln2 (d) ln2, ln2 12. A Carnot engine absorbs 1000 J of heat energy from a reservoir at 127°C and rejects 600 J of heat energy during each cycle. The efficiency of engine and temperature of sink will be: [Main Online April 12, 2014] (a) 20奠 and – 43°C (b) 40奠 and – 33°C (c) 50奠 and – 20°C (d) 70奠 and – 10°C 13. p A B 2p 11.

7.

8.

9.

(a)

T2 = ( T1 T4 )

(b)

T2 = T12 T4

(

)

1/3

(c)

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(

)

1/3

(d)

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(

)

1/2

1/4

(

)

; T3 = T1 T42

(

)

; T3 = T12 T4

(

)

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)

; T3 = T12 T4

; T3 = T1 T43

1/3

1/3

1/3

1/4

Two Carnot engines A and B are operated in series. The first one, A receives heat at T1 (= 600 K) and rejects to a reservoir at temperature T2. The second engine B receives heat rejected by the first engine and in turn, rejects to a heat reservoir at T3 (= 400 K). Calculate the temperature T2 if the work outputs of the two engines are equal: [Main 9 Jan. 2019 II] (a) 600 K (b) 400 K (c) 300 K (d) 500 K A Carnot's engine works as a refrigerator between 250 K and 300 K. It receives 500 cal heat from the reservoir at the lower temperature.The amount of work done in each cycle to operate the refrigerator is: [Main Online April 15, 2018] (a) 420 J (b) 2100 J (c) 772 J (d) 2520 J An engine operates by taking n moles of an ideal gas through the cycle ABCDA shown in figure. The thermal efficiency of the engine is : (Take Cv =1.5 R, where R is gas constant) [Main Online April 8, 2017] (a) 0.24 (b) 0.15

2P0

(d) 0.08

C

B

P P0

(c) 0.32

10.

0

D

A

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2V0

V

A Carnot freezer takes heat from water at 0°C inside it and rejects it to the room at a temperature of 27°C. The latent heat of ice is 336 × 103 J kg–1. If 5 kg of water at 0°C is converted into ice at 0°C by the freezer, then the energy consumed by the freezer is close to : [Main Online April 10, 2016] (a) 1.51 × 105 J (b) 1.68 × 106 J 7 (c) 1.71 × 10 J (d) 1.67 × 105 J

p0

D

C

v0 2v0 v The above p-v diagramrepresents the thermodynamic cycle of an engine, operating with an ideal monatomic gas. The amount of heat, extracted from the source in a single cycle is [Main 2013] æ 13 ö (a) p 0 v 0 (b) ç ÷ p0 v0 è2ø æ 11 ö (c) ç ÷ p0 v0 (d) 4p0v0 è2ø 14. Two identical containers A and B with frictionless pistons contain the same ideal gas at the same temperature and the same volume V. The mass of the gas in A is mA, and that in B is mB. The gas in each cylinder is now allowed to expand isothermally to the same final volume 2V. The changes in the pressure in A and B are found to be DP and 1.5 DP respectively. Then [1998S - 2 Marks] (a) 4mA = 9mB (b) 2mA = 3mB (c) 3mA = 2mB (d) 9mA = 4mB 3

Numeric Answer

15. If minimum possible work is done by a refrigerator in converting 100 grams of water at 0°C to ice, how much heat (in calories) is released to the surroundings at temperature 27°C (Latent heat of ice = 80 Cal/gram) to the nearest integer? [Main 3 Sep. 2020 (II)] 16. A Carnot engine operates between two reservoirs of temperatures 900 K and 300 K. The engine performs 1200 J of work per cycle. The heat energy (in J) delivered by the engine to the low temperature reservoir, in a cycle, is _______. [Main 7 Jan. 2020 I]

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EBD_8341 P-118

Physics

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Answer Key Topic-1 : First Law of Thermodynamics ҈
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�

㡰



ᔴᨨ 矄 㡰 䰘
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ᆠ㡰





ᆠ㡰

Topic-2 : Specific Heat Capacity and Thermodynamical Processes ҈
҈҈

҈
҈




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



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҈
䰘
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䰘
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ᆠ㡰 㡰 㡰 㡰

Topic-3 : Carnot Engine, Refrigerators and Second Law of Thermodynamics ҈
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
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㡰

黨
㡰 ҈黨
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҈

㡰

WWW.JEEBOOKS.IN

P-119

Kinetic Theory

Kinetic Theory

12

Topic-1 : Kinetic Theory of an Ideal Gas and Gas Laws 1 ҈


ᨨ盐
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MCQs with One Correct Answer

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矄
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l


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l


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  dV ⏌ dP
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ᨨ

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和桌

��ᨨ
ᨨ
b = -

ᔴᨨ 矄  

㡰

㡰


P

P

WWW.JEEBOOKS.IN

EBD_8341 P-120

Physics

3 ᆠ㡰

䰘






Numeric Answer

ᔼ ��泐

�矄
ᨨ
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҈
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�

⧤

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ᨨ
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
ᆠ  �
�
盐鼀
ᨨ
鼀ᨨ矄


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ᨨ
  矄盐� �
�矄

鼀盐





⧤

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�
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҈
�矄
��
泐



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�ᨨ

⏌҈
�矄
힘힘힘힘힘힘
᲼㝐�
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
ᆠ  �
�
 
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ᨨ
 
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ᨨ
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矄鼀
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DP
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�

 ᆠᨨ 矄 
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�矄
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矄鼀
矄
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泐
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矄鼀
⼄盐 �泐
DT 
ᆠᨨ 矄  矄矄盐


� ��
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㡰

P P

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꾀
 

犨�
�ᆠ�ᨨ 矄矄
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 
矄鼀
�
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 
矄鼀
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矄鼀
ᨨ盐鼀



鼀矄矄
ᨨ
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�
꾀
�矄
鼀꾀

   
�


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鼀

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ᨨ
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矄鼀
� 
ᨨ盐鼀



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�
 
矄矄盐
�

꾀
 


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ᨨ

D
  ҈
�
D
矄ᆠ�泐



᲼҈黨黨
和
㜠


㝐├矄桌 㡰
鼀꾀
㦨
黨鼀 㡰
鼀꾀
㦨
鼀 ᆠ㡰 鼀꾀
㦨

鼀 㡰 黨鼀꾀
㦨

鼀

2 Integer Value Answer 꾀
 鼀泐
�矄ᨨ
ᆠ泐� �ᆠ
ᆠᨨ矄
矄矄
ᨨ
�� 


鼀 �矄
├
�ᆠ泐

ᔼ
�矄
��
� ᨨ
犨ᨨ
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矄
泐
 � 鼀�ᆠ
 ᆠ
 鼀
ᆠᨨ 盐ᆠᨨ㡰
�ᆠ�ᨨ 矄矄
��ᨨ

ᨨ
鼀矄矄



├�

盐矄
 
��ᨨ
�矄

� ��泐


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鼀
ᨨ鼀
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ᨨ

矄
矄 ᨨ犨
�
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矄ᆠ 鼀�ᆠ
��盐
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쟼ᆠ
ᨨ
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犨ᨨ
矄
ᨨ
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矄矄
ᆠᨨ � 矄


҈
鼀ᨨ
ᨨ


�
�矄

鼀盐



⧤

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��ᨨ
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ᨨ犨
矄  
鼀ᨨ矄
犨� ᨨ盐
 泐
�矄
├� �
ᨨ鼀
ᨨ 

ᨨ
  矄矄
ᨨ
 
ᨨ 

␔ 
⼄盐���盐鼀
�矄
ᆠ 

  �矄 ᆠ
ᨨ
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��ᨨ
ᨨ鼀
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ᨨ
�
鼀㡰
犨�

힘힘힘힘힘힘힘 ├
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盐
ᨨ
��泐
㦨҈

鼀矄

 
  盐 �矄
�矄
ᆠᨨ 矄 
㦨



鼀ᨨ҈⧤҈㡰


犨



⧤
 


鼀

矄ᆠ�泐

ᔼ
 DT = C  DP 

 

盐
ᨨ
C
�
⧤⏌鼀
㡰
�矄
힘힘힘힘힘힘힘힘힘힘
᲼㝐�
和










ᔼᔼ㡰桌 4

Fill in the Blanks

҈҈
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Topic-2 : Speed of Gas, Pressure and Kinetic Energy 1 ҈






MCQs with One Correct Answer

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P-121

Kinetic Theory

5.

(a) 800 K (b) 3 × 105 K (c) 104 K (d) 650 K A mixture of 2 moles of helium gas (atomic mass = 4u), and 1 mole of argon gas (atomic mass = 40u) is kept at 300 K in a container. The ratio of their rms speeds

6.

é Vrms ( helium ) ù ê ú is close to : [Main 9 Jan. 2019 I] ë Vrms ( argon ) û (a) 3.16 (b) 0.32 (c) 0.45 (d) 2.24 N moles of a diatomic gas in a cylinder are at a temperature T. Heat is supplied to the cylinder such that the temperature remains constant but n moles of the diatomic gas get converted into monoatomic gas. What is the change in the total kinetic energy of the gas? [Main Online April 9, 2017] 1 nRT (b) 0 2 3 5 nRT nRT (d) (c) 2 2 In an ideal gas at temperature T, the average force that a molecule applies on the walls of a closed container depends on T as Tq. A good estimate for q is: [Main Online April 10, 2015] 1 1 (a) (b) 2 (c) 1 (d) 2 4 At room temperature a diatomic gas is found to have an r.m.s. speed of 1930 ms–1. The gas is: [Main Online April 12, 2014] (a) H2 (b) Cl2 (c) O2 (d) F2 In the isothermal expansion of 10g of gas from volume V to 2V the work done by the gas is 575J. What is the root mean square speed of the molecules of the gas at that temperature? [Main Online April 25, 2013] (a) 398m/s (b) 520m/s (c) 499m/s (d) 532m/s A mixture of 2 moles of helium gas (atomic mass = 4 amu) and 1 mole of argon gas (atomic mass = 40 amu) is kept at 300 K in a container. The ratio of the rms speeds æ vrms (helium) ö [2012] çè v (argon) ÷ø is rms (a) 0.32 (b) 0.45 (c) 2.24 (d) 3.16 The ratio of the speed of sound in nitrogen gas to that in helium gas, at 300 K is [1999S - 2 Marks]

(a)

7.

8.

9.

10.

11.

(a) 12.

13.

(2 / 7

(b)

(1 / 7)

(c) ( 3 ) / 5 (d) ( 6 ) / 5 A closed compartment containing gas is moving with some acceleration in horizontal direction. Neglect effect of gravity. Then the pressure in the compartment is [1999S - 2 Marks] (a) same everywhere (b) lower in the front side (c) lower in the rear side (d) lower in the upper side A vessel contains 1 mole of O2 gas (relative molar mass 32) at a temperature T. The pressure of the gas is P. An identical

vessel containing one mole of He gas (relative molar mass 4) at a temperature 2T has a pressure of [1997 - 1 Mark] (a) P/8 (b) P (c) 2P (d) 8P 14. The temperature of an ideal gas is increased from 120 K to 480 K. If at 120 K the root-mean-square velocity of the gas molecules is v, at 480 K it becomes [1996 - 2 Marks] (a) 4v (b) 2v (c) v/2 (d) v/4 15. Three closed vessels A, B and C are at the same temperature T and contain gases which obey the Maxwellian distribution of velocities. Vessel A contain only O2, B only N2 and C a mixture of equal quantities of O2 and N2. If the average speed of the O2 molecules in vessel A is v1 that of the N2 molecules in vessel B is v2, the average speed of the O2 molecules in vessel C is [1992 - 2 Marks] 1

v1 + v2 (b) v1 (c) (v1.v2 ) 2 (d) 2 where M is the mass of an oxygen molecule.

(a)

3

3kT M

Numeric Answer

16. Nitrogen gas is at 300°C temperature. The temperature (in K) at which the rms speed of a H2 molecule would be equal to the rms speed of a nitrogen molecule, is _____________. (Molar mass of N2 gas 28 g); [Main Sep. 05, 2020 (II)] 5

True / False

17. The root mean square (rms) speed of oxygen molecules (O2) at a certain temperature T (degree absolute) is V. If the temperature is doubled and oxygen gas dissociates into atomic oxygen, the rms speed remains unchanged. [1987 - 2 Marks] 18. The ratio of the velocity of sound in Hydrogen gas 7 5 ( g = ) to that in Helium gas ( g = ) at the same 5 3 21 temperature is . [1983 - 2 Marks] 5 19. Two different gases at the same temperature have equal root mean square velocities. [1982 - 2 Marks] 20. The root-mean square speeds of the molecules of different ideal gases, maintained at the same temperature are the same. [1981- 2 Marks] 6

MCQs with One or More than One Correct Answer

21. Let v, v rms and vp respectively denote the mean speed. root mean square speed, and most probable speed of the molecules in an ideal monatomic gas at absolute temperature T. The mass of a molecule is m. Then [1998S - 2 Marks] (a) no molecule can have a speed greater than

2 v rms

(b) no molecule can have a speed less than v p / 2 (c) v p < v < v rms 3 2 (d) the average kinetic energy of a molecule is mv p . 4

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EBD_8341 P-122

Physics

Topic-3 : Degree of Freedom, Specific Heat Capacity and Mean Free Path 1 ҈


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






ᔼ桌



MCQs with One Correct Answer

�  黨 RT 㡰 㡰 RT ᆠ㡰 RT 㡰 RT




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






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t

t

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T

t

t

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WWW.JEEBOOKS.IN

P-123

Kinetic Theory

҈



Cp �

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҈
桌 ᔴ - ᔴ ᔴ - ᔴ 㡰 = 㡰 = ᔴ - ᔴ ᔴ - ᔴ

㡰


ᆠ㡰 =

㡰

ᔴ ᔴ

㡰 =

ᔴ  ᔴ ᔴ  ᔴ

WWW.JEEBOOKS.IN

EBD_8341 P-124







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



Physics

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㝐├矄桌 6

MCQs with One or More than One Correct Answer


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P-125

Kinetic Theory

42.

Cv and Cp denote the molar specific heat capacities of a gas at constant volume and constant pressure, respectively. Then [2009] (a) Cp – Cv is larger for a diatomic ideal gas than for a monoatomic ideal gas (b) Cp + Cv is larger for a diatomic ideal gas than for a monatomic ideal gas (c) Cp / Cv is larger for a diatomic ideal gas than for a monoatomic ideal gas (d) Cp . Cv is larger for a diatomic ideal gas than for a monoatomic ideal gas 9

43.

the product of its pressure and its volume. [2007] because Statement-2 : The molecules of a gas collide with each other and the velocities of the molecules change due to the collision. (a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1 (b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (c) Statement-1 is True, Statement-2 is False (d) Statement-2 is False, Statement-2 is True

Assertion and Reason Type Questions

Statement-1 : The total translational kinetic energy of all the molecules of a given mass of an ideal gas is 1.5 times

Topic-4 : Miscellaneous (Mixed Concepts) Problems 1 1.

2. 3.

Two moles of ideal helium gas are in a rubber balloon at 30°C. The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to 35°C. The amount of heat required in raising the temperature is nearly (take R = 8.31 J/mol.K) [2012] (a) 62 J (b) 104 J (c) 124 J (d) 208 J An ideal gas is expanding such that PT2 = constant. The coefficient of volume expansion of the gas is – [2008] (a) 1/T (b) 2/T (c) 3/T (d) 4/T When an ideal diatomic gas is heated at constant pressure, the fraction of the heat energy supplied which increases the internal energy of the gas is [1990 - 2 Marks] 2 3 3 5 (a) (b) (c) (d) 5 5 7 7 3

4.

M1

MCQs with One Correct Answer

Numeric Answer

One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is 100 K and the universal gas constant R = 8.0 J mol –1 K –1, the decrease in its internal energy, in Joule, is__________.

M2

a

S

6 6.

MCQs with One or More than One Correct Answer

As shown schematically in the figure, two vessels contain water solutions (at temperature T) of potassium permanganate (KMnO4) of different concentrations n1 and n 2 (n 1 > n 2 ) molecules per unit volume with Dn = (n1 - n2 )
F4 㡰 F 1> F2 > F 3 >
F4 ᆠ㡰 F 1= F2 = F 3 㦨
F4 㡰 F 1> F 3 ; F 2

F4

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P-161

Electric Charges and Fields

3 ҈�


Numeric Answer

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EBD_8341 P-162

Physics

(a) electric field near A in the cavity = electric field near B in the cavity (b) charge density at A = charge density at B (c) potential at A = potential at B (d) total electric field flux through the surface of the cavity is q/e0

ur 22. The magnitude of electric field E in the annular region of a charged cylindrical capacitor. [1996 - 2 Marks] (a) is same throughout (b) is higher near the outer cylinder than near the inner cylinder (c) varies as 1/r, where r is the distance from axis (d) varies as 1/r2 where r is the distance from the axis.

Topic-4 : Miscellaneous (Mixed Concepts) Problems 1 1.

2.

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EBD_8341 P-164

Physics

16

Electrostatic Potential and Capacitance

Topic-1 : Electrostatic Potential and Equipotential Surfaces 1 ҈






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MCQs with One Correct Answer

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WWW.JEEBOOKS.IN

P-165

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
盐 � ᨨ鼀
� 
ᆠ �
�矄
├
ᨨ �
   �矄
ᨨ
 
ᆠ泐� 矄 㡰 ᨨ
ᨨ �
�  ᆠ
矄
犨
 
犨ᨨ
ᆠ泐� 矄 犨 
矄鼀
ᆠ �
 矄�泐
�矄
��
ᨨ
ᨨ
 
ᆠ泐� 矄


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EBD_8341 P-166















Integer Value Answer


�
꾀
�ᆠ

ᨨ
鼀矄矄
҈

kg
 
ᆠ �
҈


C

�矄
� ��泐
 矄

꾀
�鼀

㦨



 
�ᆠ
ᆠᨨ鼀矄
盐 
 
� 盐 ᆠ r ᨨ

ᆠ�ᆠ
�
쟼
㡰
㦨
쟼


矄�
w �ˆ

犨 
쟼

㦨
҈

ᔴ҈
 
w 㦨҈



矄҈


ᔴᨨ 矄�
 
 ᆠ
ᨨ
ᨨ 泐
  ᆠ�ᆠ
ᨨᆠ
ᨨ
 
�ᆠ


 
鼀 �鼀盐鼀
矄
�
鼀矄҈

� 
泐
 
�ᆠ

矄盐矄⼄盐 
�鼀矄
�矄 힘힘힘힘힘힘힘힘힘힘힘힘






᲼꾀



҈
桌 4 Fill in the Blanks



ᆠ�ᆠ
ᨨ �
V

 泐
ᨨ� 
x

y

z

�
鼀矄㡰
�

矄ᆠ
�矄
��
泐
V
㦨

x

ᨨ矄


ᆠ�ᆠ
�

  ᨨ� 
҈鼀






鼀㡰
�矄





















⏌鼀

᲼҈黨黨

㜠
҈
㝐├桌
䰘
��盐
矄 ᨨ犨矄
� 
ᨨ
ᆠᨨ 矄 
ᨨ �
�

��ᨨ
�
犨 �ᆠ 
ᆠ�ᆠ
�
�矄
矄 


盐矄
ᨨ
 
ᨨ �
 犨�
�
ᆠ├矄

졠
 
ᨨ� 矄
A

B
 
C

 
鼀� �盐 ᨨ
 
ᆠ�ᆠ
�
�矄
�矄

 
ᨨ� 



᲼҈黨

㜠


㝐├矄桌 꾀 


 ҈


  


҈


㡰 ␔�
S҈
ᆠᨨ矄
V҈㦨҈�


V

㦨



 㡰 ␔�
S
ᆠᨨ矄

V҈㦨
V

㦨

�
 ᆠ㡰 ␔�
S҈
 
S

ᆠᨨ矄

V҈ = V
=
㡰 ␔�
S҈
 
S
ᆠᨨ矄

V҈ = 
V
V
=

V 꾀
ᆠ �
졔q
�矄
� 

ᆠ
ᨨ
 
ᨨ� 矄
x
㦨
x



x
㦨
x

x
㦨
�x







x = ¥
ᨨ
 
x
 �矄

 

ᆠ �
q
�矄
� 
 ᆠ
ᨨ
 
ᨨ� 矄
x
㦨

x


x
㦨

x


x
㦨

x






x = ¥

 x

�矄

ᨨ矄��
ᆠᨨ 矄 

├
 
ᆠ�ᆠ
ᨨ �

 ᨨ� 
盐
ᨨ

ᆠ �
Q


�矄 ᆠ
r
ᨨ鼀
�
ᨨ

Q⏌ 
pe
r㡰



 
ᨨ �

 
ᨨ���
盐
ᨨ
 
ᨨ 矄泐矄鼀
ᨨ
ᆠ �矄
�矄 ᲼҈黨黨
和
㜠


㝐├矄桌 q ln
q 㡰
㡰
pe x ln
ᆠ㡰 ¥ 㡰
pe x



꾀
矄ᨨ�
ᆠᨨ 盐ᆠ� �
矄 
� �

ᆠ �
Q
�矄 矄盐ᨨ盐 
泐

盐 ᆠ �
ᆠᨨ ᆠ �ᆠ
ᆠᨨ 盐ᆠ� � ᨨᨨ犨
矄 �ᆠ
矄 


 
ᨨ �
�  ᆠ
犨

 
矄盐 ᆠ
ᨨ
 
矄ᨨ�
矄 
 
 
ᨨ
 
ᨨ盐
矄盐 ᆠ ᨨ
 
ᨨᨨ犨
矄 

V

ᔼ
 
矄 
�矄
ᨨ犨
��

ᆠ � ᨨ

Q

 
犨
ᨨ �
�  ᆠ
犨
 
矄鼀 ᲼҈黨
黨
㜠


㝐├矄桌 犨ᨨ
矄盐 ᆠ矄
�矄

㡰 V 㡰
V ᆠ㡰
V 㡰 

V 꾀
ᨨᨨ犨
鼀
矄 
ᨨ
�盐矄
�
ᆠ鼀矄
�矄

ᆠ �
矄盐ᆠ  
 
ᨨ �
ᨨ
�矄
矄盐 ᆠ
�矄
҈

ᨨ矄



ᨨ � 
 
ᆠ 
ᨨ
 
矄 
�矄 ᲼҈黨

㜠
҈
㝐├桌 㡰 ᨨ 㡰 ҈

ᨨ矄 ᆠ㡰 矄鼀
矄


ᨨ� 
�
ᆠ鼀矄
犨泐
ᨨ鼀
 
矄盐 ᆠ 㡰 矄鼀
矄


ᨨ� 

�
ᆠ鼀矄
犨泐
ᨨ鼀
 
矄盐 ᆠ

犨ᨨ
�
ᆠ�ᆠ盐
�矄ᆠ矄
矄
泐

�矄 ᆠ
ᨨ



҈
鼀 
ᆠᨨ

ᆠ
ᨨ

泐
�

矄犨�ᆠ
矄
矄 ᨨ犨
�
  ��盐

ᔴ �
ᨨ�
ᨨ矄
ᨨ
 矄�泐
黨


├�
鼀

矄  ᨨ盐�

� 泐
ᨨ

 
ᆠ 
ᨨ
 
ᨨ
�矄ᆠ

졠 ᆠ
矄ᨨ鼀 ᨨ�
ᨨ矄
ᆠ �
鼀� 
ᨨᆠ�泐

 
矄犨�ᆠ
�矄
ᆠᨨ矄
ᨨ 泐

ᨨ�
ᨨ





ᆠᨨ矄矄
 
�矄ᆠ矄

꾀矄

矄盐


ᨨ� ᨨ
ᨨ
�盐矄


맔҈
䰘
鼀
矄ᨨ矄
鼀ᨨ� �
�ᆠ泐
 
ᨨ矄 犨
 
�矄ᆠ矄


盐鼀
ᨨ
ᆠᨨ 矄
矄 
�
 �矄 ᨨ�
ᨨ
�矄
힘힘힘힘힘힘힘힘

 �ᆠ
 
盐ᨨ泐 ᆠ泐
ᨨᆠ

├ ᆠᆠ�ᨨ
盐
ᨨ
��泐
㦨҈

鼀矄

 
ᆠ �
ᨨ


ᆠᨨ
㡰
㦨
҈

맔҈
҈黨
ᔴ㡰







᲼꾀





桌

2

�







꾀
盐 � ᨨ鼀
ᆠ�ᆠ
�
ᨨ� � �
�
ᨨ矄��
x㜠�ᆠ�ᨨ
 �矄矄 �

��ᨨ


A 
 
ᨨ���

B 
 
ᨨ� 
ᨨ
 
x㜠 �矄
 x 㦨
졔҈
ᆠ鼀
 
C 
 
ᨨ� 
ᨨ
 
y㜠 �矄

y 㦨
졔҈
ᆠ鼀



 
ᨨ �矄

 
ᨨ� 矄
A

B  
C 矄�矄 泐
᲼


҈和桌 㡰 VA

VB 㡰 VA >
VB ᆠ㡰 VA

VC 㡰 VA >
VC ᨨ
 
ᆠ�ᆠ盐�
矄 ᨨ犨
�
��盐

犨 �ᆠ
ᨨ
 
ᨨᨨ犨� � 矄鼀 矄
�矄
盐䁸 ᲼҈黨黨黨
㜠


㝐├矄桌

ᔴ





҈黨


Physics

6 MCQs with One or More than One Correct Answer


犨ᨨ
ᨨ 㜠ᆠᨨ 盐ᆠ� �
矄 矄
ᨨ
��
҈
 


  ᆠ泐� �
盐 � ᨨ鼀
ᨨ盐鼀
ᆠ �
 矄��矄
졔r
 
r
矄ᆠ�泐


ᆠ
矄盐ᆠ
 
 泐
�泐
ᨨ
矄
矄 ᨨ犨
�
 
��盐

꾀

ᨨ� 矄
�
 
ᨨ� � ��ᨨ






᲼꾀



҈桌 r

–r


҈

㡰 
ᆠᨨ矄�ᆠ
�
�矄
ᨨ 㡰 
ᆠᨨ矄�ᆠ
ᨨ �
�矄
ᆠᨨ 矄  ᆠ㡰 
ᆠᨨ矄�ᆠ
�
�矄
ᆠᨨ 矄 
�
鼀� �盐 㡰 
ᆠᨨ矄�ᆠ
�
矄
矄鼀
�ᆠ�ᨨ


黨
和�
ᨨ� 
ᆠ �矄

├

 
�ᆠ矄
ᨨ

�盐  �ᨨ
ᨨ
矄�

 
ᆠ 
O

矄
矄 ᨨ犨
�
 
��盐

෌�

҈ q

犨 �ᆠ
ᨨ
 
ᨨᨨ犨� �
矄鼀 矄㡰
�矄
pe 0 
㡰
ᆠᨨᆠ䁸
᲼

҈
桌

 
K =

졔q



E–q P

Sw�tch

A 졔
q

S

T

O




҈
鼀

D –
q







R B졔q

– qC

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P-167

Electrostatic Potential and Capacitance





҈






㡰 
ᆠ�ᆠ
�

O
�矄

⧤
ᨨ �
OD 㡰 
ᨨ �

O
�矄
ᨨ ᆠ㡰 
ᨨ �


ᨨ� 矄
ᨨ
 
� 
PR
�矄
矄鼀 㡰 
ᨨ �


ᨨ� 矄
ᨨ
 
� 
ST
�矄
矄鼀 ␔ �ᆠ
ᨨ
 
ᨨᨨ犨� �
矄鼀 矄㡰
�矄⏌
ᆠᨨᆠ䁸
᲼

҈҈桌 㡰 ᔼ
 
ᆠ�ᆠ
�
盐
ᨨ

ᨨ� 
ᆠ �
�矄
矄
 

� � 矄
ᨨ
 


 
 
෌盐矄矄
犨
犨�
矄�

�
㡰 
෌盐矄矄
犨
ᆠ

盐矄
ᨨ
ᆠᆠ盐
 
� �矄�盐�ᨨ
ᨨ盐 

ᆠ�ᆠ
�ᨨ
ᆠ㡰 ᔼ
 
ᆠ�ᆠ
�
犨
犨ᨨ
ᨨ� 
ᆠ �矄
�矄
ᨨ 矄ᨨ鼀犨 

 
 
矄��
ᨨ
 
犨ᨨ
ᆠ �矄
�矄
 
矄鼀
㡰 
犨ᨨ├
ᨨ 
泐
 
  
ᨨᆠ
�
鼀ᨨ� �
 盐 �
ᨨ矄��
ᆠ �
ᨨ鼀
ᨨ� 
꾀

ᨨ �
꾀 ᨨ
ᨨ� 


ᨨ �

�矄

- 꾀㡰
꾀
矄 �ᆠ
鼀
矄 
꾀
ᨨ
�盐矄
꾀
 

矄ᨨ�
鼀 矄 

ᨨ
�盐矄

꾀㡰

├


 
ᆠ
�矄 ��
ᆠ �
�졔턼

ᨨ犨
 泐

ᆠᨨ

ᆠ
泐

 �
鼀 犨�



᲼

҈҈桌 㡰 E �ns�de 㡰 턼꾀
>
턼 =
A s A RB ᨨn surface ᨨn surface ᆠ㡰 = 㡰 E A < EB s B RA 꾀
矄 �ᆠ
矄泐鼀鼀�ᆠ
ᆠ �
矄泐矄鼀
�矄
ᆠ 

ᨨ���
෌�

쟼ᆠ�ᆠ
ᨨ � ᲼



和
㜠
�
㝐├矄桌 Q Q V= r £ R
㡰
V = r > R
㡰
pe
R

pe
r V

㡰 㡰 ᆠ㡰 㡰 







␔� �
r
㦨

R

ᨨ
 ᆠᨨ矄

ᆠ �
�矄
Q 쟼ᆠ�ᆠ
�
�矄
�矄ᆠᨨ � 盐

r
㦨
R
ᔴ �
�矄
ᨨ 泐
矄 

r
㦨
R
쟼ᆠᨨ矄�ᆠ
 �泐
�矄
ᨨ
ᨨ
r


R


9 Assertion and Reason Type Questions 和T꾀TE㝐ENT㜠҈


ᨨ
ᆠ�ᆠ
盐ᨨ矄矄

 

�矄 盐矄
矄

  ᆠ

ᨨ
ᨨ �
�
ᆠ�ᆠ
ᆠ�ᆠ盐�矄
 

和T꾀TE㝐ENT㜠




ᆠ�ᆠ
ᨨ �
ᨨ

矄 
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WWW.JEEBOOKS.IN

P-169

Electrostatic Potential and Capacitance

҈


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10 Subjective Problems ҈



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Topic-3 : Capacitors, Grouping of Capacitors and Energy Stored in a Capacitor 1 ҈






MCQs with One Correct Answer

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[Main Sep. 03, 2020 (I)]

C
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WWW.JEEBOOKS.IN

EBD_8341 P-170

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䰘






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[Main Sep.
02, 2020 (II)] 㡰 ҈

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WWW.JEEBOOKS.IN

P-171

Electrostatic Potential and Capacitance

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WWW.JEEBOOKS.IN

EBD_8341 P-172

Physics ᔴ �

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d⏌
� ��泐

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e
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t
d R d⏌
e
R ᆠ㡰 �d - vt ҈�d - 黨vt 㡰 e
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d
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t
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- V

C V҈
+ V

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C V҈ + V
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

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( (

) )

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졔
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( (

) )

和


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ᆠᨨ 矄 矄
k ҈

k

 
k 
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k
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A⏌


K҈

K


d⏌


d K

A

㡰 졔



mᔴ ᆠ㡰 졔



mᔴ 㡰 졔



mᔴ

҈ ҈ ҈ 㡰 ҈ = ҈ + ҈ + ҈ 㡰 = + + K K K K
K K҈ K

K ҈
 K҈ K
ᆠ㡰 K = +
K 㡰 K
㦨
K҈
졔
K

졔

K K҈ + K


WWW.JEEBOOKS.IN

P-173

Electrostatic Potential and Capacitance





䰘






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d = N æ mö ᆠᨨ 矄 
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A ö æ �
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泐
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힘힘힘힘힘힘힘힘
[Main Sep.
02, 2020 (I)]


꾀



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泐
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C

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힘힘힘힘힘힘힘힘힘힘힘힘







᲼꾀



҈
桌

V


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ᆠᆠ�ᨨ矄
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ᨨ
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ᆠᨨ

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
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ᨨ
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
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ᨨ

S


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ö çè ÷ø mF

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䁸 ᲼҈黨黨

㜠


㝐├矄桌 4 㡰

㡰

ᆠ㡰

㡰

2




Integer Value Answer

꾀
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ᆠᆠ�ᨨ
ᨨ
ᆠᆠ� ᆠ
C 矄
矄ᆠ� �
d 犨
犨ᨨ
矄
� �

A


��ᨨ
犨

C

C


Fill in the Blanks




犨ᨨ


ᆠᆠ�ᨨ矄
ᨨ
ᆠᆠ� ᆠ矄
C
 

C
 ᆠᨨ

ᆠ
�

 
ᆠ �
ᨨ

ᨨ �
�  ᆠ V


泐
�矄
 
�矄ᆠᨨ

ᆠ
 
 
��ᨨ
犨

 
矄
ᨨ
 
ᆠᆠ�ᨨ
C
�矄
ᆠᨨ鼀泐
�
犨�

鼀� ᨨ
�ᆠ�ᆠ
ᆠᨨ 矄 
K


ᨨ �
�  ᆠ矄
ᆠᨨ矄矄  
ᆠᆠ�ᨨ矄
ᨨ犨
ᆠᨨ鼀矄












᲼҈黨


㜠


㝐├矄桌
�
�
� �ᆠ
ᆠᆠ�ᨨ
矄

ᆠ
ᨨ

A


 � 矄盐ᆠ
 
ᆠ 
矄



�矄 ᆠ
d


  矄

ᆠᨨ

ᆠ
ᨨ

矄ᨨ盐ᆠ
ᨨ
鼀
V
矄
矄 ᨨ犨
�
  ��盐 ᲼҈黨

㜠


㝐├矄桌

WWW.JEEBOOKS.IN

EBD_8341 P-174

Physics




ᔼ
 
ᆠ�ᆠ盐�
矄 ᨨ犨
�
 
��盐

 

犨ᨨ
 
ᆠᆠ�ᨨ矄
ᆠ
ᨨ
ᆠᆠ� ᆠ
C. 
矄犨�ᆠ
S҈ �矄 矄矄
�矄
ᨨ
盐泐
ᆠ �
 
ᆠᆠ�ᨨ
C҈  
 

矄


矄犨�ᆠ
S
�矄
 
矄矄
ᨨ
ᆠ �
  ᆠᆠ�ᨨ
C
. 꾀 
矄ᨨ鼀
�鼀

S
�矄
矄
 
 
S �矄 矄矄

꾀 
矄ᨨ鼀
�鼀





᲼꾀



҈桌

– + ҈

�

6




MCQs with One or More than One Correct Answer



ᔼ
 
ᆠ�ᆠ盐�
矄 ᨨ犨

� ��泐
 
�矄
ᨨ
ᆠ �
ᨨ

ᆠᆠ�ᨨ矄
 
├泐矄
和҈
 
和


ᨨ


盐矄
ᨨ
  ᆠᆠ�ᨨ矄

ᔴ҈
㦨
҈
mF

ᔴ

㦨

mF
 
C
㦨
C

㦨


mF
᲼꾀



҈黨桌


黨



䰘


␔ �ᆠ
ᨨ
 
矄鼀 矄㡰
�矄⏌
ᆠᨨᆠ䁸 㡰 ᔼ
├泐
和 ҈
�矄
├
ᆠᨨ矄
ᨨ
ᨨ �
�鼀
矄盐ᆠ
  ᆠᆠ�ᨨ矄

盐泐
ᆠ �

 
ᨨ�
�  ᆠ 犨
ᨨ� 矄

 
턼
犨�


҈

㡰 
├泐
和҈
�矄
├
ᆠᨨ矄
ᨨ
ᨨ �
�鼀
矄盐ᆠ
 
ᆠᆠ�ᨨ矄 
盐泐
ᆠ �

ᨨ犨
├泐
和

�矄
ᆠᨨ矄


 �矄
�鼀

  � 矄  ᨨ盐矄
ᆠ盐 
ᆠᨨ矄矄

矄�矄ᨨ
犨
ᨨ� 矄 
 
턼㡰
犨�




꾀
ᨨ盐 
ᨨ
ᨨ
҈矄ᆠ�鼀
ᆠ㡰 ᆠ㡰 꾀
�鼀
t
㦨



 
├泐
和҈
�矄
ᆠᨨ矄

 
� 矄  ᨨ盐矄 ᆠ盐 
�
 
ᆠᨨ矄
ᆠ�ᆠ盐�
犨�


�鼀꾀
㡰 ᔼ
├泐
和 ҈
�矄
├
ᆠᨨ矄
ᨨ
ᨨ �
�鼀
矄盐ᆠ
  ᆠᆠ�ᨨ矄

盐泐
ᆠ �

 
ᨨ�
ᆠᨨ矄矄
  ᆠᆠ�ᨨ矄
ᔴ҈
犨�



꾀


ᆠᆠ�ᨨ
矄

�ᆠ�ᆠ
矄
ᨨ
�ᆠ�ᆠ ᆠᨨ 矄 
K 犨
�矄
矄
 
ᆠᨨ矄
҈⏌
ᨨ
 
 ᨨ
�矄
矄

矄
矄 ᨨ犨
�
 
��盐


ᨨ
ᆠᆠ� ᆠ ᨨ
 
ᆠᆠ�ᨨ
�矄
C
犨 �
 
ᨨ
 
ᨨ�ᨨ
犨�
�ᆠ�ᆠ �
犨
�矄
C҈

␔ 
 
ᆠᆠ�ᨨ
�矄
ᆠ �

 
 
ᆠᨨ
泐
 
�ᆠ�ᆠ
�矄
ᆠ �
Q҈
 
 
矄 ᨨ
 

�矄
ᆠ �
Q



ᆠ�ᆠ
�
�
 
�ᆠ�ᆠ �矄
E҈
 
 
�
 
ᨨ 
ᨨ�ᨨ
�矄
E


ᔴ ᨨᨨ矄
 
ᆠᨨᆠ ᨨ�ᨨ ⏌ᨨ�ᨨ 矄

�� ᨨ� �
�
 ᆠ矄






᲼꾀



҈
桌 E҈ 㡰 E = ҈ Q҈
E

Q҈  ᆠ㡰 Q = K


C
+ K 㡰 C = K ҈

�



�҈


҈

E҈ ҈ 㡰 E = K


�



Q


E


和


和҈


ᆠ �
ᨨ

҈
�矄






 
ᨨ



�矄







ᔴ҈

和

ᔴ


ᔴ

㡰 
ᆠ �
ᨨ
 
盐

ᨨ
C҈ �矄

CV
㡰 
ᆠ �
ᨨ
 
盐

ᨨ
C҈ �矄
CV
ᆠ㡰 
ᆠ �
ᨨ
 
盐

ᨨ
C
�矄

㡰 
ᆠ �
ᨨ
 
盐

ᨨ
C
�矄
CV
꾀
�ᆠ�ᆠ
矄
ᨨ
 �ᆠ├ 矄矄
d
�矄
� 矄
�

 
ᆠᆠ�ᨨ
犨 ᨨ矄
��

�矄

x
㦨


 
ᨨ矄�� 
�矄

x
㦨
d


矄
�矄
⼄盐��矄 
ᨨ鼀
 
矄

 ᆠᆠ�ᨨ
�矄
��
矄ᨨ鼀
ᆠ �

꾀矄
ᨨ 
�ᨨ矄
ᨨ鼀


ᨨ
d
᲼҈黨黨
和
㜠


㝐├矄桌 㡰  
鼀� �盐
ᨨ
 
ᆠ�ᆠ
�
鼀� 矄
 
矄鼀
㡰  
�ᆠ�ᨨ
ᨨ
 
ᆠ�ᆠ
�
鼀� 矄
 
矄鼀
ᆠ㡰  
ᆠ�ᆠ
ᨨ �
� ᆠ矄矄
ᆠᨨ � 盐ᨨ盐矄泐
㡰  
ᆠ�ᆠ
ᨨ �
� ᆠ矄矄

�矄

 
ᆠ矄矄  
��
� ᆠ矄矄
꾀


ᆠᆠ�ᨨ
ᨨ


A
 

矄�ᨨ


�矄
ᆠ �
ᨨ
ᨨ �
�  ᆠ
V
 
 
 
泐 �矄
�矄ᆠᨨ

ᆠ

꾀
矄
ᨨ
�ᆠ�ᆠ
ᆠᨨ 矄 
K
�矄
 

� 矄
犨
 
矄
ᨨ
 
ᆠᆠ�ᨨ
矄ᨨ
矄
ᨨ
�  
矄ᆠ
犨
 
矄

ᔼ
Q

E
 
W
 ᨨ 矄ᆠ�泐

 
鼀� �盐
ᨨ
ᆠ �
ᨨ
ᆠ


  ᆠ�ᆠ
�
犨
 
矄
 
 
矄
�矄
� 矄㡰
 
犨ᨨ├
ᨨ 
ᨨ
 
矄泐矄鼀

�
⼄盐矄�ᨨ

�
 
ᨨᆠ矄矄 ᨨ
� 矄� �
 
矄

 

᲼҈黨黨҈
㜠


㝐├矄桌 e
AV e
KAV 㡰 Q
㦨
㡰 Q
㦨
d d V e
AV
é ҈ù ᆠ㡰 E
㦨
㡰 W
㦨
҈- ú ê Kd
d ë K û 꾀


ᆠᆠ�ᨨ
�矄
ᆠ �
 
 
ᆠ �� � 泐
�矄
 
�矄ᆠᨨ

ᆠ

ᔼ
 
矄
ᨨ
 
ᆠᆠ�ᨨ 
鼀ᨨ
 

泐
鼀 矄
ᨨ
� 矄盐� �
 矄
᲼҈黨
䰘
㜠


㝐├矄桌 㡰  
ᆠ �
ᨨ
 
ᆠᆠ�ᨨ
� ᆠ矄矄
㡰  
ᨨ�
ᆠᨨ矄矄
 
矄
� ᆠ矄矄
ᆠ㡰  
ᆠᆠ� ᆠ
� ᆠ矄矄
㡰  
ᆠᨨ矄�ᆠ
 �泐
矄ᨨ
�
 
ᆠᆠ�ᨨ � ᆠ矄矄 꾀


�
ᆠᆠ�ᨨ
�矄
ᆠᨨ

ᆠ
ᨨ

泐



⼄盐 ��矄
ᆠ �

ᨨ�

ᆠ�ᆠ
�
 
 �泐 矄矄ᨨᆠ�
犨�
 �矄
ᆠᆠ�ᨨ

��
泐
Q


V


E

  U

矄ᆠ�泐

꾀
�ᆠ�ᆠ
矄
�矄
ᨨ犨
� ᨨ盐ᆠ
ᨨ
�  
矄ᆠ
犨
 
矄
犨�
泐
矄�
�

ᆠᨨ

ᆠ�ᨨ


ᆠᨨ矄ᨨ � �
⼄盐 ��矄
ᨨ犨
��
泐 Q

V

E
 
U


ᨨ
 
�ᨨ盐矄
ᨨ 
矄 ᲼҈黨
�
㜠


㝐├矄桌 㡰 Q > Q
㡰 V > V
ᆠ㡰 E > E
㡰 U > U


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P-175

Electrostatic Potential and Capacitance

10 Subjective Problems �


犨ᨨ
ᆠᆠ�ᨨ矄
A
 
B
犨�
ᆠᆠ��矄

mF
 


mF
 ᆠ �
ᨨ

ᨨ �
�  ᆠ
ᨨ
҈



 
҈


 矄ᆠ�泐


矄
ᨨ
 
ᆠᆠ�ᨨ矄

ᆠᨨ

ᆠ 矄
矄 ᨨ犨
�
 
��盐
犨�
ᨨ 
犨�
ᨨ鼀
ᆠ
ᆠᆠ�ᨨ 


盐

ᨨ
A
�矄
ᨨ矄��
 
 
ᨨ
B
�矄

��

꾀
盐 ᆠ �


mF
ᆠᆠ�ᨨ
C
犨�

犨�矄 矄
ᨨ
 

 矄
ᨨ
ᆠᨨ鼀
 
ᆠ�ᆠ盐�

ᔴᆠ盐 ᲼҈黨黨䰘
㜠
�
㝐├矄桌

�

犨ᨨ


ᆠᆠ�ᨨ矄
A
 
B

 
矄鼀 矄�ᨨ
d
㦨



�
맔
҈


鼀
犨
 
矄


 
ᨨ
A
 
B






鼀

 




鼀

矄ᆠ�泐

꾀
矄 ᨨ
�ᆠ�ᆠ
ᆠᨨ 矄 
�
鼀���泐㡰
K
㦨
黨
矄 �鼀 矄�ᨨ 矄
矄盐ᆠ
 
�
ᆠ
 ᆠ泐
�
 
矄ᆠ
犨

 
矄
ᨨ
ᆠᆠ�ᨨ
B

᲼҈黨黨
㜠


졔

졔


㝐├矄桌 꾀

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㝐├矄桌

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MCQs with One Correct Answer

MCQs with One or More than One Correct Answer

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黨桌

WWW.JEEBOOKS.IN

EBD_8341 P-176

Physics

ᔴᨨ盐鼀 㜠ᔼ

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WWW.JEEBOOKS.IN

P-177

Electrostatic Potential and Capacitance

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㡰 Topic-3 : Capacitors, grouping of capacitors and energy stored in a capacitor

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Topic-4 : Miscellaneous (Mixex Concepts) Problems ҈


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WWW.JEEBOOKS.IN

EBD_8341 P-178

Physics

17

Current Electricity

Topic-1 : Electric Current, Drift of Electrons, Ohm's Law, Resistance and Resistivity 1 ҈









MCQs with One Correct Answer

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P-179

Current Electricity

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EBD_8341 P-180

Physics

outer radius is R2. If a voltage V0 is applied between its two ends, a current I flows in it. In addition, it is observed that a transverse voltage DV develops between its inner and outer surfaces due to purely kinetic effects of moving electrons (ignore any role of the magnetic field due to the current). Then (figure is schematic and not drawn to scale) [Adv. 2020]

(c)

DV

(d) 19.

20.

A piece of copper and another of germanium are cooled from room temperature to 80° K. The resistance of [1988 - 1 Mark] (a) each of them increases (b) each of them decreases (c) copper increases and germanium decreases (d) copper decreases and germanium increases The temperature coefficient of resistance of a wire is 0.00125 per °C. At 300 K, its resistance is 1 ohm. This resistance of the wire will be 2 ohm at. [1980] (a) 1154 K (b) 1100 K (c) 1400 K (d) 1127 K 5

21.

True / False

The current –voltage graphs for a given metallic wire at two different temperatures T1 and T2 are shown in the figure. [1985 - 3 Marks]

T1 I

T2

V The temperature T2 is greater than T1. 22.

Electrons in a conductor have no motion in the absence of a potential difference across it. [1982 - 2 Marks] 6

23.

MCQs with One or More than One Correct Answer

Shown in the figure is a semicircular metallic strip that has thickness t and resistivity r. Its inner radius is R1 and

R2 R1

I

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24. Read the following statements carefully: [1993-2 Marks] Y : The resistivity of a semiconductor decreases with increase of temperature. Z : In a conducing solid, the rate of collisions between free electrons and ions increases with increase of temperature Select the correct statement(s) from the following; (a) Y is true but Z is false (b) Y is false but Z is true (c) Both Y and Z are true (d) Y is true and Z is the correct reason for Y 10 Subjective Problems 25. If a copper wire is stretched to make it 0.1奠 longer what is the percentage change in its resistance? [1978]

Topic-2 : Combination of Resistances 1 1.

MCQs with One Correct Answer

In the given circuit diagram, a wire is joining points B and D. The current in this wire is: [Main 9 Jan. 2020, (I)] (a) 0.4A

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EBD_8341 P-184

Physics

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P-185

Current Electricity

5 ҈



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True / False

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桌 6

10 Subjective Problems

MCQs with One or More than One Correct Answer

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Topic-4 : Heating Effect of Current 1 ҈









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WWW.JEEBOOKS.IN

EBD_8341 P-186

5.

6.

7.

8.

Physics

Two electric bulbs, rated at (25 W, 220 V) and (100 W, 220 V), are connected in series across a 220 V voltage source. If the 25 W and 100 W bulbs draw powers P1 and P2 respectively, then: [Main 12 Jan. 2019 (I)] (a) P1=16 W, P2=4 W (b) P1=16 W, P2=9 W (c) P1=9 W, P2=16 W (d) P1=4 W, P2=16 W Two equal resistances when connected in series to a battery, consume electric power of 60 W. If these resistance are now connected in parallel combination to the same battery, the electric power consumed will be : [Main 11 Jan. 2019 (I)] (a) 60 W (b) 240 W (c) 120 W (d) 30 W A current of 2 mA was passed through an unknown resistor which dissipated a power of 4.4 W. Dissipated power when an ideal power supply of 11 V is connected across it is: [Main 10 Jan. 2019 (II)] (a) 11 × 10–5 W (b) 11 × 10–3 W (c) 11 × 10–4 W (d) 11 × 105 W The figure shows three circuits I, II and III which are connected to a 3V battery. If the powers dissipated by the configurations I, II and III are P1, P2 and P3 respectively, then : [Main Online April 9, 2017]

12. Which of the four resistances P, Q, R and S generate the greatest amount of heat when a current flows from A to B? [Main Online April 23, 2013] P = 2W

Q = 4W

A

B

R = 1W

S = 2W

(a) Q (b) S (c) P (d) R 13. Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, 100 W, 60 W and 40 W bulbs have filament resistances R100, R60 and R40, respectively, the relation between these resistances is [2010] 1

(a)

R100

=

1 1 + R40 R60

(b) R100 = R40 + R60 1

1W

1W

1W 1W

1W

3V

1W

1W

1W

1W

3V 1W

1W 1W

(I)

1W

1W

1W

3V

(II)

(a) P1 > P2 > P3 (c) P2 > P1 > P3

(b) P1 > P3 > P2 (d) P3 > P2 > P1

9.

1 1 > R100 R60 R40 14. Figure shows three resistor configurations R1, R2 and R3 connected to 3V battery. If the power dissipated by the configuration R1, R2 and R3 is P1, P2 and P3, respectively, then – [2008]

(c) R100 > R60 > R40

1W

>

(d)

(III)

R

3V

3V

R

r

R1

In the circuit shown, the resistance r is a variable resistance. If for r = fR, the heat generation in r is maximum then the value of f is : [Main Online April 9, 2016] 1 1 3 (a) (b) 1 (c) (d) 2 4 4 10. In a large building, there are 15 bulbs of 40 W, 5 bulbs of 100 W, 5 fans of 80 W and 1 heater of 1 kW. The voltage of electric mains is 220 V. The minimum capacity of the main fuse of the building will be: [Main 2014] (a) 8 A (b) 10 A (c) 12 A (d) 14 A 11. Four bulbs B1, B2, B3 and B4 of 100 W each are connected to 220 V main as shown in the figure. [Main Online April 19, 2014]

(a)

P1 > P2 > P3

(b)

P1 > P3 > P2

R2

3V

(c) P2 > P1 > P3 (d) P3 > P2 > P1 R3

15. The three resistance of equal value are arranged in the different combinations shown below. Arrange them in increasing order of power dissipation. [2003S] i

220 V

B1

B2

B3

B4

i

Ammeter

The reading in an ideal ammeter will be: (a) 0.45 A (b) 0.90 A (c) 1.35 A

(d) 1.80 A

(I)

(II)

WWW.JEEBOOKS.IN

P-187

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EBD_8341 P-188




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WWW.JEEBOOKS.IN

P-189

Current Electricity

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WWW.JEEBOOKS.IN

EBD_8341 P-190

Physics

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矄�矄 ᆠ
�矄
盐
� 矄�

 ᆠᨨ矄盐
鼀� � 


��  鼀盐


盐
ᨨ� 
ᆠ

ᨨ� 

 
矄鼀 ᨨ� 
矄
 ᨨ
泐
ᆠ矄� �
 
盐
ᨨ
 
矄  矄�矄 ᆠ
和T꾀TE㝐ENT㜠



矄�矄 ᆠ
ᨨ

鼀
� ᆠ矄矄
犨� � ᆠ矄
�
鼀盐
᲼



桌 㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
盐錐
和鼀 㜠

�矄

ᆠᨨᆠ
  �ᨨ
ᨨ
和鼀 㜠҈ 㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
盐錐
和鼀 㜠

�矄
졠

ᆠᨨᆠ
  �ᨨ
ᨨ
和鼀 㜠҈ ᆠ㡰 和鼀 
㜠҈
�矄
盐

和鼀 㜠

�矄
矄 㡰 和鼀 㜠҈
�矄
矄

和鼀 㜠

�矄
盐

B

C




꾀
 �
盐 � ᨨ鼀
犨�
AB
ᨨ
 �
҈鼀


盐 ├ ᨨ犨

矄�矄 ᆠ
X
 

矄�矄 ᆠ
ᨨ
҈

W

ᆠᨨ

ᆠ
泐  �ᆠ├
ᆠᨨ 盐ᆠ� �
矄�矄

矄
矄 ᨨ犨
�
 
��盐

꾀
泐  

� ᨨ鼀
犨�

矄�� �
ᨨᆠ├泐
ᆠᨨ

ᆠ
ᨨ �㡰

矄ᨨ
�

ᔴᨨ

ᆠ�ᨨ 矄

ᨨ

鼀
ᨨ 鼀矄盐
 
盐 ├ ᨨ犨

矄�矄 ᆠ
X
盐矄� �
 
� ᆠ�
ᨨ ␔ 矄ᨨ 
��

꾀 矄犨
 
ᨨᨨ犨� �
⼄盐矄�ᨨ 矄
᲼




㜠
�
㝐├矄桌

҈
W

X A

B

C

D

㡰 꾀
 
ᨨ矄��
 
��
鼀� 矄
ᨨ
  � ᨨ鼀䁸 㡰 ᔴᨨ泐
 
��盐
�
泐ᨨ盐
 矄犨
ᨨᨨ├
 
矄 ᨨ犨
  泐
 
 
� ᨨ鼀
犨�
ᨨᆠ├泐㡰 ᆠᨨ

ᆠ

ᨨ�
ᨨ� 矄


10 Subjective Problems
䰘
















R
㦨
R҈
ᨨ
R

ᨨ
R

R G

꾀
盐 ├ ᨨ犨
矄�矄 ᆠ
X
�矄
ᨨ

鼀� 
盐矄� � 矄�矄 ᆠ矄
R҈

R

ᨨ
R

�
ᆠᨨ矄ᨨ � �
盐
ᨨ� 矄

Topic-6 : Miscellaneous (Mixed Concepts) Problems ҈


㡰 㡰 ᆠ㡰 㡰

MCQs with One Correct Answer

1

꾀
ᆠ�ᆠ盐�
�矄
ᆠᨨ

ᆠ
矄
矄 ᨨ犨
�
 
��盐
犨�
  矄犨�ᆠ
S
ᨨ

␔ 
 
矄犨�ᆠ
�矄
ᆠᨨ矄

 
ᨨ
鼀ᨨ盐  ᨨ
ᆠ �
 
ᨨ犨矄
ᨨ鼀
Y
ᨨ
X
�矄 ᲼


䰘桌 mF


m F

X




S W


W





Y

黨






㡰
㡰 �

mᔴ ᆠ㡰
䰘mᔴ 㡰
҈
mᔴ ᔼ

矄泐
ᆠ盐 
ᔼ
�矄
ᨨ犨� �
 ᨨ盐�

ᆠ泐� �ᆠ
鼀  꾀ᔴ

ᔴ ᨨᨨ矄
 
ᆠᨨᆠ
�ᨨ 矄 �



᲼



桌 A

B


r

r

I

l⏌


C

�


V AB =
VBC

ᨨ犨
ᆠᨨ矄矄
BC
�矄


�鼀矄
 
ᨨ犨
ᆠᨨ矄矄
AB ᔴ盐 
 矄��矄
�
AB
 
BC

⼄盐 쟼ᆠ�ᆠ
�
盐
ᨨ
ᆠ盐 
� 矄�
AB
 
BC
 ⼄盐 꾀
�
�矄
�矄
�
�

ᆠᨨ矄
���
 
 鼀泐
� 矄盐 ᆠᨨ � 

꾀
ᆠᨨ�
ᨨ
҈


W
矄�矄ᨨ
ᆠ泐� �
ᆠ盐 
҈
꾀
ᨨ �
鼀� 盐矄
矄盐�矄

ᨨ
 
�矄


ᆠ  �
�
�    �泐
ᨨ
 
�矄
�矄 ᲼


�和桌 㡰 ҈

├ 㡰 

├ ᆠ㡰


├ 㡰

├ 꾀


mF
ᆠᆠ�ᨨ


矄�矄 ᆠ
ᨨ


�
MW
�矄
�
矄�矄
犨� ҈


泐

� 
 
�鼀
 

犨 �ᆠ
 
ᨨ � �  ᆠ
ᆠᨨ矄矄
 
ᆠᆠ�ᨨ
�矄

�鼀矄
 
ᨨ � �  ᆠ
ᆠᨨ矄矄
 
矄�矄ᨨ

᲼෌�
n
㡰
㦨



黨桌
᲼


�和桌 㡰 ҈


矄 㡰

黨矄 ᆠ㡰 䰘矄 㡰 ҈
矄 꾀
ᆠᆠ�ᨨ
�矄
ᆠ �
盐矄� �

  
泐
犨�
 矄�矄 ᆠ
x
�
矄�矄


矄 
� 
矄 ᨨ犨矄
 
��ᨨ

ᨨ

ᔼ
犨�
矄ᆠ
ᨨ
�鼀

ᔼ
 
矄�矄 ᆠ
�矄
ᆠ  �
ᨨ
x

 
犨
�
犨�
 ᲼



和桌

l⏌


WWW.JEEBOOKS.IN

P-191

Current Electricity

㡰 P 㡰 Q

S R

҈
n ᔼ

ᆠ㡰


R



Q

P 㡰 S t 和 ᨨ犨
�
��盐
�矄

ᨨ矄
졠 �ᆠ
ᨨ

ᔼ
ᨨ
ᨨ
ᆠᆠ盐  
盐
ᨨ
  
矄�矄 ᆠ

�
矄 ᨨ盐

ᆠᨨ

ᆠ ᲼



和桌 犨

C B A

D

C'

䰘


R �


R ҈


Integer Value Answer

҈

꾀
� ᨨ鼀
��矄
盐
矄ᆠ
 ᆠ�ᨨ
犨�






꾀 ᆠ盐 

泐
ᆠᨨ

ᆠ� �
�
ᨨ


黨黨

W
矄�矄 ᆠ

�
ᆠ
 ᆠᨨ 
� ᨨ

ᨨ鼀
ᨨ
 �







ᔼ
ᆠᨨ

ᆠ
n W 矄�矄 ᆠ

�
ᆠᨨ鼀矄

鼀鼀
ᨨ
 �


 ᨨ



黨 ҈
�꾀


盐
ᨨ
n
�矄 ᲼꾀



҈
桌 ҈҈
꾀
�鼀
t 㦨




泐
ᨨ
҈


�矄
ᆠᨨ

ᆠ
ᆠᨨ矄矄 ᨨ� 矄
꾀
 

�
 
��
ᆠ�ᆠ盐�

ᔼ
 
ᆠᆠ�ᨨ矄 
ᨨ
ᆠ �
� ��泐


犨 
�鼀
�
矄ᆠᨨ 矄㡰
ᨨ矄  
ᨨ�
ᆠᨨ矄矄
 鼀
ᆠᨨ鼀


䁸
᲼ ├


l �
㦨҈


l 
㦨
҈
҈桌 ᲼

҈
桌

R



R




m


㝐W

B'

㡰 BἘ
 
CἘ 㡰 A
 
D ᆠ㡰 C
 
D 㡰 B
 
D ᔼ
 
��
ᆠ�ᆠ盐�

�
�矄
ᨨ矄
 
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ᆠ盐 
I
�矄 �   
ᨨ
 
盐
ᨨ
 
矄�矄 ᆠ
R



  矄�矄 ᆠ
盐矄
鼀盐矄
矄�矄 泐 ᲼


҈和桌

I


2

R 




꾀
㝐W

6


m

MCQs with One or More than One Correct Answer

҈

ᔼ
 
ᆠ�ᆠ盐�
矄 ᨨ犨
ᨨ犨

 
├泐
�矄
矄矄

�鼀

㦨


␔ �ᆠ
ᨨ
 
ᨨᨨ犨� �
矄鼀 矄㡰
�矄㡰
盐䁸 ᲼꾀



҈
桌


m


�
├W

R



㡰 R҈R
R�
㦨
RR
R
㡰

㡰 R҈R
㦨
R
R

㦨
R�R


黨


�

W

1 1 1 1 + = + R 5 R6 R1 + R2 R3 + R 4

ᆠ㡰 R҈R

㦨
R
R



꾀

꾀
犨�
ᨨ
 �
L  

� �ᆠ
ᆠ矄
ᨨ
����
�   矄�矄 ᆠ矄

ᆠᨨ

ᆠ
�
矄�矄

盐
ᨨ
 
ᆠ盐 
 
鼀盐
ᨨ
 
犨�
�矄
�矄
泐
DT
�

�鼀
t

꾀

盐鼀
N ᨨ
矄�鼀�
ᆠ矄
�矄
ᨨ犨
ᆠᨨ

ᆠ
�
矄�矄
犨�
 犨�
ᨨ
 
矄鼀
鼀�
 
ᆠᨨ矄矄㜠矄ᆠ�ᨨ
盐
ᨨ
 �
L


鼀盐
ᨨ
 
犨�
�矄
�矄
泐
 
矄鼀
鼀ᨨ盐  DT
�
 
矄鼀
�鼀
t

 
盐
ᨨ
N �矄 ᲼


҈和桌 㡰
㡰
ᆠ㡰
㡰 黨


ᆠᨨ矄�ᆠ
�
盐
ᨨ

ᨨ� 
ᆠ �
 矄
ᨨ

҈  
�矄 ᆠ
r
矄



ᔼ �ᆠ
犨 �ᆠ
ᨨ
 
ᨨᨨ犨� � r ⼄盐 ��矄
矄 ᨨ犨矄
矄鼀
  ᆠ
ᨨ
r


᲼҈黨

桌 㡰 ᔼ  矄�泐
ᨨ
�� 
ᨨ鼀

ᨨ� 
矄ᨨ盐ᆠ
㡰 쟼ᆠᨨ矄�ᆠ
ᨨ �
盐
ᨨ

ᨨ� 
ᆠ �
ᆠ㡰 쟼ᆠᨨ矄�ᆠ
ᨨ �


�矄 ᆠ

ᨨ鼀
 
ᆠ 
ᨨ 
ᆠ �
鼀�ᆠ
矄 


෌�
r


�盐矄
ᨨ
  矄 
㡰 ᨨ 
ᨨ
 矄

 졔


m

졔 ⧤泐



�


㡰


ᨨ鼀
�矄泐矄
�
矄
矄ᨨᨨ
矄
 
├泐
�矄 矄矄

 
�矄泐矄
졔�
 

ᨨ �
�鼀 㡰 
ᨨ鼀
犨�
�矄泐



�鼀

㦨



矄ᆠᨨ 矄 ᆠ㡰 
ᆠ盐 
�
 
鼀鼀
ᆠᨨ鼀矄
҈⏌
ᨨ
 
� �� 盐
 
҈
矄ᆠᨨ  㡰 
ᆠ盐 
�
 
鼀鼀
ᆠᨨ鼀矄
ᨨ
 

ᨨ � �鼀
҈
꾀
� ᆠ 矄ᆠ 
盐
矄

 �
�鼀 
ᨨ
盐 �矄
  �矄

ᨨ
��
鼀盐
泐
矄矄� �

ᆠ�ᆠ ᆠ盐 


ᨨ
�鼀 
鼀�矄
ᆠ├ᨨ泐
��ᨨ

 �鼀 
�矄
ᨨ矄
ᨨ
├
盐

 ᨨ鼀
ᨨᆠ�ᨨ 矄
  
矄盐 �ᆠ� 泐
ᨨ �
�鼀
ᨨ
ᨨ�ᨨ
盐
ᨨ
ᨨ 盐 � ᨨ鼀 ᨨ�ᨨ
ᨨ
盐 �矄
ᨨ鼀
 
�鼀 

ᔼ
 
盐
�矄 ᨨ犨

ᆠᨨ 矄 
ᨨ�

犨 �ᆠ
ᨨ
 
ᨨᨨ犨� � 矄鼀 矄㡰
�矄㡰
盐䁸 ᲼꾀



҈
桌 㡰 
鼀盐
�矄�盐�ᨨ
ᨨ
 
�鼀 
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EBD_8341 P-192

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P-193

Current Electricity


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EBD_8341 P-194

Physics

Moving Charges and Magnetism

18

Topic-1 : Motion of Charged Particle in Magnetic Field 1 ҈






MCQs with One Correct Answer

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WWW.JEEBOOKS.IN

P-195

Moving Charges and Magnetism

9.

10.

11.

A proton and an α -particle (with their masses in the ratio of 1 : 4 and charges in the ratio 1 : 2) are accelerated from rest through a potential difference V. If a uniform magnetic field (B) is set up perpendicular to their velocities, the ratio of the radii rp : ra of the circular paths descrfibed by them will be: [Main 12 Jan 2019, I] (a) 1: 2 (b) 1: 2 (c) 1: 3 (d) 1: 3 In an experiment, electrons are accelerated, from rest, by applying a voltage of 500 V. Calculate the radius of the path if a magnetic field 100 mT is then applied. [Charge of the electron = 1.6 × 10–19 C Mass of the electron = 9.1 × 10–31 kg] [Main 11 Jan 2019, I] (a) 7.5 × 10–3 m (b) 7.5 × 10–2 m (c) 7.5 m (d) 7.5 ×10–4 m The region between y = 0 and y = d contains a magnetic r field B = Bzˆ . A particle of mass m and charge q enters the

mv region with a velocity vr = viˆ . if d = , the acceleration 2qB of the charged particle at the point of its emergence at the other side is : [Main 11 Jan 2019, II] qv B æ 1 ˆ 3 ˆö qv B æ 3 ˆ 1 ˆ ö j÷ i + j÷ ç iç (b) m è 2 2 ø m è2 2 ø qvB æ - ˆj + iˆ ö qvB æ iˆ + ˆj ö ç ÷ ç ÷ (c) (d) m è 2 ø m è 2 ø An electron, a proton and an alpha particle having the same kinetic energy are moving in circular orbits of radii re, rp, ra respectively in a uniform magnetic field B. The relation between re, rp, ra is : [Main 2018] (a) re > rp = ra (b) re < rp = ra (c) re < rp < ra (d) re < ra < rp A negative test charge is moving near a long straight wire carrying a current. The force acting on the test charge is parallel to the direction of the current. The motion of the charge is : [Main Online April 9, 2017] (a) away from the wire (b) towards the wire (c) parallel to the wire along the current (d) parallel to the wire opposite to the current Consider a thin metallic sheet perpendicular to the plane of the paper moving with speed 'v' in a uniform magnetic field B going into the plane of the paper (See figure). If charge densities s1 and s2 are induced on the left and right surfaces, respectively, of the sheet then (ignore fringe [Main Online April 10, 2016] effects):

(a)

12.

13.

14.

15.

- Î0 vB Î vB , s2 = 0 2 2

(a)

s1 =

(b)

s1 =Î0 vB, s 2 = - Î0 vB

(c)

s1 =

Î0 v B - Î0 vB , s2 = 2 2

field occupies a region of space by width ‘d’. If a be the angle of deviation of proton from initial direction of motion (see figure), the value of sin a will be : [Main Online April 10, 2015]

B a d (a)

qV

Bd 2m

B qd 2 mV

(b)

B q q (d) Bd d 2mV 2mV 16. A particle of charge 16 × 10–16 C moving with velocity 10 ms–1 along x-axis enters a region where magnetic field ur of induction B is along the y-axis and an electric field of magnitude 104 Vm–1 is along the negative z-axis. If the charged particle continues moving along x-axis, the ur magnitude of B is : [Main Online April 23, 2013] (a) 16 × 103 Wb m–2 (b) 2 × 103 Wb m–2 3 –2 (c) 1 × 10 Wb m (d) 4 × 103 Wb m–2 ur 17. A magnetic field B = B0 Jˆ , exists in the region a < x < 2a, uv and B = – B0 ˆj , in the region 2a < x < 3a, where B0 is a positive constant. A positive point charge moving with a r velocity v = v0iˆ , where v0 is a positive constant, enters the magnetic field at x = a. The trajectory of the charge in this region can be like [2007]

(c)

B0 0

2a

a

x

3a

- B0 z

(a)

z

a

2a

3a

x

(b)

z

(c)

a

2a

x

3a

z

a

2a

3a

x

a

(d)

2a

3a

x

v

18. An electron travelling with a speed u along the positive x-axis enters into a region of magnetic field where B = –B kˆ B

s1 s2 (d) s1 = s 2 =Î0 vB A proton (mass m) accelerated by a potential difference V flies through a uniform transverse magnetic field B. The

0

(x > 0). It comes out of the region with speed v then [2004S] (a) v = u at y > 0 ×B y

(b) v = u at y < 0 (c) v > u at y > 0

e–

u

x®

(d) v > u at y < 0

WWW.JEEBOOKS.IN

EBD_8341 P-196

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㜠


㝐├矄桌 C

B A

D

5

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㝐├矄桌 6

MCQs with One or More than One Correct Answer

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
⏌


WWW.JEEBOOKS.IN

P-197

Moving Charges and Magnetism




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WWW.JEEBOOKS.IN

EBD_8341 P-198

Physics

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WWW.JEEBOOKS.IN

P-199

Moving Charges and Magnetism





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EBD_8341 P-200

Physics

m0 l m0 l 6 [ 3 - 1] 6 [ 3 + 1] (b) 4 pa 4 pa m0 l m0l 3 [ 3 - 1] 3 [2 - 3] (c) (d) 4 pa 4pa An infinitely long hollow conducting cylinder with inner radius R/2 and outer radius R carries a uniform current density along its length. The magnitude of the magnetic r field, B as a function of the radial distance r from the axis is best represented by [2012]

(a)

2R

A

P

C

16.

R

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11.

12.

15.

8 N m0 I 3/ 2

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4 N m0 I 1/ 2

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plgL tan q µ0

L q

plgL µ0 cos q plgL (d) 2sin q I I µ0 cos q Consider two thin identical conducting wires covered with very thin insulating material. One of the wires is bent into a loop and produces magnetic field B1, at its centre when a current I passes through it. The ratio B1 : B2 is: [Main Online April 12, 2014] (a) 1 : 1 (b) 1 : 3 (c) 1 : 9 (d) 9 : 1 An electric current is flowing through a circular coil of radius R. The ratio of the magnetic field at the centre of the (c)

14.

3/ 2

(b)

5 R 5 R 5 R 51/ 2 R Two identical wires A and B, each of length 'l', carry the same current I. Wire A is bent into a circle of radius R and wire B is bent to form a square of side 'a'. If BA and BB are the values of magnetic field at the centres of the circle and B square respectively, then the ratio A is: [Main 2016] BB p2 p2 p2 p2 (a) (b) (c) (d) 8 16 8 2 16 2 Two long current carrying thin wires, both with current I, are held by insulating threads of length L and are in equilibrium as shown in the figure, with threads making an angle 'q' with the vertical. If wires have mass l per unit length then the value of I is : (g = gravitational acceleration) [Main 2015] pgL tan q (a) 2 µ0 (b)

13.

4 N m0 I

sin q

coil and that at a distance 2 2R from the centre of the coil and on its axis is : [Main Online April 9, 2013] (a) 2 2 (b) 27 (c) 36 (d) 8 A symmetric star shaped conducting wire loop is carrying a steady state current I as shown in the figure. The distance between the diametrically opposite vertices of the star is 4a. The magnitude of the magnetic field at the center of the loop is [Adv. 2017] I

4a

(a)

r B

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(c)

r B

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r B

r B

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(b)

(c)

m0 NI æbö ln ç ÷ 2(b - a) èaø

æb+aö m0 NI ln ç ÷ 2(b - a) è b - a ø m0 NI æbö ln ç ÷ 2b èaø

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(

)

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(

m 0 I xiˆ - yjˆ

(

2

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2

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)

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P-201

Moving Charges and Magnetism






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EBD_8341 P-202

29.

Physics

A current I flows along the length of an infinitely long, straight, thin-walled pipe. Then [1993-2 Marks] (a) the magnetic field at all points inside the pipe is the same, but not zero. (b) the magnetic field at any point inside the pipe is zero (c) the magnetic field is zero only on the axis of the pipe (d) the magnetic field is different at different points inside the pipe.

Y-axis such that OS = OR = 0.02 m. Find the magnitude and direction of the magnetic induction at the origin O. [1989 - 6 Marks] Y

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10 Subjective Problems 30.

L

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A square loop of side 2a, and carrying current I, is kept in XZ plane with its centre at origin. A long wire carrying the same current I is placed parallel to the z-axis and passing through the point (0, b, 0), (b >> a). The magnitude of the torque on the loop about z-axis is given by : [Main Sep. 05, 2020 (I)] (a)

m0 I 2a2 2 pb

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2.

A small circular loop of conducting wire has radius a and carries current I. It is placed in a uniform magnetic field B perpendicular to its plane such that when rotated slightly about its diameter and released, it starts performing simple harmonic motion of time period T. If the mass of the loop is m then : [Main 9 Jan 2020, II]

4.

Bo

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P-203

Moving Charges and Magnetism

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EBD_8341 P-204

҈



Physics

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3

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14. A moving coil galvanometer has 50 turns and each turn has an area 2 ×10–4 m2. The magnetic field produced by the magnet inside the galvanometer is 0.02 T. The torsional constant of the suspension wire is 10–4 N m rad–1. When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by 0.2 rad. The resistance of the coil of the galvanometer is 50 W. This galvanometer is to be converted into an ammeter capable of measuring current in the range 0 –1.0 A. For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ohms, is __________. [Adv. 2018] 6

MCQs with One or More than One Correct Answer

15. Two identical moving coil galvanometers have 10 resistance and full scale deflection at 2mA current. One of them is converted into a voltmeter of 100mV full scale reading and the other into an Ammeter of 1mA full scale current using appropriate resistors. There are then used to measure the voltage and current in the Ohm’s law experiment with R – 1000W resistor by using an ideal cell. Which of the [Adv. 2019] following statement(s) is/are correct? (a) The resistance of the voltmeter will be 100k (b) The measured value of R will be 978


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EBD_8341 P-208

Physics

6 䰘






MCQs with One or More than One Correct Answer

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P-209

Moving Charges and Magnetism

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P-211

Moving Charges and Magnetism

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WWW.JEEBOOKS.IN

EBD_8341 P-212

Physics

Magnetism and Matter

19

Topic-1 : Magnetism, Gaus’s Law, Magnetic Moment, Properties of Magnet ҈






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P-213

Magnetism and Matter 

6

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EBD_8341 P-214

Physics

(ii) The ur atom is placed in a uniform magnetic induction B such that the plane-normal of the electron-orbit makes an angle of 30° with the magnetic induction. Find the torque experienced by the orbiting electron.

19. A bar magnet is placed with its north pole pointing north and its south pole pointing south. Draw a figure to show the location of neutral points. [1979] 20. A bar magnet with poles 25 cm apart and of strength 14.4 amp-m rests with centre on a frictionless pivot. It is held in equilibrium at an angle of 60° with respect to a uniform magnetic field of induction 0.25 Wb/m2, by applying a force F at right angles to its axis at a point 12 cm from pivot. Calculate F. What will happen if the force F is removed? [1978]

Topic-2 : The Earth Magnetism, Magnetic Materials and their Properties 1 1.

2.

3.

4.

MCQs with One Correct Answer

5.

An iron rod of volume 10–3 m3 and relative permeability 1000 is placed as core in a solenoid with 10 turns/cm. If a current of 0.5 A is passed through the solenoid, then the magnetic moment of the rod will be : [Main Sep. 05, 2020 (II)] (a) 50 × 102 Am2 (b) 5 × 102 Am2 (c) 500 × 102 Am2 (d) 0.5 × 102 Am2 A paramagnetic sample shows a net magnetisation of 6 A/ m when it is placed in an external magnetic field of 0.4 T at a temperature of 4 K. When the sample is placed in an external magnetic field of 0.3 T at a temperature of 24 K, then the magnetisation will be :[Main Sep. 04, 2020 (II)] (a) 1 A/m (b) 4 A/m (c) 2.25 A/m (d) 0.75 A/m A perfectly diamagnetic sphere has a small spherical cavity at its centre, which is filled with a paramagnetic substance. r The whole system is placed in a uniform magnetic field B. Then the field inside the paramagnetic substance is : [Main Sep. 03, 2020 (II)]

r (a) B P (b) zero r r (c) much large than | B | and parallel to B r r (d) much large than | B | but opposite to B Magnetic materials used for making permanent magnets (P) and magnets in a transformer (T) have different properties of the following, which property best matches for the type of magnet required? [Main Sep. 02, 2020 (I)] (a) T : Large retentivity, small coercivity (b) P : Small retentivity, large coercivity (c) T : Large retentivity, large coercivity (d) P : Large retentivity, large coercivity

The figure gives experimentally measured B vs. H variation in a ferromagnetic material. The retentivity, co-ercivity and saturation, respectively, of the material are: [Main 7 Jan. 2020 II] (a) 1.5 T, 50 A/m and 1.0 T (b) 1.5 T, 50 A/m and 1.0 T (c) 150 A/m, 1.0 T and 1.5 T (d) 1.0 T, 50 A/m and 1.5 T 6.

7.

A paramagnetic material has 1028 atoms/m3. Its magnetic susceptibility at temperature 350 K is 2.8 × 10–4 . Its susceptibility at 300 K is: [Main 12 Jan. 2019 II] (a) 3.267 × 10–4 (b) 3.672 × 10–4 (c) 3.726 × 10–4 (d) 2.672 × 10 –4 A paramagnetic substance in the form of a cube with sides 1 cm has a magnetic dipole moment of 20 × 10–6 J/T when a magnetic intensity of 60 × 103 A/m is applied. Its magnetic susceptibility is: [Main 11 Jan. 2019 II] (a) 3.3 × 10–2 (b) 4.3 × 10 –2 (c) 2.3 × 10–2 (d) 3.3 × 10–4

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P-215

Magnetism and Matter

8.

9.

A bar magnet is demagnetized by inserthing it inside a solenoid of length 0.2 m, 100 turns, and carrying a current of 5.2 A. The coercivity of the bar magnet is: [Main 9 Jan. 2019 I] (a) 285 A/m (b) 2600 A/m (c) 520 A/m (d) 1200 A/m The B-H curve for a ferromagnet is shown in the figure. The ferromagnet is placed inside a long solenoid with 1000 turns/cm.. The current that should be passed in the solenoid to demagnetise the ferromagnet completely is: [Main Online April 15, 2018]

12. The coercivity of a small magnet where the ferromagnet gets demagnetized is 3 ´ 103 Am -1. The current required to be passed in a solenoid of length 10 cm and number of turns 100, so that the magnet gets demagnetized when inside the solenoid, is: [Main 2014] (a) 30 mA (b) 60 mA (c) 3 A (d) 6 A 13. Three identical bars A, B and C are made of different magnetic materials. When kept in a uniform magnetic field, the field lines around them look as follows:

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These materials are used to make magnets for elecric generators, transformer core and electromagnet core. Then it is proper to use : [Main 2016] (a) A for transformers and B for electric generators. (b) B for electromagnets and transformers. (c) A for electric generators and trasformers. (d) A for electromagnets and B for electric generators. A fighter plane of length 20 m, wing span (distance from tip of one wing to the tip of the other wing) of 15m and height 5m is lying towards east over Delhi. Its speed is 240 ms–1. The earth's magnetic field over Delhi is 5 × 10–5T with the declination angle ~0° and dip of q such that sin q 2 = . If the voltage developed is VB between the lower and 3 upper side of the plane and VW between the tips of the wings then VB and VW are close to: [Main Online April 10, 2016] (a) VB = 40 mV; VW = 135 mV with left side of pilot at higher voltage (b) VB = 45 mV; VW = 120 mV with right side of pilot at higher voltage (c) VB = 40 mV; VW = 135 mV with right side of pilot at higher voltage (d) VB = 45 mV; VW = 120 mV with left side of pilot at higher voltage

Make the correspondence of these bars with their material being diamagnetic (D), ferromagnetic (F) and paramagnetic (P): [Main Online April 11, 2014] (a) A « D, B « P, C « F (b) A « F, B « D, C « P (c) A « P, B « F, C « D (d) A « F, B « P, C « D 14. The mid points of two small magnetic dipoles of length d in end-on positions, are separated by a distance x, (x > > d). The force between them is proportional to x–n where n is: [Main Online April 9, 2014] N

S

N

S

x

(a) 1 (b) 2 (c) 3 (d) 4 15. Two short bar magnets of length 1 cm each have magnetic moments 1.20 Am2 and 1.00 Am2 respectively. They are placed on a horizontal table parallel to each other with their N poles pointing towards the South. They have a common magnetic equator and are separated by a distance of 20.0 cm. The value of the resultand horizontal magnetic induction at the mid-point O of the line joining their centres is close to (Horizontal component of earth.s magnetic induction is 3.6× 10.5Wb/m2) [Main 2013] (a) 3.6 × 10.5 Wb/m2 (b) 2.56 × 10.4 Wb/m2 (c) 3.50 × 10.4 Wb/m2 (d) 5.80 × 10.4 Wb/m2 16. The earth’s magnetic field lines resemble that of a dipole at the centre of the earth. If the magnetic moment of this dipole is close to 8 × 1022 Am2, the value of earth’s magnetic field near the equator is close to (radius of the earth = 6.4 × 106 m) [Main Online April 25, 2013] (a) 0.6 Gauss (b) 1.2 Gauss (c) 1.8 Gauss (d) 0.32 Gauss

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EBD_8341 P-216

Physics

Topic-3 : Magnetic Equipment 1 ҈


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WWW.JEEBOOKS.IN

P-217

Magnetism and Matter

Answer Key Topic-1 : Magnetism, Gaus's Law, Magnetic Moment, Properties of Magnet ҈
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WWW.JEEBOOKS.IN

EBD_8341 P-218

Physics

Electromagnetic Induction

20

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WWW.JEEBOOKS.IN

P-219

Electromagnetic Induction





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桌

WWW.JEEBOOKS.IN

EBD_8341 P-220

Physics

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2

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Numeric Answer




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ᔼ桌 4

㡰

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 �
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ᨨ
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A
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ᆠ盐
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矄盐ᆠ
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R

␔ 

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㡰

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Fill in the Blanks




ᔼ
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᲼҈黨黨
㜠
҈
㝐├矄桌

WWW.JEEBOOKS.IN

P-221

Electromagnetic Induction

5
�







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ᆠᨨ�
ᨨ
鼀
犨�
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├
矄�ᨨ 泐
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
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
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桌 6


䰘


ᆠ㡰 

�矄
矄盐 泐
矄ᨨ

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T


ᆠ盐 

True / False

MCQs with One or More than One Correct Answer

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├
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ᨨᨨ

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矄鼀 矄㡰
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桌 㡰 
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ᆠ盐  ᆠ矄矄
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ᆠ盐  ᆠ矄矄
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矄泐
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10 Subjective Problems






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鼀

AB ᆠ
矄�
ᨨ

犨ᨨ

 �ᆠ├
鼀�ᆠ �矄
矄
泐

�矄 ᆠ l

꾀
矄�矄 ᆠ
R  


� 盐ᆠ ᆠ
L 
ᆠᨨ

ᆠ ᨨ
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�矄
矄
矄 ᨨ犨
�
  ��盐

꾀
ᨨ �
矄�� 
犨� ᆠ泐� �

ᆠᨨ 矄 
ᆠ盐  I

�矄
ᆠ
�
 
 
ᨨ  
�矄
 
 �ᆠ盐
ᨨ
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矄
矄 ᨨ犨



AB �矄


矄


�矄 ᆠ
x

ᨨ鼀
 
ᨨ �
犨�

꾀
t
㦨


�
�矄
鼀
ᨨ
矄�
ᨨ
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犨泐
ᨨ鼀
 
犨�

꾀 矄犨  
ᨨᨨ犨� �
⼄盐矄�ᨨ 矄
᲼




㜠
�
㝐├矄桌 㡰 � 

�ᨨ
鼀ᨨ �
�



d� df
 


犨 
�
�矄
  dt dt

� 4

 ᨨ盐�
矄�矄 ᆠ
R �矄
ᨨ盐 
ᨨ

1

�鼀

T

�   
盐
ᨨ


L
�
鼀矄
ᨨ
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ᨨ 
��
⼄盐 ��矄
R


黨
꾀
ᆠ �盐
鼀
ABCD

鼀
ᨨ




盐 � ᨨ鼀
鼀
犨�

矄

矄��  ᆠᨨ

ᆠ�ᨨ
犨
E  
F 鼀
ᨨ  
矄鼀
犨�

矄
矄 ᨨ犨
�
��


AEFD �矄

矄⼄盐
ᨨ
矄�
҈鼀

 
EB 㦨
FC 㦨

�鼀


 �
ᆠ�ᆠ盐�
�矄
ᆠ
�

矄�泐
� ᆠ矄� �

盐 � ᨨ鼀
鼀� �ᆠ �
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ᨨ
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鼀� �ᆠ
�
�矄
҈ ⏌矄

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
盐 �
 �
ᨨ
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犨�
�矄
҈W ⏌ 鼀

� 
  鼀� �盐矄
 
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ᨨ
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ᆠ盐 矄
�
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矄�鼀 矄 AE, BE  
EF
᲼҈黨黨㜠�
㝐├矄桌 


� �ᆠ
ᆠᨨ矄
ᆠᨨ�矄
A

B  
C 
ᆠ
犨�  �
 矄

ᨨ
ᨨ 
 ᨨ 

ᔴᨨ�矄
A  
C ᆠ泐 ⼄盐
ᆠ盐 矄
矄
矄 ᨨ犨
�
��

ᔴᨨ�矄
B  
C 
� 
�

ᨨ矄��ᨨ
 
ᆠᨨ�

A

�矄
鼀ᨨ
ᨨ犨矄
B
犨�
盐 � ᨨ鼀 鼀ᨨ�ᨨ

ᔼ矄
 
 泐
� 盐ᆠ
ᆠ盐 
�
B 䁸
ᔼ
ᨨ

�� 矄ᨨ 矄

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泐矄
鼀├
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�ᆠ�ᨨ
ᨨ
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� 盐ᆠ
ᆠ盐 
�

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��鼀
᲼҈黨


㜠


㝐├矄桌

҈
꾀
ᆠ盐 
ᨨ鼀
A ᨨ
B �矄
� ᆠ矄� �
�
鼀� �盐

␔ 
�矄  
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ᨨ
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ᆠ盐 

�
 泐

�
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ᨨᨨ
矄 矄 ᨨ犨
�
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ᆠ盐 
�
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ᆠ�ᆠ盐�
 
f
�矄
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盐
ᨨ
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鼀� �ᆠ �
盐
ᨨ
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ᨨ �
犨�
 ᨨ盐�
 
ᆠ�ᆠ盐�
㡰 ᔼ
�矄
ᨨ矄
 

�鼀
t
㦨
T

 
鼀

AB �矄
 
�矄 ᆠ
ᨨ

x

ᨨ鼀
 
ᨨ �
犨�
 
 
矄�矄 ᆠ R
ᆠ�矄

ᆠ盐 
�҈

졠�

 矄矄�ᨨ
ᨨ
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 ᆠ �
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矄
ᨨ犨
 ᨨ盐�
矄�矄 ᆠ
R ᨨ鼀
t
㦨

ᨨ
t
㦨
T


Topic-2 : Motional and Static EMI and Application of EMI 1 ҈


MCQs with One Correct Answer

꾀
� � �泐
ᨨ �
矄�� 
犨�
ᆠ泐� �
ᆠ盐 
I

ᨨ 
矄� ᨨ 
ᆠ �盐
ᨨᨨ
 

ᆠᨨ 盐ᆠᨨ
C
犨�

矄�� � ᆠᨨ

ᆠᨨ

ᨨᆠ
�
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矄鼀
 

矄
矄 ᨨ犨
�
  ��盐


ᆠᨨ

ᆠᨨ
矄
 �
l
 
矄�矄 ᆠ
R

ᔼ

矄�矄
ᨨ
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�� 
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
ᨨᆠ�泐
v


矄�矄 ᆠ
ᨨ
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 
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矄
� 盐ᆠ ᆠ
ᨨ
 
ᨨᨨ


����


� 盐ᆠ
ᆠ盐 
�
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ᨨᨨ

矄

盐 ᆠ�ᨨ

ᨨ
矄�ᨨ
r

犨
 
ᆠᨨ

ᆠᨨ
 
 
矄��  犨�
�矄

᲼㝐�
和


�






ᔼᔼ㡰桌

WWW.JEEBOOKS.IN

EBD_8341 P-222

Physics

ᨨ 
矄�
ᨨ 
ᨨ � ᆠᨨ 盐ᆠ� �
犨�
ᨨᨨ

C I

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e
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R

v

l

r


m
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Ivl 㡰 ᆠ㡰 㡰
p Rr
p Rr p Rr p Rr 꾀
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桌

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


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l
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rx㡰
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r

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WWW.JEEBOOKS.IN

P-223

Electromagnetic Induction

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㝐├矄桌

WWW.JEEBOOKS.IN

EBD_8341 P-224

Physics

5




True / False

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㝐├矄桌 㡰 犨⏌鼀 㡰 ᨨ㜠矄ᆠᨨ ⏌鼀 ᆠ㡰 ᨨ盐⏌鼀㡰
㡰 ᨨ 鼀㜠矄ᆠᨨ 

WWW.JEEBOOKS.IN

P-225

Electromagnetic Induction


�


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㝐├矄桌

WWW.JEEBOOKS.IN

EBD_8341 P-226

҈


Physics

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 
ᨨᨨ
矄ᨨ
矄
ᨨ


矄泐
ᆠ盐 
ᨨ
҈
鼀��鼀
�

 

ᨨᨨ
䁸

෌�
 
�ᆠ�ᨨ
ᨨ
ᆠ盐 
�
 
ᨨᨨ
᲼҈黨

㜠


㝐├矄桌 


犨ᨨ
�矄
ᨨ

�犨泐
ᆠ├

� 矄盐
ᨨ鼀
ᆠ
ᨨ   
 
�ᨨ盐 


ᆠᨨ

ᆠ
ᨨ

鼀��
ᨨ鼀

␔ 
�矄  
� �
ᨨ
 
鼀��
ᨨ鼀
犨 

�
矄

 矄
ᨨ
҈


├鼀⏌ ᨨ盐
ᨨ �
 
ᆠ├

��
 
 
�ᆠ ᆠᨨ鼀ᨨ  
ᨨ
 矄
鼀� �ᆠ
�
�矄




맔
҈


犨⏌鼀


 
�矄

矄
泐
҈
鼀䁸

᲼҈黨
҈㜠


㝐├矄桌

Topic-3 : Miscellaneous (Mixed Concepts) Problems 1 ҈









MCQs with One Correct Answer

꾀
� � �泐
ᨨ �
ᆠ泐� 
�矄
├

ᨨ

盐 � ᨨ鼀 鼀� �ᆠ
�
B �ᆠ
ᨨ �
ᨨ矄��
㜠 �矄

 �ᆠ�ᨨ
ᨨ
� 盐ᆠ
ᆠ盐 
矄
矄
ᨨ鼀
 
㜠 �矄
犨� 
᲼


�和桌 㡰 ᨨ 㡰  �ᆠᨨᆠ├犨�矄
ᨨ
 
졔

 �矄 ᆠ㡰 ᆠᨨᆠ├犨�矄
ᨨ
 
졔

 �矄 㡰 ᨨ �
 
鼀� �ᆠ
� 꾀
矄 ᨨ㜠ᆠ�ᆠ盐�
ᆠᨨ�
�矄
ᆠ
�

�鼀㜠泐� �
鼀� �ᆠ �

쟼ᆠ�ᆠ
ᨨ犨
�矄
�矄矄�
盐
ᨨ
 
ᆠ盐  � 盐ᆠ
�
 
ᆠᨨ�

ᔼ
 
盐鼀
ᨨ
盐 矄
犨
ᨨ
 ⼄盐盐
 
 
犨�
�盐矄


 
ᆠ�ᆠ
ᨨ犨 �矄矄�
犨ᨨ盐

᲼



和桌 㡰  㡰  
矄鼀 ᆠ㡰 ᨨ盐 㡰 ⼄盐盐 꾀矄
矄 ᨨ犨
�
 
��盐

P  
Q 
犨ᨨ
ᆠᨨ �
ᆠᨨ 盐ᆠ� � ᨨᨨ矄
矄
泐
矄ᨨ鼀
�矄 ᆠ

␔ 
 
矄犨�ᆠ
S
�矄 ᆠᨨ矄


ᆠᨨᆠ├犨�矄
ᆠ盐 
IP ᨨ犨矄
�
P 矄
矄
泐
E㡰  

� 盐ᆠ
ᆠ盐 
IQ1
ᨨ犨矄
�
Q


矄犨�ᆠ
鼀� 矄 ᆠᨨ矄
ᨨ

ᨨ �
�鼀

␔ 
S
�矄
ᨨ 


ᆠ盐 
IQ
ᨨ犨矄
�
Q


 
�ᆠ�ᨨ
IQ҈
 
IQ

矄
矄
泐
E㡰 
᲼



和桌





�


㡰 矄ᆠ�泐
ᆠᨨᆠ├犨�矄
 
 �㜠ᆠᨨᆠ├犨�矄 㡰 ᨨ
ᆠᨨᆠ├犨�矄 ᆠ㡰 ᨨ
 �㜠ᆠᨨᆠ├犨�矄 㡰 矄ᆠ�泐
 �㜠ᆠᨨᆠ├犨�矄
 
ᆠᨨᆠ├犨�矄 꾀
ᆠᨨ�
ᨨ
犨�
� �
� 盐ᆠ ᆠ
 
矄�矄 ᆠ
矄
 ᆠᨨ 盐ᆠ� �
� �
ᆠ
ᆠᨨ �泐
犨� �
�


ᆠᨨ�
�矄 ᆠᨨ

ᆠ
ᨨ

泐

�鼀
t
㦨



矄ᨨ
 

�鼀㜠   ᆠ盐 
҈t㡰
矄矄
ᨨ犨� �
 ᨨ盐�
 
ᆠᨨ�

ᔼ
I
t㡰
�矄
  ᆠ盐 
� 盐ᆠ
�
 
� �

 
Bt㡰
�矄
 
鼀� �ᆠ
�
  
 �矄
ᨨ
 
ᆠᨨ�
盐
ᨨ
I҈t㡰

 
矄

盐 ᆠ�ᨨ
ᨨ
�鼀 t
>

㡰

 
ᨨ盐ᆠ
I
t㡰
Bt㡰
᲼



和桌 㡰 � ᆠ矄矄
犨�
�鼀 㡰 ᆠ矄矄
犨�
�鼀 ᆠ㡰 ᨨ矄
ᨨ
泐
犨�
�鼀 㡰 矄矄矄
 ᨨ盐�

鼀 �鼀盐鼀 꾀
盐 � ᨨ鼀
盐
�鼀㜠泐� �
鼀� �ᆠ
�
Bt㡰
 �矄矄
�
 ᆠ�ᆠ盐
��ᨨ
ᨨ
�盐矄

 
�矄
�ᆠ
� ᨨ
 
 
ᨨ  


矄
矄 ᨨ犨


鼀� �盐
ᨨ
 
� 盐ᆠ
ᆠ�ᆠ �

ᨨ� 



�矄 ᆠ

ᨨ鼀
 
ᆠ 
ᨨ
 
ᆠ�ᆠ盐 ��ᨨ


᲼



和桌 㡰 �矄
ᨨ 㡰 ᆠ矄矄
矄
҈⏌r ᆠ㡰 � ᆠ矄矄
矄
r 㡰 ᆠ矄矄
矄
҈⏌r






犨ᨨ
� �ᆠ
ᆠ�ᆠ盐
ᨨᨨ矄
ᨨ
鼀
犨�

泐� �
ᨨ
 
犨� ᨨ盐
ᨨ盐ᆠ � �
ᆠ
ᨨ 

ᨨᨨ㜠꾀
ᆠ�矄

ᆠ盐  犨 �ᆠ
� ᆠ矄矄
犨�
�鼀

ᔼ
矄ᨨ 矄

 
ᨨᨨ㜠 㡰 鼀� 矄
矄�ᨨ 泐 ᲼҈黨黨黨和
㜠


㝐├矄桌 㡰 �矄
ᆠ
泐
 
ᨨᨨ㜠꾀 ᆠ㡰 �矄

泐
 
ᨨᨨ㜠꾀 㡰 ᨨ矄
ᨨ盐
�矄
ᔴ㝐

犨�
ᔴ㝐
� 

WWW.JEEBOOKS.IN

P-227

Electromagnetic Induction

6 䰘


MCQs with One or More than One Correct Answer

꾀
矄ᨨ盐ᆠ
ᨨ
ᆠᨨ 矄 
ᨨ�

�矄
ᆠᨨ

ᆠ
ᨨ

矄�矄 ᆠ
 
犨ᨨ
�
� 盐ᆠᨨ矄
҈
 


 ᨨ盐�

矄犨�ᆠ
和
矄 矄 ᨨ犨


�矄
ᨨ
鼀盐盐
� 盐ᆠ ᆠ
犨
 
犨ᨨ � 盐ᆠᨨ矄


矄犨�ᆠ
和
�矄
� ��泐
ᨨ

꾀

㦨



 
矄犨�ᆠ �矄
ᆠᨨ矄
 
ᆠ盐 
�� 矄
ᨨ
ᨨ犨

␔ �ᆠ
ᨨ
 
ᨨᨨ犨� � ᨨ�ᨨ 矄
�矄⏌
ᆠᨨᆠ䁸 ᲼꾀



҈䰘桌 黨


和 졔  

҈




㡰
꾀 

ᨨ �
�鼀

 
ᆠ盐 
 ᨨ盐�
҈
犨�
  
҈ + 


㡰
꾀 

ᨨ �
�鼀

 
ᆠ盐 
 ᨨ盐�


犨�
  ҈ ҈ + 


ᆠ㡰

�ᨨ
ᨨ
 
ᆠ盐 矄
 ᨨ盐�
҈
 


�矄
� 
 
�鼀矄

>

㡰  꾀
ᆠ�ᆠ盐
� 矄盐
ᆠᨨ
犨�
ᨨᨨ
�矄
犨�矄
ᨨ
ᨨ鼀 犨ᨨ
ᨨᨨ矄
ᨨ

꾀
 

꾀
矄
矄 ᨨ犨
�
 
��盐

꾀
  ᨨ� 
ᨨ
ᆠᨨ矄矄� �
 
犨�矄
鼀�
ᆠ�ᆠ泐
� 矄盐 ᨨ鼀
ᆠ
ᨨ 


 �
ᨨᨨ
�矄
�
 
 
ᨨ
 

㡰
꾀

㦨



 
ᆠ盐 
 ᨨ盐�
 
矄�矄 ᆠ

�矄




®

㡰

꾀
盐 � ᨨ鼀
鼀� �ᆠ
�

ᨨ� 矄
� ᨨ
 
  ᨨ
 


꾀

㦨



 
ᨨᨨ
矄矄
ᨨ� �
ᨨ盐
  ᆠᨨ鼀鼀ᨨ
�鼀
矄
 �矄
犨�

ᆠᨨ 矄 
 �盐
ᨨᆠ�泐 w
�
 
鼀� �ᆠ
�

␔ �ᆠ
ᨨ
 
ᨨᨨ犨� �
ᨨ�ᨨ 矄
�矄⏌ 
ᆠᨨᆠ䁸 ᲼꾀



҈䰘桌

맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔

맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔

맔 맔 맔 맔 맔  맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔
꾀 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 

꾀 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 w

맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔

㡰 
鼀
� 盐ᆠ
�
 
ᨨᨨ
�矄
ᨨᨨ�ᨨ 
ᨨ
  矄盐鼀
ᨨ
 
矄
ᨨ
 
犨ᨨ
ᨨᨨ矄

㡰 
鼀�盐
ᨨ
 
鼀 �鼀盐鼀

鼀
� 盐ᆠ
盐
ᨨ ᨨ
 
ᨨᨨ矄
�矄
⼄盐
ᨨ
 
鼀�盐
ᨨ
鼀 �鼀盐鼀 鼀
� 盐ᆠ
�
 
矄鼀
ᨨᨨ
ᨨ  ᆠ㡰 

鼀
� 盐ᆠ
盐
ᨨ
ᨨ
 
ᨨᨨ矄
�矄 ᨨᨨ�ᨨ 
ᨨ
ᆠᨨ矄
w 㡰 

ᨨ
ᆠ  �
ᨨ
 
盐
�矄
鼀 �鼀盐鼀
犨 
   
ᨨ
 
ᨨᨨ矄
�矄
 �ᆠ盐
ᨨ
 
ᨨ
   꾀
�鼀
t 㦨



鼀� 
꾀 B D �
 
ᆠ�ᆠ盐�
矄 ᨨ犨
�
  A ��盐
�矄
ᆠᨨ

ᆠ
ᨨ
B
泐 
├泐
 

 � � �

V C
㦨



m ᆠ盐 
It㡰
㦨
I
ᆠᨨ矄
w㡰
犨�
I

㦨
҈
꾀
 
w
㦨
�


矄҈
矄矄
ᨨ犨� �
�

R
㦨
҈

W

䰘p

w  
├泐
�矄
矄犨�ᆠ 
ᨨ鼀
B ᨨ
D

ᨨ犨
ᨨ 犨矄
ᨨ 泐
A
  D

ᆠᨨ

ᆠ

꾀
ᨨ
ᆠ �
Q ᨨ犨矄
ᨨ鼀
 
泐 ᨨ
ᆠ �
 
ᆠᆠ�ᨨ
盐泐

ᔼ
C 㦨



m


㦨
҈

W
 
  泐
�矄
�
犨�
鼀
ᨨ
�



� � 泐
 
ᆠᨨᆠ 矄鼀 矄㡰
᲼꾀



҈
桌 㡰 㝐� �盐
ᨨ
 
鼀 �鼀盐鼀
ᆠ �
ᨨ
 
ᆠᆠ�ᨨ 䰘p  ᨨ
t =
�矄
҈
맔
҈

ᔴ
w 㡰 
ᆠ盐 
�
 
 

ᨨ
 
ᆠ�ᆠ盐�
盐矄
 ᨨ 䰘p t=
�矄
ᆠᨨᆠ├犨�矄
w ᆠ㡰 ᔼ鼀鼀�泐
 
A
�矄
ᆠᨨ

ᆠ
ᨨ
D

 
ᆠ盐 
�
R �矄
҈

A 㡰 Q
㦨


맔
҈

ᔴ ҈

L

C
 
R 矄 
 
 泐矄�ᆠ
⼄盐 ��矄

� 盐ᆠ ᆠ
ᆠᆠ� ᆠ
 
矄�矄 ᆠ
矄ᆠ�泐


ᆠᨨ鼀� �ᨨ 矄㡰 犨 �ᆠ

 
�鼀 矄�ᨨ 矄
ᨨ
⼄盐 ᆠ泐
 ᲼҈黨

㜠


㝐├矄桌 㡰 ҈⏌RC 㡰 R/L ᆠ㡰 ҈⏌ LC 㡰 C/L �
犨�
 
� ��
�ᆠ�ᨨ
矄 ᨨ犨
�
 
��盐

꾀
t =

8

Comprehension/Passage Based Questions 矄矄�

꾀
ᨨ� 
ᆠ �
턼
�矄
鼀ᨨ� �
�

ᆠ�ᆠ盐
ᨨ�
ᨨ
�盐矄

�
  㜠泐
 
犨�

 �盐
ᨨᆠ�泐
w

�矄
ᆠ

ᆠᨨ 矄�
矄 턼w ⼄盐� 
ᨨ

ᨨᨨ
ᆠ泐� �

矄泐
ᆠ盐 


꾀
盐 � ᨨ鼀
p 鼀� �ᆠ
�
ᨨ �
 
ᨨ矄��
㜠 �矄
�矄
ᨨ犨
矄犨�ᆠ 
ᨨ
犨 �ᆠ
� ᆠ矄矄


ᆠᨨ 矄 

ᨨ鼀


ᨨ

�
ᨨ 
矄ᆠᨨ 
꾀矄矄盐鼀
 
 
�盐矄
ᨨ
 
ᨨ�
鼀� 矄
ᆠᨨ 矄 

 �ᆠ�ᨨ
ᨨ
 
鼀� �ᆠ
�
� 盐ᆠ矄

鼀
�
 
ᨨ�

 � 盐ᆠ
鼀
�矄
 � 
矄
 
犨ᨨ├
ᨨ 
泐

� 盐ᆠ
ᆠ�ᆠ �
�
鼀ᨨ� �

盐 �
ᨨ矄��
ᆠ �
ᨨ盐 

ᆠᨨ矄
ᨨᨨ

ᔼ
�矄 ├ ᨨ犨
 

ᨨ

ᨨ�� �
ᆠ �

 
鼀� �ᆠ
�ᨨ
鼀ᨨ鼀  �矄
ᨨᨨ�ᨨ 
ᨨ
 
 �盐
鼀ᨨ鼀 盐鼀
犨�

ᨨᨨ�ᨨ �泐 ᆠᨨ 矄 
g
᲼
꾀



҈桌

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EBD_8341 P-228

҈҈


Physics


鼀� �盐
ᨨ
 
� 盐ᆠ
ᆠ�ᆠ
�
�
 
ᨨ�
  泐
� 矄 
ᨨ
�鼀
盐� �
 
�鼀
� 
ᨨ
 
鼀� �ᆠ �
ᆠ  �
�矄   㡰 ᆠ㡰  㡰





ᆠ  �
�
 
鼀� �ᆠ
�ᨨ
鼀ᨨ鼀 
矄矄ᨨᆠ� 犨�
 
ᨨ�


 
 
ᨨ
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�鼀
� 
ᨨ
 
鼀� �ᆠ �
ᆠ  �

�矄

㡰

҈



㡰 g턼
ᆠ㡰 g 9 ҈


턼



턼



㡰

-g

㡰

g 턼


Assertion and Reason Type Questions

和鼀 㜠҈


꾀
�ᆠ
�ᨨ
ᨨ
矄
ᆠᨨ�
ᨨ




犨�
犨ᨨ盐 
ᨨ
�

 
ᨨᨨ鼀
 

꾀

 � �
ᆠ盐 
ᨨ犨矄
�
 
ᆠᨨ�

 ᨨ
�ᨨ矄
 ᨨ盐�

ᆠᨨ 盐ᆠ� �
� �
矄 矄 ᨨ犨
�
 
��盐


� �
ᆠ
ᨨ

 ᆠ�
�� 
ᨨ
 
ᆠᨨ�
和鼀 㜠



ᔼ
 
ᨨ
矄�盐�ᨨ

 ᆠ盐 
�矄
� 盐ᆠ
�
 
� �
犨 �ᆠ
� ᆠ矄
犨�
  ᨨ�ᨨ 
ᆠᨨ鼀ᨨ  
ᨨ
 
鼀� �ᆠ
�
ᨨ
ᨨ盐ᆠ


�
ᨨᆠ
�
 
盐犨
�ᆠ�ᨨ
᲼


䰘桌 㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
盐錐
和鼀 㜠
�矄

ᆠᨨᆠ
  �ᨨ
ᨨ
和鼀 㜠҈ 㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
盐錐
和鼀 㜠
�矄
졠

ᆠᨨᆠ
  �ᨨ
ᨨ
和鼀 㜠҈ ᆠ㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
矄 㡰 和鼀 㜠҈
�矄
矄

和鼀 㜠

�矄
盐
10 Subjective Problems

҈



꾀
ᨨ �
矄ᨨ ᨨ�
ᨨ
�盐矄

 
盐鼀
ᨨ
盐 矄

盐 �  �

�矄
 ᆠᨨ矄
泐
ᆠ泐� �ᆠ
矄 
ᨨ
�盐矄
R
 �ᆠ├ 矄矄
d d



R
㡰
 
 �
L

A
�
ᆠ盐 
� 㦨
�
矄� w
ᨨ犨矄
 ᨨ盐�
 
ᆠᨨ�

ᔼ
 
矄�矄��泐
ᨨ
 
鼀� ᨨ
ᆠ泐� �ᆠ
矄 
�矄
r

� 
 
� 盐ᆠ
ᆠ盐 
�
  矄 

᲼


�
㜠



㝐├矄桌

҈�


꾀
矄⼄盐
ᨨᨨ
ᨨ
矄�
�a
犨�

ᆠᆠ�ᨨ
ᨨ
ᆠᆠ� ᆠ
C �矄
ᨨᆠ
犨
犨ᨨ
ᆠ盐 
ᆠ泐� �
ᨨ �

犨�矄 矄
矄 ᨨ犨


盐
ᨨ
I
�
 
犨�矄
�矄
��
矄
I
㦨
I
矄� wt
᲼



㜠


㝐├矄桌

㡰 ᔴᆠ盐
鼀 �鼀盐鼀
ᆠ盐 
�
 
矄⼄盐
ᨨᨨ
㡰 犨

�
犨
ᆠ �矄
ᨨ
 
盐

ᨨ  
ᆠᆠ�ᨨ
矄
�鼀
҈

꾀
ᆠ �盐
ᨨᨨ
PQRS 鼀
ᨨ鼀

盐 � ᨨ鼀
犨�
矄  �
a

犨�
b  
鼀矄矄
鼀

ᔼ
�矄

ᨨ
ᨨ
ᨨ盐
  鼀
PQ

犨 �ᆠ
鼀� 矄
� �
ᨨ �

ᨨ�ᨨ 
� 
├

矄
 
泐㜠 �矄
矄
��盐㡰

├
 
�ᆠ泐
盐犨 �ᆠ�ᨨ
矄
 
㜠 �矄

꾀
盐 � ᨨ鼀
鼀� �ᆠ
� ur B = �ˆ +
kˆ㡰 B
 �矄矄
�
 
��ᨨ


ᨨᨨ
�矄

�




 
㜠泐
 
 

ᆠ盐 
ᔼ
�矄
矄矄
 ᨨ盐�
�

 ᨨᨨ
�矄
ᨨ犨
矄
 
�矄
ᨨ盐 
ᨨ
矄泐
�
 
ᨨ�ᨨ  ᨨ矄��ᨨ
�
⼄盐���盐鼀
᲼




㜠
�
㝐├矄桌 z

P

Q y

a x

S

R b 㡰 ␔ 
�矄
 
�ᆠ�ᨨ
ᨨ
 
ᆠ盐 
I �
PQ䁸 㡰 � 
 
鼀� �ᆠ
ᨨᆠ
ᨨ
 
鼀
RS
ᆠ㡰 � 
 
 矄矄�ᨨ
ᨨ
I
�
鼀矄
ᨨ
B


a

b
 
m
҈䰘
犨ᨨ
ᨨ �

ᨨ�ᨨ 
�矄


�矄 ᆠ
d

  ᆠ
� �

矄�矄 ᆠ
l

盐 �
 �


ᨨ� 
 ᨨ 
 
泐

矄�矄 ᆠ
R

꾀
 ᆠ泐
ᆠᨨ 盐ᆠ� �
ᨨ
MN ᨨ
鼀矄矄
鼀
�矄

ᨨ
矄�
ᨨ �
 
�矄
犨� ᨨ盐
�ᆠ�ᨨ

矄
��盐㡰


�矄

盐 � ᨨ鼀
鼀� �ᆠ
�
ᨨ
� 盐ᆠ�ᨨ

B
ᨨ鼀
ᨨ
 
 
ᨨ
 

 
�ᆠ
� ᨨ
  

꾀
�
ᨨᆠ
F
�矄
�
ᨨ
 
ᨨ
MN 矄盐ᆠ
 
矄
 
ᨨ
鼀ᨨ矄


ᆠᨨ 矄 
ᆠ盐 
ᨨ犨矄
 ᨨ盐�
R
᲼҈黨


㜠


㝐├矄桌

�㡰 � 
 
ᨨᆠ�泐
ᨨ
 
ᨨ
 
 
�
ᨨᆠ
F 矄 盐 ᆠ�ᨨ
ᨨ
 
�矄 ᆠ
x ᨨ
 
ᨨ
ᨨ鼀
R
��㡰 ␔ 
ᆠ�ᨨ
ᨨ
 
犨ᨨ├
ᨨ 

矄ᆠᨨ 
泐
F �矄 ᆠᨨ 
� ᨨ

䁸

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P-229

Electromagnetic Induction

Answer Key Topic-1 : Magnetic Flux, Faraday's and Lenz's Law ҈
㡰



㡰


㡰



㡰

�
㡰



㡰

䰘
㡰



㡰

黨
㡰

҈҈
㡰

҈

㡰

҈
㡰

҈

㡰

҈�
㡰

҈

㡰

҈䰘
ᆠ㡰

҈

㡰

҈黨

㡰


҈
҈�㡰 ҈





҈
㡰



҈䰘�


㡰


 
ᨨ
�� 


䰘


ᆠ㡰




㡰


�
矄


盐

҈

ᨨ 盐矄㡰


�㡰

Topic-2 : Motional and Static EMI and Application of EMI ҈
㡰 ҈҈
㡰


盐



㡰


㡰

҈

㡰 ҈
㡰
҈




㡰




㡰



㡰

�
ᆠ㡰



㡰

҈

㡰 ҈�
㡰 ҈

㡰



㡰




ᆠ
㡰
�

ᆠ㡰

䰘
ᆠ㡰



ᆠ㡰

҈䰘
㡰

҈

㡰

黨
㡰

҈

㡰

҈黨



맔
҈


矄

҈

꾀

Topic-3 : Miscellaneous (Mixed Concepts) Problems ҈
㡰



㡰


㡰

҈҈
㡰

҈

㡰

҈
㡰



㡰

�
㡰



ᆠ㡰

䰘




ᆠ㡰



㡰

黨
ᆠ

㡰 ҈



ᆠ㡰

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EBD_8341 P-230

Physics

Alternating Current

21

Topic-1 : Alternating Current, Voltage and Power 1 ҈






MCQs with One Correct Answer

꾀
 � �
ᨨ�
vt㡰
㦨




矄�
҈

Àt
ᨨ
�矄
� ᨨ

盐泐
矄�矄�
ᨨ
ᨨ
�
W


�鼀
├
ᨨ
  ᆠ盐 
ᨨ
�矄
ᨨ鼀

ᨨ
 
├
盐
ᨨ
 
├ 盐
�矄

᲼㝐�


꾀�


҈黨
ᔼ桌 㡰 �
鼀矄 㡰



鼀矄 ᆠ㡰 䰘


鼀矄 㡰 

鼀矄 꾀
矄鼀
ᆠ�ᆠ盐
ᨨᨨ
ᨨ
犨�
ᨨ
�盐矄
a
�矄
ᨨᆠ

  ᆠ 
ᨨ

鼀盐ᆠ
�
ᆠ�ᆠ盐
犨�
ᨨᨨ
ᨨ
�盐矄
b

 犨ᨨ
ᨨᨨ矄

�
 
矄鼀
 


ᨨ盐
ᨨᨨ
ᨨ
�盐矄 b
ᆠ�矄

 � �
ᆠ盐 
ᔼ
㦨
ᔼᨨ
ᆠᨨ矄
w㡰


鼀 � 盐ᆠ
�
 
矄鼀
�


ᨨᨨ
�矄
泐

᲼㝐�
졠 � 
꾀�





҈䰘桌

㡰


pm
ᔼᨨ 

w 矄� w㡰 㡰 pm
ᔼᨨ
 w ᆠᨨ矄 w㡰



ᆠ㡰 pm
ᔼᨨ 



w 矄� w㡰 

㡰

pm
ᔼ ᨨ 
w ᆠᨨ矄  w㡰 

꾀
矄� 盐矄ᨨ�
ᨨ�
㡰
㦨
҈


矄�
�

㡰
�矄
�
ᆠᨨ矄矄 
盐
� 盐ᆠ ᆠ
ᨨ

㦨








ᆠ盐 
 ᨨ盐�
  ᆠᨨ�
�矄
᲼㝐�
졠 � 
꾀�
҈




҈
桌 㡰 ҈

ᆠᨨ矄
�


㡰 㡰 
҈

ᆠᨨ矄
�

㡰 ᆠ㡰 ҈

矄�
�

㡰 㡰 
҈

矄�
�

㡰

Topic-2 : AC Circuit, LCR Circuit, Quality and Power Factor 1 ҈






꾀
꾀ᔴ
ᆠ�ᆠ盐�
矄
R
㦨
҈


W

C
㦨


m
 
L
㦨



鼀
ᆠᨨ

ᆠ
�
矄�矄


⼄盐�泐
ᆠᨨ
ᨨ
 
ᆠ�ᆠ盐�
�矄

᲼㝐�
和










ᔼ㡰桌 㡰
㡰

� ᆠ㡰

㡰


꾀
矄�矄
L㜠R
ᆠ�ᆠ盐�
�矄
ᆠᨨ

ᆠ
ᨨ

泐
ᨨ
鼀
V

ᔼ  
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P-231

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EBD_8341 P-232

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P-233

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WWW.JEEBOOKS.IN

EBD_8341 P-234

Physics

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P-235

Alternating Current

10 Subjective Problems 



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Topic-3 : Transformers and LC Oscillations 1 ҈


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EBD_8341 P-236

3.

Physics

A power transmission line feeds input power at 2300 V to a step down transformer with its primary windings having 4000 turns. The output power is delivered at 230 V by the transformer. If the current in the primary of the transformer is 5A and its efficiency is 90奠, the output current would be: [Main 9 Jan. 2019 II] (a) 50 A (b) 45 A (c) 35 A (d) 25 A

A power transmission line feeds input power at 2300 V to a step down transformer with its primary windings having 4000 turns, giving the output power at 230 V. If the current in the primary of the transformer is 5 A, and its efficiency is 90奠, the output current would be: [Main Online April 16, 2018] (d) 25 A (a) 20 A (b) 40 A (c) 45 A

4.

Topic-4 : Miscellaneous (Mixed Concepts) Problems 6 1.

A series R– C circuit is connected to AC voltage source. Consider two cases; (A) when C is without a dielectric medium and (B) when C is filled with dielectric of constant 4. The current IR through the resistor and voltage VC across the capacitor are compared in the two cases. Which of the following is/are true? [2011] (a) 7

2.

MCQs with One or More than One Correct Answer

I RA > I RB

(b)

I RA < I RB

VCA > VCB

(c)

(d)

VCA < VCB

Match the Following

You are given many resistances, capacitors and inductors. These are connected to a variable DC voltage source (the first two circuits) or an AC voltage source of 50 Hz frequency (the next three circuits) in different ways as shown in Column II. When a current I (steady state for DC or rms for AC) flows through the circuit, the corresponding voltage V1and V2 , (indicated in circuits) are related as shown in Column I. Match the two [2010] Column I Column II V1

(A) I ¹ 0, V1 is proportional to I

(p)

V2 3m F

6mH

V V1

V2

6mH

(B) I ¹ 0, V2 > V1

2W

(q) V

(C) V1 = 0, V2 = V

V1

V2

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(r) V

V1

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(s)

V2

6mH

3m F

V1

V V 2

3m F

1kW

(t) V

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P-237

Alternating Current

Comprehension/Passage Based Questions Passage 1 In the given circuit the capacitor (C) may be charged through resistance R by a battery V by closing switch S1. Also when S1 is opened and S2 is closed the capacitor is connected in series with inductor (L). 8

5.

Given that the total charge stored in the LC circuit is Q0, for t ³ 0, the charge on the capacitor is [2006 – 5M, –2] t ö æp (a) Q = Q0 cos ç + ÷ è2 LC ø

(c) Q = - LC

At the start, the capacitor was uncharged. When switch S1 is closed and S2 is kept open, the time constant of this circuit is t. Which of the following is correct [2006 – 5M, –2] CV (a) after time interval t, charge on the capacitor is 2 (b) after time interval 2t, charge on the capacitor of CV (1 – e–2) (c) the work done by the voltage source will be half of the heat dissipated when the capacitor is fully charged (d) after time interval 2t, charge on the capacitor is CV (1 – e–1) When the capacitor gets charged completely, S1 is opened and S2 is closed. Then, [2006 – 5M, –2] (a) at t = 0, energy stored in the circuit is purely in the form of magnetic energy (b) at any time t > 0, current in the circuit is in the same direction (c) at t > 0, there is no exchange of energy between the inductor and capacitor (d) at any time t > 0, instantaneous current in the circuit

3.

4.

may be V

C L

d 2Q

t ö æp (b) Q = Q0 cos ç ÷ è2 LC ø

(d) Q = -

dt 2

1 d 2Q LC dt 2

Passage 2 A thermal power plant produces electric power of 600 kW at 4000 V, which is to be transported to a place 20 km away from the power plant for consumers' usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method , a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers' end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with power factor unity. All the currents and voltages mentioned are rms values. [Adv. 2013] 6. If the direct transmission method with a cable of resistance 0.4 W km–1 is used, the power dissipation| (in 奠) during transmission is (a) 20 (b) 30 (c) 40 (d) 50 7. In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is 1 : 10. If the power to the consumers has to be supplied at 200 V, the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is (a) 200 : 1 (b) 150 : 1 (c) 100 : 1 (d) 50 : 1

Answer Key Topic-1 : Alternating Current, Voltage and Power 1.

(d)

2. (a)

3. (b) Topic-2 : AC Circuit, LCR Circuit, Quality and Power Factor

1. (a) 11. (a) 21. (c) 31.

(400)

39.

3.465 s

2. (a) 12. (d) 22. (c)

3. (d) 13. (a) 23. (c)

4. (a) 14. (b) 24. (d)

32. (55)

33. (b, d)

5. (b) 15. (a) 25. (d)

6. (c) 7. (d) 16. (b) 17. (b) 26. (b) 27. (b) 4 (r - r ) 34. (b, c) 35. (a,c) 36. 3 1 2

8. (Bonus) 9. 18. (b) 19. 28. (a) 29. p 37. 20 A, 38. 4

(b) (c) (8)

10. (a) 20. (c) 30. (33)

12e–5t V

Topic-3 : Transformers and LC Oscillation 1.

(a)

2. (c)

3. (b)

4. (c)

Topic-4 : Miscellaneous (Mixed Concepts)Problems 1.

(a,c)

2. A-r,s,t; B-q,r,s,t; C-p,q; D-q,r,s,t

3. (b)

4. (d)

5. (c)

6. (a)

7. (b)

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EBD_8341 P-238

Physics

Electromagnetic Waves

22

Topic-1 : Electromagnetic Waves, Conduction and Displacement Current 1 ҈


MCQs with One Correct Answer

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ᆠᨨ矄ᨨ � �
ᆠ�ᆠ
�
�矄

᲼㝐�
和









ᔼ㡰桌 ur 㡰 E = 黨矄� ᲼


p  y + ct 㡰桌kˆ
⏌鼀 ur 㡰 E = -҈
-
矄� ᲼


p  y + ct 㡰桌kˆ
⏌鼀 ur ᆠ㡰 E =  ´ ҈

矄� ᲼


p  y + ct 㡰桌kˆ
⏌鼀 ur 㡰 E = -黨矄� ᲼


p  y + ct 㡰桌kˆ
⏌鼀 �



ᆠ�ᆠ
�
ᨨ

 
ᆠᨨ鼀� �ᆠ
犨 ᨨ�� �
ᨨ �
 
x
�ᆠ�ᨨ
�
ᆠ盐盐鼀
�矄 r r E = E
ˆ ᆠᨨ矄wt - kx 㡰


鼀� �ᆠ
�
B


  鼀ᨨ鼀 
t
㦨


�矄

᲼㝐�
和









ᔼᔼ㡰桌 r r E
㡰 B = ᆠᨨ矄kx㡰kˆ

㡰

B = E
m
e
ᆠᨨ矄kx㡰 ˆ m
e


r r ᆠ㡰 B = E
m
e
ᆠᨨ矄kx㡰kˆ
㡰

B =



E
m
e


ᆠᨨ矄kx㡰 ˆ

꾀
 
ᆠᨨ鼀� �ᆠ
犨

矄
⼄盐 ᆠ泐
ᨨ




맔
҈
҈

 
�矄
 �泐
 矄�泐
�矄
҈



맔
҈


⏌鼀
�
ᆠ盐盐鼀



鼀�盐
ᨨ
 
鼀� �ᆠ
�
ᨨ
 
犨
�矄
ᆠᨨ矄
ᨨ

Nm
҈ = 黨 ´҈
黨

 
矄
ᨨ
�� 
㦨

맔
҈


鼀矄҈㡰


pe
C ᲼㝐�
和










ᔼ㡰桌 㡰 ҈�



㡰 ҈




ᆠ㡰 ҈




㡰 ҈黨





WWW.JEEBOOKS.IN

P-239

Electromagnetic Waves

䰘


ᔼ

 
ᆠᨨ鼀� �ᆠ
犨

 
�ᆠ�ᨨ 矄
ᨨ
ᆠ�ᆠ �
 
鼀� �ᆠ
�

矄 
泐
kˆ
 

�ˆ -
ˆ
矄ᆠ�泐

␔ 
�矄
 
盐 �
ᆠᨨ
ᨨ �
�ᆠ�ᨨ
ᨨ ᨨ��ᨨ
ᨨ
 
犨
᲼㝐�
和










ᔼᔼ㡰桌 㡰

҈ ˆ ˆ � +  㡰


㡰

҈


 ˆ + kˆ㡰

҈ ˆ ˆ ҈ ˆ 
� +  㡰 � +
ˆ 㡰 㡰 � � 꾀
 
ᆠᨨ鼀� �ᆠ
犨
�矄
ᨨ�� �
ᨨ �
  �ˆ + ˆ

犨�
�矄
ᨨ��ᨨ
ᨨ �
 
�ᆠ�ᨨ

�ᆠ�ᨨ

kˆ


ᆠᨨᆠ
ᨨ鼀
ᨨ
 
鼀� �ᆠ
�
ᨨ
 
犨 犨ᨨ盐

 


�矄

ᨨ�
ᆠᨨ 矄 㡰
᲼㝐�
黨







ᔼᔼ㡰桌 æ �ˆ - ˆ �ˆ + ˆ ö ᆠᨨ矄 ç wt - k 㡰 
è

÷ø ᆠ㡰





㡰 


�矄
v
kˆ

 
ᨨᆠ
ᆠ� �
ᨨ
�
盐
ᨨ
 
犨
�矄
᲼㝐�
䰘







ᔼᔼ㡰桌 �ˆ + ˆ 㡰 
ᨨ
㡰 ᨨ
�ˆ + ˆ ᆠ㡰  �
ᨨ
㡰 
ᨨ
kˆ
҈

꾀
ᆠᨨ鼀� �ᆠ
犨
�矄
矄 
泐
 
ᆠ�ᆠ ur �
E = E n$ 矄� ᲼w  + 
泐 -
㡰桌

├� �
盐 �
ᆠᨨ矄
�




x, y
 
z
�ᆠ�ᨨ 矄
ᨨ

$�
$
k$

 
�ᆠ�ᨨ
ᨨ

҈


æ �ˆ + ˆ ö ᆠᨨ矄 ç wt + k è

÷ø

ˆ - �ˆ

æ �ˆ + ˆ ö ˆ ᆠ㡰 
k ᆠᨨ矄 ç wt - k è
÷ø æ �ˆ + ˆ ö ᆠᨨ矄 ç wt - k è

÷ø 꾀
 
ᆠᨨ鼀� �ᆠ
犨
ᨨ
⼄盐 ᆠ泐

�
෌
�矄 ᨨ�� �
�
ᆠ盐盐鼀
ᨨ �
 
z㜠�ᆠ�ᨨ

꾀
 �ᆠ盐
ᨨ� 
�
矄ᆠ
 
�鼀

 
鼀� �ᆠ
�
�矄 r ��
泐 B = � ´ ҈
-
ˆ T



ᆠᨨ矄ᨨ � �
ᆠ�ᆠ r �
E
�矄
矄
ᨨ
�� 
c
㦨

´



鼀矄㡰 ᲼㝐�









ᔼᔼ㡰桌 㡰 ҈



´
҈
҈

�ˆ
⏌鼀 㡰 
҈



´
҈
҈


�ˆ
⏌鼀

㡰 


黨


҈



�ˆ + ˆ

҈



ᆠ㡰 ҈�
�ˆ
⏌鼀 㡰 ҈�
�ˆ
⏌鼀 ᔼ
 
鼀� �ᆠ
�
�

 
ᆠᨨ鼀� �ᆠ
犨
�矄 ur ��
泐
B
㦨

´
҈


矄�



´
҈
x
졔



´
҈
҈
㡰
ˆ

 

犨 
犨�

 矄矄�ᨨ
ᨨ
ᆠ�ᆠ
�䁸 ᲼㝐�
䰘







ᔼ㡰桌 ur  㡰 E
㦨



矄�
҈


´ 
x 졔



´ 
҈
㡰 kˆ ⏌鼀㡰 ur 㡰 E
㦨
黨
矄�
҈


´ 
x 졔



´ 
҈
㡰 kˆ ⏌鼀㡰 ur ᆠ㡰 E
㦨

´ 


矄�



´ 
x
졔



´ 
҈
㡰 kˆ ⏌鼀㡰 ur 㡰 E
㦨

´ 

矄�



´ 
x 졔



´ 
҈
㡰 kˆ ⏌鼀㡰 ҈҈

ᆠ�ᆠ
�
ᨨ

 
ᆠᨨ鼀� �ᆠ
犨
�矄 ��
泐 �ˆ + ˆ r E
㦨
E

ᆠᨨ矄kz + wt 㡰 꾀
t 㦨




ᨨ矄��泐
ᆠ �
�ᆠ
�矄

 
ᨨ� 

pö æ x

y

z㡰
㦨
çè



÷ø

ᔼ
�矄
� 矄  ᨨ盐矄
ᨨᆠ�泐

t 㦨

㡰 k

҈�


҈



ᨨᨨ��ᨨ
s$
�矄









᲼㝐�
҈

꾀�


҈黨

ᔼ㡰桌 $i -
$j -
k$ +  $j 㡰 s$ = 㡰 s$ = � � $j - k$ $ +
k$ ö æ   㡰 s$ = ᆠ㡰 s$ = ç ÷÷ ç � � è ø 꾀
 
ᆠᨨ鼀� �ᆠ
犨
� �

⼄盐 ᆠ泐 v
㦨


黨
෌
ᨨ�矄
ᨨ �
 
ᨨ矄��
z㜠�ᆠ�ᨨ
�


矄ᆠ


├
盐
ᨨ
 
쟼ᆠ�ᆠ
�
�矄



⏌鼀
␔ �ᆠ
鼀ᨨ �
 
ᨨᨨ犨� �
�矄
 
ᆠᆠ
鼀� �ᆠ �
ᆠᨨ鼀ᨨ  
�
 
ᆠᨨ鼀� �ᆠ
犨
䁸 ᲼㝐�
҈

꾀�


҈黨

ᔼᔼ㡰桌 ur 㡰 B =
´ ҈
䰘 矄� 

� ´ ҈
 z + ҈
� ´ ҈
҈҈ t 㡰$� ur 㡰 B =
´ ҈
-䰘 矄� 

� ´ ҈
 z - ҈
� ´ ҈
҈҈ t 㡰$� ur ᆠ㡰 B =

矄� 

� ´ ҈
 x + ҈
� ´ ҈
҈҈ t 㡰 k$ ur -䰘
҈҈ $ 㡰 B =
´ ҈
矄� ҈
� ´ ҈
x +

� ´ ҈
t 㡰 


ᆠ�ᆠ
�
ᨨ

 
ᆠᨨ鼀� �ᆠ
犨
�矄
��

ur 泐
쟼 = 쟼
$� ᆠᨨ矄├㡰 ᆠᨨ矄 w 㡰


ᆠᨨ矄ᨨ � �
鼀� �ᆠ
�
�矄
 
��
泐

᲼㝐�
҈

꾀�


҈黨

ᔼ㡰桌 ur 쟼
$ 矄� ├㡰 矄� w 㡰 㡰  = ᔴ ur 쟼
$ 矄� ├㡰 ᆠᨨ矄  w 㡰 㡰  = ᔴ ur 쟼
$ ᆠᨨ矄 ├㡰 矄� w 㡰 ᆠ㡰  = ᔴ ur 쟼
├$ 矄� ├㡰 ᆠᨨ矄 w 㡰 㡰  = ᔴ �� 
�矄
� ᆠ� 
ᨨ鼀泐
ᨨ

ᆠᨨ鼀泐
矄ᨨ� � 矄盐 ᆠ
犨�

 �泐
盐
ᨨ

�
␔ᆠ鼀


ᔼ
 
矄盐 ᆠ
矄 

ᨨ

�
ᆠ鼀


 
鼀ᨨ鼀 盐鼀
 矄 
ᨨ
 
矄盐 ᆠ �



鼀�
�鼀
盐�ᨨ
犨�


᲼㝐�
҈

꾀�


҈黨

ᔼᔼ桌 㡰

맔҈


矄 㡰 ҈

맔҈


矄 ᆠ㡰 �

맔҈

矄 㡰 
�맔҈


矄 꾀
 
ᆠᨨ鼀� �ᆠ
犨
ᨨ
⼄盐 ᆠ泐
�

㝐
矄 �

矄ᆠ
ᨨ �
 
ᨨ矄��
x㜠�ᆠ�ᨨ

꾀

�ᆠ盐 r ᨨ� 
�
矄ᆠ
 
�鼀

쟼 =

 ˆ  ⏌ 鼀

 r ᆠᨨ矄ᨨ � �
鼀� �ᆠ
�



 
ᨨ� 
犨�

᲼㝐�
黨
꾀�


҈黨
ᔼ桌 
㡰 ҈

黨
맔
҈


├

ˆ ˆ 㡰

҈
맔
҈


├



ˆ ˆ ᆠ㡰


맔
҈


├

㡰 ҈

黨
맔
҈

├

WWW.JEEBOOKS.IN

EBD_8341 P-240

҈䰘


҈



҈黨


Physics

�

␔⏌m

 �泐
 矄�泐
ᨨ
矄盐 �� 
�矄
ᨨ鼀泐
� ᆠ�  ᨨ
 
矄盐 ᆠ
ᨨ

矄ᨨ
 

和ᨨ鼀

ᨨ
� ᆠ�   �泐

�奠㡰
�矄
 ᆠ
ᨨ鼀
 
矄盐 ᆠ
 
 
矄
�矄 矄ᨨ


ᨨᆠ
 
ᨨ
҈鼀

矄盐 ᆠ

犨�
 ᆠᨨ矄
ᨨ
ᆠ
㦨

맔
҈


鼀⏌矄㡰




᲼㝐�
黨
꾀�


҈黨

ᔼᔼ桌 㡰 ҈�
맔
҈


 㡰


맔
҈


 ᆠ㡰 ҈

맔
҈


 㡰 �
맔
҈


 꾀
 
ᆠᨨ鼀� �ᆠ
犨
矄
�

矄ᆠ
ᨨ � the x㜠�ᆠ�ᨨ


ᆠ�ᆠ
�
ᆠᨨ鼀ᨨ  
ᨨ
 
犨 

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ᨨ� 
ᨨ
矄ᆠ
 
�鼀
�矄
쟼
㦨


鼀 ҈
ᨨ � y㜠�ᆠ�ᨨ

ᔼ矄
ᆠᨨ矄ᨨ � �
鼀� �ᆠ
�
ᆠᨨ鼀ᨨ  

犨ᨨ盐

᲼㝐�


꾀�


҈黨
ᔼ桌 㡰

맔
҈



ᨨ �
z㜠�ᆠ�ᨨ

㡰

맔
҈



ᨨ �
x㜠�ᆠ�ᨨ

ᆠ㡰

맔
҈



ᨨ �
z㜠�ᆠ�ᨨ

㡰

맔
҈



ᨨ �
y㜠�ᆠ�ᨨ


鼀� �ᆠ
�
ᨨ

ᆠᨨ鼀� �ᆠ
犨
�矄
��
泐
ur Wb  = ҈

´ ҈

ᆠᨨ矄
´ ҈
䰘 z +
´ ҈
҈� t
�ˆ + ˆ
m


矄矄ᨨᆠ�
ᆠ�ᆠ
�
犨�


᲼㝐�


꾀�


҈黨

ᔼᔼ桌

(

)(

)

(

)

V ur 㡰 쟼
㦨




맔
҈


ᆠᨨ矄

맔
҈
䰘
z



맔
҈
҈�
t㡰

�ˆ + ˆ m ur V 㡰 쟼 =


´ ҈

ᆠᨨ矄
´ ҈
䰘 z -
´ ҈
҈� t㡰 -
$j + $i㡰 m V ur ᆠ㡰 쟼 㦨




맔
҈


ᆠᨨ矄

맔
҈
䰘
z
졔


맔
҈
҈�
t㡰
 �ˆ +
ˆ m V ur 㡰 쟼
㦨




맔
҈


ᆠᨨ矄

맔
҈
䰘
z
졔


맔
҈
҈�
t㡰
�ˆ 
ˆ m


鼀
�  矄�泐
ᨨ
��ᨨ
ᨨ
 
矄盐 ᆠ
ᨨ
 
和盐

�矄
ᨨ盐
҈


␔⏌鼀



鼀矄
盐
ᨨ
 
ᆠᨨ矄ᨨ � � 鼀� �ᆠ
�
�矄
ᆠᨨ矄矄
ᨨ

᲼㝐�
҈




҈黨

ᔼᔼ桌 㡰 ҈


㡰҈




ᆠ㡰 ҈


㡰 ҈




꾀
ᆠᨨ鼀� �ᆠ
犨
ᨨ
�  矄�泐
�

␔鼀

 矄
�
 鼀�盐鼀
ᨨ
 ᆠ�
� 
� 
犨� ᨨ盐
 泐
ᨨ矄矄


�ᨨ ᨨ
 
鼀� �盐矄
ᨨ
ᆠ�ᆠ
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 
 
�ᨨ
ᨨ
  鼀� �盐矄
ᨨ
鼀� �ᆠ
�矄
ᨨ
 
犨
 ᨨ
 
   � �
� ᨨ
 
鼀�盐鼀

矄ᆠ�泐

��
泐

᲼㝐�
҈҈



҈黨

ᔼ桌 æ ҈ ҈ ö





㡰 ç 㡰 ÷ è ø ҈ ö æ æ ҈ ö
÷ ᆠ㡰 ç
㡰 ç ÷

ø è è

ø 꾀

䰘
鼀␔
矄
鼀
矄

ᆠᨨ矄矄㜠矄ᆠ�ᨨ 

ᨨ
҈

鼀鼀




鼀� �盐
ᨨ
 
鼀 �鼀盐鼀
ᆠ�ᆠ
�
�
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泐

᲼෌�
鼀���泐
ᨨ
矄ᆠ
Î

㦨
黨
맔
҈

҈

和ᔼ
盐 �矄

和 ᨨ
�� 
ᆠ
㦨

맔
҈


鼀⏌矄桌



᲼㝐�
҈҈



҈黨

ᔼᔼ桌 㡰

├⏌鼀 ᆠ㡰

䰘
├⏌鼀 㡰 ҈
├⏌鼀 㡰 ҈


├⏌鼀 ᔼ
 
鼀� �ᆠ
�
ᨨ

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 
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�� 
㦨

맔
҈


鼀⏌矄㡰

(

(







҈


(










)

)

)

é xö ù ҈� æ 
㦨
҈


맔
҈


矄�
ê
p ´
´ ҈
çè  - ÷ø ú ᆠ û ë  
 
鼀 �鼀盐鼀
ᆠ�ᆠ
�
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犨�
�
�矄
᲼㝐�
҈





҈黨
ᔼ桌

㡰

맔
҈


⏌ᔴ 㡰 
맔
҈


⏌ᔴ
ᆠ㡰

맔
҈

⏌ᔴ 㡰

�
҈


⏌ᔴ



ᆠ�ᆠ
�
ᨨ

 
ᨨ�
ᆠᨨ鼀� �ᆠ 犨
�

矄ᆠ

�鼀

㦨


�矄
��
泐

 矄矄�ᨨ

ur 쟼  x
y 㡰 = ҈
ˆ ᆠᨨ矄 ᲼
xur+
z㡰桌


鼀� �ᆠ
�
  x
z
t 㡰
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泐

ᆠ
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  ᨨᆠ�泐
ᨨ
�� 㡰 ᲼㝐�
҈




҈黨

ᔼᔼ桌 ҈ ˆ ˆ ᆠᨨ矄 [ 
x -
z + ҈
ᆠt 㡰 ] 㡰 
├ +
�㡰 ᆠ ҈ ˆ ˆ ᆠᨨ矄 [
x +
z - ҈
ᆠt 㡰 ] 
├ -
�㡰 㡰 ᆠ ҈ ˆ ˆ ᆠᨨ矄 [ 
x +
z - ҈
ᆠt 㡰 ] 
├ +
�㡰 ᆠ㡰 ᆠ ҈ ˆ ˆ ᆠᨨ矄 [
x +
z + ҈
ᆠt 㡰 ] 
├ -
�㡰 㡰 ᆠ
�
꾀
쟼㝐
犨
ᨨ鼀
�
 矄
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鼀�盐鼀


ᆠ�ᆠ
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E҈ = E
҈ xˆ ᆠᨨ矄 ê
pv ç - t ÷ú
�
�
  è c øû ë r E
= E

xˆ ᆠᨨ矄 [ k 
z - ct 㡰]
�
鼀�盐鼀

犨 
 
犨

盐鼀
k
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v
 
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 �
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�
�

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ᔼ
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ᨨ
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ᨨ
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犨 �ᆠ
ᨨ
  ᨨᨨ犨� �
ᨨ�ᨨ 矄
�矄
ᆠᨨᆠ䁸






᲼㝐�
黨



҈黨

ᔼ桌 Îr Îr ҈ ҈ =
=
㡰 㡰 Îr Îr


ᆠ㡰

Îr Îr

҈




=




҈


㡰

Îr Îr

҈


=

҈






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�


矄ᆠ





᲼㝐�
黨



҈黨

ᔼᔼ桌

 㡰 쟼
>

ᆠ㡰 쟼


 㡰 쟼
㦨

䰘
꾀
鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ
鼀
ᨨ
�� 
矄

⼄盐 ᆠ泐  v= ´ ҈
҈

 
�矄
ᨨ�� �
ᨨ �
 
�ᆠ�ᨨ


p �ˆ + ˆ


ᔼ
�矄
ᨨ�
ᨨ �
 
kˆ
�ᆠ�ᨨ


ᆠᆠ
ᨨ鼀
ᨨ
 
鼀� �ᆠ
�
�矄
᲼㝐�
졠 � 
꾀�
҈�



҈
桌 㡰  쟼 =

é
æ �ˆ - ˆ ö r E æ �ˆ - ˆ ö ҈
ù 㡰 k
ç ÷ ᆠᨨ矄 ê҈
ç ÷
r -  ´ ҈
㡰t ú C è
ø êë úû è
ø 㡰

é
æ �ˆ + ˆ ö r E
æ �ˆ - ˆ ö ҈
ù ç ÷ ᆠᨨ矄 ê҈
ç ÷
r -  ´҈
㡰t ú C è
ø êë úû è
ø

ᆠ㡰

é æ �ˆ + ˆ ö r E
ˆ ҈
ù k ᆠᨨ矄 ê҈

ç ÷
r +  ´҈
㡰t ú C è
ø ëê ûú

㡰

é æ �ˆ + ˆ ö r E
�ˆ + ˆ + kˆ㡰 ҈
ù ᆠᨨ矄 ê҈

ç ÷
r +  ´ ҈
㡰t ú C  è
ø ëê ûú

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Electromagnetic Waves







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 =  
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�
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 
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㡰
( 泐ˆ + ˆ )
 
 
鼀� �ᆠ
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㡰 ( 泐ˆ + ˆ )






҈













㡰 ᨨ

ᆠᨨ鼀� �ᆠ
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㡰 ( 泐ˆ - ˆ )


r  
 
鼀� �ᆠ
�
�矄
 = ҈  泐 
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ᨨ
 
ᆠᨨ鼀� �ᆠ
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�
  㜠�ᆠ�ᨨ

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ᨨ 
ᨨ
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ᨨᨨ犨� �
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ᆠᨨᆠ
ᨨ矄矄�
�ᆠ�ᨨ
ᨨ
쟼
 

� 矄ᆠ�泐䁸 ᲼㝐�
졠 � 
꾀�
҈҈



҈�桌 㡰 (2$i + 3$j)  
($i + 2$j) 㡰 (-2$i - 3$j)  
(3$i - 2 $j) ᆠ㡰 (3$i + 4 $j)  
(4$i - 3$j) 㡰 ($i + 2$j)
 
(2$i - $j) 盐� �
 
ᨨ��ᨨ
ᨨ
ᆠᨨ鼀� �ᆠ
犨矄
�
 鼀�盐鼀




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桌 㡰 쟼ᆠ�ᆠ
 �泐
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ᨨ
 
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�矄
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ᨨ
 
鼀� �ᆠ
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�矄
⼄盐
ᨨ
 
鼀� �ᆠ
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㡰 ᨨ
ᆠ�ᆠ
 
鼀� �ᆠ
 �泐
 矄��矄

ᨨ
꾀
鼀
鼀�矄
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�� 
盐 � ᨨ鼀泐
�
 �ᆠ�ᨨ 矄


鼀
�矄
奠
 �ᆠ� 
�
ᆠᨨ � �
ᆠ�ᆠ ᨨ犨
ᨨ
ᆠᨨ鼀� �ᆠ
犨矄
 
ᆠᨨ 矄盐鼀矄
҈


␔
ᨨ ᨨ犨


鼀�盐
ᨨ
 
ᆠ�ᆠ
�
矄矄ᨨᆠ�
犨�  
ᆠᨨ鼀� �ᆠ
��ᨨ


�矄 ᆠ
ᨨ
�
鼀
ᨨ鼀
  ᲼㝐�
졠 � 
꾀�
҈




҈
桌 鼀
犨�

泐
㡰 ҈


⏌鼀 㡰




⏌鼀 ᆠ㡰




⏌鼀 㡰 �


⏌鼀


鼀� �ᆠ
�
�

� �
ᆠᨨ鼀� �ᆠ
犨 矄

├
盐
ᨨ








├
盐
ᨨ
ᆠ�ᆠ
� 矄 �
�矄





᲼㝐�


҈桌 㡰 
⏌鼀 㡰

⏌鼀 ᆠ㡰 黨
⏌鼀 㡰 ҈

⏌鼀 꾀
 
ᆠᨨ鼀� �ᆠ
犨
�

ᨨ 㜠鼀� �ᆠ
�ᆠ�ᆠ ur ur 鼀�盐鼀
�矄
��
泐
E = E

´ ҈
-䰘 x - �
t 㡰
犨� �矄 ᆠ
� �
�
鼀
 
�鼀
�
矄ᆠᨨ 矄


�ᆠ�ᆠ ᆠᨨ 矄 
ᨨ
 
鼀�盐鼀
�矄

᲼㝐�
졠 � 
꾀�






҈桌 㡰


㡰�

ᆠ㡰


㡰




Topic-2 : Electromagnetic Spectrum 1 ҈









㡰 㡰 ᆠ㡰 㡰

MCQs with One Correct Answer

ᔴ ᨨ矄矄
 
ᆠᨨᆠ
ᨨ�ᨨ
� �
犨 � 矄
ᨨ �  
矄
ᨨ
ᆠᨨ鼀� �ᆠ
犨
矄ᆠ盐鼀

᲼㝐�
和










ᔼ㡰桌 㡰 l �矄� < l 鼀�ᆠᨨ
犨矄 < l �ᨨ
犨矄 < l X -泐矄 㡰 l �ᨨ
犨矄 > l 鼀�ᆠᨨ
犨矄 > l �矄� > l x-泐矄 ᆠ㡰 l x- 泐矄 < l 鼀�ᆠᨨ
犨矄 < l �ᨨ
犨矄 < l �矄� 㡰 l �矄� > l x-泐矄 > l �ᨨ
犨矄 > l 鼀�ᆠᨨ
犨矄 꾀 �
 
ᨨᨨ犨� �
ᆠᨨ鼀� �ᆠ
��ᨨ 矄
 ⼄盐 盐鼀
�
 
ᨨ
ᨨ
� ᆠ矄� �
 �泐


᲼㝐�


҈
桌 꾀


盐
��  


Ӹᨨ犨
��  ᔴ


㜠泐 


�ᨨ犨
㡰 ᔴ

꾀



 㡰 

꾀



ᔴ ᆠ㡰 



꾀

ᔴ 㡰 꾀





ᔴ 㝐�ᆠᨨ犨
ᨨ
ᆠ矄
ᨨ
 
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졠 � 
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桌





�


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ᨨ
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鼀ᨨᆠ盐矄 ��� �
 矄�ᨨ 
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ᨨ
犨
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鼀ᨨᆠ盐矄  矄 � �
ᆠᨨ 矄
ᨨ鼀
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泐
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ᔼ

෌

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犨 �ᆠ ᨨ
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졠 � 
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桌 㡰

㝐

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෌  ᨨᨨ 矄
ᨨ
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 �泐 ҈҈
├
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ᨨ
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ᨨ
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矄ᆠ盐鼀 ᨨ矄
�
ᨨ �
䁸
᲼㝐�
졠 � 
꾀�
黨



҈桌 㡰 㜠泐
��ᨨ

㡰 
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ᆠ㡰 ᔼ 
��ᨨ

㡰 �矄�
��ᨨ

WWW.JEEBOOKS.IN

EBD_8341 P-242

Physics

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ᆠᨨᆠ
鼀ᆠ
犨
 
 �矄
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W �h 꾀㡰 㝐�ᆠᨨ犨 �㡰 ҈


鼀 㡰 ෌鼀鼀
泐矄 ��㡰 ҈
҈�
鼀 ᔴ㡰 꾀
㝐

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犨矄 ���㡰 ҈
҈

鼀 㡰 㜠泐矄 �㡰 ҈

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㡰㜠�㡰

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和



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᲼㝐�


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桌





䰘






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犨矄 �ᨨ
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҈
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
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
㡰 �㡰 ���㡰 ��㡰 �㡰 㡰 �㡰 ��㡰 �㡰 ���㡰 ᆠ㡰 ���㡰 ��㡰 �㡰 �㡰 㡰 �㡰 ��㡰 ���㡰 �㡰 㝐ᆠ
 
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�矄㜠ᔼᔼ

ᨨ

黨


ᆠᨨ鼀� �ᆠ
矄ᆠ盐鼀㡰
 
矄ᆠ
 
ᆠᨨᆠ
ᆠᨨ
ᨨ鼀  
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 �矄
�矄矄
᲼㝐�
졠 � 
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҈҈



҈
桌

Match the Following

7

ᔼ ᔼᔼ

�矄
ᔼ  ᨨ 盐 
ᨨ
矄ᨨ �盐 鼀

�矄
ᔼᔼ 꾀 㡰  �矄� 
 ��ᨨ

␔   �
ᆠᨨ 矄 ᨨ  � �
 ᨨ
 鼀 盐
矄矄ᨨ ᆠ�
犨 �
 
�矄ᨨᨨ  �ᆠ
 � �ᨨ
�� �

矄ᆠ 

 㡰 㝐 �ᆠ ᨨ犨  

ᔴ 㡰 和 ᨨ 
�ᨨ
犨

ᔼᔼᔼ ␔   �
鼀 � 
 泐
ᨨ 鼀�ᆠ
泐 ᨨ �
�
� 矄 
矄  ᆠ

ᔼ ␔   �
ᨨ
� �ᨨ
 㡰  㜠泐矄 �矄� �
ᨨ 鼀
犨ᨨ
ᆠᨨ矄
 � 泐
 矄
�
泐 ᨨ �

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ᔼᔼ㡰㜠㡰

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ᔼᔼ㡰㜠㡰

ᔼᔼᔼ㡰㜠ᔴ㡰

ᔼ㡰㜠ᔴ㡰 ᆠ㡰 ᔼ㡰㜠㡰

ᔼᔼ㡰㜠ᔴ㡰

ᔼᔼᔼ㡰㜠꾀㡰

ᔼ㡰㜠㡰 㡰 ᔼ㡰㜠㡰

ᔼᔼ㡰㜠꾀㡰

ᔼᔼᔼ㡰㜠㡰

ᔼ㡰㜠꾀㡰 ҈

㝐ᆠ
�矄
ᔼ
␔ �
 �
ᨨ
ᆠᨨ鼀� �ᆠ 矄ᆠ盐鼀㡰
犨�
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ᔼᔼ
㝐 ᨨ
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桌 �矄
ᔼ

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ᔼᔼ

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ᨨ
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ᨨ
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҈
鼀鼀  
鼀ᨨᆠ盐矄

㡰 ҈
鼀
ᨨ




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
矄 
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ᨨ鼀矄
鼀ᨨ� �
ᨨ鼀
ᨨ 
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ᨨ

ᨨ犨



㡰

҈

鼀 ���㡰 �ᨨᆠ�
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ᨨ
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盐ᆠ盐矄

㡰 ҈
鼀鼀
ᨨ


҈
鼀

㡰 㡰 ᆠ㡰 㡰

�㡰 㝐� ᨨ



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Answer Key Topic-1 : Electromagnetic Waves, Conduction and Displacement Current ҈
҈҈

҈
҈


㡰 ᆠ㡰 ᆠ㡰 ᆠ㡰



҈








ᆠ㡰 ᆠ㡰 㡰 㡰


҈





㡰 㡰 㡰 㡰



҈








㡰 㡰 㡰 㡰

�
ᆠ㡰 ҈�
ᆠ㡰
�
ᆠ㡰



㡰 ҈

㡰


㡰

䰘
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㡰
䰘
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㡰


ᆠ㡰

黨
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ᆠ㡰
黨
㡰

҈

㡰


㡰 

㡰



㡰

黨
㡰

҈

㡰

Topic-2 : Electromagnetic Spectrum ҈


㡰



ᆠ㡰


ᆠ㡰



ᆠ㡰

�
㡰



㡰

䰘
ᆠ㡰

WWW.JEEBOOKS.IN

P-243

Ray Optics and Optical Instruments

Ray Optics and Optical Instruments

23

Topic-1 : Plane Mirror, Spherical Mirror and Reflection of Light 1

MCQs with One Correct Answer





҈
졠ᆠ

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


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ᔼ
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矄ᆠ 鼀�ᆠ
 
ᨨ ᨨ
矄ᆠ㡰 [Main Sep.
02, 2020 (I)] 㡰 ᔼ 


 
鼀� � � 㡰 쟼ᆠ

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ᨨᆠ
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ᨨ




鼀


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鼀 㡰 ҈



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

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ᔼ桌 㡰  㡰
  和 ᆠ㡰  d  㡰








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ᨨ
�  鼀� � �ᆠ�ᨨ
m㡰
矄盐矄
�矄 ᆠ
ᨨ
 
ᨨᆠ
ᨨ鼀
  鼀�ᨨ
x㡰
�矄
ᆠᨨᆠ泐
��
泐 ෌ 矄

犨
矄ᆠ 鼀�ᆠ泐
 

ᨨ
ᨨ
矄ᆠ㡰 ᲼㝐�






҈黨
ᔼᔼ桌

㡰

㡰

ᆠ㡰

㡰

꾀
�ᆠ
�矄
ᨨ矄ᆠ�� �
ᨨ
 
㜠 �矄
犨�

鼀�盐

ᆠ鼀
ᨨ盐
 
ᨨ� 
x

㦨
҈

ᆠ鼀
犨�

⼄盐 ᆠ泐
w

꾀
ᆠᨨ ᆠ 鼀�ᨨ
ᨨ
ᨨᆠ
 �
�
ᆠ鼀
�矄
ᆠ

 
ᨨ���
矄
��盐㡰 ᔼ � 泐
 
ᆠᨨᆠ
矄鼀 矄
᲼㝐�
졠 � 
꾀�
҈�



҈
桌 꾀㡰 
�鼀�
 ᆠ盐矄
�ᨨ�ᆠ
鼀ᨨ�ᨨ

㡰 
�鼀�
 ᆠ盐矄
ᨨ 㜠�ᨨ�ᆠ
鼀ᨨ�ᨨ

ᔴ㡰 
盐 � �
ᨨ� 矄
ᨨ
 
�鼀�

矄泐鼀鼀�ᆠ
犨

  
�鼀�
ᨨ
 
ᨨ� 

x
㦨
҈

ᆠ鼀 㡰 
�矄 ᆠ
犨
 
盐 � �
ᨨ� 矄
ᨨ
  ҈

x

㦨
҈

ᆠ鼀 ᨨ矄ᆠ��ᨨ
ᨨ
 
�鼀�
�矄

҈ x
㦨

㡰 㡰

㡰 㡰 㡰

ᔴ㡰 ᆠ㡰 꾀㡰

ᔴ㡰

㡰 㡰 꾀㡰

㡰

WWW.JEEBOOKS.IN

EBD_8341 P-244

Physics

Ӹᨨ盐

矄├
ᨨ
矄��

矄 � �
鼀�ᨨ
矄矄盐鼀� �
 
 矄ᨨ
├矄
�
҈

ᆠ鼀
ᨨ鼀
�矄
ᆠ
 
�犨矄
 
鼀� � � �鼀�
ᨨ
 
ᆠ

 
ᆠᨨ矄矄
ᆠᨨ鼀 ᨨ
�矄 ᆠ
ᨨ
�
ᆠ鼀


�盐矄
ᨨ
ᆠ盐盐
ᨨ
 
鼀�ᨨ
犨ᨨ盐
 

᲼㝐�
졠 � 
꾀�
҈




҈�桌 㡰


ᆠ鼀 㡰 


ᆠ鼀 ᆠ㡰 



ᆠ鼀 㡰


ᆠ鼀 ҈

꾀
泐
ᨨ
�� 
� �
�
 
�ᆠ�ᨨ
�ˆ +  ˆ 㡰 �矄
� ᆠ� 
ᨨ

 
鼀�ᨨ

꾀 
 ᆠ�ᨨ

�
矄
ᨨ �
 
�ᆠ�ᨨ

҈ ˆ � -  ˆ 㡰


 �
ᨨ
� ᆠ� ᆠ
�矄 ᲼꾀



҈桌
㡰 
ᆠ㡰

㡰
� 㡰 䰘� 黨
ᔼ

 �鼀 
ᨨ
鼀� 
 
ᨨᆠ
 � f 㡰
ᨨ

ᆠᨨ ᆠ
鼀�ᨨ
泐
 
盐
㜠

鼀 ᨨ


矄盐 
ᆠ矄  
ᨨᆠ
�

꾀
ᨨ
 
� ᆠ�
 �矄


�矄 ᆠ

ᨨ鼀
  ᨨ
P


矄盐 
ᨨᨨ├矄

 
�
 
�矄
� 
�鼀� ᨨ鼀

�矄 ᆠ
├� �
�矄⏌ 
泐
�
� 
犨�
PA

␔ 

 
矄盐 
矄 � 矄
�矄⏌ 
泐
ᨨ犨矄
 

 
�鼀�
矄 ᨨ
 
�� 
ᨨ
 
ᨨᆠ
�






























᲼


䰘桌 㡰 x




f 㡰 f




x



f ᆠ㡰 x

㦨


f 㡰 x

>


f ҈

꾀
ᨨ� 
矄ᨨ盐ᆠ
ᨨ
�� 
B
�矄
ᆠ

 B �矄 ᆠ
L
�
ᨨ 
ᨨ
 
ᆠ 
ᨨ
 d 鼀�ᨨ
ᨨ
犨�
ἘdἘ
盐 �
�ᆠ泐

ᨨ
 犨

꾀
鼀
犨├矄
�
ᨨ 
ᨨ
 
鼀�ᨨ






L ᨨ �

� 

ᨨ
 
鼀�ᨨ

 2L �矄 ᆠ

L
ᨨ鼀
�
矄
矄 ᨨ犨
�
��

 �矄
�矄 ᆠ
ᨨ
犨 �ᆠ

ᆠ
矄
 
�鼀�
ᨨ
  �� 
矄ᨨ盐ᆠ
�
 
鼀�ᨨ
�矄 ᲼



和桌 㡰 d⏌
㡰 d ᆠ㡰
d 㡰 d ҈҈
꾀
矄 ᨨ
� 
ᨨᆠ
ᨨ
 �

�矄
ᨨ �
 
 �矄
ᨨ
 ᆠᨨ ᆠ
鼀�ᨨ
ᨨ
ᨨᆠ
 �



�矄 
盐
ᨨ鼀
  ᨨ
ᨨ
 
鼀�ᨨ


矄�
ᨨ
 
�鼀�
�矄
ᨨ �鼀泐 ⼄盐
ᨨ ᲼҈黨


㜠


㝐├桌 䰘


㡰

æu - f ö bç è f ÷ø

ᆠ㡰

æu- f ö bç è f ÷ø

2 ҈



æ f ö bç è u - f ø÷




㡰

Integer Value Answer

ᔼ鼀�
ᨨ

ᨨᆠ
ᨨᆠ � �

ᆠᨨ 
鼀�ᨨ
ᨨ
�盐矄 ᨨ
ᆠ盐盐



鼀
ᨨ �
�矄
ᨨ�ᆠ
 �矄
�矄
ᨨ矄
ᨨ
� �
鼀ᨨ
ᨨ鼀 鼀 ᨨ 鼀
�


矄ᆠᨨ 矄

␔ 
�矄
 
矄  䰘 ᨨ
 
ᨨᆠ
�
├鼀

ᨨ盐䁸 ᲼

҈
桌 3

҈


æ f ö bç è u - f ø÷

҈⏌


㡰

҈⏌


Numeric Answer

␔ 

ᨨᆠ
�矄
├


�矄 ᆠ
ᨨ


ᆠ鼀
ᨨ鼀

ᆠᨨ ᆠ 鼀�ᨨ

 
�鼀�
�矄
ᨨ鼀


�矄 ᆠ
ᨨ
҈

ᆠ鼀
ᨨ鼀
  鼀�ᨨ

ᔼ
 
ᨨᆠ
�矄
鼀ᨨ
犨�

矄
ᨨ
黨 ᆠ鼀矄҈

 
矄
�
ᆠ鼀矄҈㡰
犨�
犨 �ᆠ
�鼀�
鼀ᨨ矄
  
� 矄 
�矄
힘힘힘힘힘힘힘힘
᲼㝐�
和









ᔼᔼ㡰桌

4

Fill in the Blanks

f ҈

꾀
 �
ᨨ
ᨨ
 �

�矄
ᆠ
ᨨ �
 
ᨨ�ᆠ
 �矄
ᨨ
  ᆠᨨ ᆠ
鼀�ᨨ
ᨨ
ᨨᆠ
 �

矄盐ᆠ
 
�矄
�鼀�
犨 �ᆠ �矄

 
ᨨ �

盐矄
ᨨ盐ᆠ 矄
 
ᨨ

 鼀� � �ᆠ�ᨨ
�矄




















᲼҈黨黨҈
㜠
҈

㝐├桌

6

MCQs with One or More than One Correct Answer

҈�
꾀
犨�
�矄
 
�
 
矄 
ᨨ

�� 
 �
� �
 
�矄 ᆠ
�
ᨨ 
ᨨ

ᆠᨨ ᆠ
鼀�ᨨ
ᨨ
ᨨᆠ
 �
f

矄 矄 ᨨ犨
�
 
��盐

␔ �ᆠ
ᨨ
 
��盐矄
矄 ᨨ犨
�
  ᨨ盐
ᨨ�ᨨ 矄
⼄盐��泐
矄 矄㡰
 
矄 
ᨨ
  �鼀�
ᨨ
 
 
犨�䁸
 矄
��盐矄

ᨨ
ᨨ
矄ᆠ
㡰 ᲼꾀



҈
桌

ᨨ


� f


f

¥

㡰

a

ᨨ a >

�




㡰


¥

ᆠ㡰


a

ᨨ a






�


㡰


҈

꾀
矄盐 
 ᨨ鼀
 
 �鼀 
ᨨ
鼀� �ᨨ
ᨨ ᨨᆠ
 �
ᨨ

ᆠᨨ ᆠ
鼀�ᨨ
泐
u㜠v 鼀 ᨨ
盐矄� �


ᨨ�ᆠ
 ᆠ
ᨨ
 �
҈
�
鼀


ᨨᆠ
 �
ᨨ
  鼀�ᨨ
盐矄
�矄



ᆠ鼀


鼀 �鼀盐鼀
ᨨ
�
 
ᨨᆠ�ᨨ
ᨨ  
�鼀�
ᆠ





ᆠ鼀


�
矄矄
ᨨ
u, v 㡰
盐矄
ᆠᨨ 泐
 
矄盐 

�
ᆠ鼀㡰






�
㡰







㡰







㡰





㡰

䰘


黨㡰



矄矄㡰  
ᆠ

ᨨ
ᆠᨨ鼀
ᨨ鼀
 �鼀 
 
�矄
㡰
� ᆠᨨᆠ泐 ᆠᨨ

�矄
㡰 ᲼


黨桌 㡰 



�
㡰 㡰 





㡰 ᆠ㡰 



㡰 㡰 䰘


黨㡰 9

Assertion and Reason Type Questions

҈䰘
和T꾀TE㝐ENT㜠҈

ᨨ鼀盐
ᆠᨨ

ᆠ� �
u

v
 
f
ᨨ
 矄 �ᆠ
鼀�ᨨ
�矄
�
ᨨ
鼀�ᨨ矄
犨 ᨨ矄
矄�矄

泐 矄鼀
ᆠᨨ鼀
ᨨ
 �
��
ᨨ
ᆠ盐盐
和T꾀TE㝐ENT㜠

犨矄
ᨨ
 ᆠ�ᨨ

矄�ᆠ泐
�
ᨨ  
矄盐 ᆠ矄

盐
ᨨ
ᨨ
�
矄 �ᆠ
矄盐 ᆠ矄
᲼


䰘桌 㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
盐錐
和鼀 㜠
�矄

ᆠᨨᆠ
  �ᨨ
ᨨ
和鼀 㜠҈ 㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
盐錐
和鼀 㜠

�矄
졠

ᆠᨨᆠ
  �ᨨ
ᨨ
和鼀 㜠҈ ᆠ㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
矄 㡰 和鼀 㜠҈
�矄
矄

和鼀 㜠

�矄
盐

WWW.JEEBOOKS.IN

P-245

Ray Optics and Optical Instruments

Topic-2 : Refraction of Light at Plane Surface and Total Internal Reflection 1 ҈


MCQs with One Correct Answer

꾀
矄矄
ᨨ



�矄

�
犨�

�⼄盐�
ᨨ
 ᆠ� � 



 
 
盐

犨�
 ᨨ 
�⼄盐�
ᨨ  ᆠ�
� 




�⼄盐�矄

�鼀鼀�矄ᆠ�

  

ᨨ
 
�


矄盐 ᆠ
ᨨ
 
ᨨᨨ鼀
ᨨ
矄矄 犨�

᲼㝐�
黨







ᔼ桌 h h 㡰 㡰

+ ҈㡰


h












 h
ᆠ㡰 㡰 




�矄

矄鼀
矄ᨨ盐ᆠ
ᨨ
�� 

矄ᨨ鼀

ᨨ犨
 
矄盐 ᆠ
ᨨ
犨
 ᆠ�
� 
㦨
㡰
�

 ├
ᨨ
�  ᆠᨨ矄矄
矄ᆠ�ᨨ 
矄盐 ᆠ


�ᆠ� �
 泐
 ᆠ�ᨨ

ᨨ鼀
 
ᨨᨨ鼀
 
矄ᨨ�ᨨ
泐
犨

ᆠ �
ᨨ �� 
 
鼀�矄
ᨨ盐
ᨨ
矄盐 ᆠ
�矄
 泐㡰
᲼矄
 
ᆠ
 
矄盐 ᆠ

ᨨ

矄 �ᆠ
ᆠ
ᨨ
��  h
 
�盐矄
ᨨ
ᆠ盐盐
r
�矄

prh桌
᲼㝐�

黨







ᔼᔼ桌 㡰
҈奠 㡰 
奠 ᆠ㡰 ҈䰘奠 㡰 �
奠


ᆠ��ᆠ
 �
ᨨ

鼀�盐鼀
ᨨ

矄ᆠ� �ᆠ
犨 �

�  
鼀�盐鼀
矄
�
鼀���泐

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WWW.JEEBOOKS.IN

EBD_8341 P-246

10.

Physics

A point source S is placed at the bottom of a transparent block of height 10 mm and refractive index 2.72. It is immersed in a lower refractive index liquid as shown in the figure. It is found that the light emerging from the block to the liquid forms a circular bright spot of diameter 11.54 mm on the top of the block. The refractive index of the liquid is [Adv. 2014]

Liquid

Block

11.

S (a) 1. 21 (b) 1. 30 (c) 1. 36 (d) 1. 42 A light ray falls on a square glass slab as shown in the diagram. The index of refraction of the glass, if total internal reflection is to occur at the vertical face, is equal to : [Main Online April 23, 2013]

Air r Water

r

Glass

i

(a) (4/3)sini (b) 1/sini (c) 4/3 (d) 1 16. Two plane mirrors A and B are aligned parallel to each other, as shown in the figure. A light ray is incident at an angle 30° at a point just inside one end of A. The plane of incidence coincides with the plane of the figure. The maximum number of times the ray undergoes reflections (including the first one) before it emerges out is [2002S] B

2 3m

0.2m 30°

45° Incident ray

A

(a) 28 (b) 30 (c) 32 (d) 34 17. An observer can see through a pin-hole the top end of a thin rod of height h, placed as shown in the figure. The beaker height is 3h and its radius h. When the beaker is filled with a liquid up to a height 2h, he can see the lower end of the rod. Then the refractive index of the liquid is [2002S] 5 3 3 ( 2 + 1) (b) (c) (d) 2 2 2 2 A ball is dropped from a height of 20 m above the surface of water in a lake. The refractive index of water is 4.3. A fish inside the lake, in the line of fall of the ball, is looking at the ball. At an instant, when the ball is 12.8 m above the water surface, the fish sees the speed of ball as [Take g = 10 m/s2.] [2009] (a) 9 m/s (b) 12 m/s (c) 16 m/s (d) 21.33 m/s A ray of light traveling in water is incident on its surface open to air. The angle of incidence is q, which is less than the critical angle. Then there will be [2007] (a) only a reflected ray and no refracted ray (b) only a refracted ray and no reflected ray (c) a reflected ray and a refracted ray and the angle between them would be less than 180° – 2q (d) a reflected ray and a refracted ray and the angle between them would be greater than 180° – 2q A point object is placed at the centre of a glass sphere of radius 6 cm and refractive index 1.5. The distance of virtual image from the surface is [2004S] (a) 6 cm (b) 4 cm (c) 12 cm (d) 9 cm A ray of light is incident at the glass-water interface at an angle i, it emerges finally parallel to the surface of water, then the value of m g would be [2003S]

(a)

12.

13.

14.

15.

3h h

2h

5 3 5 3 (b) (c) (d) 2 2 2 2 18. A ray of light passes through four transparent media with refractive indices m1, m2, m3 and m4 as shown in the figure. The surfaces of all media are parallel. If the emergent ray CD is parallel to the incident ray AB, we must have [2001S] (a)

m1

m2

D

m3

m4

C B

A

(a) m1 = m2 (c) m3 = m4

(b) m2 = m3 (d) m4 = m1

WWW.JEEBOOKS.IN

P-247

Ray Optics and Optical Instruments

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i

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2


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D



D



Numeric Answer




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[Main Sep. 03, 2020 (I)]

WWW.JEEBOOKS.IN

EBD_8341 P-248

Physics


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P-249

Ray Optics and Optical Instruments

10 Subjective Problems 



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Topic-3 : Refraction at Curved Surface Lenses and Power of Lens





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WWW.JEEBOOKS.IN

EBD_8341 P-250

= u

v f

v

Physics

(c)

u

f v

v f

= u

8. (d)

f 4.

5.

(b) 9

(c) 5

(d)

17

One plano-convex and one plano-concave lens of same radius of curvature ‘R’ but of different materials are joined side by side as shown in the figure. If the refractive index of the material of 1 is m1 and that of 2 is m2, then the focal length of the combination is : [Main 10 Apr. 2019 I]

(a)

R m1 - m2

(b)

2R 2(m1 - m2 )

(d)

2R m1 - m2

R 2 - (m1 - m2 ) The graph shows how the magnification m produced by a thin lens varies with image distance v. What is the focal length of the lens used ? [Main 10 Apr. 2019 II]

(c) 6.

u

A thin lens made of glass (refractive index = 1.5) of focal length f = 16 cm is immersed in a liquid of refractive index 1.42. If its focal length in liquid is fl ,then the ratio fl /f is [Main 7 Jan. 2020 II] closest to the integer: (a) 1

4 3 (b) (c) (d) 3 2 3 2 An upright object is placed at a distance of 40 cm in front of a convergent lens of focal length 20 cm. A convergent mirror of focal length 10 cm is placed at a distance of 60 cm on the other side of the lens. The position and size of the final image will be : [Main 8 April 2019 I] (a) 20 cm from the convergent mirror, same size as the object (b) 40 cm from the convergent mirror, same size as the object (c) 40 cm from the convergent lens, twice the size of the object (d) 20 cm from the convergent mirror, twice the size of the object A convex lens (of focal length 20 cm) and a concave mirror, having their principal axes along the same lines, are kept 80 cm apart from each other. The concave mirror is to the right of the convex lens. When an object is kept at a distance of 30 cm to the left of the convex lens, its image remains at the same position even if the concave mirror is removed. The maximum distance of the object for which this concave mirror, by itself would produce a virtual image [Main 8 Apr. 2019 would be : II] (a) 30 cm (b) 25 cm (c) 10 cm (d) 20 cm What is the position and nature of image formed by lens combination shown in figure? (f1, f2 are focal lengths) [Main 12 Jan. 2019 I]

(a)

9.

10.

2 cm A

20 cm

B

f1 = + 5 cm

f2 = –5 cm

(a) 70 cm from point B at left; virtual (b) 40 cm from point B at right; real 20 cm from point B at right, real 3 (d) 70 cm from point B at right; real 11. Formation of real image using a biconvex lens is shown below : [Main 12 Jan. 2019 II]

(c)

a b b2 b2c (b) (c) (d) c c ac a A thin convex lens L (refractive index = 1.5) is placed on a plane mirror M. When a pin is placed at A, such that OA = 18 cm, its real inverted image is formed at A itself, as shown in figure. When a liquid of refractive index µi is put between the lens and the mirror, the pin has to be moved to A’, such that OA’ = 27 cm, to get its inverted real image at A’ itself. The value of µi will be:[Main 9 Apr. 2019 II]

(a)

7.

2f 2f

f

screen

f

If the whole set up is immersed in water without disturbing the object and the screen positions, what will one observe on the screen ? (a) Image disappears (b) Magnified image (c) Erect real image (d) No change

WWW.JEEBOOKS.IN

P-251

Ray Optics and Optical Instruments

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EBD_8341 P-252






Physics

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WWW.JEEBOOKS.IN

P-253

Ray Optics and Optical Instruments

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�鼀�
᲼


和桌 㡰 ҈

�
ᆠ鼀 㡰

�
ᆠ鼀 ᆠ㡰 ҈

�
ᆠ鼀 㡰

ᆠ鼀 

꾀
ᨨᨨ犨
ᨨ盐
ᆠᨨ ᆠ
 矄
�矄
鼀
ᨨ
泐
 �

 矄 
鼀�

ᔼ
ᆠ

�
犨�
�
ᨨ
� 
ᨨ
犨ᨨ �⼄盐�矄
L҈
ᨨ
L

� �
 ᆠ�
� �ᆠ矄
m҈
 
m
矄ᆠ�泐
m

>
m҈
>
҈㡰


 矄
犨�
��

 鼀
ᨨ
�� 
�
�
�矄
�
犨� ᲼



和桌 㡰 �
 
ᆠ
�
� 㡰 �
 
�鼀鼀矄
�
L҈ ᆠ㡰 L҈
 
�鼀鼀矄
�
L
㡰 L

 
�鼀鼀矄
�
L҈ �
꾀
ᆠᨨ ᆠ
 矄
ᨨ
�矄矄

 ᆠ�
� 
҈
�
矄
ᨨ 矄盐 ᆠ矄
ᨨ
矄鼀
�盐矄
ᨨ
ᆠ盐盐
R

졠
�鼀鼀矄�ᨨ
�
 鼀�盐鼀
ᨨ
 ᆠ�
� 
҈
䰘�

�
犨�
 
矄
 ᲼҈黨黨黨和
㜠


㝐├矄桌 㡰 ᆠᨨ � 
 矄
ᨨ
ᨨᆠ
 �

�
R 㡰 ᆠᨨ � 
 矄
ᨨ
ᨨᆠ
 �



R ᆠ㡰 �� 
 矄
ᨨ
ᨨᆠ
 �

�
R 㡰 �� 
 矄
ᨨ
ᨨᆠ
 �



R

҈�ᆠ鼀










᲼
꾀



҈�桌 �
ᆠ鼀

2

Integer Value Answer

黨

ᨨᆠ
 �
ᨨ

 �
�ᆠᨨ 
 矄
�矄



ᆠ鼀

␔ 


ᨨᆠ
�矄
鼀ᨨ
ᨨ鼀

�矄 ᆠ
ᨨ

�
ᆠ鼀
�
ᨨ 
ᨨ
�
ᨨ
�
ᆠ鼀

 
鼀� � �ᆠ�ᨨ
ᨨ
�矄
�鼀�
ᆠ  �矄
ᨨ鼀
鼀
�
ᨨ m
� 鼀�



�ᨨ
�矄 ᲼

҈
桌 m�
3

Numeric Answer





�矄 ᆠ
犨

ᨨᆠ
 

矄ᆠ
�矄
҈


ᆠ鼀

꾀  矄
ᆠ
ᨨ盐ᆠ

�鼀�
ᨨ
 
ᨨᆠ
ᨨ
 
矄ᆠ
ᨨ 犨ᨨ
�  
ᨨ矄��ᨨ 矄
犨
 
矄ᆠ
 
 
ᨨᆠ



�矄 ᆠ
犨
 矄
犨ᨨ
ᨨ矄��ᨨ 矄
�矄



ᆠ鼀

ᔼ
  æ N ö ᨨ犨
ᨨ
 
 矄
�矄
ᆠᨨ矄
ᨨ
ç D
犨 
N
�矄

� �

è ҈

÷ø  
盐
ᨨ
N
�矄
힘힘힘힘힘힘힘힘힘힘힘
᲼㝐�
和










ᔼᔼ㡰桌
҈
꾀
ᨨ� 
ᨨᆠ
�
�
�矄
�
ᨨ 
ᨨ
 
ᆠ盐
矄盐 ᆠ
ᨨ
 planᨨ㜠cᨨnvex
 矄


�盐矄
ᨨ
ᆠ盐盐
ᨨ
 
ᆠ盐 矄盐 ᆠ
�矄


ᆠ鼀
 
 
 ᆠ�
� 
ᨨ
 
 矄
鼀� �矄
҈
�

 
 
ᨨᆠ
 �
ᨨ
 
 矄

�
ᆠ鼀㡰 �矄힘힘힘힘힘힘힘힘힘힘
᲼㝐�









ᔼ桌 4

Fill in the Blanks




犨ᨨ
 �
 矄矄

犨 
�
ᆠᨨ ᆠ

ᨨ盐ᆠ

ᆠᨨ鼀� �ᨨ

ᨨ
ᨨ犨
졔҈

�ᨨ矄

␔ 
 泐




�
鼀


  ᨨ犨
盐ᆠ矄
ᨨ
졔

�ᨨ矄


ᨨᆠ
 �
ᨨ
   矄矄






m
 




m
᲼҈黨黨䰘
㜠


㝐├矄桌

WWW.JEEBOOKS.IN

EBD_8341 P-254











�







Physics

꾀
 �
 矄
ᨨ
 ᆠ�
� 
҈
�
矄

ᨨᆠ
 �
ᨨ
҈�
ᆠ鼀 �
�

␔ 
 
 矄
�矄
ᆠ
�

鼀�盐鼀
ᨨ
 ᆠ�
� 


�矄
ᨨᆠ
 �
犨�
ᆠᨨ鼀








ᆠ鼀
᲼҈黨
䰘
㜠


㝐├矄桌  꾀
ᆠᨨ 
 矄
A
ᨨ
ᨨᆠ
 �



ᆠ鼀
 

ᆠᨨ ᆠ
 矄 B
ᨨ
ᨨᆠ
 �
�
ᆠ鼀

├
ᨨ �
 
矄鼀
 �矄
犨�
 �矄 ᆠ
d
犨
 鼀

ᔼ


鼀
ᨨ
�� 
� � ᨨ
A

矄
B
矄


鼀

 
d
�矄
⼄盐
ᨨ







ᆠ鼀
᲼҈黨
�
㜠


㝐├矄桌

5 True / False 꾀

鼀
ᨨ
犨 �
�� 

ᨨ

ᆠᨨ鼀� �ᨨ
ᨨ
 ᆠᨨ ᆠ
 

ᆠᨨ 
 矄

ᨨ
ᨨ
 
矄鼀
鼀�


�
ᨨᆠ
 � 矄

҈�
ᆠ鼀
 


ᆠ鼀
矄ᆠ�泐
ᨨ  
鼀
犨 �
�
犨 �
�� 

졠
 
ᨨ 
矄�
ᨨ  
 矄
矄泐矄鼀

ᨨ 
矄矄
ᆠᨨᨨ盐
 矄
犨�
�ᨨ ᆠᨨᨨ盐

 
ᨨ盐
�
᲼҈黨


㜠


㝐├矄桌 꾀
ᆠᨨ 
 矄
ᨨ
ᨨᆠ
 �
҈
鼀
 


ᆠᨨ ᆠ
 矄 ᨨ
ᨨᆠ
 �



�
鼀

├


䰘�
鼀


꾀
 鼀
ᨨ
�� 

�矄
矄矄矄
 ᨨ盐�
 

ᆠᨨ 
 矄

 

 ᨨ盐�
 
ᆠᨨ ᆠ
 矄
 
ᆠᨨ鼀矄
ᨨ

ᨨᆠ盐矄


�
鼀 犨泐
ᨨ鼀
 
ᆠᨨ ᆠ
 矄
᲼҈黨

㜠


㝐├矄桌 6


䰘


MCQs with One or More than One Correct Answer


꾀
├
ᨨ
�盐矄
r
�矄
�
犨�
犨
 ᆠ�
� 
㡰
盐  ᨨ

�� 
H
矄
矄 ᨨ犨
�
 
��盐
ᨨ
 
 


├
�矄 ├
ᨨ

ᨨ�ᨨ 

ᨨ� �
犨�
 �盐
矄
w

�矄 鼀├矄
 
犨
矄盐 ᆠ
ᆠ盐
矄ᨨ
 
 
�  ᆠ
�
  �� 
ᨨ
犨


 
ᆠ 
 

 
ᆠ�ᆠ盐鼀  ᆠ
ᨨ  
├
�矄 hh = H
h = r 㡰

矄
矄 ᨨ犨
�
 
��盐
ᨨ

 
�� 

├
 �矄
矄盐 ᆠ
ᨨ

ᨨ �鼀泐
矄 �ᆠ
犨� 
�盐矄
ᨨ
ᆠ盐盐
R

␔ �ᆠ
ᨨ
 
ᨨᨨ犨� �
�矄⏌
ᆠᨨᆠ䁸 g
�矄
 
ᆠᆠ�ᨨ
盐
ᨨ
��泐㡰 w

h

H

㡰 R =

h
+ r

h

r

h ᆠ㡰 꾀 

ᨨ
 
ᨨᨨ鼀
ᨨ
 
├
�矄
ᆠᨨ矄




꾀
 ᨨᆠᨨ 
 矄
�矄
鼀
ᨨ

鼀�
ᨨ
 ᆠ� � 


␔ 

矄鼀
ᨨᆠ
�矄
ᆠ


ᆠ鼀
犨泐
�
ᨨ  ᨨ
 
ᆠ盐
矄盐 ᆠ
ᨨ
 
 矄


�鼀�
ᨨ
ᨨ盐
  矄�
ᨨ
 
ᨨᆠ
�矄
ᨨ盐ᆠ

盐
ᨨ
 ᆠ�ᨨ
ᨨ鼀
  ᆠᨨ 
矄盐 ᆠ
ᨨ
 
 矄

 ᨨ 
� 
�鼀�
�矄
ᨨ矄 

�矄 ᆠ
ᨨ
҈

ᆠ鼀
犨泐
ᨨ鼀
 
 矄

␔ �ᆠ
ᨨ
  ᲼꾀



҈
桌 ᨨᨨ犨� �
矄鼀 矄㡰
�矄㡰
盐䁸 㡰 
 ᆠ�
� 
ᨨ
 
 矄
�矄


� 㡰 
�盐矄
ᨨ
ᆠ盐盐
ᨨ
 
ᆠᨨ 
矄盐 ᆠ
�矄

�
ᆠ鼀 ᆠ㡰 
� 
�鼀�
�矄
ᆠ
 
 㡰 
ᨨᆠ
 �
ᨨ
 
 矄
�矄



ᆠ鼀
黨
꾀
矄 �ᆠ
矄盐 ᆠ
ᨨ
�盐矄
ᨨ
ᆠ盐盐

矄矄
�  ᆠ�
� 
҈

㡰
ᨨ鼀
�矄矄
 ᆠ�
� 
҈
�㡰

 ᆠ 
ᨨ
ᆠ盐盐
�矄
�
 
�矄矄

꾀
ᨨ� 
ᨨᆠ

ᆠ �
�
�矄
ᨨ盐 
ᨨ



�鼀�
Q
�
 
�矄矄


�  PQ
ᆠ盐矄
 
矄盐 ᆠ


ᨨ� 
O

 
PO
㦨
OQ

 ᲼҈黨黨

㜠


㝐├矄桌 �矄 ᆠ
PO
�矄
⼄盐
ᨨ 㡰 �R 㡰 R ᆠ㡰
R 㡰 ҈
�R �

꾀

�鼀�
ᨨ

�矄 
ᨨᆠ
�矄
ᨨ鼀
泐

 ᨨ㜠ᆠᨨ   矄
ᨨ
�矄
� ᆠ�
 �矄

和 �ᆠ
�ᨨ

㡰 �矄
矄 
᲼ ҈黨黨

㜠


㝐├矄桌 㡰 �矄
矄鼀
�
 
ᆠ盐
矄盐 ᆠ
ᨨ
 
 矄
ᆠ矄
 
ᨨᆠ
ᆠ㡰 �矄
矄鼀
�
 
 
矄盐 ᆠ
ᨨ
 
 矄
ᆠ矄
 
ᨨᆠ
㡰 �矄
 
矄鼀
犨 �ᆠ 
矄�
ᨨ
 
 矄
ᆠ矄
 
ᨨᆠ �҈
␔ �ᆠ
ᨨ
 
ᨨᨨ犨� �
ᨨ鼀矄㡰

�盐
 
ᆠ
�鼀� ᨨ

ᨨ矄��ᨨ 矄
ᨨ
 
ᨨᆠ
䁸 ᲼҈黨黨

㜠


㝐├矄桌 㡰 ᔴᨨ 
 矄 㡰 ᔴᨨ ᆠ
 矄 ᆠ㡰 ᔴᨨ 
鼀�ᨨ 㡰 ᔴᨨ ᆠ
鼀�ᨨ
�

꾀
ᆠᨨ �� �
 矄
�矄
盐矄
ᨨ
ᨨ鼀

�鼀�
ᨨ

矄ᆠ
␔ 
 
盐

ᨨ
 
 矄
�矄
ᆠᨨ
泐

ᨨ⼄盐 矄ᆠ

᲼҈黨


㜠


㝐├矄桌 㡰 
 
�鼀�
犨�
�矄
㡰 ᆠᨨ鼀
�鼀�
犨�

ᨨ鼀
ᆠ㡰 �  矄�泐
ᨨ
 
�鼀�
犨�
� ᆠ矄
㡰 �  矄�泐
ᨨ
 
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P-255

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EBD_8341 P-256

60.

Physics

In Fig., S is a monochromatic point source emitting light of wavelength l = 500nm. A thin lens of circular shape and focal length 0.10 m is cut into two identical halves L1 and L2 by a plane passing through a diameter. The two halves are placed symmetrically about the central axis SO with a gap of 0.5 mm. The distance along the axis from S to L1 and L2 is 0.15 m while that from L1 and L2 to O is 1.30 m. The [1993 - 5+1 Marks] screen at O is normal to SO. L1

A

0.5mm

S

O

62. A plano convex lens has a thickness of 4 cm . When placed on a horizontal table, with the curved surface in contact with it, the apparent depth of the bottom most point of the lens is found to be 3 cm. If the lens is inverted such that the plane face is in contact with the table, the apparent depth of the centre of the plane face is found to be 25/8 cm. Find the [1984- 6 Marks] focal length of the lens. 63. The convex surface of a thin concavo-convex lens of glass of refractive index 1.5 has a radius of curvature 20 cm. The concave surface has a radius of curvature 60 cm. The convex side is silvered and placed on a horizontal surface. [1981- 6 Marks]

Screen

L2 0.15m

1.30m

(i)

61.

If the third intensity maximum occurs at the point A on the screen, find the distance OA. (ii) If the gap between L1 and L2 is reduced from its original value of 0.5mm, will the distance OA increase, decrease, or remain the same? Monochromatic light is incident on a plane interface AB between two media of refractive indices n1 and n2 (n2 > n1) at an angle of incidence q as shown in fig. The angle q is infinitesimally greater than the critical angle for the two media so that total internal reflection takes place. Now if a transparent slab DEFG of uniform thickness and of refractive index n3 is introduced on the interface (as shown in the figure), show that for any value of n3 all light will ultimately be reflected back again into medium II. Consider [1986 - 6 Marks] separately the cases (i)

n3 < n1 and

E

MEDIUM I

(n 3 )

G

F

B

A q

Where should a pin be placed on the optic axis such that its image is formed at the same place? (ii) If the concave part is filled with water of refractive index 4/3, find the distance through which the pin should be moved so that the image of the pin again coincides with the pin. 64. What is the relation between the refractive indices m1 and m2, if the behaviour of light rays is as shown in the figure? [1979] m1

m

m1

m2

m m2

n3 > n1 .

(ii) MEDIUM I (n1 )

D

(i)

65. A pin is placed 10 cm in front of a convex lens of focal length 20 cm, made a material of refractive index 1.5. The surface of the lens farther away from the pin is silvered and has a radius of curvature are of 22 cm. Determine the position of the final image. Is the image real as virtual? [1978]

MEDIUM II

(n 2 )

Topic-4 : Prism and Dispersion of Light 1 1.

MCQs with One Correct Answer

The variation of refractive index of a crown glass thin prism with wavelength of the incident light is shown. Which of the following graphs is the correct one, if D m is the angle of minimum deviation ? [Main 11 Jan. 2019, I]

Dm

(a)

1.535 1.530

400 500 600 700

n2

l (nm)

1.525 1.520 1.515 1.510

400

500

600

700

l (nm)

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P-257

Ray Optics and Optical Instruments 鼀





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EBD_8341 P-258

9.

Physics

A beam of light consisting of red, green and blue colours is incident on a right-angled prism on face AB. The refractive indices of the material for the above red, green and blue colours are 1.39, 1.44 and 1.47 respectively. A person looking on surface AC of the prism will see [Main Online May 26, 2012] A

(a) are parallel to each other (b) are diverging (c) make an angle 2 [sin–1 (0.72) – 30°] with each other (d) make an angle 2 sin –1 (0.72) with each other 14. A glass prism of refractive index 1.5 is immersed in water (refractive index 4/3). A light beam incident normally on the face AB is totally reflected to reach on the face BC if [1981- 2 Marks] (a) (b)

45°

C (a) no light (b) green and blue colours (c) red and green colours (d) red colour only Two beams of red and violet colours are made to pass separately through a prism (angle of the prism is 60°). In the position of minimum deviation, the angle of refraction [2008] will be (a) 30° for both the colours (b) greater for the violet colour (c) greater for the red colour (d) equal but not 30° for both the colours An equilateral prism is placed on a horizontal surface. A ray PQ is incident onto it. For minimum deviation [2004S]

B

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EBD_8341 P-260


�


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






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 
ᆠᨨ鼀ᨨ盐  鼀�ᆠᨨ矄ᆠᨨ
矄 ᨨ盐

᲼㝐�
졠 � 
꾀�
҈҈



҈
桌 㡰


㡰 ҈

ᆠ㡰


㡰 ҈�
ᔼ

ᆠᨨ鼀ᨨ盐 
鼀�ᆠᨨ矄ᆠᨨ

 
� 鼀�
�鼀�
�矄 ᲼



和桌 㡰 �盐

ᆠ
 
鼀� � � 㡰 

ᆠ
 
鼀� � � ᆠ㡰 

� 
 
鼀� � � 㡰 �盐

ᆠ
 
盐ᆠ

WWW.JEEBOOKS.IN

P-261

Ray Optics and Optical Instruments






ᨨᆠ
 � 矄
ᨨ
 
ᨨᆠ�
 
 
泐
�ᆠ
ᨨ
 ᆠᨨ鼀ᨨ盐 
鼀�ᆠᨨ矄ᆠᨨ





ᆠ鼀
 



ᆠ鼀

矄ᆠ�泐



�矄 ᆠ
犨
 
ᨨᆠ�
 
 
泐
�ᆠ
�矄 ҈�


ᆠ鼀


� 
�鼀�
ᨨ鼀
泐
 
泐
�ᆠ
�矄
 � � �泐


犨ᨨ
 矄矄

 �


�矄 ᆠ
�
ᆠ鼀
ᨨ
  ᨨᆠ
 
 
�鼀�
ᨨ盐ᆠ
泐
 
ᨨᆠ�

鼀矄盐 ᨨ鼀
 
ᨨᆠ�
 矄


矄ᆠ�泐 ᲼҈黨黨�和桌 㡰



 
҈


㡰



 
҈�

ᆠ㡰



 
҈


㡰



 


3

黨


҈



Numeric Answer

꾀
ᆠᨨ鼀ᨨ盐 
鼀�ᆠᨨ矄ᆠᨨ
ᆠᨨ 矄�矄矄
ᨨ

ᨨᆠ�
 矄
ᨨ ᨨᆠ
 �
҈
ᆠ鼀
 

泐
�ᆠ
ᨨ
ᨨᆠ
 �
�
ᆠ鼀 犨�

矄�ᨨ
ᨨ
҈

ᆠ鼀



�矄 ᆠ
犨

ᨨᆠ
 
 
ᨨᆠ�
 矄

 n 犨 �ᆠ
 
矄�
ᨨ
 
泐
�矄
鼀� �鼀盐鼀
�矄
ᆠ鼀





盐
ᨨ
n �矄
힘힘힘힘힘힘
᲼㝐�
和


�






ᔼ㡰桌 ᔼ

ᆠᨨ鼀ᨨ盐 
鼀�ᆠᨨ矄ᆠᨨ

 
鼀� � �
�盐
�鼀�
�矄 ᨨ鼀


�矄 ᆠ
ᨨ

�
ᆠ鼀
ᨨ鼀
 
泐㜠�ᆠ


ᨨᆠ  �
ᨨ
�矄
ᨨᆠ�
 矄
�矄
҈
ᆠ鼀

ᔼ
 
鼀� � �ᆠ�ᨨ
�矄 ҈


 
 
盐
 �
ᨨ
 
鼀�ᆠᨨ矄ᆠᨨ
�矄



ᆠ鼀

 

 
ᨨᆠ
 �
ᨨ
 
泐㜠�ᆠ
 矄
�
ᆠ鼀㡰
�矄 [Main Sep.
04, 2020 (I)] 힘힘힘힘힘힘힘힘힘힘


4

Fill in the Blanks

҈҈

矄ᨨ� �
ᨨ犨
ᨨ
ᆠᨨ
鼀�ᆠᨨ矄ᆠᨨ
�矄
�� 
   
ᨨ

ᨨ�ᆠ
鼀�ᆠᨨ矄ᆠᨨ
ᆠ盐矄
 
犨 �
ᨨ ᆠᨨ 矄
�矄






















 
 
犨 �
ᨨ
�矄� �� 
᲼҈黨黨

㜠
҈
㝐├桌 6

MCQs with One or More than One Correct Answer

҈

꾀
 
�矄
ᨨ矄
泐

矄ᨨ ᨨ鼀�ᆠ

 ᆠ� � 矄ᆠᨨ
� �

ᨨᆠ�
ᨨ
ᨨᆠ
 �
҈

鼀
 


泐�ᆠ
ᨨ
ᨨᆠ
 �


ᆠ鼀
᲼҈黨黨

㜠


㝐├矄桌 㡰 
�矄 ᆠ
犨
 
ᨨᆠ�
 
 
泐�ᆠ �矄
҈




鼀 㡰 
 �盐
鼀� � �ᆠ�ᨨ
ᨨ
 
 
�矄




ᆠ㡰 
�鼀�
ᨨ
 
 
�矄
�  㡰 
ᨨᆠ�
�矄
�
 
 
泐�ᆠ ҈
꾀
矄ᨨ ᨨ鼀�ᆠ
矄ᆠᨨ
矄

 �盐
鼀� � �ᆠ�ᨨ
ᨨ 鼀� �盐
�
ᨨ
�矄 
ᨨᆠ矄


矄�ᨨ
犨

 
ᨨᆠ�
 
 
泐�ᆠ
�矄


ᆠ鼀
 
 
� 
�鼀� �矄
ᨨ鼀

� � �泐


ᨨᆠ
 �
f

ᨨ
 
ᨨᆠ�
   
ᨨᆠ
 �
f

ᨨ
 
泐�ᆠ
 ᲼҈黨
黨
㜠


㝐├矄桌 㡰 f

㦨

�
ᆠ鼀
 
fe
㦨
黨
ᆠ鼀 㡰 f

㦨
�

ᆠ鼀
 
fe
㦨
҈

ᆠ鼀 ᆠ㡰 f

㦨
䰘


ᆠ鼀
 
fe
㦨
�
ᆠ鼀 㡰 f

㦨


ᆠ鼀
 
fe
㦨


ᆠ鼀


Topic-6 : Miscellaneous (Mixed Concepts) Problems 1 ҈


��ᨨ
ᔼ

MCQs with One Correct Answer

꾀
�� 
泐
� �
�
�矄矄
鼀�盐鼀
�矄
� ᆠ� 
ᨨ
�矄矄㜠 �
�  ᆠ


 �
ᨨ
� ᆠ� ᆠ
q


 ᆠ
 㡰  
 矄鼀�
 㡰
�  矄��矄

ᨨ
矄
盐 ᆠ�ᨨ
ᨨ
q

 ᨨ


ᆠᨨᆠ
矄├ᆠ
�矄 ᲼

҈҈桌

��ᨨ
ᔼᔼ








ᔼ 矄�泐




鼀





㡰


ᆠ鼀




ᔼ  矄�泐

ᔼ  矄�泐





㡰 矄� ҈
⏌
㡰 㡰 矄� ҈
҈⏌
㡰 ҈ ᆠ㡰 矄�
҈⏌
㡰 㡰 矄� ҈
҈⏌㡰 꾀
ᆠᨨ � 
�矄
�
犨�
犨

m
㦨
҈
㡰
盐ᨨ

�� 
ᨨ 

�
ᆠ鼀

꾀
ᆠᨨ ᆠ
鼀�ᨨ
�矄
ᆠ
҈�
ᆠ鼀
ᨨ
 
犨 
 
 
�鼀�
ᨨ

ᨨᆠ
ᆠ

 
ᨨᨨ鼀
�矄 ᨨ鼀

�
ᆠ鼀
ᨨ犨
 
犨


ᨨᆠ
 �
ᨨ
  鼀�ᨨ
�矄 ᲼


�和桌
㡰 ҈�
ᆠ鼀

㡰

㡰



҈

奠 ᔼ  矄�泐

ᆠ㡰

��ᨨ
ᔼ






鼀




㡰

��ᨨ
ᔼᔼᔼ

҈�
ᆠ鼀





















黨
q





꾀
�� 
鼀
�矄
� �
ᨨ鼀
��ᨨ
ᔼ
ᨨ
ᔼ
 ��盐㡰





 ᆠ�
� 
�
��ᨨ 矄
ᔼ

ᔼᔼ

ᔼᔼᔼ
 
ᔼ









 

矄ᆠ�泐


 �
ᨨ
� ᆠ� ᆠ
q
ᨨ
犨 �ᆠ
 
鼀
盐矄
鼀�矄矄矄
 � �
��ᨨ
ᔼ
�矄
 ᲼



桌

ᆠ㡰 ҈

҈
ᆠ鼀

m㦨҈




�
ᆠ鼀




�
ᆠ鼀
㡰 ҈

ᆠ鼀 졠

ᔼ

WWW.JEEBOOKS.IN

EBD_8341 P-262





�






䰘


Physics

꾀
ᨨ� 
ᨨᆠ
�矄
ᆠ

 
ᆠ 
ᨨ

�矄矄
矄 
ᨨ �盐矄


ᆠ鼀
 
 ᆠ�
� 
҈
�


�矄 ᆠ
ᨨ
�盐 ᲼



和桌 �鼀�
ᨨ鼀
 
矄盐 ᆠ
�矄 㡰


ᆠ鼀 㡰

ᆠ鼀 ᆠ㡰 ҈

ᆠ鼀 㡰 黨
ᆠ鼀 꾀
矄ᨨ盐ᆠ
鼀�矄
矄ᨨ盐 
ᨨ
⼄盐 ᆠ泐





� 矄�
犨



⼄盐 ᆠ泐

�
�
犨�

⼄盐
ᨨ
ᨨᆠ�泐 ᨨ
矄ᨨ盐 
�
犨
㦨
҈�


鼀⏌矄

ᨨᆠ�泐
ᨨ
矄ᨨ盐 
�
� 㦨



鼀⏌矄㡰 ᲼



和桌 㡰 



 㡰 ҈


 ᆠ㡰



 㡰




 꾀
鼀
ᨨ
犨 �
�� 
�矄
� ᆠ� 
ᨨ
�矄矄
�
�  ᆠ ᨨ鼀
�矄矄
ᨨ
�
矄盐ᆠ
 
�
�� 
盐矄
矄盐 矄
ᨨ �  
 ᆠ�ᨨ


ᆠᨨᨨ矄
ᨨ
 
�� 
犨 �ᆠ
犨�
ᆠᨨ鼀 ᨨ盐
ᨨ
�
 ᲼



和桌 㡰 �ᨨ

ᔼ ��ᨨ

盐 㡰 꾀
ᆠᨨᨨ矄
 ᆠ
�

ᆠ㡰 Ӹᨨ犨

졠 �

 㡰 ␔ �
��  ␔ �ᆠ
ᨨ 
ᨨ
 
ᨨᨨ犨� �
矄 �ᆠ
 矄矄
ᨨ矄
ᨨ  ��
�矄矄�ᨨ 䁸

��
ᨨ
ᆠ盐盐
ᨨ
 
矄盐 ᆠ矄 ᲼



和桌 ᨨ
 
 矄矄

矄
��
�
 
��鼀矄


㡰

҈

ᆠ㡰



2



҈





㡰




㡰




꾀

ᔴ


=


ᔴ



 =




졠

҈�
ᆠ鼀







ᆠ鼀

5 True / False ҈҈

�  矄�泐
ᨨ
�� 


�矄 ᆠ
�r
ᨨ鼀
 
 �矄
ᨨ
 ᨨ �
ᆠ泐� �ᆠ
矄ᨨ盐ᆠ
�矄
� 矄泐
ᨨᨨ�ᨨ 
ᨨ
�r
᲼҈黨
҈㜠


㝐├矄桌 ҈


矄� �
矄盐
矄
�� 
�
 
矄├泐
 
�
泐
�矄
᲼҈黨

桌 6

MCQs with One or More than One Correct Answer

҈

�矄矄
ᆠ泐� 矄
ᨨ
⼄盐
�� 

㦨


ᆠ鼀
 
矄鼀  ᆠ�
� 
n
㦨
҈
�

ᆠ
ᨨ

ᨨ�ᨨ 
矄盐 ᆠ 矄
矄 ᨨ犨
�
��盐

ᔴ泐� 
ᔼ
矄


ᨨ

ᆠ泐� 
ᔼᔼ 矄

ᆠᨨ 
ᨨ
 
ᆠ泐� 
ᔼᔼᔼ
矄

ᆠᨨ ᆠ
ᨨ

 ��
ᨨ
ᆠ盐盐
ᨨ
 
犨ᨨ
ᆠ盐
ᨨ矄

矄鼀

㦨
 鼀㡰

ᔼ
H҈

H


 
H

 
 
 矄
ᨨ

ᨨ� 
X ᨨ
 
ᨨᨨ鼀
ᨨ
 
 
ᆠ泐� 矄

矄ᆠ�泐

  ᲼꾀



҈黨桌 ᆠᨨᆠ
矄鼀 矄㡰
�矄⏌



Integer Value Answer

和盐 �� 
ᨨ
�  矄�泐
҈

├␔
鼀

�矄
� ᆠ� 
ᨨ鼀泐
ᨨ
  �
ᆠᨨ 
 矄
ᨨ
ᨨᆠ
 �



ᆠ鼀

ᔼ� ᨨ
 
 �泐 ᨨ矄矄
ᨨ
�� 
盐
ᨨ
 
 矄
 
矄矄盐鼀
 
 
 矄 盐
矄�
�矄
鼀盐ᆠ
矄鼀
 
�矄
ᨨᆠ
 �

 �
�  矄�泐
ᨨ
�� 

�
├␔
鼀




�矄 ᆠ



ᆠ鼀 ᨨ鼀
 
 矄
ᨨ
 
ᨨ 
矄�
�矄
힘힘힘힘힘힘힘힘힘힘
᲼꾀



҈
桌 4

黨





҈ = ҈



Fill in the Blanks

ᔼ
e

 
m



矄ᆠ�泐

 
ᆠ�ᆠ
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WWW.JEEBOOKS.IN

P-263

Ray Optics and Optical Instruments

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Comprehension/Passage Based Questions

8

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鼀�矄

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sin q҈ n

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sin q
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Meta㜠material q2

WWW.JEEBOOKS.IN

EBD_8341 P-264

Physics


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q1

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Meta㜠material

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q2






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⼄盐
ᨨ
R
᲼


㜠

㝐├矄桌 m҈ < m
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m


m




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҈


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A



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+ ҈桌҈⏌



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P
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Q
矄�ᨨ
㡰 ᆠᨨ � � �

��ᨨ 矄
ᨨ
犨 � 矄





处
 
�



处 犨 �ᆠ

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泐
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ml㡰
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l 犨 
l
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处
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b
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矄�矄 � ᨨ
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矄� q =




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턼



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㡰 � 
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盐
ᨨ
b
㡰 � 
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��ᨨ
ᨨ
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鼀矄
 矄鼀�
 ᨨ盐�  
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�矄
盐矄
ᨨ
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 矄鼀� 鼀矄
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ᔼ
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ᨨ
 矄鼀�矄矄�ᨨ

ᨨ鼀
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ᆠ
AC



҈
 
I
矄ᆠ�泐

� 
  矄盐 
�  矄�泐

 
ᨨᆠ盐矄


WWW.JEEBOOKS.IN

P-265

Ray Optics and Optical Instruments

和ᆠ
和
�矄
�盐鼀� 
泐
犨ᨨ
ᨨ� 
矄ᨨ盐ᆠ矄
A
 
B
꾀 ᨨ 
矄ᨨ盐ᆠ
C
矄 矄



鼀
ᨨ
�� 
ᨨ犨矄 ᨨ� 
P
ᨨ
 
矄ᆠ

矄
��盐㡰

� 
AP
�矄
ᨨ鼀
ᨨ
  矄ᆠ
 
 
� 矄
AP

BP  
CP

�
ᨨ 
 

 �矄 ᆠ
AP

BP
 
CP


鼀

҈
�
鼀
 
҈
�
鼀 矄ᆠ�泐


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ᨨ
矄ᨨ盐ᆠ矄
A
  B

黨

犨矄

 
҈

犨矄
矄ᆠ�泐

 鼀
ᨨ鼀

ᔴ
�矄
ᨨ 60° P �  矄�泐



犨矄⏌鼀

A× 60° ᔴᆠ盐
 
�  矄�泐
 P
ᨨ
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矄ᆠ
᲼҈黨


㜠
�
㝐├矄桌


䰘


B




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ᨨᆠ
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҈
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ᨨ
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҈

ᆠ鼀

꾀
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矄
ᨨ
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
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� 
҈
�
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ᨨ
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ᨨ
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矄
ᨨ鼀
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ᨨ
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҈
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桌
黨
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ᨨ
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ᨨ
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� 
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鼀� �鼀盐鼀
盐
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ᨨ
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ᨨ
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ᨨ
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ᆠ矄
ᨨ
  ᲼҈黨䰘黨桌 ᨨᆠ├


S

Answer Key Topic-1 : Plane Mirror, Spherical Mirror and Reflection of Light ҈
㡰 ҈҈
㡰



ᆠ㡰 ҈

㡰


ᆠ㡰 ҈
҈㡰



㡰 �
ᆠ㡰 ҈

҈
�㡰 ҈�
㡰



ᆠ㡰 䰘
ᆠ㡰 ҈

ᆠ

㡰 ҈䰘
ᆠ㡰



㡰

黨
㡰

҈

㡰

Topic-2 : Refraction of Light at Plane Surface and Total Internal Reflection ҈
҈҈

҈




㡰

ᆠ㡰 㡰 ҈

ᆠ㡰 㡰


㡰 




处

�
맔
҈
҈
㡰


҈


҈


㡰

ᆠ㡰 �
㡰

ᆠ㡰 ҈

㡰 ҈�
㡰 ҈


㡰



㡰
�

㡰




맔
҈

鼀⏌矄



맔
҈
䰘鼀㡰 


ᆠ
㡰 


㡰 䰘
ᆠ㡰 㡰 ҈䰘
㡰 ҈�
㡰
䰘
黨
㡰 ᆠ

㡰 



ᆠ㡰



҈




�


㡰 㡰 �
㡰 㡰

黨
㡰 ҈

ᆠ㡰 ҈黨
㡰


㡰
黨





处㡰

Topic-3 : Refraction at Curved Surface Lenses and Power of Lens ҈
҈҈

҈
҈

҈
�

�



㡰

㡰 
㡰

㡰 ҈

㡰 ҈
㡰 ҈

㡰


㡰

ᆠ㡰


㡰 

㡰 
㡰 



㡰




҈
�鼀



�鼀㡰

㡰 �҈

ᆠ㡰 �


㡰 �
꾀㜠

⼄



矄錐
㜠⼄錐
ᔴ㜠

⼄



矄錐
㜠

⼄



矄

㡰 �
㡰 ҈�
㡰
�
㡰 �


ᆠ鼀㡰


㡰 �



㡰

㡰 ҈

ᆠ㡰


㡰 

҈�ᆠ鼀㡰
�
㡰 ��


㡰 䰘
㡰

ᨨ盐 矄㡰 黨
㡰 ҈䰘
㡰 ҈

㡰 ҈黨
ᆠ㡰
䰘
㡰


㡰
黨
ᆠ㡰 䰘
ᆠ㡰 

䰘㡰 黨


盐錐


矄錐
䰘

㡰


꾀㜠

錐
㜠⼄
矄
錐
ᔴ㜠



㜠⼄
矄

ᆠ㡰 ҈

㡰


㡰 


㡰




㡰
黨


㡰 㡰 ᆠ㡰 
䰘

҈黨㡰 㡰

Topic-4 : Prism and Dispersion of Light ҈
㡰 ҈҈
㡰
҈
ᆠ㡰



ᆠ㡰 ҈

ᆠ㡰


㡰


ᆠ㡰 ҈
ᆠ㡰

㡰



ᆠ㡰 ҈

㡰

�
㡰 ҈�

㡰



ᆠ㡰 ҈

҈
�㡰

䰘
ᆠ㡰 ҈䰘
҈� 㡰



ᆠ㡰 ҈



㡰

黨
㡰 ҈

㡰 ҈黨
盐錐




ᆠ

㡰



㡰

黨
�
㡰 ҈





㡰

Topic-5 : Optical Instruments ҈
㡰

㡰 
㡰 ҈҈
矄鼀㡰҈



ᆠ
㡰 ҈ 㡰



ᆠ㡰

�
ᆠ㡰



㡰

䰘
ᆠ㡰

Topic-6 : Miscellaneous (Mixed Concepts) Problems

҈
ᆠ㡰



㡰

҈


ᆠ鼀㡰 ҈҈
盐錐


㡰


ᆠ㡰 ҈

盐錐



㡰

�
ᆠ㡰



ᆠ㡰

҈

㡰 ҈





ᆠ㡰҈�
㡰

䰘
ᆠ㡰



҈
㡰

҈





ᆠ㡰҈䰘
㡰

黨


me 㡰 m
e


҈

ᆠ㡰

҈黨
ᆠ㡰

WWW.JEEBOOKS.IN

EBD_8341 P-266

Physics

Wave Optics

24

Topic-1 : Wavefront, Interference of Light, Coherent and Incoherent Sources 1 ҈


MCQs with One Correct Answer

ᔼ
 
��盐
ᨨ犨


 
턼

犨ᨨ
⼄盐泐
�  矄 ᆠᨨ  
矄ᨨ盐ᆠ矄
鼀�� �
��ᨨ
ᨨ
犨 �



鼀



矄�ᨨ
犨

 
턼
�矄
�
鼀
 
 
 矄
ᨨ
 �矄
 
ᨨ
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ᨨ
턼
泐
黨
º

꾀


 
ᔴ

 
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WWW.JEEBOOKS.IN

P-267

Wave Optics





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6

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矄
泐
 矄鼀
�矄 ᆠ
ἘἘ
矄
矄 ᨨ犨


� �矄
ᨨ� 
ᨨ
  矄ᆠ
犨�





᲼㝐�


҈桌  和҈ 和


和ᆠ



㡰 ᨨ� 矄 ᆠ㡰 矄鼀�㜠ᆠ�ᆠ矄 ҈䰘


҈



҈黨








҈







㡰 矄�� 
� 矄 㡰 ᆠᨨ ᆠ �ᆠ
ᆠ�ᆠ矄


�

l
l
�矄
犨 �  ᨨ
�� 
盐矄㡰
 
 ᆠ�
� 
m
㦨
҈
�
�矄
� 矄 犨
ᨨ 
ᨨ
 
矄�矄
 
 
矄ᆠ
�
Ӹᨨ盐 �矄
ᨨ盐 矄�
 �鼀 

꾀

ᨨ� 
ᨨ
 
矄ᆠ
⼄盐��矄 
ᨨ鼀  
矄�矄

 
�ᨨ
ᨨ
 
�  矄��矄
 ᨨ
 
 
  � ᨨ盐ᆠ�ᨨ
ᨨ
 
�矄矄

�矄

᲼㝐�
졠 � 
꾀�

�



҈桌 ᆠ㡰



҈ 㡰



 㡰



҈ 㡰 ҈



ᔼ
 
Ӹᨨ盐 �矄
ᨨ盐
矄�
 �鼀 
盐矄� �
 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ
�� 
ᨨ
犨 �
l,  

�  ᆠ �
鼀矄
ᨨ

� �
n㡰
ᆠᨨ矄ᨨ � �
ᨨ
 泐
ᨨ� 
� � 
 
├
�  矄�泐
�矄 ᲼꾀



҈桌 l l l l 㡰 
n + ҈㡰

㡰

n +҈㡰

ᆠ㡰

n + ҈㡰

㡰

n +҈㡰 ҈



Ӹᨨ盐 �矄
ᨨ盐
矄�
 �鼀 
�矄
ᆠ�
ᨨ盐
泐
盐矄� � �


 
盐
�� 

ᨨ 
ᆠᨨᨨ


�鼀


� � 犨� 矄
ᆠᨨ

G

R
 
B

矄ᆠ�泐


᲼

҈
桌 㡰 G
>
B
>
R 㡰 B
>
G
>
R ᆠ㡰 R
>
B
>
G 㡰 R
>
G
>
B ᔼ
Ӹᨨ盐 �矄
ᨨ盐
矄�
 �鼀 
�  矄�泐


ᨨ� 
�矄
҈⏌
㡰
ᨨ
 
鼀 �鼀盐鼀
�  矄�泐

꾀 �盐
ᨨ矄��ᨨ
ᨨ
 �矄
ᨨ�  �矄 ᲼


�和桌 㡰 矄� ҈l⏌㡰 㡰 矄� ҈l⏌
㡰 ᆠ㡰 矄� ҈l⏌㡰 㡰 矄� ҈l⏌
㡰 㝐ᨨ ᨨᆠ ᨨ鼀�ᆠ
�� 
ᨨ
犨 �




鼀
 
�


鼀 
� ᆠ� 
矄�鼀盐 ᨨ盐矄泐
 
ᨨ鼀泐
ᨨ
ᨨ盐
矄�矄 盐矄
犨 ᨨ矄
矄�矄
矄�ᨨ
�矄


҈
鼀鼀
 
矄ᆠ

�矄 ᆠ
�矄
҈鼀

�矄 ᆠ
犨
矄
ᨨ
ᨨ
├ 矄矄 犨�
 ᲼



和桌 㡰

鼀鼀 㡰 �


鼀鼀 ᆠ㡰 ҈

鼀鼀 㡰


鼀鼀 ᔼ
 
�
ᨨ盐㜠矄�
 �鼀 

犨 

�矄矄㜠  ᆠ�
� 
҈
�㡰
ᨨ
 �ᆠ├ 矄矄

�矄
� ᨨ盐ᆠ
�
 

꾀
 �
�矄矄

ᨨ
 �ᆠ├ 矄矄
�矄



ᨨ
ᨨ 
ᨨ
 
�  � �
鼀矄
犨㜠 � 
l 㡰

  �  矄�泐

 
ᨨ矄��ᨨ
犨 
 
ᆠ 
鼀 �鼀盐鼀 ᨨᆠᆠ盐
�ᨨ盐矄泐
鼀� 矄
盐 ᆠ  �


鼀� �鼀盐鼀  �ᆠ├ 矄矄
ᨨ
 
�矄矄㜠
�矄 ᲼



和桌 㡰
l 㡰
l ⏌ ᆠ㡰 l ⏌ 㡰 l

ᔼ

Ӹᨨ盐 �矄
ᨨ盐
矄�
 �鼀 

҈

� �矄

ᨨ矄 ᨨ

ᨨ鼀
�

ᆠ�
矄�鼀 
ᨨ
 
矄ᆠ
犨 
��  ᨨ
犨 �




鼀
�矄
盐矄

ᔼ
 
犨 �
ᨨ
�� 
�矄 ᆠ  �
ᨨ




鼀

盐鼀
ᨨ
� �矄
ᨨ矄
�
  矄鼀
矄�鼀 
ᨨ
 
矄ᆠ
�矄
��
泐 ᲼


҈和桌 㡰 ҈
㡰 ҈
ᆠ㡰

㡰 



ᔼ

ᨨ盐
矄�
 �鼀 
� 矄
ᨨ
├� �
矄�矄
ᨨ
⼄盐 犨� 矄

ᨨ 
矄�
�矄
鼀
犨�ᆠ
矄
犨�
矄
 
ᨨ 



�

 
�   ᆠ


᲼



和桌 㡰  
�  矄��矄
ᨨ
ᨨ
 
鼀 �鼀
 
 
鼀� �鼀 � ᆠ矄 㡰  
�  矄�泐
ᨨ
 
鼀 �鼀
� ᆠ矄矄
 
 
鼀� �鼀 矄
ᨨ
�  矄�泐 ᆠ㡰  
�  矄�泐
ᨨ
 
鼀 �鼀
ᆠ矄矄
 
 
ᨨ
  鼀� �鼀
� ᆠ矄矄 㡰  
�  矄�泐
ᨨ
 
鼀 �鼀
ᆠ矄矄
 
 
鼀� �鼀 矄
ᨨ
�  矄�泐
�
ᔼ

�   ᆠ
 �鼀 
矄�鼀�
ᨨ
Ӹᨨ盐 �矄
ᨨ盐㜠 矄�
 �鼀 

 
矄�矄
S҈
 
S


�盐鼀� 
犨� ᆠᨨ  
鼀�ᆠᨨ犨
矄ᨨ盐ᆠ矄

ᆠ
ᨨ
⼄盐 ᆠ泐
҈






矄ᨨ盐ᆠ矄

矄泐 ᆠ ᨨ �
ᨨ

ᨨ
 矄 �  ᆠ


矄�矄

矄
泐

�矄 ᆠ

㦨
҈�


鼀


�  矄�泐
I
q㡰
�矄
鼀矄盐
矄

盐 ᆠ�ᨨ
ᨨ
q

犨  q
�矄
 � 
矄
矄 ᨨ犨

ᔼ
I

�矄
 
鼀 �鼀盐鼀
�  矄�泐

 

I
q㡰
ᨨ

£ q £ 黨
°
�矄
��
泐 ᲼҈黨黨�和桌 㡰

I q㡰 = I
⏌
fᨨr q = 
°

㡰

I q㡰 = I
⏌
fᨨr q = 黨
°

ᆠ㡰

I q㡰 = I
fᨨr q =
°

㡰

I q㡰
�矄
ᆠᨨ 矄 
ᨨ


和҈ ⏌


⏌
和


盐矄
ᨨ
q



ᔼ
Ӹᨨ盐 �矄
ᨨ盐㜠矄�
 �鼀 

 
矄�ᨨ
犨

 
矄�矄
�矄

 
 
�矄 ᆠ
犨
 
矄�矄
   
矄ᆠ
�矄
ᨨ盐


� �
犨�
�矄


᲼҈黨
҈㜠


㝐├矄桌 㡰 盐 ᆠ  �
㡰 
ᆠ㡰 ᨨ盐 㡰 ⼄盐盐 2

Integer Value Answer


䰘
꾀
Ӹᨨ盐 �Ἐ矄
ᨨ盐
矄�
�   ᆠ
 �鼀 
犨�
矄�矄
S҈
 
S

�矄
�鼀鼀矄
�
犨
 ᆠ�
� 
㦨 㡰
矄  矄 ᨨ犨
�
 
��盐

 ᨨ矄��ᨨ 矄
ᨨ
鼀 �鼀盐鼀
ᨨ

 
矄盐 ᆠ
ᨨ
犨
 S ��
泐
x

㦨
p
m
l

 d 꾀� x d


犨 
l
�矄
  犨 �
ᨨ
�� 
�
� d  ᆠ�
� 
㦨
҈㡰


d
�矄 S ␔  
矄�ᨨ
犨

 
矄�矄
 
m
�矄


� �


盐
ᨨ
p
�矄 ᲼꾀



҈�桌 ҈




WWW.JEEBOOKS.IN

EBD_8341 P-270

Physics

Numeric Answer

3





黨


꾀
泐ᨨ盐 �矄
ᨨ盐㜠矄�
 �鼀 
�矄
 ᨨ鼀
盐矄� � 鼀ᨨ ᨨᆠᨨ鼀�ᆠ
�� 
ᨨ
犨 �
l


�

 矄�泐
ᨨ �� 


ᨨ� 
ᨨ
 
矄ᆠ

犨 
 

�  ᆠ
�矄 l

�矄
⧤
盐 �矄


�  矄�泐
ᨨ
�� 


ᨨ� 
犨 
  l

⧤ 
�  ᆠ
�矄

�矄
��
泐


犨 

�矄

� 㜠
҈
�


盐
ᨨ
n
�矄
힘힘힘힘힘힘
᲼㝐�

和










ᔼᔼ㡰桌 ᔼ

Ӹᨨ盐 �矄
ᨨ盐
矄�
 �鼀 
҈�
� �矄

ᨨ矄 ᨨ

矄鼀
ᨨ�ᨨ
ᨨ
 
矄ᆠ
犨 
�� 
ᨨ
犨 � �


鼀
�矄
盐矄


� �矄

ᨨ矄
ᨨ
 
矄鼀 矄ᆠ�ᨨ
ᨨ
 
矄ᆠ
犨 
 ᨨ 
�� 
矄ᨨ盐ᆠ
ᨨ 犨 �
l
�矄
盐矄


 
盐
ᨨ
l
�矄
�
鼀㡰 힘힘힘힘힘힘







᲼㝐�

黨






ᔼᔼ桌 Fill in the Blanks

4 



ᔼ
Ӹᨨ盐 �矄
ᨨ盐㜠矄�
 �鼀 

 
犨ᨨ
矄�矄
ᆠ
矄 ᆠᨨ  
矄ᨨ盐ᆠ矄
ᨨ
⼄盐
鼀�盐
�A
 
ᨨ
犨 � � l 

ᔼ
 ᨨ 
 �鼀 
犨�
 
矄鼀
矄㜠盐
 
犨ᨨ 矄�矄

矄ᨨ盐ᆠ矄
ᨨ
⼄盐
鼀�盐
�A
 
犨 � � l 

盐

� ᆠᨨ  


�ᨨ
ᨨ
 
�  矄�泐
ᨨ
��  
 
鼀�ᨨ� 
ᨨ
 
矄ᆠ
�
 
�矄
ᆠ矄
ᨨ
 
�
  矄ᆠᨨ 
ᆠ矄
�矄













᲼҈黨


㜠


㝐├矄桌 True / False

5 ҈






ᔼ

Ӹᨨ盐 �矄
ᨨ盐
矄�
 �鼀 
 ᨨ鼀
犨�

矄ᨨ盐ᆠ ᨨ
犨 �
�� 

ᨨ 泐
ᆠ├
 
犨 �
� �矄

ᨨ矄
᲼҈黨
䰘
㜠


㝐├矄桌


犨ᨨ
矄�矄
�

Ӹᨨ盐 �矄
ᨨ盐
矄�
 �鼀 
 �盐鼀� 
泐
犨ᨨ
�  
矄ᨨ�盐鼀
鼀矄
鼀�� �
�� 
ᨨ  
矄鼀
犨 �

ᨨ
�   ᆠ

犨�
 ᨨ矄
ᨨ
 
矄ᆠ
᲼҈黨

㜠


㝐├矄桌

MCQs with One or More than One Correct Answer

6

 졠 和҈

和






和ᆠ


␔ �
ᆠᨨ 盐ᆠ� �
 
Ӹᨨ盐 �矄
ᨨ盐
矄�
 �鼀 

 矄盐 
ᆠ
 
犨ᨨ
矄�矄
犨�

�
ᨨ⼄盐

�

 
㜠泐
 
ᆠᨨ � � �
犨ᨨ
矄鼀
ᨨ矄
 
ᆠ
矄
犨ᨨ ᆠᨨ  
ᨨ� 
矄ᨨ盐ᆠ矄
和҈

和
㡰
鼀�� �
�� 
ᨨ
犨 �



鼀


矄盐 
鼀�矄├ 泐
ᆠ
 
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

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ᔴ
 






WWW.JEEBOOKS.IN

P-271

Wave Optics

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EBD_8341 P-272

Physics

fig. Calculate the power (in watts) received at the focal spot F of the lens. The lens is symmetrically placed with respect to the apertures. Assume that 10奠 of the power received by each aperture goes in the original direction and is brought to the focal spot. [1989 - 8 Mark]

A

F

B

45. A beam of light consisting of two wavelengths, 6500Å and 5200Å, is used obtain interference fringes in a Young’s double slit experiment : [1985 - 6 Marks] (i) Find the distance of the third bright fringe on the screen from the central maximum for wavelength 6500Å. (ii) What is the least distance from the central maximum where the bright fringes due to both the wavelengths coincide? The distance between the slits is 2 mm and the distance between the plane of the slits and the screen is 120 cm.

Topic-3 : Diffraction, Polarisation of Light and Resolving Power 1 1.

2.

3.

4.

5.

MCQs with One Correct Answer

A beam of plane polarised light of large cross-sectional area and uniform intensity of 3.3 Wm–2 falls normally on a polariser (cross sectional area 3 × 10–4 m2) which rotates about its axis with an angular speed of 31.4 rad/s. The energy of light passing through the polariser per revolution, is close to : [Main Sep. 04, 2020 (I)] –5 (a) 1.0 × 10 J (b) 1.0 × 10–4 J –4 (c) 1.5 × 10 J (d) 5.0 × 10–4 J The aperture diameter of telescope is 5m. The separation between the moon and the earth is 4 × 105 km. With light of wavelength of 5500 Å, the minimum separation between objects on the surface of moon, so that they are just resolved, is close to: [Main 9 Jan. 2020 I] (a) 60 m (b) 20 m (c) 200 m (d) 600 m A polarizer - analyser set is adjusted such that the intensity of light coming out of the analyser is just 10奠 of the original intensity. Assuming that the polarizer analyser set does not absorb any light, the angle by which the analyser need to be rotated further to reduce the output intensity to be zero, is: [Main 7 Jan. 2020 I] (a) 71.6° (b) 18.4° (c) 90° (d) 45° The value of numerical aperature of the objective lens of a microscope is 1.25. If light of wavelength 5000 Å is used, the minimum separation between two points, to be seen as distinct, will be : [Main 12 April 2019 I] (a) 0.24 µm (b) 0.38 µm (c) 0.12 µm (d) 0.48 µm A system of three polarizers P1, P2, P3 is set up such that the pass axis of P3 is crossed with respect to that of P1. The pass axis of P2 is inclined at 60o to the pass axis of P3. When a beam of unpolarized light of intensity Io is incident on P1, the intensity of light transmitted by the three polarizers is I. The ratio (Io/I) equals (nearly) : [Main 12 April 2019 II]

6.

7.

8.

9.

(a) 5.33 (b) 16.00 (c) 10.67 (d) 1.80 Diameter of the objective lens of a telescope is 250 cm. For light of wavelength 600 nm. Coming from a distant object, the limit of resolution of the telescope is close to: [Main 9 April 2019 II] (a) 1.5 ´ 10 -7 rad(b) 2.0 ´ 10-7 rad (c) 3.0 ´ 10 -7 rad (d) 4.5 ´ 10-7 rad Calculate the limit of resolution of a telescope objective having a diameter of 200 cm, if it has to detect light of wavelength 500 nm coming from a star. [Main 8 April 2019 II] (a) 305 × 10–9 radian (b) 610 × 10–9 radian (c) 152.5 × 10–9 radian (d) 457.5 × 10–9 radian In a double-slit experiment, green light (5303Å) falls on a double slit having a separation of 19.44 µm and a width of 4.05 µm. The number of bright fringes between the first and the second diffraction minima is : [Main 11 Jan 2019 II] (a) 10 (b) 05 (c) 04 (d) 09 Consider a tank made of glass (refractive index 1.5) with a thick bottom. It is filled with a liquid of refractive index m. A student finds that, irrespective of what the incident angle i (see figure) is for a beam of light entering the liquid, the light reflected from the liquid glass interface is never completely polarized. For this to happen, the minimum value of m is: [Main 9 Jan. 2019 I]

(a)

5 3

(b)

3 5

(c)

5 3

(d)

4 3

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P-273

Wave Optics

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EBD_8341 P-274

23.

24.

Physics

Abeam of unpolarised light of intensity I0 is passed through a polaroidAand then through another polaroid B which is oriented so that its principal plane makes an angle of 45° relative to that of A. The intensity of the emergent light is [Main 2013] (a) I0 (b) I0/2 (c) I0/4 (d) I0/8 A person lives in a high-rise building on the bank of a river 50 m wide. Across the river is a well lit tower of height 40 m. When the person, who is at a height of 10 m, looks through a polarizer at an appropriate angle at light of the tower reflecting from the river surface, he notes that intensity of light coming from distance X from his building is the least and this corresponds to the light coming from light bulbs at height 'Y' on the tower. The values of X and Y are 4 ) 3 [Main Online April 9, 2013]

respectively close to (refractive index of water ;

L

40 m

10 m

Y

X

50 m

25.

(a) that the central maximum is narrower (b) more number of fringes (c) less number of fringes (d) no diffraction pattern 26. Consider Fraunhoffer diffraction pattern obtained with a single slit illuminated at normal incidence. At the angular position of the first diffraction minimum the phase difference (in radians) between the wavelets from the opposite edges of the slit is [1995S] (b) p / 2 (c) 2 p (a) p / 4 (d) p 27. A beam of light of wave length 600 nm from a distance source falls on a single slit 1 mm wide and a resulting diffraction pattern is observed on a screen 2m away. The distance between the first dark fringes on either side of [1994 - 1 Mark] central bright fringe is (a) 1.2 cm (b) 1.2 mm (c) 2.4 cm (d) 2.4 mm 28. A parallel monochromatic beam of light is incident normally on a narrow slit. A diffraction pattern is formed on a screen placed perpendicular to the direction of the incident beam. At the first minimum of the diffraction pattern, the phase difference between the rays coming from the two edges of the slit is [1998 - 2 Marks] (a) 0 (b) p/2 (c) p (d) 2p

(a) 25 m, 10 m (b) 13 m, 27 m (c) 22 m, 13 m (d) 17 m, 20 m Yellow light is used in a single slit diffraction experiment with slit width of 0.6 mm. If yellow light is replaced by X– rays, then the observed pattern will reveal, [1999S - 2 Marks]

3

Numeric Answer

29. Orange light of wavelength 6000 × 10–10 m illuminates a single slit of width 0.6 × 10–4 m. The maximum possible number of diffraction minima produced on both sides of the central maximum is _____________. [Main Sep. 04, 2020 (II)]

Topic-4 : Miscellaneous (Mixed Concepts) Problems 4 1.

A point source emits sound equally in all directions in a non-absorbing medium. Two points P and Q are at a distance of 9 meters and 25 meters respectively from the source. The ratio of amplitudes of the waves at P and Q is ............ [1989 - 2 Marks] 7

2.

8

Fill in the Blanks

Match the Following

A simple telescope used to view distant objects has eyepiece and objective lens of focal lengths fe and fo, respectively. Then [2006 - 6M] Column I Column II (A) Intensity of light received (p) Radius of aperture by lens (B) Angular magnification (q) Dispersion of lens (C) Length of telescope (r) Focal length of objective lens and eyepiece lens (D) Sharpness of image (s) Spherical aberration

Comprehension/Passage Based Questions

Passage The figure shows a surface XY separating two transparent media, medium-1 and medium-2. The line ab and cd represent waveforms of a light wave travelling in medium-1 and incident on XY. The lines ef and gh represent wavefronts of the light wave in medium-2 after refraction. [2007] b d

X

a c

medium-1 Y f h medium-2

e g

3.

Light travels as a (a) parallel beam in each medium (b) convergent beam in each medium (c) divergent beam in each medium (d) divergent beam in one medium and convergent beam in the other medium.

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P-275

Wave Optics

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EBD_8341 P-276

Physics

Dual Nature of Radiation and Matter

25

Topic-1 : Matter Waves, Cathode and Positive Rays 1 ҈






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WWW.JEEBOOKS.IN

P-277

Dual Nature of Radiation and Matter

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WWW.JEEBOOKS.IN

EBD_8341 P-278

Physics

Topic-2 : Photon, Photoelectric Effect X-rays and Davisson-Germer Experiment 1 ҈


MCQs with One Correct Answer

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WWW.JEEBOOKS.IN

P-279

Dual Nature of Radiation and Matter

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l m鼀㡰 

ᨨ㡰






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桌 㡰



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
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᲼Main

҈�桌

WWW.JEEBOOKS.IN

EBD_8341 P-280

Physics

�矄㜠ᔼᔼ �矄㜠ᔼ 꾀



 ᆠ├㜠 �㡰


�ᆠ
盐
ᨨ
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쟼 �鼀  


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�矄ᆠ
 �泐
矄
ᨨ
ᨨ鼀





 �鼀 
ᔴ


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␔
盐
ᨨ
ᆠᨨ







 �鼀  �㡰

和盐ᆠ盐
ᨨ
ᨨ鼀
















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҈




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桌 㡰

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҈

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矄矄
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u


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├�
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  �  
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� �
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fq
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fr
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
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᲼Main


黨桌 ᔼ

ᔼ  ⼄








㡰


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l ᨨ
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矄鼀 矄
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�矄 ᲼


䰘桌

WWW.JEEBOOKS.IN

P-281

Dual Nature of Radiation and Matter

㡰

l
=


mcl
h
m
c
l 

㡰

l
=


h mc

㡰 l
= l h
҈
Ka
犨 �
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犨�
犨 �

l. ᲼


�和桌 㡰 Z
㦨

㡰 Z
㦨

ᆠ㡰 Z
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

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和桌 ᆠ㡰

l
=

ᔼ ᔴ

 꾀 








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 
B
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矄ᆠᨨ 
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㡰 � ´ ҈

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lC
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ᔼ


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lC
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├
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㜠泐
盐

⧤㜠矄 
ᆠᨨ 矄
ᨨ
盐 �矄

䰘

�
├  �泐

㜠泐矄
鼀�
泐
 
盐
ᆠᨨ �
ᨨ 泐
᲼



和桌 㡰 
ᆠᨨ � 盐ᨨ盐矄
㜠泐
矄ᆠ盐鼀
鼀矄矄 盐 �㡰
犨� 
鼀� �鼀盐鼀
犨 �
ᨨ


҈��处 㡰 
ᆠᨨ � 盐ᨨ盐矄
㜠泐
矄ᆠ盐鼀
鼀矄矄 盐 �㡰
犨� 
犨 � 矄 ᆠ㡰  
ᆠ ᆠ�矄�ᆠ
㜠泐
矄ᆠ盐鼀
ᨨ
盐 �矄

㡰 
ᆠᨨ � 盐ᨨ盐矄
㜠泐
矄ᆠ盐鼀
鼀矄矄 盐 �㡰
犨� 
鼀� �鼀盐鼀
犨 �
ᨨ


҈��处
 
  ᆠ ᆠ�矄�ᆠ
㜠泐
矄ᆠ盐鼀
ᨨ
盐 �矄





犨ᨨ├
盐 ᆠ�ᨨ
ᨨ

矄盐矄 ᆠ
�矄







ᨨ �矄 犨 �
ᨨ
�� 
 
ᆠ
ᆠ盐矄
 ᨨᨨᆠᨨ
鼀�矄矄�ᨨ

ᨨ鼀
 �矄
矄盐矄 ᆠ
�矄
ᨨ �鼀泐
᲼҈黨黨
和
㜠


㝐├矄桌 㡰 �


鼀 㡰



鼀 ᆠ㡰 ҈

鼀 㡰



鼀 䰘
㜠泐矄

ᨨ盐ᆠ
�

㜠泐
盐
ᨨ� �


��

ᆠᆠ� �
ᨨ�


犨 �
ᨨ
 
ᆠᨨ � 盐ᨨ盐矄 㜠泐矄
矄
盐矄
ᨨ鼀 ᲼҈黨黨
和
㜠


㝐├矄桌 㡰
ᨨ ¥ 㡰 l 鼀� ᨨ ¥ 犨  l 鼀� >
ᆠ㡰
ᨨ l 鼀 犨  l 鼀 < ¥ 㡰 l 鼀� ᨨ l 鼀 犨 
< l 鼀� < l 鼀 < ¥ 


鼀 �鼀盐鼀
├� �ᆠ
 �泐
ᨨ
 ᨨᨨᆠᨨ 矄
鼀� ᨨ鼀

矄盐 ᆠ
犨 
 ᨨᨨ 矄
ᨨ
 �泐




ᨨ
�
�矄




矄ᨨ� �
ᨨ �

�
ᨨ

�矄 ᲼҈黨黨䰘
㜠
҈
㝐├桌 㡰
㡰
ᆠ㡰
㡰 ҈
黨
꾀
 �泐
ᨨ






�矄
⼄盐�
ᨨ
鼀ᨨ
ᨨ 
ᨨ
  ᆠᨨ 矄
ᨨ鼀

盐
�盐鼀
ᨨ鼀


 �泐
�
㡰 ⼄盐�
ᨨ
鼀ᨨ
ᨨ
 
ᆠᨨ 矄
ᨨ鼀

盐
�盐鼀 ᨨ鼀
�矄 ᲼҈黨黨�和桌 㡰 


㡰
黨

ᆠ㡰 �҈

㡰 䰘黨





㜠泐
鼀
ᆠᨨ鼀� �
ᨨ鼀

㜠泐
盐
犨�
 㡰 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ ᲼҈黨
�
㜠


㝐├矄桌 㡰 � �

犨 � 矄
矄鼀
 

ᆠ�
鼀 �鼀盐鼀 犨 � ᆠ㡰 � �

犨 � 矄
�
 

ᆠ�
鼀� �鼀盐鼀 犨 � 㡰 � �

犨 � 矄
泐� �
犨

鼀� �鼀盐鼀  

鼀 �鼀盐鼀
犨 �
҈

矄 ᨨ矄
犨 �
ᨨ
㜠泐矄
鼀�
ᨨ鼀

㜠泐 盐
 矄
ᨨ

᲼҈黨


㜠

㝐├矄桌 㡰
 
ᆠ盐 
�
 
盐 㡰
 
ᨨ�
�
ᨨ
 
盐 ᆠ㡰
 
盐
ᨨ
 
�矄
�
盐 㡰
 
ᨨ鼀�ᆠ
盐鼀
ᨨ
 
�
鼀� 2

Integer Value Answer




꾀
 ᆠ泐
 ᆠ� �
鼀�ᨨ
ᨨ
鼀矄矄
㝐
鼀ᨨ盐 
ᨨ
 矄� �
ᆠᨨ 矄�盐矄

矄� �㜠鼀矄矄
矄泐矄鼀
ᨨ
 �盐 ⼄盐 ᆠ泐

矄盐ᆠ
 
㦨
҈


m㜠

犨�
h 矄
 ᆠ├矄
ᆠᨨ 矄 
N
 ᨨᨨ 矄
ᨨ
犨 �
l
㦨

p
맔
҈
㜠

鼀
矄�├
 
鼀�ᨨ 矄�鼀盐 ᨨ盐矄泐

ᨨ鼀
� ᆠ� ᆠ
矄盐ᆠ
 
 
鼀�ᨨ �矄
�矄ᆠ
泐
҈m鼀

ᔼ
 
盐
ᨨ
N
�矄
x
맔
҈
҈


 

 
盐
ᨨ
x
�矄
힘힘힘힘힘힘 ᲼ᔴᨨ 矄�
 
矄� �
矄
鼀矄矄矄矄桌







᲼꾀



҈黨桌




犨ᨨ├
盐 ᆠ�ᨨ 矄
ᨨ
和�
 
和ᨨ�盐鼀





 


 

矄ᆠ�泐


�ᨨ
ᨨ
 
矄ᨨ
ᨨ
 
矄ᨨ� � ᨨ �
矄盐矄
⼄盐 ᆠ泐
ᨨ
ᨨ
和�
ᨨ
 
ᨨ
和ᨨ�盐鼀 �矄 ᲼꾀



҈㜠ᔼ桌

WWW.JEEBOOKS.IN

EBD_8341 P-282






Physics

꾀
矄�
矄 
ᨨ
�盐矄
҈
ᆠ鼀
 
犨ᨨ├
盐 ᆠ�ᨨ


䰘

�矄 矄盐矄 
ᨨ鼀

� 矄盐� �
 
�
矄ᆠ

ᔼ
�矄 盐 
ᆠᨨ � 盐ᨨ盐矄
�盐鼀� �ᨨ
ᨨ




鼀
犨 �
�� 
꾀矄
 ᨨᨨᆠᨨ 矄

鼀�

 
矄 
�矄
ᆠ �  
ᆠ⼄盐�矄

ᨨ �


鼀 �鼀盐鼀
盐鼀
ᨨ  ᨨᨨᆠᨨ 矄
鼀�
ᨨ鼀
 
矄 
�矄
꾀
´ ҈


犨  ҈


꾀


҈
㡰


盐
ᨨ

�
�矄 ᲼

҈҈桌 3


�








䰘


Numeric Answer

꾀
鼀
ᨨ
ᆠᨨ 矄
ᨨ
 �泐
쟼
矄ᆠ矄
ᨨ鼀

� � �
ᨨ鼀�ᆠ
矄ᆠ� �
ᨨ
҈处


�矄
鼀 �鼀盐鼀
�  矄�泐 ᨨᆠᆠ盐矄

q
㦨


º


쟼
�
㡰
�矄
힘힘힘힘힘힘
 ├
ᆠᨨ 矄 
h
㦨





맔
҈


矄

҈

㦨
҈


맔
҈
҈黨

ᆠᨨ
鼀矄矄
m
㦨
黨
҈
맔
҈
҈
├�㡰᲼㝐�
和


�






ᔼ㡰桌


矄盐 ᆠ
ᨨ

鼀
�矄
�盐鼀� 
 泐
犨�  ᨨᨨ 矄
ᨨ
 ��矄
E҈
㦨



 
E

㦨


�

矄ᆠ�泐



�ᨨ
ᨨ
鼀 �鼀盐鼀
矄矄
ᨨ
 
 ᨨᨨᆠᨨ 矄
鼀� �
 
犨ᨨ
ᆠ矄矄
�矄




犨ᨨ├
盐 ᆠ�ᨨ
ᨨ
 
鼀
�
㡰 �矄
힘힘힘힘힘힘힘힘힘힘힘힘힘힘힘
᲼㝐�
和


�






ᔼᔼ㡰桌 ␔ 
��ᨨ
ᨨ
犨 �
l
�矄
盐矄
ᨨ
�盐鼀� 
 鼀�ᆠ
矄盐 ᆠ

 
矄ᨨ� �
ᨨ �
�矄
V

␔ 
  矄鼀
矄盐 ᆠ
�矄
�盐鼀� 
犨�
��ᨨ
ᨨ
犨 �

V l

 
矄ᨨ� �
ᨨ �
�矄


ᔼ
 
 矄 ᨨ
犨 �
ᨨ
 
鼀�ᆠ
矄盐 ᆠ
�矄
nl
 
盐
ᨨ
n
犨�
 힘힘힘힘힘힘힘힘힘
[Main Sep.
02, 2020 (I)]


꾀
鼀
ᨨ
ᆠᨨ鼀� �ᆠ
��ᨨ
ᨨ
�  矄�泐




맔 ҈
�
␔⏌ᆠ鼀

�矄
ᆠᨨ鼀�矄
ᨨ
犨 �

l
㦨
҈

鼀

ᔼ 矄
ᨨ鼀泐
ᨨ

鼀
犨ᨨ├
盐 ᆠ�ᨨ
j 㦨

㡰
ᨨ
矄盐 ᆠ 
ᨨ
҈
ᆠ鼀


ᔼ
ᨨ 
�
҈

 ᨨᨨ 矄
ᆠ矄

ᆠᨨ

ᨨ

盐鼀
ᨨ
ᆠᨨ 矄
ᆠ
�
҈
矄
�矄
҈
x

hc
㦨
҈



 鼀

 
㦨
҈


맔
҈
҈黨
㡰

 
x
�矄
힘힘힘힘힘힘
᲼Main 䰘







ᔼ桌
黨
ᔼ

 ᨨᨨᆠ�ᆠ
 �鼀 


鼀
ᨨ 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ
�� 
犨�
ᨨ犨
ᨨ




W
�矄
� ᆠ� 
ᨨ
  ᆠ泐
矄ᨨ� �
ᆠ ᨨ
ᨨ
犨ᨨ├
盐 ᆠ�ᨨ



�


 ⼄盐 ᆠ泐
ᨨ
�� 
�矄
盐矄
ᨨ
 
 矄 ᨨ
⼄盐 ᆠ泐 矄ᨨ
 
 
 ᨨᨨᆠᨨ 矄

鼀�
犨�
���� ├� �ᆠ
 �泐

꾀矄矄盐鼀
 
 
 ᨨᨨᆠᨨ
鼀�矄矄�ᨨ

 �ᆠ� ᆠ泐
�矄
҈

奠

꾀
ᨨ �
�  ᆠ
ᨨ
�


V
�矄
� 犨
 
ᆠ ᨨ
 
 
 ᨨ

꾀
 
鼀� ᆠᨨ 矄

� ᆠ� 
ᨨ鼀泐
ᨨ
 
 ᨨ
 
 矄ᨨ


 ᨨ
 � ᆠ矄

ᨨᆠ

㦨

맔
҈


 盐
ᨨ
 
�鼀ᆠ
ᨨ
 
ᆠᨨ 矄


盐
ᨨ
n
�矄 힘힘힘힘힘힘힘힘힘힘

㝐矄矄
ᨨ
 
ᆠᨨ
m
㦨
黨
맔҈
҈
kg
  ҈




㦨
҈


맔҈
҈黨
J









᲼꾀



҈
桌

4 �



�҈


Fill in the Blanks


犨 �
ᨨ
⧤a
㜠泐矄
ᨨ盐ᆠ
泐

㜠泐
盐 �矄


䰘
处


ᨨ鼀�ᆠ
盐鼀
ᨨ
 
 ᨨ
鼀�
ᨨ
  盐
�矄




᲼҈黨黨

㜠


㝐├矄桌 꾀
ᨨ �
�  ᆠ
ᨨ



├
�矄
�
ᆠᨨ矄矄

㜠泐 盐


鼀� �鼀盐鼀
犨 �
ᨨ
㜠泐矄
�  �矄










处
᲼҈黨黨

㜠


㝐├矄桌

�

ᔼ

㜠泐
盐

ᆠᨨ 矄
ᆠᆠ
 ᨨ盐�

ᨨ � �  ᆠ
ᨨ
҈�





ᨨ矄
矄�├

ᆠᨨ
�


矄 ᨨ
 
鼀�
㜠泐
� 矄�
 
盐
�矄




















鼀⏌矄 ᲼҈黨黨

㜠
҈
㝐├桌 �

犨 �
ᨨ
 
ᆠ ᆠ�矄�ᆠ
㜠泐
⧤ a
�  鼀�
泐

泐ᨨ�
�├
鼀 
�矄




处

 犨 �
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⧤ b
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᲼҈黨黨

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␔ 
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ᨨ
ᆠᨨ 矄
矄�├� �
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 ᨨ
ᨨ


㜠泐
盐
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 









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鼀�
㜠泐矄 � ᆠ矄矄

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矄矄
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ᆠᨨ 矄
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 ᨨ

� ᆠ矄

 
ᆠ盐㜠ᨨ
犨 �
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㜠泐矄








᲼҈黨


㜠


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
鼀 �鼀盐鼀
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  � ᆠ� 
��ᨨ
᲼҈黨

㜠


㝐├矄桌 �

ᨨ
ᨨ盐ᆠ
ᆠ ᆠ�矄�ᆠ
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盐矄� �
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� �
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� ᨨ

 
ᆠᆠ� �
ᨨ�
矄 ᨨ盐
 �

 
힘힘힘힘힘힘힘힘힘힘힘
ᨨ矄

 
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 �泐
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힘힘힘힘힘
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� � �
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�

鼀ᨨ矄
ᆠᨨ
�
盐 �矄

᲼҈黨

㜠


㝐├矄桌 �矄



├㡰
5

True / False

�䰘
ᔼ

 ᨨᨨᆠ�ᆠ
鼀�矄矄�ᨨ
ᨨᆠ矄矄
 
鼀 �鼀盐鼀
 �泐 ᨨ
 
 ᨨᨨ㜠ᆠᨨ 矄
� ᆠ矄矄
犨�
� ᆠ矄� �
�  矄�泐 ᨨ
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� ᆠ� 
�� 
᲼҈黨


㜠

㝐├矄桌 �

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├� �ᆠ
 �泐
ᨨ
 ᨨᨨᆠᨨ 矄
鼀�
泐
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矄盐 ᆠ
 矄
ᨨ
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�  矄�泐
ᨨ
  � ᆠ� 

��ᨨ
᲼҈黨
҈㜠


㝐├矄桌 6

MCQs with One or More than One Correct Answer

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ᔼ

㜠泐
盐

ᆠᨨ 矄
鼀�
ᨨ鼀
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ᆠ ᨨ㡰 ᆠ泐� �
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I
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
�
 ᨨ㡰
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d
ᨨ鼀  
ᆠ ᨨ

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�
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V
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 

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ᆠ ᨨ
矄盐� �
�
鼀�矄矄�ᨨ
ᨨ
ᆠᨨ � 盐ᨨ盐矄
  ᆠ ᆠ�矄�ᆠ
㜠泐矄

ᔼ
 
�鼀 
ᆠ盐 
I
�矄
ᆠ矄 I ᨨ


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ᨨ �
�  ᆠ
V
�矄
� ᆠ矄
ᨨ

V

 
 
d 矄�ᨨ
�矄 ᆠ
d
�矄
盐ᆠ
ᨨ


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㡰  
ᆠ盐㜠ᨨ
犨 �
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盐ᆠ
ᨨ


 
  犨 � 矄
ᨨ
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鼀�

 
矄鼀 㡰  
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犨 �
矄
犨
矄
 
犨 � 矄
ᨨ  
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鼀�
 
矄鼀 ᆠ㡰  
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犨 �
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盐ᆠ
ᨨ


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  �  矄��矄
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
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㜠泐矄
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ᆠ矄 㡰  
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ᆠ矄


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ᔼ
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l

犨 �ᆠ
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矄鼀 矄㡰
�矄
㡰
盐䁸







᲼꾀



҈
桌

WWW.JEEBOOKS.IN

P-283

Dual Nature of Radiation and Matter �� 

쟼ᆠᨨ 矄

 


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ᆠ
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ᨨ
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 ᨨᨨᆠᨨ 矄 ᨨ鼀
鼀


 
鼀


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 ᨨᨨ 矄
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 ᨨᨨᆠᨨ 矄
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TA

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ᨨ
 ᨨᨨᆠᨨ 矄
�
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鼀
B
泐  ᨨᨨ 矄
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 �泐


䰘

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 ᨨᨨᆠᨨ 矄
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l B =
l A
 

᲼҈黨黨

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 㡰 TB
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䰘�
 㡰 㡰





㝐






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᲼҈黨黨

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᲼҈黨


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 ᨨᨨᆠ�ᆠ
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矄盐ᨨ矄
⼄盐 盐鼀
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ᨨ
��  ᆠ盐矄 ᲼҈黨
䰘
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ᨨ
ᨨ

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ᨨ鼀 
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�矄
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处

 ᨨᨨᆠᨨ 矄
犨�
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鼀�
犨 

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鼀�
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ᨨ鼀
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㜠

㝐├矄桌 㡰
�

犨
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盐�ᨨ
鼀 㡰
҈㜠犨
盐�ᨨ
鼀 9

Assertion and Reason Type Questions




和T꾀TE㝐ENT㜠҈ ᲼


䰘桌 ᔼ
 
ᆠᆠ� �
ᨨ �
�

㜠泐
盐
�矄
� ᆠ矄
 
犨 � 矄
ᨨ
 
ᆠ ᆠ�矄�ᆠ
㜠泐矄
ᨨ
ᨨ
ᆠ  �
和T꾀TE㝐ENT㜠
␔ 

ᆠᨨ
鼀
矄�├矄
 
�
�

㜠泐
盐
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 
├� �ᆠ
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�矄
ᆠᨨ 
� ᨨ
㜠泐
 �泐
㡰 和鼀 㜠҈
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EBD_8341 P-284

Physics

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6

MCQs with One or More than One Correct Answer

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Answer Key Topic-1 : Matter Waves, Cathode and Positive Rays ҈
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Topic-2 : Photon, Photoelectric Effect X-rays and Davisson-Germer Experiment ҈
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WWW.JEEBOOKS.IN

P-285

Atoms

Atoms

26

Topic-1 : Atomic Structure and Rutherford's Nuclear Model 1 ҈


MCQs with One Correct Answer

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Topic-2 : Bohr's Model and the Spectra of the Hydrogen Atom 1 ҈


MCQs with One Correct Answer

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WWW.JEEBOOKS.IN

EBD_8341 P-286

6.

7.

8.

9.

10.

Physics

Taking the wavelength of first Balmer line in hydrogen spectrum (n = 3 to n = 2) as 660 nm, the wavelength of the 2nd Balmer line (n = 4 to n = 2) will be; [Main 9 April 2019 I] (a) 889.2 nm (b) 488.9 nm (c) 642.7 nm (d) 388.9 nm Radiation coming from transitions n = 2 to n = 1 of hydrogen atoms fall on He+ ions in n = 1 and n = 2 states. The possible transition of helium ions as they absorb energy from the radiation is : [Main 8 April 2019 I] (a) n = 2 ® n = 3 (b) n = 1 ® n = 4 (c) n = 2 ® n = 5 (d) n = 2 ® n = 4 A hydrogen atom, initially in the ground state is excited by absorbing a photon of wavelength 980Å. The radius of the atom in the excited state, in terms of Bohr radius a0, will be: [Main 11 Jan 2019 I] (a) 25a0 (b) 9a0 (c) 16a0 (d) 4a0 In a hydrogen like atom, when an electron jumps from the M-shell to the L-shell, the wavelength of emitted radiation is l. If an electron jumps from N-shell to the Lshell, the wavelength of emitted radiation will be: [Main 11 Jan 2019 II] (a)

27 l 20

(b)

16 l 25

(c)

25 l 16

(d)

20 l 27

An electron from various excited states of hydrogen atom emit radiation to come to the ground state. Let ln, lg be the de Broglie wavelength of the electron in the nth state and the ground state respectively. Let Ln be the wavelength of the emitted photon in the transition from the nth state to the ground state. For large n, (A, B are constants)[Main 2018] B (a) L n » A+ 2 (b) Ln » A + Bln ln (c) L 2n » A + B l 2n

11.

12.

(d) L 2n » l

If the series limit frequency of the Lyman series is v1, then the series limit frequency of the P-fund series is : [Main 2018] (a) 25 nL (b) 16 nL (c) nL/16 (d) nL/25 The energy required to remove the electron from a singly ionized Helium atom is 2.2 times the energy required to remove an electron from Helium atom. The total energy required to ionize the Helium atom completely is: [Main Online April 15, 2018] (a) 20 eV

13.

(b) 79 eV

(c) 109 eV

(d) 34 eV

Some energy levels of a molecule are shown in the figure. The ratio of the wavelengths r = l1/l2, is given by [Main 2017]

(a) r =

3 4

(b) r =

1 3

(c) r =

4 3

(d) r =

2 3

14. According to Bohr’s theory, the time averaged magnetic field at the centre (i.e. nucleus) of a hydrogen atom due to the motion of electrons in the n th orbit is proportional to : (n = principal quantum number) [Main Online April 8, 2017] (a) n –4 (b) n –5 (c) n –3 (d) n –2 15. A hydrogen atom makes a transition from n = 2 to n = 1 and emits a photon. This photon strikes a doubly ionized lithium atom (z = 3) in excited state and completely removes the orbiting electron. The least quantum number for the excited state of the ion for the process is : [Main Online April 9, 2016] (a) 2 (b) 4 (c) 5 (d) 3 16. As an electron makes a transition from an excited state to the ground state of a hydrogen - like atom/ion : [Main 2015] (a) kinetic energy decreases, potential energy increases but total energy remains same (b) kinetic energy and total energy decrease but potential energy increases (c) its kinetic energy increases but potential energy and total energy decrease (d) kinetic energy, potential energy and total energy decrease 17. If one were to apply Bohr model to a particle of mass ‘m’ and charge ‘q’ moving in a plane under the influence of a magnetic field ‘B’, the energy of the charged particle in the nth level will be : [Main Online April 10, 2015]

æ hqB ö (a) n çè ÷ 2pm ø

æ hqB ö (b) n çè ÷ 8pm ø

hqB ö (c) n æç è 4pm ÷ø

æ hqB ö (d) n ç è pm ÷ø

18. The radiation corresponding to 3 ® 2 transition of hydrogen atom falls on a metal surface to produce photoelectrons. These electrons are made to enter a magnetic field of 3 × 10–4 T. If the radius of the largest circular path followed by these electrons is 10.0 mm, the work function of the metal is close to: [Main 2014] (a) 1.8 eV (b) 1.1 eV (c) 0.8 eV (d) 1.6 eV

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P-287

Atoms

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 �泐
ᆠ矄矄

ᨨ �
 �泐 � ᆠ矄矄
 
�矄
ᨨ
 �泐
鼀� 矄
 
矄鼀
ᆠ㡰 ᔼ矄
├� �ᆠ
 
ᨨ
 ��矄
ᆠ矄
 
�矄
ᨨ �  �泐
� ᆠ矄矄
㡰 ᔼ矄
├� �ᆠ

ᨨ �
 
ᨨ
 ��矄
ᆠ矄
҈
ᔼ鼀�� 

ᨨ鼀
鼀
盐
ᨨ

ᨨᨨ
 

泐ᨨ �ᆠ �ᆠ
ᨨ
ᨨ盐
 
鼀矄矄
ᨨ
 
ᆠᨨ
盐
� �
  矄鼀
ᆠ �
矄
 
ᆠᨨ

꾀泐
 
ᨨ 
ᨨ鼀
鼀ᨨ  
ᆠᨨ 矄�

ᨨ矄矄�
 矄��ᨨ 矄
ᨨ
 �矄
泐ᨨ �ᆠ �ᆠ
ᨨ
 
�矄
 ᆠ�



ᨨ �矄
犨 �  ᨨᨨ
 
犨�

鼀�
矄
犨 �
l
��
�

鼀矄
ᨨ
 
泐�
ᆠᨨ 矄 
R
ᨨ
 
泐ᨨ�
ᨨ鼀㡰 ⼄盐
ᨨ ᲼



和桌 㡰 黨⏌�R㡰 㡰 
⏌�R㡰 ᆠ㡰 ҈
⏌�R㡰 㡰
⏌R

WWW.JEEBOOKS.IN

EBD_8341 P-288












Physics

ᔼ
泐ᨨ�
矄ᆠ盐鼀
 
犨 �
ᨨ
a
� 
�矄


�


鼀

犨 矄
�
 
矄ᆠ盐鼀

ᨨ

�矄 
� 泐

a
�  犨 �
�矄
䰘


鼀

쟼矄�鼀
矄
ᨨ
 
� 泐
犨� 矄ᆠ
ᨨ

�矄
᲼҈黨黨黨和
㜠


㝐├矄桌 㡰

맔
҈


鼀⏌矄 㡰

맔
҈
䰘
鼀⏌矄 ᆠ㡰

맔
҈


鼀⏌矄 㡰

맔
҈
�
鼀⏌矄 꾀矄

ᨨ 
鼀ᨨ

 
鼀� �鼀盐鼀
 �泐
�
㡰
⼄盐� ᨨ
鼀ᨨ

ᆠᨨ
ᨨ鼀
 
�ᨨ盐 
矄
ᨨ
ᨨ盐泐 �ᨨ �
�
ᨨ鼀
Z 㦨
㡰
�矄 ᲼҈黨黨䰘
㜠
҈
㝐├桌 㡰 ҈
�҈ 㡰 ҈

ᆠ㡰



㡰 ҈



ᔴᨨ 矄�
 
矄ᆠ
� 
矄盐� �
ᨨ鼀
 

 矄��ᨨ

n =
® n = ҈ �
 
ᨨ鼀矄
 
�ᨨ 矄
��
ᨨ犨

 矄 ᨨ矄
犨 �
�矄
ᨨ盐ᆠ
泐
᲼҈黨

㜠
҈
㝐├桌 㡰 泐ᨨ�
ᨨ鼀 㡰 盐�盐鼀
ᨨ鼀 ᆠ㡰 和� �泐
�ᨨ �
�盐鼀 㡰 ᨨ盐泐
�ᨨ �矄
� �盐鼀 2

�






Integer Value Answer

ᔴᨨ 矄�

泐ᨨ� 㜠�├
�ᨨ �
ᨨ鼀
犨�
ᨨ鼀�ᆠ
盐鼀 
犨�

矄� �
ᆠᨨ

ᔼ
 
鼀�矄矄�ᨨ
矄ᆠ盐鼀
ᨨ
 �矄 ᨨ鼀

 
 ᨨᨨ
鼀�
�
 

㦨


ᨨ

㦨
҈
 矄��ᨨ

矄
 �泐
䰘




�� 
 
 
 ᨨᨨ
鼀�
�
 


㦨

ᨨ

㦨


 矄��ᨨ


�ᨨ ��ᨨ
 �泐
ᨨ
  泐ᨨ�
ᨨ鼀
�矄
҈





盐
ᨨ
Z
�矄
힘힘힘힘힘힘힘힘힘힘



᲼꾀



҈
桌 꾀
ᆠᨨ
�

泐ᨨ�
ᨨ鼀
盐 �ᨨ矄

 矄��ᨨ

ᨨ鼀

ᨨ�
犨�
⼄盐 盐鼀
盐鼀
�
ᨨ
 ᨨ 
犨� ⼄盐 盐鼀
盐鼀


�
 


矄ᆠ�泐
 
� ��

�
㦨



�
  
 
矄鼀矄
ᨨ矄矄�

�矄
᲼꾀



҈䰘桌 䰘
꾀
泐ᨨ�
ᨨ鼀
�
�矄
�ᨨ盐 
矄
�矄
��
泐
��  ᨨ
犨 �
黨䰘

处

├� �
ᆠ⏌
㦨
҈

䰘
맔
҈



鼀
   
�ᨨ盐 
矄
 �泐
ᨨ
泐ᨨ�
ᨨ鼀
矄
҈




 

盐鼀
ᨨ
� 矄
矄 
�
 
鼀�矄矄�ᨨ
矄ᆠ盐鼀
�矄 ᲼꾀



҈
桌 

꾀
ᆠᨨ
�矄

 ᆠ�
矄
ᨨ
�
졔
�ᨨ
矄
 �盐 鼀ᨨ鼀 盐鼀
 ⏌
p



ᨨ��
犨 �
ᨨ
 
ᆠᨨ

�
 �矄
矄
�矄
p 

犨 


�矄
 
ᨨ 
�盐矄㡰

 盐
ᨨ
p
�矄


᲼꾀



҈�桌 黨
ᔴᨨ 矄�

泐ᨨ�
ᨨ鼀
犨�
�矄
ᆠᨨ
�
 
n
ᨨ�
꾀
ᆠᨨ鼀� �ᆠ
��ᨨ
ᨨ
犨 �
黨

鼀
�矄
盐矄 ᨨ
�ᨨ �
 
ᨨ鼀

ᔼ
 
├� �ᆠ
 �泐
ᨨ
 
ᆠ ᆠᨨ
�矄
҈





 
 
盐
ᨨ
n
�矄
hc
㦨
҈




nm㡰 ᲼꾀



҈�桌

 
� 
ᨨ �
 ��矄
ᨨ
 
ᆠᨨ

ᔼ


3




Numeric Answer

꾀
�ᆠ
ᨨ
鼀矄矄




㝐⏌ᆠ

ᆠᨨ�矄
犨�

泐ᨨ�

ᨨ鼀

矄

和ᨨᨨ
 
 
ᆠᨨ�矄�ᨨ
 
�ᆠ
ᆠᨨ鼀矄
ᨨ 矄

 
 
ᨨ鼀
ᆠᨨ�矄
 
�ᨨ矄
ᨨ
�矄
�矄
 ᆠ�
矄



� ��
├� �ᆠ
 �泐
ᨨ
 
�ᆠ
�
㡰
�矄


盐
ᨨ

�矄



᲼㝐�
和


�






ᔼ㡰桌

෌�
 
鼀矄矄
ᨨ
 
泐ᨨ�
ᨨ鼀
ᨨ

҈
෌⏌ᆠ
㡰
҈
ᔼ
 
� 
矄ᆠ
ᨨ
泐ᨨ�
ᨨ鼀

�  ᆠ
犨

 
�矄
 
 
矄 ᨨ矄
犨 � 矄
ᨨ
 
泐鼀

矄�矄
�矄



处


ᆠᨨ矄ᨨ � �
�  ᆠ
ᨨ
  矄ᆠ 
矄�矄
�
处
�矄


힘힘힘힘힘힘힘힘힘힘
[㝐�
Sep.
04, 2020 (I)]



�矄
鼀鼀
ᨨ
 
鼀
矄�矄
ᨨ
泐ᨨ�
ᨨ鼀 矄

犨 �
ᨨ

�
҈
处


犨 �
ᨨ
  矄ᆠᨨ 
鼀鼀
ᨨ
 
鼀
矄�矄
�
鼀㡰
�矄
힘힘힘힘힘힘
᲼㝐�










ᔼᔼ桌 4

Fill in the Blanks



ᔼ
 
ᨨ 
鼀ᨨ
ᨨ
 
泐ᨨ�
ᨨ鼀

 
�ᨨ
ᨨ
  ├� �ᆠ
 �泐
ᨨ
 
ᨨ
 �泐
ᨨ
 
ᆠᨨ
�
 ⼄盐 盐鼀
矄
n
�矄




















᲼҈黨黨

㜠
҈
㝐├桌



ᨨ 
�盐矄
ᨨ
 
� 
ᆠᨨ
ᨨ
 ᨨ矄 ᨨᨨ盐矄
ᨨ鼀 ᨨ鼀�ᆠ
盐鼀
㦨
҈�㡰
ᆠ� �
矄

ᨨ 
�
矄��ᆠᨨ
� �ᆠ�ᆠ
ᆠᨨ 矄 
㦨
҈
㡰
�矄





















处


᲼҈黨黨҈
㜠
҈
㝐├桌 6

MCQs with One or More than One Correct Answer


�

꾀
�ᆠ
ᨨ
鼀矄矄
m
鼀ᨨ矄
�
ᆠ�ᆠ盐
ᨨ�矄
犨�
ᨨ �  �泐
Vr㡰
㦨
Fr

犨 
F
�矄

ᨨ矄��
ᆠᨨ 矄 
 
r
�矄
�矄 �矄 ᆠ
ᨨ鼀
 
ᨨ���

ᔼ矄
 ��矄

ᆠᆠ盐
盐矄� �  
ᨨ 
鼀ᨨ

ᔼ
 
�盐矄
ᨨ
 
�ᆠ矄
ᨨ�
�矄  ᨨ
泐
R
 
�矄
矄
 
 �泐

 ᨨ
泐
v  
E

矄ᆠ�泐

 
ᨨ
 
n
ᨨ�
 
h
�矄
   ᆠ├矄
ᆠᨨ 矄 㡰







᲼꾀





桌 㡰 R µ n҈⏌ 
 
v µ n
⏌  㡰 R µ n
⏌   
v µ n҈⏌   æ n
h
F
ö ᆠ㡰 E = ç ÷
è
p
m ø

҈⏌ 

҈⏌ 

æ n
h
F
ö
E = 㡰 ç ÷ è
p
m ø


꾀

泐ᨨ�
ᨨ鼀
 
矄ᨨ� �

 ᨨᨨ
ᨨ 犨 �
l
�矄
 ᆠ�
ᨨ鼀
 
矄
n㦨
҈
ᨨ
 
矄 n 㦨



ᔼ鼀鼀�泐
 
 
 
ᆠᨨ
盐鼀矄
ᨨ
n 㦨
m 矄
泐
鼀�� �

 ᨨᨨ
ᨨ
犨 �
l


  ᆠ  �
�
鼀ᨨ鼀 盐鼀
ᨨ
ᨨ鼀
盐
ᨨ
 
矄ᨨ�ᨨ
  ҈  
鼀�矄矄�ᨨ

Dpa
 
Dpe

矄ᆠ�泐

ᔼ

l a ⏌ le =
� 犨 �ᆠ
ᨨ
 
ᨨ�ᨨ 矄㡰
�矄⏌
ᆠᨨᆠ䁸 ᲼矄
hc
㦨
҈




鼀錐
҈
鼀
㦨
҈
黨
鼀

h
 
c

 ᆠ├矄 ᆠᨨ 矄 
 
矄
ᨨ
�� 

矄ᆠ�泐桌



᲼꾀



҈黨桌 ҈ 㡰 Dpa ⏌ Dpe =
㡰 
�ᨨ
ᨨ
├� �ᆠ
 �泐
ᨨ
 
ᆠᨨ
�
 
矄 ҈ n
㦨
m ᨨ
 
矄
n
㦨
҈
�矄

ᆠ㡰 m
㦨

㡰 l
㦨

҈

鼀

WWW.JEEBOOKS.IN

P-289

Atoms

�� 泐
 ᆠ�
矄矄
ᨨ
泐ᨨ� �├
ᨨ鼀矄
矄ᨨ
ᆠ 泐�
矄矄㡰
犨�
盐ᆠ
ᆠ �


 � 
泐  �
� ᆠ�
⼄盐 盐鼀
盐鼀


犨 
>>҈

␔ �ᆠ
ᨨ  
ᨨᨨ犨� �
矄鼀 矄㡰
�矄㡰
盐䁸









᲼꾀



҈
桌 㡰 �
ᆠ  �
�
 
��
ᨨ
犨ᨨ
ᆠᨨ 矄ᆠ盐� ᨨ�矄
ᨨ矄
ᨨ
 
ᨨ
 㡰 �
ᆠ  �
�
 
��
ᨨ
犨ᨨ
ᆠᨨ 矄ᆠ盐� ᨨ�矄
�矄
矄
҈⏌

ᆠ㡰 �
ᆠ  �
�
 
 �泐
ᨨ
犨ᨨ
ᆠᨨ 矄ᆠ盐� ᨨ�矄
�矄
矄
҈⏌
 㡰 �
ᆠ  �
�
 
 �盐
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ᨨ ᆠᨨ 矄 8

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Comprehension/Passage Based Questions

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






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WWW.JEEBOOKS.IN

EBD_8341 P-290


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10 Subjective Problems
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WWW.JEEBOOKS.IN

P-291

Atoms

Topic-3 : Miscellaneous (Mixed Concepts) Problems 10 Subjective Problems �� 
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Answer Key Topic-1 : Atomic Structure and Rutherford's Nuclear Model ҈


ᆠ㡰



ᆠ㡰 Topic-2 : Bohr's Model and the Spectra of the Hydrogen Atom

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㡰 㡰 㡰 �҈㡰 

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WWW.JEEBOOKS.IN

EBD_8341 P-292

Physics

27

Nuclei

Topic-1 : Composition and Size of the Nuclei ҈







1 MCQs with One Correct Answer


�盐矄
R
ᨨ

盐ᆠ盐矄
ᨨ
鼀矄矄
盐鼀
A
ᆠ
 矄�鼀
泐
 
ᨨ鼀盐
R
㦨
҈

맔
҈
҈�㡰A҈⏌
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ᔼ
ᨨᨨ犨矄  
 
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 矄�泐
ᨨ

盐ᆠ盐矄
�矄
ᨨ
 
ᨨ
ᨨ

 M ᨨ
@ M 盐
; ҈

䰘 ´ ҈
-
䰘 ├�㡰 ᲼㝐�
和









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├�
鼀 㡰 ҈
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├�
鼀 ᆠ㡰 ҈



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鼀 㡰 ҈
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µ V 㡰 m
µ ҈⏌V ᆠ㡰 m
µ V 㡰 m
µ V
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맔
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҈
├�⏌鼀 㡰 ҈
҈҈├�⏌鼀





5 True / False

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�矄 ҈


├�
鼀

᲼҈黨
黨
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㝐├矄桌





6 䰘


MCQs with One or More than One Correct Answer

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鼀矄矄
盐鼀
ᨨ

盐ᆠ盐矄
�矄 ᲼҈黨


㜠


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矄矄
 
�矄
ᨨ鼀�ᆠ
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鼀ᨨ
 
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 
 
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⼄盐
ᨨ
�矄 ᨨ鼀�ᆠ
盐鼀

Topic-2 : Mass-Energy Equivalence and Nuclear Reactions 1 ҈







º
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MCQs with One Correct Answer

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
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 
鼀矄矄
ᨨ
䰘 �
㦨
䰘

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㝐矄矄
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㦨
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泐
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WWW.JEEBOOKS.IN

P-293

Nuclei

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
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ø è 
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㡰 �

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Binding Energy/nucleon in Mev





Y 8.5 8.0 7.5

X W

5.0

Z 0

90 30 60 Mass number of nuclei

120

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矄 �� � ᲼҈黨黨

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 鼀
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4

Fill in the Blanks

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 
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WWW.JEEBOOKS.IN

EBD_8341 P-294

҈



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Physics

盐� �
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MCQs with One Correct Answer

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[Main Sep. 05, 2020 (I)] ᔼ




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[Main Sep. 03, 2020 (I)] �

҈



 䰘  㡰 ᆠ㡰 㡰 � �



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P-295

Nuclei

T 㡰
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EBD_8341 P-296

26.

27.

28.

Physics

A radioactive material decays by simultaneous emission of two particles with respective half-lives 1620 and 810 years. The time, in years, after which one-fourth of the [1995S] material remains is (a) 1080 (b) 2430 (c) 3240 (d) 4860 Consider a particles, b particles and g - rays, each having an energy of 0.5 MeV. In increasing order of penetrating powers, the radiations are: [1994 - 1 Mark] (a) a, b, g (b) a, g , b (c) b, g, a (d) g, b, a The equation

35. To determine the half life of a radioactive element, a student dN (t ) dN (t ) is the rate versus t. Here dt dt of radioactive decay at time t. If the number of radioactive nuclei of this element decreases by a factor of p after 4.16 years, the value of p is [2010]

plots a graph of ln

4 11 H + ¾¾ ® 42 He 2 + + 2e – + 26 MeV represents [1983 - 1 Mark]

29.

30.

(a) b -decay (b) g-decay (c) fusion (d) fission Beta rays emitted by a radioactive material are (a) electromagnetic radiations [1983 - 1 Mark] (b) the electrons orbiting around the nucleus (c) charged particles emitted by the nucleus (d) neutral particles The half life of radioactive Radon is 3.8 days. The time at 1 th of the radon sample will remain the end of which 20 undecayed is (given log10 e = 0.4343 ) [1981- 2 Marks] (a) 3.8 days (b) 16.5 days(c) 33 days (d) 76 days. Integer Value Answer is an isotope of Iodine that B decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with 131I is injected into the blood of a person. The activity of the amount of 131I injected was 2.4 × 105 Becquerel (Bq). It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After 11.5 hours, 2.5 ml of blood is drawn from person's body, and gives an activity of 115 Bq. The total volume of blood in the person¡¦s body, in liters is approximately (you may use ex » 1 + x for |x| < < 1 and ln2 » 0.7). [Adv. 2017] For a radioactive material, its activity A and rate of change dN dA and R = – of its activity R are defined as A = , dt dt where N(t) is the number of nuclei at time t. Two radioactive sources P (mean life t) and Q (mean life 2t) have the same activity at t = 0. Their rates of change of activities at t = 2t R n are RP and RQ, respectively. If P = , then the value of RQ e n is [Adv.2015] A freshly prepared sample of a radioisotope of half-life 1386 s has activity 103 disintegrations per second. Given that ln2 = 0.693, the fraction of the initial number of nuclei (expressed in nearest integer percentage) that will decay in the first 80 s after preparation of the sample is [Adv. 2013-I] The activity of a freshly prepared radioactive sample is 1010 disintegrations per second, whose mean life is 109 s. The mass of an atom of this radioisotope is 10–25 kg. The mass (in mg) of the radioactive sample is [2011] 2

31.

32.

33.

34.

131I

4

Fill in the Blanks

( )

36. When Boron nucleus 105 B is bombarded by neutrons, a -particles are emitted. The resulting nucleus is of the element ........ and has the mass number ...... [1986 - 2 Marks] 37. In the Uranium radioactive series the initial nucleus is

238

92

U

206

and the final nucleus is 82 Pb . When the Uranium nucleus decays to lead, the number of a-particles emitted is ......and the number of b-particles emitted is ..... [1985 - 2 Marks] 38. The radioactive decay rate of a radioactive element is found to be 103 disintegration/second at a certain time. If the half life of the element is one second , the decay rate after one second is _______ and after three seconds is _______. [1983 - 2 Marks] 6

MCQs with One or More than One Correct Answer

39. In a radioactive decay chain,

232 90Th nucleus

decays to

212 82 Pb nucleus. Let Na and Nb be the number of a and b particles, respectively, emitted in this decay process.

Which of the following statements is (are) true? [Adv. 2018] (a) Na = 5 (b) Na = 6 (c) Nb = 2 (d) Nb = 4 40. The half-life period of a radioactive element X is same as the mean-life time of another radioactive element Y. Initially both of them have the same number of atoms. Then [1999S - 3 Marks] (a) X and Y have the same decay rate initially (b) X and Y decay at the same rate always (c) Y will decay at a faster rate than X (d) X will decay at a faster rate than Y 41. The half-life of 131I is 8 days. Given a sample of 131I at time t = 0, we can assert that [1998S - 2 Marks] (a) no nucleus will decay before t = 4 days (b) no nucleus will decay before t = 8 days (c) all nuclei will decay before t = 16 days (d) a given nucleus may decay at any time after t = 0

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P-297

Nuclei















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 
鼀�矄矄�
矄  


鼀� �鼀盐鼀
�鼀
 
犨 �ᆠ
�
犨ᨨ盐

ᨨ矄矄�
ᨨ 犨ᨨ├
矄 泐
犨�
 �矄
矄ᨨ盐ᆠ
�矄 ᲼҈黨


㜠


㝐├矄桌 㡰

 㡰 ҈

 ᆠ㡰


 㡰 ҈


 盐� �

��

ᆠ泐 ᲼҈黨
䰘
㜠


㝐├矄桌 㡰

ᨨ鼀�ᆠ
ᆠᨨ
�矄
ᆠ 㡰 
ᆠᨨ
犨 �ᆠ
�矄
泐
矄 
犨� �
 
盐ᆠ盐矄 �矄
ᆠ ᆠ㡰 
盐ᨨ
�
 
盐ᆠ盐矄
ᆠ泐矄
鼀�� �

ᆠᨨ

㡰 

ᨨ
 
� � �
 �泐
ᨨ
 
盐ᆠ盐矄
�矄 ᆠᨨ 
� ᨨ

ᆠᨨ

7


�


Match the Following

㝐ᆠ
 
盐ᆠ
ᨨᆠ矄矄矄
��
�
ᆠᨨ盐鼀
ᔼ
犨�
  ᨨ�
ᨨ�ᨨ 矄㡰
�
ᆠᨨ盐鼀
ᔼᔼ





᲼꾀



҈�桌 ᔴᨨ盐鼀
ᔼ ᔴᨨ盐鼀
ᔼᔼ 꾀㡰 盐ᆠ
盐矄�ᨨ

㡰 꾀矄ᨨ�ᨨ
ᨨ
 鼀 㡰 �矄矄�ᨨ
�

盐ᆠ ᆠᨨ ᔴ㡰 b㜠ᆠ泐






㡰 쟼 �泐
ᨨ盐ᆠ�ᨨ
�
矄矄 �
泐ᨨ�
ᆠᨨ 矄�ᨨ

ᨨ
�盐鼀 㡰 g㜠泐
鼀�矄矄�ᨨ

矄㡰 泐
犨 㡰 盐� ᨨ
鼀�矄矄�ᨨ

㝐ᆠ
�矄
ᔼ
ᨨ
 
盐ᆠ
ᨨᆠ矄矄矄
犨�
�矄
ᔼᔼ
ᆠᨨ � � �  
盐ᆠ盐矄
 
ᨨ 
ᨨ
 
 
ᨨ盐ᆠ矄
ᨨ
ᆠ
ᨨᆠ矄矄  
 
矄ᆠ
 
ᆠᨨᆠ
 矄犨
盐矄� �
 
ᆠᨨ矄
��

ᨨ犨
 
�矄矄

᲼꾀



҈㜠ᔼᔼ桌 �矄
ᔼ �矄
ᔼᔼ 



䰘


⼄㡰


�

盐ᨨ 矄
泐
黨


ᔴᨨ
盐ᆠ盐矄
䰘

꾀 
ᆠ泐

҈


턼
b졔
ᆠ泐






�矄矄�ᨨ




҈� ҈�
졠 ® 䰘 졠 +


҈


黨
 ® 黨
+


҈
� ҈


 � ®

 +



黨 ҈

黨
盐 ® �䰘  +




和
ᨨᨨ
鼀�矄矄�ᨨ



ᔴᨨ矄
 턼 和 㡰

҈  㡰 ҈ 

ᆠ㡰
҈
 㡰

҈ ෌�
ᨨ犨

ᆠ�
鼀ᆠ � �
泐
⼄盐矄�ᨨ 矄

犨  犨ᨨ
ᆠᨨ盐鼀 矄
ᆠ
� �


�鼀矄㡰

��

ᔼ鼀鼀�泐  
 
ᆠᨨ盐鼀 矄
 
鼀ᆠ � �
��
�矄
��

犨 
ᆠ �鼀
ᨨ
ᔴᨨ盐鼀
ᔼ
矄
ᨨ

鼀ᆠ 
犨�
 
�鼀矄
ᨨ ᔴᨨ盐鼀
ᔼᔼ

泐
 ᆠ�ᆠ� �
 
ᆠᨨᆠ
鼀ᆠ 矄㡰

ᨨ
  
�鼀
ᨨ
ᆠᨨ盐鼀
ᔼ
ᆠ
鼀ᆠ
犨�
鼀ᨨ
 
ᨨ 
�鼀
ᨨ ᆠᨨ盐鼀
ᔼᔼ

꾀
 
�鼀矄
ᨨ
ᆠᨨ盐鼀
ᔼᔼ
鼀盐矄

鼀ᆠ 
㝐ᆠ
 
ᨨᨨ犨� �


᲼




㜠

㝐桌

ᔴᨨ盐鼀
ᔼ 꾀㡰 盐ᆠ
盐矄�ᨨ

㡰 盐ᆠ
�矄矄�ᨨ

ᔴ㡰 b㜠ᆠ泐 㡰 쟼 ᨨ 鼀�ᆠ
盐ᆠ ᆠ�ᨨ

8

ᔴᨨ盐鼀
ᔼᔼ 㡰 ᔴᨨ 矄
矄ᨨ鼀
鼀
� ᨨ  �泐 ⼄㡰 ෌ 泐
ᨨ矄矄�
ᨨ

盐ᆠ�
犨�
ᨨ犨
ᨨ鼀�ᆠ

盐鼀 㡰 ෌ 泐
ᨨ矄矄�
ᨨ

盐ᆠ�
犨�
�� 
ᨨ鼀�ᆠ

盐鼀 矄㡰 쟼矄矄 �泐
ᨨᆠ矄
泐 犨├
盐ᆠ
ᨨᆠ矄

Comprehension/Passage Based Questions

矄矄�㜠҈


鼀矄矄
ᨨ

盐ᆠ盐矄
꾀  
�矄
矄矄
 
 
矄盐鼀
ᨨ
 
鼀矄矄矄
ᨨ 꾀㜠㡰
盐鼀
ᨨ
盐ᨨ 矄
 

盐鼀
ᨨ
ᨨᨨ 矄
�
 
盐ᆠ盐矄



 �泐
⼄盐� 
ᨨ
 
ᆠᨨ矄ᨨ � �
鼀矄矄
�  ᆠ
�矄 ├ ᨨ犨
矄
 
� � �
 �泐
ᨨ
 
盐ᆠ盐矄

꾀
泐
盐ᆠ盐矄 ᨨ
鼀矄矄
㝐
ᆠ
├
� ᨨ
犨ᨨ
�� 
盐ᆠ�
ᨨ
鼀矄矄矄
鼀҈
 
鼀
ᨨ 泐
�
鼀҈
졔
鼀
㡰


㝐

꾀矄ᨨ
犨ᨨ
�� 
盐ᆠ�
ᨨ
鼀矄矄矄
鼀
 
鼀
ᆠ
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盐矄�ᨨ
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ᨨ鼀

泐
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鼀
졔
鼀
㡰
>
㝐Ἐ


鼀矄矄矄
ᨨ
矄ᨨ鼀
盐
ᨨ鼀矄
 ��
�
 

ᨨ犨
҈ ҈

҈


䰘

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盐


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҈
҈


盐

 ҈



҈

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盐



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盐


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
盐

䰘  �

䰘

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盐

䰘




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黨
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
和


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黨҈
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盐

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


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黨
黨


䰘

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㝐⏌ᆠ
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҈桌



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├㡰
ᨨ
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 
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犨 
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҈


盐ᆠ盐矄


ᨨ

矄
盐 �ᨨ矄
 
ᆠ泐

�矄 㡰 �҈黨 㡰 �


ᆠ㡰 �䰘
䰘 㡰 �
҈

黨

ᆠᨨᆠ
矄鼀 
�矄

㡰 
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 �
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㡰 
盐ᆠ盐矄


ᨨ
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ᆠ㡰 盐ᨨ
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㡰 
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䰘


 

和
ᆠ
盐 �ᨨ
ᆠᨨ鼀
盐矄�ᨨ

10 Subjective Problems �

꾀
�ᨨᆠ�
矄鼀
ᨨ



ᆠ泐矄
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 ᨨ盐�
 ᨨᆠ矄矄
ᨨ
犨 �ᆠ
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 㜠� 
�矄


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黨
泐矄

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盐鼀
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

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
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黨
泐矄

෌�

㡰҈⏌
㦨
҈



᲼




㜠



㝐├矄桌 �҈
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矄鼀
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n
b㜠�ᆠ矄
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n
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
㜠


㝐├矄桌

WWW.JEEBOOKS.IN

EBD_8341 P-298

�



�


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Physics

꾀
�ᨨᆠ�
盐ᆠ盐矄
X
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ᨨ
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Y 犨�
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lx
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҈
矄㜠҈

Y
盐 
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矄

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Z
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lY
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矄㜠҈

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盐ᆠ�
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盐鼀
�矄
Nᨨ
㦨
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
⼄盐�ᨨ 矄
ᨨ
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ᨨ盐�ᨨ 矄
ᨨ
X

Y
 
Z


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Y
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泐
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X
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㜠


㝐├矄桌 �

꾀
盐ᆠ盐矄
X

� ��泐

矄

盐 �ᨨ矄
 
ᆠ泐 ᆠᆠᨨ� �
ᨨ
 
⼄盐�ᨨ
A 黨
X

®


Z Y +a




᲼҈黨黨҈
㜠


졔



㝐├矄桌

㡰 � 
 
盐矄
ᨨ
A
 
Z
�
 
ᨨ
ᨨᆠ矄矄
㡰 
 
�ᆠ
ᨨ盐ᆠ
�
 
ᨨ
ᨨᆠ矄矄
�矄 ᨨ盐 
ᨨ
鼀ᨨ
�

ᆠ�ᆠ盐
ᆠ├
ᨨ
�盐矄


҈҈
鼀
�
 盐 � ᨨ鼀
鼀� �ᆠ
�
ᨨ

矄

� 
 
 �泐
ᔼ

㝐㡰
矄
盐� �
 
ᨨᆠ矄矄
 
 
� � �  �泐
ᨨ
 
 
盐ᆠ盐矄
X


( ) ҈

෌�
 


mY㡰
㦨






u錐
m
n
㦨
҈


黨
u


( )

( )

m



㦨





u錐
m ҈҈ 
㦨
҈




u

Topic-4 : Miscellaneous (Mixed Concepts) Problems 1 ҈






꾀
ᆠᆠ� 
�

盐ᆠ
ᨨᨨ泐
矄盐
�
ᨨ矄��ᨨ

ᨨ

ᆠ�
鼀ᨨ盐 
ᨨ
�ᨨᆠ�
鼀�
ᨨ
 㜠� 
҈
泐矄
� 矄�
 
ᨨᨨ泐

矄矄

 
 
��ᨨ

犨矄



�鼀矄
鼀ᨨ
 
 
鼀�矄矄�

⼄盐�
ᨨ 矄 
ᨨ�ᨨ
ᨨ
 
ᨨᨨ泐

␔ 
�矄
 
鼀� �鼀盐鼀

盐鼀
ᨨ
泐矄
 
犨 �ᆠ

 
ᨨᨨ泐
ᆠ
 ᆠᨨ 矄�
矄 
ᨨ
盐矄䁸








᲼꾀



҈
桌 㡰

㡰 黨
ᆠ㡰 ҈

㡰 ҈




ᆠᨨ矄�ᆠ
 �泐
ᨨ

ᨨᨨ 矄
盐 � ᨨ鼀泐
�矄�盐  ᨨ盐� ᨨ盐

矄 �ᆠ
盐ᆠ盐矄
ᨨ
�盐矄

�矄
��
泐 쟼㦨




�矄
盐泐
盐
ᨨ
 
ᆠᨨ矄�ᆠ
 �泐

 
�盐矄
ᨨ � 
ᨨ
 
盐ᆠ�
�矄
҈
鼀
㦨
҈
҈�
鼀㡰




᲼꾀



҈
桌 㡰


�
鼀 㡰 


鼀 ᆠ㡰 



鼀 㡰 



鼀


盐ᆠ盐矄

 
矄ᨨ� �
 �泐

ᆠ泐矄
� ᨨ
犨ᨨ a㜠�ᆠ矄
 

盐 ├ ᨨ犨
盐ᆠ盐矄


盐 ├ ᨨ犨

盐ᆠ盐矄
�矄 ᲼҈黨黨黨和
㜠


㝐├矄桌 㡰 �ᨨ� 㡰 ᆠᨨ ᆠ㡰 ᨨᨨ

㡰 ᨨ 泐�






�矄ᨨᨨ
҈
�  � �

鼀矄矄
҈


҈

盐
盐 �ᨨ矄

MCQs with One Correct Answer

҈
bᆠ泐
ᨨ
҈

ᔴ

ᔴ
矄

 ᆠ�
矄
ᨨ
 
盐ᆠ盐矄

( ҈

ᔴã)





҈
㝐
ᨨ
�矄
�ᨨ盐 
矄

ᔼ
҈
� 쟼 ᆠ泐矄


    㜠 ҈㡰  �
πε


ᨨ
҈

ᔴ ã

 
鼀 �鼀盐鼀
├� �ᆠ
 �泐
ᨨ
 
b�ᆠ
�

҈�


鼀矄盐
鼀矄矄矄
ᨨ
 
盐ᨨ
҈҈ 
҈� 䰘   
졠


҈





�
盐

҈


䰘

�
盐

҈�



҈
黨
盐
 
҈�




�
盐
҈� 矄ᆠ�泐

෌�
 
 
��
ᨨ
ᨨ
 
҈� 䰘   
졠

盐ᆠ�

矄鼀

҈
盐
㦨
黨҈
�
㝐
⏌ᆠ

ᆠ
�矄
 
矄
ᨨ �� 㡰
 

⏌

pe
㡰
㦨
҈



㝐
鼀

꾀矄矄盐鼀� �

 
  ҈� �  ᆠ
犨
 
� � �
 ��矄
ᨨ
҈� 䰘   
졠

�


盐 �矄
ᨨ
㝐
�矄
҈
盐
㦨
黨҈
�
㝐⏌ᆠ


犨 
ᆠ
�矄
 
矄
ᨨ �� 
�
ᆠ盐盐鼀㡰





᲼꾀



҈
桌 꾀
盐ᆠ
ᨨ犨
 
矄盐泐� �
ᆠ�ᆠ
ᨨ犨
ᨨ
 ��
盐矄矄

�ᨨᆠ�
鼀�
ᨨ

� 
T
泐矄
矄
  盐


鼀ᨨ盐 
ᨨ
盐

 
��

� �
�矄
矄盐ᆠ
 
  ᨨ
ᨨ犨
⼄盐�鼀 
ᨨ
 
��
�矄
҈

�奠
ᨨ
  ᆠ�ᆠ
ᨨ犨
�
ᨨ鼀
 
 

 
�鼀

ᔼ
   
�矄

ᨨ
鼀
 
ᨨ
ᨨ犨
矄
ᨨ
 
�� ᨨ

鼀 �鼀盐鼀
�ᨨ
ᨨ
nT
泐矄

 
 
盐
ᨨ
n
�矄 ᲼꾀



҈�桌

WWW.JEEBOOKS.IN

P-299

Nuclei

Fill in the Blanks

4



꾀ᨨ鼀矄
� �
 
矄鼀












盐
�  












 ᆠ
�矄ᨨᨨ矄
᲼҈黨


㜠


㝐├矄桌

MCQs with One or More than One Correct Answer

6 䰘


҈

꾀
�矄矄�ᨨ
ᆠ�ᨨ
�矄
��
泐


黨
 ® �
 +

黨

和

+ x + y


犨 
x
 
y

犨ᨨ
�ᆠ矄

ᔴᨨ 矄�� �


黨
 ᨨ

 矄

 
├� �ᆠ
 ��矄
ᨨ
 
ᨨ盐ᆠ矄

 ᨨ
泐 K

K和

K 

㝐㡰
 
⧤泐

㝐㡰

矄ᆠ�泐


  黨
҈

� � �
 ��矄

盐ᆠᨨ
ᨨ


黨

�

 

和 





䰘
�
㝐



�
㝐
 


�
㝐

矄ᆠ�泐

ᔴᨨ 矄�� � �  
ᆠᨨ 矄�ᨨ
犨矄

 
ᆠᨨᆠ
ᨨ�ᨨ 矄㡰
�矄㡰 ᲼꾀



҈�桌 㡰 x
㦨
n

y
㦨
n

⧤和
㦨
҈
黨
㝐

⧤
㦨



㝐 㡰 x
㦨
p

y
㦨


⧤和
㦨
҈
黨
㝐

⧤
㦨



㝐 ᆠ㡰 x 㦨
p

y
㦨
n

⧤和
㦨
҈
黨
㝐

⧤
㦨



㝐 㡰 x
㦨
n

y
㦨
n

⧤和
㦨



㝐

⧤
㦨
҈
黨
㝐 꾀
矄
� ��泐
矄
҈



盐ᨨ 矄

ᔼ
ᨨ盐ᆠ矄  �泐
�
 
ᨨᆠ矄矄矄
҈ 
+҈ 
®҈   + p

  ҈









+҈  ®
 + n

ᔼ
 
�
ᨨ犨
�

泐
 
矄
�矄
҈
҈

W

 
盐ᨨ
矄盐泐
ᨨ
 
矄
�矄  盐矄
�

�鼀
ᨨ
 
ᨨ
ᨨ ᲼҈黨黨㜠

㝐├矄桌 㡰 ҈

s


㡰 ҈

s


ᆠ㡰 ҈
҈
s
㡰 ҈
҈
s



鼀矄矄矄
ᨨ
 
盐ᆠ�

矄
ᨨᨨ犨矄

M 
㡰 =


҈
鼀盐錐

M
p㡰
㦨
҈


䰘
鼀盐錐
Mn㡰
㦨
҈




鼀盐錐 M 
㡰 =



҈鼀盐
7 黨


Match the Following

ᨨ盐
 泐矄�ᆠ
⼄盐 ��矄

�矄
�
ᔴᨨ盐鼀
ᔼ

�
盐矄 
�矄
�
ᔴᨨ盐鼀
ᔼᔼ
�

 ᨨ鼀
ᨨ

᲼҈黨
䰘
㜠


㝐├矄桌 ᔴᨨ盐鼀
ᔼ ᔴᨨ盐鼀
ᔼᔼ 㡰 鼀
 �泐
ᨨ 㡰




 �
鼀ᨨᆠ盐矄

ᨨᨨ鼀 鼀 㡰 � � �
 �泐
ᨨ  㡰

 泐
盐ᆠ�


盐ᆠᨨ

ᆠ㡰 㜠泐
 ᨨᨨ
 �泐 �㡰 ҈├
 㡰  ᨨᨨ
 �泐
ᨨ  㡰 䰘
㝐 �矄�
�� 


ᆠᨨᆠ
鼀ᆠ � �
ᨨ
ᔴᨨ盐鼀 矄
ᔼ
 
ᔼᔼ
�矄
��
泐 㡰 a  e, b  h, c  g, d  f 㡰 a  e, b  g, c  f, d  h ᆠ㡰 a  f, b e, c  g, d  h 㡰 a  f, b  h, c  e, d  g


҈

ᔴᨨ盐鼀 㜠ᔼᔼ
��矄
ᆠ�
矄泐矄鼀矄
盐 �ᨨ� �

ᨨᆠ矄矄
ᔴᨨ盐鼀 㜠ᔼ
矄盐��矄矄
ᆠ  �矄
�
矄ᨨ鼀
ᨨ
 
鼀矄 
ᨨ
 
矄泐矄鼀

㝐ᆠ
 
矄鼀 矄
�
ᔴᨨ盐鼀 㜠ᔼ ᨨ
 
ᨨ�
ᨨᆠ矄矄矄㡰
ᨨ鼀
ᔴᨨ盐鼀 㜠ᔼᔼ
᲼


黨桌 ᔴᨨ盐鼀 㜠ᔼ ᔴᨨ盐鼀 㜠ᔼᔼ 꾀㡰 
 �泐
ᨨ
  㡰 System

꾀
ᆠᆠ�ᨨ

� ��泐 矄泐矄鼀
�矄
� ᆠ矄 盐 ᆠ � Prᨨcess

ᔼ
�矄
ᆠᨨ

ᆠ ᨨ

泐 㡰 㝐ᆠ  �ᆠ
 �泐 ⼄㡰 System
꾀
�矄
�


�矄
ᨨ�
ᨨ
  ��ᆠ
ᆠᨨ � 
� 矄泐矄鼀

犨 �ᆠ
�矄 犨�

��ᆠ
�矄ᨨ

ᆠᨨ 
� ᨨ
 �泐 Prᨨcess


�矄
�矄 ᨨ
 ᨨ鼀
鼀ᨨ�ᨨ
ᨨ ᆠᨨ鼀矄矄
泐
盐矄 � � �矄
矄  
�矄ᨨ

ᔴ㡰 ᔼ  
 �泐 㡰 System

꾀
�矄
�

��� ᨨ
 
矄泐矄鼀
�矄 ᆠᨨ �  ᆠᨨ 
� ᨨ
�矄 Prᨨcess

�矄
�矄 鼀ᆠ  �ᆠ
 �泐
ᆠᨨᨨ
盐
ᨨ
ᆠᨨ 鼀ᨨ矄 
矄盐ᨨ盐 � �
� 㡰 㝐矄矄
ᨨ
 
矄泐矄鼀 矄㡰 System

꾀
泐
盐ᆠ盐矄
�矄
ᆠ矄 � ��泐

矄 Prᨨcess


盐ᆠ盐矄 �矄矄�ᨨ 矄
� ᨨ
犨ᨨ
�鼀 矄 ᨨ
泐
⼄盐
鼀矄矄矄  
矄ᨨ鼀
盐ᨨ 矄
 鼀� 㡰 System
꾀
矄�矄�
犨�
ᨨᨨ Prᨨcess


ᨨᨨ
�矄 ᆠ
�

�鼀
泐� � 鼀� �ᆠ
�  �ᆠ盐
ᨨ
�矄
  ҈҈
ᔼ
 
ᨨᨨ犨� �

ᆠᨨ盐鼀
ᔼ
�矄矄
矄ᨨ鼀
 泐矄�ᆠ
⼄盐 ��矄  
 
ᆠᨨ盐鼀
ᔼᔼ
��矄
ᨨ �鼀
 �泐
盐矄 矄矄ᨨᆠ�
犨�
矄ᨨ鼀
ᨨ
 鼀

ᔴ ᨨᨨ矄
 
ᨨ�
盐 ᨨ
 �泐
ᨨ鼀
ᆠᨨ盐鼀
ᔼᔼ
ᨨ
ᆠ
ᨨ
 
 泐矄�ᆠ
⼄盐 ��矄 �
ᆠᨨ盐鼀
ᔼ
 
犨�
 
ᆠᨨ矄ᨨ � �
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⼄



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WWW.JEEBOOKS.IN

EBD_8341 P-300

Physics

8

Comprehension/Passage Based Questions

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WWW.JEEBOOKS.IN

P-301

Nuclei ҈

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鼀 
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248 96
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P-305

Semiconductor Electronics : Materials, Devices and Simple Circuits


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EBD_8341 P-306

Physics




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P-307

Semiconductor Electronics : Materials, Devices and Simple Circuits

4.

A common emitter amplifier circuit, built using an npn transistor, is shown in the figure. Its dc current gain is 250, RC = 1 k &! and VCC = 10V. What is the minimum base current for VCE to reach saturation ? [Main 8 Apr. 2019 II]

(a)

1 -1 (W ) ri 0.01

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5.

6.

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In the figure, given that VBB supply can vary from 0 to 5.0 V, VCC = 5 V, bdc = 200, RB = 100 kW, RC = 1 KW and VBE = 1.0 V. The minimum base current and the input voltage at which the transistor will go to saturation, will be, respectively : [Main 12 Jan. 2019 II] (a) 25 µA and 3.5 V (b) 20 µA and 3.5 V (c) 25 µA and 2.8 V (d) 20 µA and 2.8 V In a common emitter configuration with suitable bias, it is given than RL is the load resistance and RBE is small signal dynamic resistance (input side). Then, voltage gain, current gain and power gain are given, respectively, by: (b is current gain, IB, IC, IE are respectively base, collector and emitter currents:) [Main Online April 15, 2018] (a)

b

RL DI E 2 RL , ,b RBE DI B RBE

R

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7.

8.

DI

R

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In a common emitter amplifier circuit using an n-p-n transistor, the phase difference between the input and the output voltages will be : [Main Online April 2, 2017] (a) 135° (b) 180° (c) 45° (d) 90° For a common emitter configuration, if a and b have their usual meanings, the incorrect relationship between a and b is : [Main 2016] b (a) a= 1 +b

(b) a=

b2 1 +b2

1 1 = +1 (d) All of these a b A realistic graph depicting the variation of the reciprocal of input resistance in an input characteristics measurement in a common emitter transistor configuration is : [Main Online April 10, 2016]

(c)

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~ v0

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1

10. An n-p-n transistor has three leads A, B and C. Connecting B and C by moist fingers, A to the positive lead of an ammeter, and C to the negative lead of the ammeter, one finds large deflection. Then, A, B and C refer respectively to: [Main Online April 9, 2014] (a) Emitter, base and collector (b) Base, emitter and collector (c) Base, collector and emitter (d) Collector, emitter and base. 11. For a given plate voltage, the plate current in a triode valve is maximum when the potential of [1985 - 2 Marks] (a) the grid is positive and plate is negative (b) the grid is zero and plate is positive (c) the grid is negative and plate is positive (d) the grid is positive and plate is positive 12. The plate resistance of a triode is 3 × 103 ohms and its mutual conductance is 1.5 × 10 –3 amp/volt. The amplification factor of the triode is [1981- 2 - 3 Marks] (a) 5 × 10–5 (b) 4.5 (c) 45 (d) 2 × 10 5 3

Numeric Answer

13. The output characteristics of a transistor is shown in the figure. When VCE is 10 V and IC = 4.0 mA, then value of bac is ______. [Main Sep. 06, 2020 (II)] (IB ) 60 mA 50 mA 40 mA 30 mA 20 mA 10 mA

8

(IC) in mA

(a) 40 mA (c) 7 mA

6 4 2 2

6

4

6 8 10 12 14 (VCE) in volts

MCQs with One or More than One Correct Answer

14. A transistor is used in the common emitter mode as an amplifier. Then [1998S - 2 Marks] (a) the base-emitter junction is forward-biased (b) the base-emitter junction is reverse-biased (c) the input signal is connected in series with the voltage applied to bias the base-emitter junction (d) the input signal is connected in series with the voltage applied to bias the base-collector junction

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EBD_8341 P-308

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Topic-3 : Digital Electronics and Logic Gates 1 ҈





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P-309

Semiconductor Electronics : Materials, Devices and Simple Circuits

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Answer Key Topic-1 : Solid Semiconductors and P-N Junction Diode ҈
҈҈

҈
҈
䰘

҈


㡰

ᆠ㡰 
㡰 ҈

㡰 ҈
ᆠ㡰


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㡰 

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졔㡰
p㜠矄�

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㡰




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ᆠ㡰 㡰 㡰 

㡰

盐 㡰



҈






黨


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



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黨
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㡰  矄

��㡰


ᆠ㡰

Topic-2 : Junction Transistor ҈
ᨨ 盐矄㡰

㡰 ҈҈
㡰 ҈

㡰


㡰 ҈
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黨
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҈

ᆠ㡰



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黨
ᆠ㡰

҈

ᆠ㡰

Topic-3 : Digital Electronics and Logic Gates ҈
㡰 ҈҈
ᆠ㡰



ᆠ㡰 ҈

㡰


㡰 ҈
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㡰 ҈

㡰

�
㡰



㡰

䰘
ᆠ㡰

WWW.JEEBOOKS.IN

EBD_8341 P-310

Physics

Communication Systems

29

Topic-1 : Communication Systems 1 ҈


MCQs with One Correct Answer

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[Main Sep.
02, 2020 (I)] � 



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�䰘
�

WWW.JEEBOOKS.IN

P-311

Communication Systems

҈҈


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WWW.JEEBOOKS.IN

EBD_8341 P-312


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Physics

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�矄
ᆠ �
盐
ᨨ  
ᨨ
�  
 ᆠ�ᨨ
ᨨ
 
㜠鼀
犨
ᨨ鼀


ᨨ�
�� 
�
 
�ᨨ ᨨ矄 
和鼀 㜠


 ᆠ�
� 
ᨨ

矄鼀
�矄
�    ᨨ
 
⼄盐 ᆠ泐
ᨨ
㜠鼀
犨矄
᲼㝐�
졠 � 
꾀�






҈桌 㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
矄
㡰 和鼀 㜠҈
�矄
矄

和鼀 㜠

�矄
盐
ᆠ㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
盐
盐 和鼀 
㜠

�矄
ᨨ
 
ᆠᨨᆠ
  �ᨨ
ᨨ 矄鼀 㜠҈
㡰 和鼀 㜠҈
�矄
盐

和鼀 㜠

�矄
盐
  和鼀 
㜠

�矄
 
ᆠᨨᆠ
  �ᨨ
ᨨ

和鼀 㜠҈
ᔼ

ᆠ�
犨
ᆠ㡰
㦨
꾀
矄�
wᆠ
�矄
鼀�盐
鼀ᨨ盐
泐 
鼀ᨨ盐ᨨ
矄�� 
鼀㡰
㦨
꾀
矄�
w鼀
 
 
⼄盐�ᨨ
ᨨ 鼀ᨨ盐
矄�� 
᲼ᔴ鼀㡰桌
 
�矄
鼀ᨨ盐�ᨨ
� 
 矄ᆠ�泐

᲼㝐�
졠 � 
꾀�
黨



҈桌 㡰 Cm t㡰
㦨
A
҈
졔
矄�
wm
t㡰
矄�
wc
t
 

㡰 Cm
t㡰
㦨
A
҈
졔
矄�
wm
t㡰
矄�
wm
t
 
҈ ᆠ㡰 Cm
t㡰
㦨
A ҈
졔
矄�
wm
t㡰
矄�
wc
t
 
҈ 㡰 Cm
t㡰
㦨
A
҈
졔
矄�
wc
t㡰
矄�
wm
t
 



Answer Key Topic-1 : Communication Systems ҈
҈҈

҈
҈


㡰 ᆠ㡰 㡰 ᆠ㡰



҈








㡰 㡰 ᆠ㡰 㡰


҈





ᆠ㡰 㡰 㡰 ᆠ㡰



㡰 ҈

ᆠ㡰


ᆠ㡰

�
㡰 ҈�
ᆠ㡰
�
㡰



㡰 ҈

㡰


㡰

䰘
ᆠ㡰 ҈䰘
㡰
䰘
㡰



ᆠ㡰 ҈

ᆠ㡰


㡰

黨
ᆠ㡰 ҈黨
ᆠ㡰
黨
㡰

҈

ᆠ㡰


㡰 

㡰

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EBD_8341 PS-2





Physics ᆠ㡰 쟼 �泐

E µ AaT b P c


�㡰 ᨨ













E = kAaT b P c 犨 
k
�矄

�鼀 矄�ᨨ 矄矄
ᆠᨨ 矄 
 
a, b
 
c

   ᨨ  矄
�鼀 矄�ᨨ
ᨨ
鼀ᨨ鼀 盐鼀

P = M ҈ L҈T -҈ �鼀 矄�ᨨ
ᨨ


A
㦨
L
�鼀 矄�ᨨ
ᨨ
�鼀

T
㦨
T ҈ 盐� �
 矄
盐
�
⼄盐�ᨨ
�㡰

犨
�


᲼ 桌
㦨
᲼a桌
᲼b桌 㝐 

㦨
㝐҈
᲼b桌 ҈



M ҈L
T -
= M c L
a + cT b - c

泐
ᆠᨨ鼀�矄ᨨ

c
㦨
҈
a 졔 c
㦨

b  c
㦨



҈


c
㦨
҈

a
㦨
҈⏌


b
㦨
҈

\
E = A҈⏌
T -҈ P҈

m
m

= = m
c e
e
m


䰘


㡰





m
c
®

㝐 
꾀

맔
 ҈ 㝐
 꾀
�鼀 矄�ᨨ 矄
ᨨ
矄�矄 ᆠ
㡰 
㦨
�Ӹ


ÞY µ




�㡰

Z


᲼ A
T
桌 Q Q
= =  W ᲼ ML
T -
桌


㦨
᲼㝐 ҈

꾀
桌 F Z =B= IL 
㦨
᲼㝐 
꾀҈桌

᲼Q
F
㦨
ILB桌

᲼ M -҈ L-
T
A
桌

é e
ù ê ú= ëê m
úû

㡰

Q F=

҈



é e =ê
m
e
êë m
e


ù ú
㦨
e C᲼LT ҈桌맔᲼e 桌 0 0 úû é ù ҈ = Cú êQ m
e
êë úû


pe
r


᲼ AT 桌
᲼ MLT -
桌 ´ ᲼ L
桌

= ᲼ A
M -҈ L-T
桌

é e ù \ ê
ú = ᲼ LT -҈ 桌 ´ ᲼ A
M -҈L-T
桌
= ᲼ M -҈L-
T  A
桌 ëê m
úû b㡰 꾀矄
犨
├ ᨨ犨


élù êë  úû = [ ]   [ ᆠ] = [ 꾀 ] é l ù é ù \ê ú= ê = é 꾀 ҈ ù ë ᆠ û ë 꾀 ûú ë û ҈҈


҈�


矄� �
�㡰㡰

e



q


Þ ᲼e
桌 =

\ V
µ h
I ҈G ҈C � ҈



æ -
ö ç ÷ ç a├ ÷ ø abè

҈䰘


é ù

\ ê a├ ú 㦨
㝐




Þ

Þ
᲼a桌
㦨
㝐҈


҈





�矄
 
 �泐
 矄�泐
ᨨ
鼀� �ᆠ
m


é B
ù 쟼 �泐 ᨨᆠ ´ �矄ᆠ鼀  = ê ú= �矄ᆠ鼀 㡰 ëê
m
ûú Vᨨlume é ML
T 
ù ҈ 
= ú = ML T













ê  êë L úû

b㡰  ├
 �
�矄

盐 �
ᨨ
 �

lp
㦨
҈

҈


黨
맔
҈
�
鼀

lp =

úû

 = 㝐


[ a ]㝐
-


㡰 
⼄盐 �泐
�








෌ ҈ �
 = -
\  µ

ᔴ�

Nᨨ 㡰 和ᨨ� �
ᨨ �
V
㡰 µ h x I y G Z C r
-҈ 
h 㦨
 ᆠ├矄
ᆠᨨ 矄 
= éë ML T ùû I
㦨
ᆠ盐 
㦨
᲼A桌 G
㦨
෌��ᨨ 
ᆠᨨ 矄 
㦨
᲼M҈LT
桌  
c
㦨
矄
ᨨ
�� 
㦨
᲼LT ҈桌 V

㦨
ᨨ �㦨
᲼ML
T A҈桌 \ ᲼ML
T A ҈桌㦨᲼ML
T ҈桌
᲼A桌泐᲼M ҈LT
桌᲼LT ҈桌r Mx  z錐
L2x졔z졔r錐
Tx2zr錐
Ay ᔴᨨ鼀� �
�鼀 矄�ᨨ
ᨨ
㝐





꾀

犨
� 泐
㦨
҈


㦨




㦨

҈



㦨
�

Þ

b㡰 ᨨᆠ
ᨨ
� ᆠ�ᨨ
犨
犨ᨨ
ᨨ鼀矄

= 和� ᆠ
 ᨨ  �
鼀矄

�鼀 矄�ᨨ 矄矄

êë

㡰 
᲼Ӹ桌
㦨
᲼桌
᲼ 桌
᲼꾀桌ᆠ ᲼㝐҈ 
桌
㦨
᲼ ҈桌
᲼㝐 
桌
᲼ 
桌ᆠ ᲼㝐҈ 
桌
㦨
᲼㝐졔졔ᆠ


ᆠ桌 ᔴᨨ鼀� �
ᨨ犨
ᨨ
矄�
ᨨ
矄�鼀�
鼀矄
犨
�

㦨
҈


졔

졔
ᆠ
㦨

҈





ᆠ
㦨


矄ᨨ� �
ᨨ
⼄盐�ᨨ 矄
犨
�

㦨





㦨
҈

ᆠ
㦨

矄ᨨ
᲼Ӹ桌
㦨
᲼
꾀
桌
㦨
᲼
꾀
桌 b㡰 �鼀 矄�ᨨ
ᨨ
᲼h桌
㦨
᲼ML
T҈桌 ᲼ᔴ桌
㦨
᲼LT ҈桌 ᲼G桌
㦨
᲼M҈L T
桌  ᆠ
�鼀 矄�ᨨ
ᨨ é hC � ù é ML
T -҈ ù × é L�T -� ù û ë û ê ú㦨 ë é M -҈LT -
ù ê G ú ë û ë û 㦨
᲼ML
T
桌
㦨
 �泐 ᆠ㡰 


µ
෌
泐
ᔴ �鼀 矄�ᨨ 矄
ᨨ
෌
㦨
᲼㝐҈ 
桌

㦨
᲼㝐
҈桌
 
ᔴ
㦨
᲼ ҈桌 ᲼ 桌
㦨
᲼㝐 ҈ 
桌 ᲼㝐
҈桌泐᲼ ҈桌 ᲼㝐

҈桌
㦨
᲼㝐  졔泐
 졔
泐졔

泐桌 泐
ᆠᨨ鼀� �
 
ᨨ犨矄
ᨨ
㝐




ᨨ
 
矄�矄 

졔
泐
㦨


Þ

㦨
泐 
졔

泐

졔

㦨


Þ
�
졔

㦨







�㡰 


泐

㦨
҈ Þ


졔

㦨
҈





��㡰 和ᨨ� �
⼄ 矄

�㡰
 
��㡰


= 泐=

-
-҈


᲼ MT A 桌 Ӹ
㦨
᲼㝐 

꾀
桌 黨


҈



X

X = ᔴᆠ� ᆠ 㦨

Y=

æ ҈ ö ç = c÷ ç m e ÷ è

ø

Þ ᲼b桌
㦨
㝐
 


hG c

㡰 
鼀矄矄


矄
㝐
µ
ᔴ泐  㝐҈


㦨
 Ἐ㡰
҈ ҈㡰泐
㝐҈
҈㡰  㝐҈


㦨
㝐 
泐
졔


졔
泐 

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PS-3

Physical World, Units and Measurements 
㦨
҈ 泐
졔


㦨

泐
㦨



҈黨


Q




泐


㦨


졔



҈
㦨


㦨
҈








㝐
㦨
᲼ ҈
ᔴ

҈桌 b㡰 ␔
├ ᨨ犨
 
矄�矄��泐

㦨

꾀 

=

ᔼ
 턼 
㦨
ᔼ㡰 꾀






᲼桌᲼ᔼ桌 ␔ ␔ Q = = é ᲼㝐 ù ⼄ �

ê ú ´ ᲼ 桌 ᲼ᔼ桌᲼ 桌 ëê ûú

ᆠ㡰 
Ҙ


犨�
e

m

c
 
h
矄
ᨨᨨ犨矄
m

㦨
keambcchd ᲼MLT
A
桌
㦨
᲼AT桌a
᲼M桌b
᲼LT҈桌c
᲼ML
T҈桌d 㦨
᲼Mb
졔
d
Lc
졔

d
Ta

c 
d
Aa桌 졠
ᆠᨨ鼀� �
ᨨ
矄�矄
犨
� a
㦨





�㡰 b
졔
d
㦨
҈


��㡰 c
졔
2d
㦨
҈


���㡰 a

c

d
㦨




�㡰 泐
⼄盐�ᨨ
�㡰

��㡰

���㡰


�㡰
犨
�
a
㦨




b
㦨



c
㦨

҈

d
㦨
҈

é h ù ú ë ce
û






u
㦨



�







㦨


é 㝐

҈꾀҈ ù ë û

�鼀 矄�ᨨ 泐

f E ¹ E


ᆠ�ᆠ
�㡰

�ᨨ
鼀ᨨ鼀 

p
㦨
ᆠ �
q㡰
맔
�矄 ᆠ
d㡰 ᆠ㡰 
ᨨ犨
ᨨ

 ᨨ  
�矄

盐鼀
ᨨ
ᆠᨨ 矄 







az = ᲼ M ° L°T °桌 kq

kq z ᲼ ML
T -
q -҈ 桌᲼q桌 é = MLT -
ù \ a
㦨
ë û ᲼ L桌 ᨨ盐 = ᲼ ML
T -
q-҈ 桌
 
z
㦨
᲼L桌 꾀矄
k = ├�

a a Þ b
㦨
꾀 

�鼀 矄�ᨨ 泐
P
㦨
b P MLT -
= ᲼ M
L
T
桌 \ ᲼b桌
㦨
ML-҈T -
F é -҈ -
ù êQ P = A = ᲼ ML T 桌ú ë û Dq e
A = 㡰 ᔴᆠ� ᆠ

C = Dt L DV Dq㡰 L DV = L \
X = e
L Dt ADV 㡰 Dt \





p é 㝐 -҈-
꾀
ù é 㝐 -҈ -
ù é 㝐҈

ù ë ûë ûë û é
꾀
ù ë û 㦨

p é㝐 -҈-҈+
-+ 
-
꾀
ù ë û é
꾀
ù ҈ ë û 㦨
㦨




ù
p é
p 㝐  꾀 ë û

턼 㝐
-
=
㦨


鼀 㝐 ᔴ � = 㝐 -҈-

꾀
ᔴᆠ� ᆠ
ᔴ
㦨


ur ur 㡰 쟼ᆠ�ᆠ
盐

f E = E
S

 ᨨ

�鼀 矄�ᨨ 泐


e
a
hc


pe
෌鼀


⼄҈⼄





























㦨
᲼㝐҈



꾀
桌 㡰 꾀 �盐
鼀ᨨ鼀 盐鼀
㦨
鼀
맔

맔

㦨
㝐

҈

\
䰘


 = é 㝐

҈꾀҈ ù ë û ҈ 
e
= é 㝐  꾀
ù ë û -҈  -
ù é 鼀 = é 㝐҈

ù ෌= 㝐 

ë û
 
 ë û


҈ ⼄҈⼄

pe



 



㦨


b㡰 
�鼀 矄�ᨨ 
ᨨ鼀盐
ᨨ

ᨨ犨



b㡰 꾀矄
犨
├ ᨨ犨

=


p
ᔴ
᲼꾀 桌
 ᆠ

e
= = 
鼀
᲼㝐 -
桌᲼
桌

㡰 
盐 �
�u

犨�
e

a


h
 
c
矄
ᨨᨨ犨矄
᲼u桌
㦨
᲼e桌a
᲼a
桌b
᲼h桌c
᲼C桌d 矄� �
�鼀 矄�ᨨ 
鼀 ᨨ
᲼M҈L
T졔
A졔
桌
㦨
᲼A҈T҈桌a᲼L桌b᲼ML2T҈桌c᲼LT҈桌d ᲼M҈L
T졔
A졔
桌
㦨
᲼Mc Lb졔
c졔d Tacd Aa桌 a
㦨



b
㦨
҈

c
㦨

҈

d
㦨

҈ \

᲼㝐 -
꾀 -
桌 �鼀 矄�ᨨ 矄 ᨨ m
㦨
�鼀 矄�ᨨ 矄 ᨨ Î ᲼㝐 -҈-
꾀
桌

Þ e
=


-


\
᲼m
桌 = ê
҈




㦨
᲼㝐


꾀

桌
㦨
᲼
桌

㦨 ᲼㝐㜠 ҈㜠   桌᲼ᔼ
桌 㦨 ᲼㝐㜠 ҈㜠  ᔼ
桌





 
矄鼀
盐
�
�  
矄泐矄鼀矄
ᨨ
盐 �矄
ᆠ㡰 �鼀 矄�ᨨ 矄
ᨨ
m
㦨
᲼㝐 
꾀
桌 �鼀 矄�ᨨ 矄
ᨨ
Î
㦨
᲼㝐҈
꾀
桌 �鼀 矄�ᨨ 矄
ᨨ

㦨
᲼㝐
꾀
桌 \

 ҈ ᔴᨨ 盐ᆠ��泐
㦨
矄�矄��泐 㦨 꾀

㦨

҈ 

�矄
�鼀 矄�ᨨ 矄矄
 盐矄
 
ᆠᨨ鼀� �ᨨ

犨ᨨ盐
p
pe ෌鼀





a
㦨


 Dq 㡰 L
DV Dq = = I

ᆠ盐 㡰 DV L
Dt Dt æ ҈ö 
ç ÷ e
E

矄 矄

 �泐
 矄�泐
�



 �泐 è
ø

=


黨


ᆠ㡰


盐 �
ᨨ盐鼀



-
é҈ ù [ 쟼 �泐] ML T \ ê e
E
ú = = = ᲼ ML-҈T -
桌 
ᨨ盐鼀 ] ë û [ L

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EBD_8341 PS-4





Physics 㡰


d µ r x S y f z = K r x S y f z
犨 
⧤
�矄








b

ᆠ㡰
꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

�鼀 矄�ᨨ 矄
ᨨ
 �盐
鼀ᨨ鼀 盐鼀 ᲼mvr桌
㦨
M
L
T

 
ᨨ
鼀矄矄
᲼m桌
㦨
M
L
T


�鼀 矄�ᨨ 矄矄
ᆠᨨ 矄 


ML
T ҈
㦨
ML
T

Þ
T
㦨
L


M
L҈T
= M x L-x M yT - yT - z 쟼⼄盐� �
�鼀 矄�ᨨ 矄
ᨨ
矄�矄

mvr M
L
T
= = L-҈ r L ҈ ҈ mv 㡰
E = mv
= = L-


m E L-
L-
P= = =
= L-
t T L -
E L F= = = L- x L

㝐ᨨ鼀 盐鼀

p
㦨
mv
㦨


M
L҈T
= M x + y L-xT - y - z

\
x
졔
y
㦨





x
㦨
҈

쟼 �泐

҈ ҈ \
x = -
 

y
㦨
 

ᨨ犨

\
n
㦨


҈


ᔴᨨ 盐ᆠ��泐

s =

J � /A �
t = = E F/q FA

ᨨᆠ 䰘


= é M ҈LT  A
ù ë û 



MLT é a ù = ᲼ P 桌 Þ ᲼a桌
㦨
᲼PV
桌
㦨

ú L
ëV û



ê

s= 


é q
ù
 -
ê ú = ᲼ Fr 桌 = ᲼U ´ r 桌 = ᲼ ML T 桌 e ëê ûú

L
㦨
ML� T 


᲼kBT桌
㦨
᲼U桌
㦨
᲼ML
T 
桌 �鼀 矄�ᨨ
ᨨ
l
㦨
᲼L桌

J � A �
t = = = ᲼M -҈L-T  A
桌 E F q Fa

㡰

\

᲼Y桌
㦨


᲼M -҈ L-
T
Q
桌 ᲼ MT -҈Q -҈ 桌













= ᲼U ´ r 桌 ´

æ ö X �� ÷ çQ Y =
Z è ø

= ᲼ M - L-
T
Q
桌 㡰

e k BT 㡰



和

㦨


nq

矄� �
E
㦨
hv Þ
 ᆠ├矄
ᆠᨨ 矄 




b㡰
෌�
ᨨ矄��ᨨ


㦨
᲼a桌 和

 ҈
㦨
᲼b桌 꾀ᆠᆠ�ᨨ

 

㦨
᲼桌 � 
鼀ᨨ鼀 盐鼀

㝐 ҈
㦨
᲼⼄桌 ᨨᆠ

㝐 

㦨
᲼桌

ᆠ㡰

㡰

和 b = = 

�鼀 a-b

᲼  桌 =

�  鼀ᨨ鼀 盐鼀 �鼀

᲼⼄ 桌 ᲼

a-b

桌

Þ 
㦨
⼄
졔
b - a Þ 
㦨
⼄
졔
b - 
b - 㡰 Þ 
㦨
⼄

b +  Þ 
졔
⼄


㦨
b  ᆠ
ᨨ�ᨨ
㡰
�矄
ᆠᨨᆠ


k B T 㡰

=




n q e 㡰

҈ ҈ = = ᲼L桌 - ᲼n桌᲼r 桌 ᲼ L 桌᲼ L桌



和

㦨


æ q
ö ҈ ҈ ´ ç ÷ ç e ÷ k BT 㡰 n
 è ø



和

=

҈ ᲼ L-
桌

҈












= ᲼ L桌 

ᆠ㡰
矄� �

C =

᲼U 桌 ᲼n桌᲼U ´ r 桌

= ᲼ L
桌

æ q
ö ҈ ҈ ´ ç ÷´ ç e ÷ k B T 㡰 n҈  è ø












= ᲼U ´ r 桌 ´ ᲼U 桌 ´





=












=

҈

a-b +
㦨
b \ a 졔

㦨
2b  ᆠ
ᨨ�ᨨ
㡰
�矄
ᆠᨨᆠ
ᨨᆠ
=
















= ᲼U ´ r 桌 ᲼U 桌

ᨨ矄��ᨨ

᲼ a 桌 = �鼀
= =  a-b 和 ᲼b 桌 꾀ᆠᆠ�ᨨ
=

҈ ´ n = ᲼n桌 ´ ᲼r 桌 ᲼U 桌












= ᲼ L- 桌᲼ L桌 = ᲼ L-҈ 桌

é
-
ù E P
S
ë ML T û é
-҈ ù = = ML T h
㦨
= ë û ҈ v éT -҈ ù ë û T

�


æ q
ö ҈ ´n ç ÷´ ç e ÷  k BT 㡰 è ø



和

㦨


᲼X桌
㦨
᲼C桌
㦨
᲼M ҈
L

T

Q
桌 ᲼Z桌
㦨
᲼B桌
㦨
᲼MT ҈
Q ҈ 桌





-


b

㡰 �鼀 矄�ᨨ 矄
ᨨ
᲼n桌
㦨
᲼L桌

᲼q桌
㦨
᲼AT桌

҈ ᲼ L-҈ 桌 ҈

m
e


҈ ҈

᲼n

桌

= ᲼r 桌 ´

᲼n

桌

= ᲼ L桌   R =

m V m

I

㦨
e

V

Þ

=




m
e


æm ö = R
= ç
÷
e
I è e
ø m
I ҈ = =

Q

V
㦨
RI㡰 㡰 e

I
㦨
m

V Þ
e
V R 㡰

҈ ҈




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PS-5

Physical World, Units and Measurements ᆠ㡰

�鼀 矄�ᨨ 泐
� ᆠᨨᆠ
I
㦨
e
C V

\

҈ V = =R \ e
C I

㡰 \

҈ e


㝐 -
=
= 㝐-҈ -҈
p  ´  -҈ 쟼 㝐
-
= 㝐-
-҈  ᆠ├
ᆠᨨ 矄 

h
㦨
= v

-҈ l

鼀
ᆠᨨ 盐ᆠ��泐
⧤ᆠᨨ 盐ᆠ��泐
㦨
꾀D

㝐
-
´  =

㦨 㝐

⧤҈

´ 
´ ⧤

ᔴᨨ �ᆠ� 
ᨨ
�矄ᆠᨨ矄�泐
᲼h桌
㦨


҈ m
e


=R

�鼀 矄�ᨨ 泐
ᆠᨨᆠ
m

C I
㦨
e

V

m
V m
R = = = ´ e
IC C e


҈ 㦨
Ҙ
҈ m
e







�鼀 矄�ᨨ 泐
� ᆠᨨᆠ
黨


ᆠ

㡰 꾀矄
 ᆠ├矄
ᆠᨨ 矄 
h

矄
ᨨ
�� 
c
 
���ᨨ 
ᆠᨨ 矄  G

盐矄
矄
矄�ᆠ
盐 �矄
ᨨ
 �
L
 
㝐矄矄
M
矄ᨨ
L
µ
hx cy Gz




�㡰   㝐µ
hp cq Gr




��㡰 �鼀 矄�ᨨ 矄
ᨨ
᲼h桌
㦨
᲼M L

T҈桌

᲼c桌
㦨
᲼L T҈桌 ᲼G桌
㦨
᲼M҈
L
T
桌 矄� �
� ᆠ�
ᨨ
ᨨ鼀ᨨ� �泐
ᨨ
�鼀 矄�ᨨ 矄 ᨨ
⼄

�㡰

᲼ M
L T
桌 = ᲼ M x L
x T  x 桌᲼ Ly T  y 桌᲼ M  z L z T 
z 桌

M

L T

㦨
Mx

z㡰
L
x
졔
y
졔
z㡰
T x  y 

z㡰 졠
ᆠᨨ鼀� �
ᨨ犨矄
ᨨ鼀
ᨨ
矄�矄

犨
� x

z
㦨




x
졔
y
졔
z
㦨
҈

x

y


z
㦨

졠
矄ᨨ� �
 矄
⼄ 矄


犨
�

҈  ҈

y
㦨


z
㦨



hG L = K  錐
K
�矄

ᆠᨨ 矄 
\ c


�





ᔼ
 
矄鼀
犨泐
矄ᨨ� �
⼄

��㡰
犨
�










b

ᆠ㡰矄� �
F
㦨








 
C =

҈

m
e

pe
㡰 r


᲼Q 桌 I T =  ᆠ

᲼e
桌
㦨

㦨
M҈L T
I

᲼ F 桌᲼ r 桌 MLT -

L
҈ ҈ ᲼ LT -
桌
= ᲼Ҙ
桌
㦨
-҈ -


᲼e
桌᲼C 桌 ᲼M L T I 桌





= ᲼ MLT -
I -
桌


҈







b

ᆠ㡰 泐 ᨨ矄
盐鼀 㦨
ᔴᨨ �ᆠ� 
ᨨ
�ᆠ�ᨨ
㦨
᲼M
L
T
桌
�



�鼀 矄�ᨨ 矄矄
ᔴ盐�
�矄
 
盐 �
ᨨ
�ᨨᆠ��泐
 盐鼀
ᨨ
ᨨ鼀矄
ᆠ泐� �
 矄ᆠᨨ 㡰
 
⼄盐 ᆠ泐
矄ᨨ
矄
 
盐 �

矄ᆠᨨ 
�.e.

᲼M
L
T҈桌











ᆠ㡰

ᨨ鼀


ᆠᨨ 矄 
᲼ K 桌 = u B

\

᲼⧤桌 = 㝐
-
⧤ -҈

q

҈ m
e


\ C
=

҈ m
e


\ m
= e
-҈L-
T


᲼Q

C
㦨
LT҈桌 F A
㦨
᲼M҈L
T҈Q҈桌 �㡰
㝐� �ᆠ
盐
f = BA = qv F ��㡰
㝐ᨨ盐盐矄
ᨨ
����泐
㦨

㦨
᲼ML҈T
桌 A

꾀矄

⼄盐矄�ᨨ

T
µ
PadbEc ᨨ
᲼T桌
㦨
᲼P桌a᲼d桌b᲼E桌c ᨨ ᲼M
L
T҈桌
㦨
᲼ML ҈T
桌a
᲼ML桌b
᲼ML
T
桌c ᨨ M
L
T҈
㦨
Ma
졔
b
졔
c
La
b
졔

c
T
a


c \ a
졔
b
졔
c
㦨







�㡰 
a

b
졔

c
㦨







��㡰 

a


c
㦨
҈





���㡰 和ᨨ� �

⼄ 矄

�㡰

��㡰
 
���㡰
犨
� a
㦨

�⏌


b
㦨
҈⏌


c
㦨
҈⏌


Topic-3 : Errors in Measurements & Experimental Physics ҈


b㡰 ෌�


ᨨ

ᨨ
��矄�ᨨ
ᨨ
ᆠ�ᆠ盐
矄ᆠ
ᨨ
矄ᆠ犨
�盐�
㦨
�
�ᆠ
㦨


�
鼀鼀 矄
ᆠᨨ盐 
ᨨ
矄ᆠ犨
�盐�

�ᆠ ᨨ

ᨨ
��矄�ᨨ
ᨨ
ᆠ�ᆠ盐
矄ᆠ

� = 鼀鼀 = ҈ ´ ҈

� 鼀
㦨
҈

m鼀 �


=

W Q
 

㦨

 
෌��ᨨ
ᨨ �
㦨
m m

�鼀 矄�ᨨ 泐
矄鼀
᲼M
L
T
桌 

㡰
ᨨ⼄盐

t
㦨
F
맔
r
맔
矄�
q  
犨ᨨ├
ᨨ 

W
㦨
F
맔
d
맔
ᆠᨨ矄q �鼀 矄�ᨨ 泐

t
㦨
W
㦨
᲼ML
T 
桌 �鼀 矄�ᨨ 泐

�� 
泐
㦨
犨 �
㦨
᲼L桌

\




쟼
㦨
ᔴ
㦨
 ҈ 㡰 ␔
├ ᨨ犨
 

C=


䰘


hc 錐
K'
�矄

ᆠᨨ 矄 
G Q҈Q


҈ q
= 錐
᲼C桌
㦨
᲼M҈L
T
Q
桌
u
u ᔼ 盐ᆠ ᆠ
L
㦨

錐᲼ L桌
㦨
᲼ML
Q
桌 � RA L
 
R =



\

᲼r桌
㦨
᲼MLT҈Q
桌 矄�矄��泐
r
㦨
l t E ᆠ㡰 矄� �

C =

犨 
ᔴ
㦨
矄
ᨨ
��  B

ᔴᆠ� ᆠ
C
㦨


x
㦨


M = KἘ

꾀 �盐
㝐ᨨ鼀 盐鼀
㦨
mvr
㦨
᲼ML
T҈桌  

᲼L桌
㦨
Q⏌m
᲼L桌
㦨
᲼L
T
桌

ᨨ⼄盐
t
㦨
F 맔
r
᲼t桌
㦨
᲼ML
T
桌





҈ æ ö çQ u = K BT ÷
è ø 


꾀 
盐
ᨨ
ᨨ
ᨨ
�矄
ᨨ矄��
㡰 꾀�
�鼀

d
㦨
�
�䰘�
鼀鼀 ��ᨨ
ᨨ


Dd
㦨



䰘黨�
鼀鼀 꾀矄
 
鼀矄盐


盐ᨨ
犨ᨨ
���矄
 
ᆠ�鼀

  ᨨ  矄犨
矄 ᨨ盐

�
犨ᨨ
���矄
 
ᆠ�鼀


\ d = �
�
±


䰘㡰 鼀鼀 a
b
⏌  㡰 ෌�


Z = cd 

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EBD_8341 PS-6

Physics ᆠ �
ᨨ
�
Z,

҈



DZ
Da
Db ҈ Dc Dd = + + + Z a  b
c d
҈ =
´
+ ´ ҈
� + ´
+  ´

� = ҈

�奠



҈҈


㦨






��矄�ᨨ

和ᨨ

ᆠᨨᆠ
鼀矄盐鼀 
�矄
鼀矄盐鼀 
ᨨ
� �
鼀盐�
ᨨ 
ᔴ
ᆠ㡰 
ᔴ

ᨨ
 �
ᆠ�矄
㦨
҈
㝐和

҈
和


䰘
��矄�ᨨ
ᨨ
 �
矄ᆠ
ᆠᨨ� ᆠ�矄
犨�

��矄�ᨨ
ᨨ 鼀�
矄ᆠ
 
 
ᨨ
ᨨ
 �
矄ᆠ
�矄
泐� �
�� 
矄�
ᨨ
  ᨨ
ᨨ
鼀�
矄ᆠ
ᨨ
ᨨ
㦨
䰘
맔


҈
㦨


䰘
鼀鼀
㦨



䰘
ᆠ鼀
 �
ᨨ
 
ᆠ泐� 
㦨
鼀矄盐
盐

ᨨ
ᨨ 㦨

҈
졔


맔



҈㡰




䰘
㦨


䰘
ᆠ鼀
㡰 ␔ 
矄ᆠ犨
ᨨ

矄ᆠ犨㜠�盐�
�矄
��
矄�
ᨨ�ᨨ 矄

� 鼀ᨨ矄
泐
鼀鼀
ᨨ
 
鼀�
矄ᆠ

=

҈



䰘


D҈
㦨
A҈
졔
B҈
졔
C҈
㦨





졔



䰘


졔

�



㦨







҈


҈

҈�


D
㦨
A
졔
B
졔
C
㦨

�


졔
҈黨


҈

졔



҈
㦨




䰘 ᨨ 
ᨨ
 
ᨨ�ᨨ
鼀ᆠ 矄
ᆠ㡰 ෌�

 �
ᨨ
矄�鼀
 盐盐鼀

l
㦨

�


ᆠ鼀

�鼀
ᨨ



ᨨ矄ᆠ��ᨨ

T
㦨
�
s

�鼀
�ᨨ
ᨨ
 盐盐鼀

l g


p
l
p
l
Þ g = g T
Dg Dl
DT = + Þ
ᆠ�ᨨ 
ᨨ
�
�
㦨
g l T Dg æ

҈ ö ҈ æ ö Þ =ç ÷ +
çè ÷ø =




g è
�

ø �
Dg \ ᆠ � ᨨ � g = ´ ҈

=


奠 g M M ᨨ 盐矄㡰
d = =  = Ml - V l

҈
æ


҈ö Dd DM Dl  + ç = +
= è

҈
÷ø
㦨


҈
├�⏌鼀
҈



d M l

㦨


�
맔
҈

㦨


�
맔
҈


鼀鼀
졔

ᨨ
㦨

맔


�
맔
҈


鼀鼀 㦨
҈
�
맔
҈


鼀鼀
㦨




҈�
鼀鼀 � �
㦨
㝐和
졔
ᔴ和

졔
ᨨ㡰 㦨
�
�
鼀鼀
졔



맔


�
맔
҈

㡰




҈� 㦨
�
�
졔









҈�
㦨
�
䰘
�
鼀鼀 ᆠ㡰

㝐 �鼀盐鼀
ᨨ
㦨
҈
�
奠
졔

҈奠㡰
㦨


�奠 㡰 㝐 �鼀盐鼀
ᆠ �
ᨨ
�
꾀

㦨
҈
�
졔


졔
҈
�
졔
҈
�
㦨


�奠 ҈



Þ T
=

黨


�ᆠ ᨨ
ᨨ ��矄�ᨨ

= 奠 ᨨ � 㡰 +
奠 ᨨ � 턼㡰 ҈ + 奠 ᨨ � 㡰 + ҈奠 ᨨ � 和㡰
҈ =  ´

� +
´ ҈ + ´  + ҈ ´ ҈
�


D

㦨
A

졔
B

졔
C

㦨

�
졔



҈黨҈
졔
҈黨
�
㦨


҈

T =
p

҈
- 

Þ


㦨



ᆠ㡰 矄
ᆠᨨ盐 
ᨨ
矄ᆠ犨
�盐�
ᔴ
㦨


D

㦨
A

졔
B

졔
C

㦨






졔
҈





졔






㦨




䰘





�ᆠ 盐鼀
ᨨ
��矄�ᨨ
ᨨ
ᆠ�ᆠ盐
矄ᆠ

� ´ ҈
-
=


�ᆠ =

ᨨ 盐矄㡰



҈ æ ҈ö ´ ҈

+
ç ÷ ´ ҈

è 
ø ��

㦨


҈

졔



䰘
㦨



奠 b㡰 矄
ᆠᨨ盐 
ᨨ
鼀�
矄ᆠ
ᨨ
矄ᆠ犨
�盐�
㦨
҈
鼀鼀 矄
ᆠᨨ盐 
ᨨ
矄ᆠ犨
�盐�

=

 =

� 鼀鼀
�ᆠ

� 鼀鼀 = \ 矄
ᆠᨨ盐 

ᔴ

= �
CSD ҈ 鼀鼀 =


҈ mm =



҈ᆠ鼀 = ҈

\

l
l
ᨨ
g =
p g T


Dg DR DT ´ ҈

= ´ ҈

+
´ ҈

g Q T

�ᆠ

҈ = =





ᆠ鼀
 ᔴ�ᆠ盐
矄ᆠ
��矄�ᨨ �


黨ö æ = ç҈ - ÷ ´ ҈ =

҈
鼀鼀
㦨



҈
ᆠ鼀 ҈
è ø





T =
p

㡰 �ᆠ├ 矄矄
㦨
㝐
和

� �
졔
ᔴ�ᆠ盐
和ᆠ
� �

ᔴ
㡰 
ᔴ
=

�


㡰 A
㦨
䰘
맔
�

黨
㦨
䰘


ᆠ鼀



矄盐
矄 ᨨ盐

 
矄�� � �ᆠ 
��盐矄

矄ᨨ A
㦨
䰘


ᆠ鼀
㡰 ␔


㡰 矄
ᆠᨨ盐 
㦨


㦨


盐
ᨨ
҈

ᨨ
鼀�
矄ᆠ 盐鼀
ᨨ
矄
ᨨ
 �
矄ᆠ




� ᆠ鼀
㦨
�
맔
҈


ᆠ鼀 �맔҈



� �
㦨


맔



�
ᆠ鼀
졔


맔
�
맔
҈


ᆠ鼀 㦨




졔



҈�
㡰
ᆠ鼀
㦨



҈�

ᆠ鼀
 �ᆠ├ 矄矄
ᨨ
犨�㡰 ҈䰘


㡰 和盐 ᆠ
 矄�ᨨ

T =

rhg ´ ҈



�
ᨨ
�
矄盐 ᆠ
 矄�ᨨ


DT Dr Dh = + +

Q
�





҈


ᆠᨨ 矄 㡰 T r h ᆠ �
ᨨ

҈

´

DT æ ҈

´


҈ ҈

´


҈ö =ç + ÷ ҈

T è ҈

� ´ ҈

҈

� ´ ҈

ø

㦨




졔




黨㡰 㦨
҈


黨㡰
㦨
҈


黨奠
@ ҈
�奠

WWW.JEEBOOKS.IN

PS-7

Physical World, Units and Measurements ҈



D ҈ D D Dᆠ D = +
+ +

  ᆠ  ҈ = ´
+
´ ҈ +  ´  +
´ � 㦨

奠
҈黨


d
㦨


�
졔


맔



҈
졔




㦨


�黨
ᆠ鼀

ᆠ㡰 ෌�


㦨
҈⏌



ᆠ



㝐 �鼀盐鼀
�
ᨨ


b㡰 \


ᨨ
 
犨
㦨
L
㦨



鼀

�鼀
├
泐
矄ᨨ 
ᨨ
ᆠ
 
ᨨᨨ鼀
ᨨ
犨


t҈
㦨


҈
ö
L æ


ç 盐矄� �

L = gt ÷
è ø g

㝐
�鼀
㦨

�


쟼ᨨ
㦨
᠀



҈
泐
 ᨨ  �
盐 ᆠ�ᨨ 㡰 㦨

mA㡰 ´




L v

ᨨ
�鼀
├
�
 
ᨨᆠ矄矄
�矄
��
泐



㦨
҈
졔


㦨
\

d
㦨



L L + g v


䰘


\






҈
dL 






















 D ҈  +  D
 +  D   +  D


+҈+  +
= = ҈
�


ᆠ㡰

D =

꾀矄
 
矄ᨨ盐�ᨨ
ᨨ
鼀矄盐� �
ᆠᨨᆠ├
�矄
҈
�
  ᨨ
 
鼀

�鼀
矄 ᨨ盐

黨

᠀
҈
� b㡰 ᨨ
ᔴ҈
 �
ᆠ�

ᔴ

㦨
҈㝐和

҈和
㦨
҈鼀鼀



黨
鼀鼀
㦨


҈
鼀鼀
㦨



҈
ᆠ鼀 ᲼Q

҈

和

㦨
黨
㝐和
㦨
黨
鼀鼀桌 � �
㦨
㝐和
졔

ᔴ

맔
和
㦨




졔



҈㡰
맔
䰘
㦨



䰘
ᆠ鼀 ᨨ
ᔴ

 �
ᆠ�

ᔴ

㦨
҈
鼀鼀

҈
҈
鼀鼀 ᲼Q

҈

和
҈҈
㝐和
㦨
҈҈
鼀鼀桌 
ᔴ
㦨



҈
鼀鼀
㦨




҈
ᆠ鼀 � �
㦨




졔
҈


䰘㡰
맔



҈
㦨




ᆠ鼀 㡰 꾀矄

�
㦨

p


L

T
Dg DL DT 和ᨨ

´҈

= ´҈

+
´҈

g L T

҈ ҈ ´҈

+
´ ´҈


㦨


䰘

;
奠 㦨


黨


҈
㦨



҈
ᆠ鼀 b㡰 矄
ᆠᨨ盐 
㦨
҈
d҈
㦨


�
졔


맔



҈
졔




㦨



҈
ᆠ鼀 d

㦨


�
졔


맔



҈
졔




㦨


�䰘
ᆠ鼀

Þ

҈
҈

� - D =







\

D = -


� ᨨ

b㡰 꾀ᆠᆠᨨ� �
ᨨ
 
⼄盐矄�ᨨ


Dt ҈ = t 黨
Dl

҈ l
㦨



᠀


҈㡰
ᨨ

= l

Dg 奠=䁸 g

t 㦨
黨

᠀
҈㡰
ᨨ





� 㡰 
ᔴ

= 㦨



҈
鼀鼀 �
ᨨ
ᨨ
㦨
�
맔



҈
㦨



�
鼀鼀
��㡰 � �
㦨


�
졔

�
맔



҈㡰
졔



�
㦨





鼀鼀


҈


b㡰 㝐矄盐
 �
ᨨ
ᨨ
㦨

�

ᆠ鼀 ᨨ
 �
和ᆠ
犨�
҈
㝐�
和ᆠ
��矄�ᨨ
㦨
҈
鼀鼀 黨
㝐�
和ᆠ
��矄�ᨨ
㦨
҈

 �
和ᆠ
��矄�ᨨ
矄
ᆠᨨ盐 
㦨
҈
㝐和
҈
和
㦨


҈
鼀鼀 㡰 ᔼ

ᨨ鼀

 µl


҈ dL dL + g
L v



dT dL ´ ҈

= ´ ҈

ᆠ�ᨨ 
ᨨ

L ҈
L 




҈ = ´ ´ ҈

; ҈奠 ҈




҈


´ 


҈㡰
㦨




鼀꾀 




㦨
├l ᨨ犨

�
�矄
��
쟼
㦨
҈
҈
ᨨ
ᨨ
l҈
㦨




ᆠ鼀  

㦨


�
ᨨ
ᨨ
l

㦨




ᆠ鼀 
 
ᨨ
�
� �
ᨨ
ᨨ鼀

D
 
҈
҈
㦨




⧤
 


�

D㡰
㦨




⧤


졠
矄盐矄�盐� �
 
��
盐矄
犨
�
d
㦨


ᆠ盐 
ᨨ�
�ᨨ
ᨨ
�ᨨ
�矄

I = e҈


V ⏌T - ҈㡰
鼀꾀
�� 㡰 ␔ 

I = �mA
e҈


V ⏌T =
mA ҈


꾀矄ᨨ

dI = e҈


V ⏌ T 㡰 ´ T

�鼀
├
泐
�鼀ᆠ
矄ᨨ盐 
ᨨ
ᆠ
 
矄ᨨ


t
=

㡰




҈ +

�䰘 +

�黨 㦨


�黨
ᆠ鼀 

꾀矄
犨
├ ᨨ犨


l
p 2l

Þ
g =
g t Dg D l D t æ ö æ

҈ +
´ ҈ ö ᨨ

= ± ç +
÷
㦨
ç ÷
㦨




䰘 黨
ø è l g t ø è

t =
p

\
黨


Dg 奠 =

䰘奠 g

b㡰 ᔼ
 
鼀矄盐鼀 

 
�鼀
ᨨ
ᆠ泐� 

㦨
㝐
和

졔

和
㡰
맔

ᔴ
㡰

ᔴ

㦨
҈㝐和

҈和㡰

æ


� ö ÷ =


҈ ᆠ鼀 è �
ø

㦨

�
҈� - �
҈
㡰 - ç 



\ 㡰


㦨
�
҈

졔



맔




҈
㦨
�
҈


ᆠ鼀


鼀 �鼀盐鼀
ᨨ矄矄�
ᨨ
�
Y
盐
ᨨ
l
 
d

DY Dl
Dd = + Y l d

矄
ᆠᨨ盐 

㦨


㦨

�ᆠ ᨨ
ᨨ ��矄�ᨨ ᨨ ᆠ�ᆠ盐 矄ᆠ



� 鼀鼀 㦨



� 鼀鼀 ҈





Dd
㦨
Dl
㦨




�
鼀鼀 쟼ᨨ
ᆠᨨ �盐�ᨨ
ᨨ
l 㦨


Dl



� 鼀鼀 ҈ 㦨 㦨


� 鼀鼀 �
l

WWW.JEEBOOKS.IN

EBD_8341 PS-8

Physics 쟼ᨨ
ᆠᨨ �盐�ᨨ
ᨨ
d

㦨


҈


ᆠ㡰

҈
Dd
맔



� 鼀鼀 㦨 㦨

� 鼀鼀 �
d

�






ᆠ㡰

�ᆠ

ᨨ

ᨨ
��矄�ᨨ 矄
ᨨ
ᆠ�ᆠ盐
矄ᆠ

� =


҈鼀鼀













㦨
�


矄
ᆠᨨ盐 
=

�鼀

r
㦨
㝐
和


졔
ᔴ
和
㡰
맔

ᔴ
㡰

ᨨ
ᨨ
㦨
�
맔
L
C.
㦨
�
맔



҈
鼀鼀
㦨



�
鼀鼀 �鼀
ᨨ

㦨
᲼ � �
ᨨ
鼀�
矄ᆠ桌
졔
᲼ � �
ᨨ
ᆠ�ᆠ盐 矄ᆠ
맔
L
C
桌

ᨨ
ᨨ 㦨


�
맔


졔

�
맔



҈




�

㦨
҈



鼀鼀



�
㦨


䰘

鼀鼀 �
҈ Dr


҈ Dr = ᨨ
´ ҈

=

䰘


䰘 r r m m ᨨ犨

 矄�泐

d = = V
æ r ö pç ÷  è
ø Dd ´ ҈

ᆠ �
ᨨ
�
 矄�泐

d ì Dm æ Dr ö ü + ç 㦨
í ÷ ý ´ ҈

è r øþ î m Dm æ Dr ö ´ ҈

+  ´ ç 㦨
÷ ´ ҈

m è r ø ҈ = 
҈҈奠 㦨

奠
졔

맔


䰘

�鼀

r
㦨


�
鼀鼀
졔



맔


\








䰘


\

㡰

黨



㡰

ᨨ盐鼀
ᨨ
ᆠ盐
V
㦨
l 㦨
҈


맔
҈


鼀㡰 㦨
҈
䰘


맔
҈


鼀 Þ
V
㦨
҈
䰘
맔
҈


鼀
꾀矄
 �
矄
犨ᨨ
矄�� � �ᆠ 
��盐矄
矄ᨨ
ᨨ盐鼀
矄
矄ᨨ 犨ᨨ
矄�� � �ᆠ 
��盐矄
Et㡰
㦨
A

eµt
㦨
A

e


t

\

ᨨ�
E
㦨


ᨨ�
A



t

졠
�  �� �
犨
�

dE dA dt =



´ t E A t 꾀矄
ᨨ矄
犨泐矄

盐










=


´ ҈


) (

⏌鼀

)

㦨



�
졔





�
㦨


҈҈
�









꾀
 


ᆠᆠ盐泐
├ ᨨ犨
\

ᆠ �
ᨨ
�
Ӹᨨ盐 �矄
鼀ᨨ盐盐矄




DY
DD Dl ᨨ犨 = + l Y D ᲼Q
 
盐
ᨨ
m

g
 
L

 ᆠ桌


҈


� + 㦨

´ 㦨


맔




�
졔





�





Þ DY
㦨
Y
맔


҈҈
�
㦨



맔
҈
҈҈
맔


҈҈
�
㦨




�
맔
҈
҈҈ =


´ ҈
҈҈ ⏌鼀
� �
ᨨ
Ӹᨨ盐 �矄
鼀ᨨ盐盐矄
㦨
Y
졔
DY㡰 \ = (
±


) ´ ҈
҈҈ ⏌鼀


dE æ dA ö æ dt ö ´ ҈

=
ç ´ ҈

÷ +


t ç ´ ҈

÷ E A t è ø è ø dE Þ 맔
҈


㦨


맔
҈

�奠
졔




맔
�
맔
҈
�奠
㦨

奠 E FL 
㡰 Ӹᨨ盐 �矄
鼀ᨨ盐盐矄
Y = A´l \




㦨
҈
鼀鼀





鼀鼀

㦨




鼀鼀

(







盐� �
 
盐矄
ᨨ
Dl
฀l

DT
 
T
犨
�
쟼҈
㦨


҈
�奠 矄
鼀ᨨ �
쟼҈

쟼

 
쟼


҈҈

ᆠ �
ᨨ
�
 矄�泐


Dr
´ ҈

= ´҈

=
奠
r ҈



l l
l
Þ T
=
p

Þ
g =
p g g T
Dg ´ ҈

ᆠ �
ᨨ
�
g
㦨 g Dl DT ´ ҈

+
´҈





























= l T

F A
mgL
´҈ ´ 黨

´
= =
l L pD
l p


´ ҈
- ´


´҈
-






�㡰


Dr = r ҈



҈
҈
´ ҈鼀鼀
㦨




鼀鼀
��矄�ᨨ 矄
ᨨ
鼀�
矄ᆠ
㦨






Y=

m m m
㦨
= l p r
v Al

ᨨ鼀
⼄盐矄�ᨨ

盐� �
 
盐矄
ᨨ Dl
㦨





ᆠ鼀

l =
ᆠ鼀錐
Dr
㦨




�
ᆠ鼀錐
r
㦨


�
ᆠ鼀
m
㦨



�錐
Dm
㦨





�
�
⼄

�㡰
犨
�

␔
├ ᨨ犨
 
矄
ᆠᨨ盐 


ᔴ

㦨
҈㝐和

҈和 b㡰 
�鼀
�ᨨ
ᨨ

矄�鼀
 盐盐鼀
�矄
��
泐

b㡰

 矄�泐
r
=

Dr Dm
Dr Dl = + + l r m r

T =
p





㡰

├� �
ᨨ�
 
�  �
ᨨ
ᨨ矄

㡰


��矄�ᨨ 矄
ᨨ
 
 �
矄ᆠ 㦨
҈

��矄�ᨨ 矄
ᨨ
鼀�
矄ᆠ \
҈
��矄�ᨨ
ᨨ
 
 �
矄ᆠ 㦨


Dg Dl DT = +
g T l

Dl
 
DT

矄
 
盐鼀
ᨨ
� �矄
├

鼀 �鼀盐鼀 �
ᨨ�ᨨ
㡰

  ᨨ
 
鼀矄盐鼀 
ᨨ
g
�矄
鼀ᨨ矄
ᆠᆠ盐


 ᆠ
ᆠᨨ �盐�ᨨ
ᨨ
 
鼀 �鼀盐鼀
ᨨ矄矄�
ᨨ
�
  鼀矄盐鼀 
ᨨ
y
盐
ᨨ
l
 
d
�矄
 
矄鼀
矄
ᆠᨨ盐 
ᨨ
矄ᆠ犨
�盐�

�ᆠ

� = = =


҈
鼀鼀 㦨 Dr ��矄�ᨨ 矄
ᨨ
ᆠ�ᆠ盐
矄ᆠ �


㡰 �
ᨨ
�
g





҈


҈

´ ҈
-� DY DL ´ ҈


㦨

奠 ´ ҈

= ´ ҈


㦨
Y l
� ´ ҈
-�

҈
�
㡰  矄�泐

r =


M M Þ r = MD - =  p V
æ Dö pç ÷  è
ø

æ Dr ö Dm æ DD ö \奠ç ÷ = + ç =
+  ´ ҈
� = ҈

�奠 è D ÷ø è rø m

WWW.JEEBOOKS.IN

PS-9

Physical World, Units and Measurements 
ᔴ

ᨨ
 �
ᆠ�
㦨
҈
㝐和

҈
和

æ Dr ö ҈
�
æ x ö 奠ç ÷ = 奠=ç 奠 è ҈

÷ø è r ø ҈



æ è

㦨
ç ҈ -

\ x = ҈
�








꾀矄
黨
㝐和
㦨
҈

和

҈

㡰

犨ᨨ
ᆠᆠ�ᨨ矄

ᔴ҈
 
ᔴ


ᆠᨨ

ᆠ
�
矄�矄

\

ᔴᔴ ҈ ҈ ҈ = + Þ ᔴ⼄ = ҈
ᔴ ⼄ ᔴ҈ ᔴ
ᔴ҈ + ᔴ


ᔴ ⼄ =


�


㦨
҈MSD -

Dᔴ Dᔴ Dᔴ ҈ ҈ ҈ = + Þ -
= -
҈ -

ᔴ⼄ ᔴ҈ ᔴ
ᔴ⼄ ᔴ҈ ᔴ


Q

҈
㝐和
㦨
҈
鼀鼀㡰 � �
ᨨ
矄�

l
㦨
㝐和
졔
和
ᔴ㡰
㦨
҈

鼀鼀
졔
҈
맔


҈
鼀鼀














㦨
҈

҈
鼀鼀
㦨
҈

҈
ᆠ鼀

é ù
Dᔴ҈ Dᔴ
Þ Dᔴ = ᔴ⼄ ê
+
ú ᔴ
úû êë ᔴ҈







é ҈
҈� ù Þ Dᔴ = ´ + ê � � ë



´






´ 


úû

ᨨ犨

 矄�泐
㦨


黨
+ ҈� ´ =

�
� 쟼 �泐
矄ᨨ
�

ᆠᆠ�ᨨ
= ҈
´

҈ ᔴ
Þ
D Þ ´ ҈

= 










和� ᆠ


\

Y
㦨


␔
├ ᨨ犨
 





�㡰
䰘


=

D

D


䰘䰘 ´ ҈

= ´҈

= ҈


奠 = ҈
黨奠



㦨


҈

´ ҈
´ ҈ æ � ´ ҈

pç ç
è





҈



� ´ ҈
-
㡰 ´ 

҈
� ´ ҈
-
㡰

Fl AY

ö ҈҈ ÷÷ ´
´ ҈
ø

;


鼀鼀


㦨





맔
҈
҈҈⏌

鼀
\

㝐 �鼀盐鼀
ᨨ矄矄�
ᨨ
�
 
鼀矄盐鼀 
ᨨ
Y DY

㦨





黨㡰Y 㦨





黨
맔





맔
҈
҈҈



















㦨
҈

黨
맔
҈
҈

⏌鼀





ᆠ鼀 㡰

҈鼀鼀 =


҈鼀鼀 ҈



ᔴ盐
矄盐 ᆠ

㦨

prl =
p

\
D

㦨



䰘䰘  ᆠ
ᆠ �
ᨨ
�
 
鼀矄盐鼀 
ᨨ
ᨨᆠ
 �




㡰␔
├ ᨨ犨
 
Dl
㦨


�
´
´ ҈҈
´ ҈
-


�鼀
㦨
MSR
졔
CSR

맔

ᔴ
㡰









㦨
҈
鼀鼀
졔

䰘
맔



҈㡰
鼀鼀
㦨
҈

䰘
鼀鼀

(

)
( 
)
(

)







X F L ´
㦨

L X pD


W
㦨
�

錐
D
㦨



�
ᆠ鼀
㦨



�
맔
҈

鼀錐 X

㦨


҈
�
ᆠ鼀
㦨


҈
�
맔
҈

鼀錐 L
㦨
҈҈

ᆠ鼀
㦨
҈҈

맔
҈

鼀

矄
ᆠᨨ盐 


ᔴ

㦨


和盐矄�盐� �
 
盐矄
�
⼄

�㡰



�

F A

æ DY ö æ DD ö DX DL =
ç + + çè ÷ è D ÷ø X Y ø 鼀 L æ



҈ö æ



҈ö æ

҈ ö 㦨

ç + + =




黨 è


� ÷ø çè

҈
� ÷ø çè ҈҈
÷ø

æ҈ ҈ö 盐
±
D盐
㦨
䰘�


�㡰
±
ç + ÷
㦨


±


�㡰
ᆠ鼀 è

ø æ҈ ҈ö 
±
D
㦨
҈�

䰘�㡰
±
ç + ÷
㦨



±


�㡰
ᆠ鼀 è

ø

+

 ᨨ盐 � �
ᨨ
ᨨ

矄�� � �ᆠ 
��盐矄㡰 ҈

黨
×
҈
҈

N⏌鼀
㡰

㝐 �鼀盐鼀
ᨨ
�
Y

㦨


�
졔




㦨
҈

奠

盐矄
 
ᆠ �
ᨨ
�
 
ᆠᆠ盐�ᨨ
ᨨ
 
 �泐
矄ᨨ �矄
҈

奠 ҈
黨㡰



�

m

䰘
g 鼀矄矄 = = =




�⏌ᆠ鼀 ᨨ盐鼀 l  ҈

҈㡰

Ӹᨨ盐 �矄
鼀ᨨ盐盐矄

Y
=

D Dᔴ D ´ ҈

= ´ ҈

+
´ ҈

 ᔴ⼄ 
´�



´҈

+
´ ´҈





�

 ҈ ҈ - =  盐 D D盐 D \+
+
=
 盐 ҈ ҈ ҈ ҈ ҈ ҈ = ᨨ犨
- = Þ  盐

-
҈ ҈ Þ = \ =

ᆠ鼀




 �
鼀├� �
ᨨ
 �
矄ᆠ
犨�
ᆠᨨ� ᆠ�
犨�
 
鼀�

矄ᆠ
ᆠ盐矄
矄
ᆠᨨ盐 
�矄


҈
鼀鼀





�
ᆠ鼀㡰 
ᔴ

㦨
҈
MSD 
҈
VSD

黨 MSD

Q

黨
MSD 㦨
҈

VSD㡰 ҈
黨ö æ æ ҈ ö ҈ MSD
㦨
ç ´ ҈÷ 鼀鼀 㦨


҈
鼀鼀 㦨
ç҈ - ÷ MSD 㦨
è ҈
ø è ҈
ø ҈






´ 






 = �


�

=

黨 ö ÷ =

҈ 鼀鼀 ҈
ø

㦨





D l
㦨
pDl




맔
҈

䰘
맔
�

鼀鼀


㦨


�
䰘


ᆠ鼀
䰘

㦨




ᆠ鼀

 ᨨ盐 � �
ᨨ
ᨨ
犨ᨨ
矄�� � �ᆠ 
��盐矄㡰 

b

㡰

鼀 =



�
+

�
+

�䰘 +

�
+

�黨 =

��
»

�
和


쟼ᨨ
�
� �
鼀

҈錐
鼀


錐
鼀

錐
鼀


錐

鼀

�

㝐
ᨨ =





+
+


҈ +



+


 =


҈
»



和


WWW.JEEBOOKS.IN

EBD_8341 PS-10

Physics

\

奠
ᨨ
�
 
鼀矄盐鼀 
ᨨ
� 

ᨨ
�鼀


㦨


矄ᆠᨨ 矄 =

ᨨ

ᨨ
ᨨ矄ᆠ��ᨨ 矄

DT DT ҈ ҈ \ = Þ = Þ DT =


�矄ᆠ T






�鼀
�ᨨ
T =





D

´ ҈

= ´ ҈

= 
�䰘奠




�
奠
ᨨ
�
 
鼀矄盐鼀 
ᨨ
�

D� D

æ D + D ö ´҈

=
´҈

+ ç ÷ ´҈

�

è - ø æ ҈ +҈ ö 㦨

( 
�䰘 ) + ç ÷ ´҈


» ҈҈奠 è

- ҈
ø


黨


ᨨ犨
ᨨ
鼀矄盐� �
ᨨ
�
�g

\ � =
p


D ҈ ´ ҈

= ´ ҈

= ҈
奠  ҈


\

ᆠ �
ᨨ
�
�g

b

ᆠ㡰  �
ᆠ�矄

҈ᆠ鼀
㦨


҈
�ᆠ鼀


�
和
㦨

㝐和 \
҈
和 =

�




´

҈
�
㦨


҈ᆠ鼀 �


ᔴ

㦨
҈㝐和

҈和 㦨


҈
�ᆠ鼀



҈ᆠ鼀
㦨




�ᆠ鼀 和ᆠ犨
�盐� ᔼ
 
�ᆠ
ᨨ
矄ᆠ犨
�盐�
�矄
犨�ᆠ
 

ᔴ
ᨨ
 �
ᆠ�矄  
�ᆠ
㦨


맔

ᔴ

ᨨ
 �
ᆠ�
㦨


맔




�
㦨



�ᆠ鼀

ᔴ

ᨨ
矄ᆠ犨
�盐�

�ᆠ

ᨨ ᨨ
ᨨ ��矄�ᨨ 矄 ᨨ ᆠ�ᆠ盐 矄ᆠ


� ᆠ鼀
㦨





�ᆠ鼀
㦨




�
鼀鼀
㦨
҈



�


ᨨ犨
�
 
矄
ᆠᨨ盐 
ᨨ
 
� 
矄ᆠ
ᨨ
 
矄ᆠ犨
�盐�
�矄 犨�ᆠ
 
矄
ᆠᨨ盐 
ᨨ
 �
ᆠ�矄
 


ᔴ

ᨨ
� 
矄ᆠ
ᨨ
矄ᆠ犨
�盐�
㦨


맔




�
㦨



�ᆠ鼀



�ᆠ
㦨


맔



�
㦨


҈ᆠ鼀




҈ \



ᔴ

ᨨ
矄ᆠ犨
�盐�
㦨
ᆠ鼀
㦨




҈ᆠ鼀
㦨



҈鼀鼀
҈

l 㡰 Q



矄�
q
㦨
l \ 
㦨
ᆠᨨ矄ᆠ q




�㡰
㡰 l = [ - ᆠᨨ矄ᆠ q ᆠᨨ q] \ q
l \ 㡰
㦨


ᆠᨨ矄ᆠq
ᆠᨨq
q



��㡰


Dg Dl DT ´ ҈

= ´ ҈

+
´ ҈

g l T Dg æ ҈ ö \ ´҈

=
+
ç ÷ ´҈


㦨
�奠 g è

ø Dr D҈ - a㡰 D ҈ + a㡰 æ ҈- a ö = + b㡰 r = ç ÷ Þ r ҈ - a㡰 ҈ + a㡰 è ҈+ a ø Dr Da Da Da ҈ + a + ҈ - a 㡰 = + = Þ r ҈- a ҈+ a ҈ - a 㡰҈ + a 㡰
Da ҈ - a 㡰
D a \ Dr = = ҈ - a 㡰҈ + a 㡰 ҈ + a 㡰 ҈ + a 㡰
ᆠ㡰 矄� �
N = N
e -lt Þ  N =  N
- l t

҈ dN =
-t N dl DN

Dl = = =



Nt



´ ҈

Þ

�



n
졔
҈㡰

和

㦨
n
㝐
和
 \

҈

和

㦨


n ö æ = ç҈ è n + ҈÷ø 㝐
和
 ҈ a 㦨
㝐
和


㦨
盐 �矄



᲼Q ҈
㝐和
㦨

盐 �矄桌 n +҈ n +҈

Topic-4 : Miscellaneous (Mixed Concepts) Problems ҈


ᆠ㡰

҈ 㦨
᲼ ҈桌

ᨨ

ᨨ
ᆠᨨ 矄

盐 �
ᨨ盐鼀


㦨
᲼桌

鼀���泐
ᨨ


꾀矄
q
� ᆠ矄矄
ᨨ鼀


ᨨ
黨


ᆠᨨ
q
ᆠ矄矄
 
  ᨨ

꾀


桌 矄ᆠ

Îᨨ
㦨
᲼㝐҈

꾀

㡰
ᆠ矄矄 

Dt DT ҈ = = t T



ᔴ �
e
㦨
ᔼ
㦨
᲼꾀 桌

 �盐
⼄盐 ᆠ泐
w
㦨


㡰
㦨
ᆠᨨ
q

q 

\

n
㝐
和
 n +҈

矄
ᆠᨨ盐 


ᔴ
㡰
㦨
҈
㝐
和


҈
和


��� �
��㡰
泐
�㡰
犨
�

�.e.

 
ᆠ�ᨨ 
ᨨ
�
d
ᆠ矄矄


ᆠ㡰

 �
ᨨ
 
矄� �
ᨨ
矄�鼀
 盐盐鼀
�矄
 ᆠ泐

鼀 �� 㡰

  ᨨ
 
ᨨ
�
 �
Dl
㦨


盐 
 
ᨨ矄矄���泐
ᨨ
ᨨ
�
鼀矄盐� �
�鼀
�矄
҈矄
�


矄
矄  
矄
ᆠᨨ盐 
ᨨ
矄ᨨ
犨ᆠ
�矄
҈矄





�  ��ᨨ
犨�
矄ᆠ
ᨨ
l

㦨


�҈


l

奠
ᨨ
�
 
鼀矄盐鼀 
ᨨ
r

҈
㝐和
㦨


�



 Þ T
=
p
l
g g

盐矄� �

T =
p

ᨨ犨
矄盐矄�盐� �
 
�鼀 矄�ᨨ 矄
ᨨ
ᨨ
⼄盐 ��矄
犨
�


Ne
㦨
m Îᨨ



L- A
T
M M -҈ L- A
T


= T -
= T -҈ 㦨
w



Ne
c m Îᨨ =
pn =
p
Þ
l =
pc l m Îᨨ Ne


b㡰

wp =

㦨

´



҈
-
´҈
-҈҈ ´  ´ ҈


㦨




鼀 䰘
´ ҈

䰘 ´ ҈

´ ҈
-҈黨 㡰


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PS-11

Motion in a Straight Line

Motion in a Straight Line

2





Topic-1 : Distance, Displacement & Uniform Motion ҈


b㡰 ෌�

v
㦨
b x ᨨ

dx = b x҈⏌
dt

ᨨ

x

t







-҈⏌
ò x dx = ò bdt

ᨨ

꾀

��
� 矄 
ᨨ �鼀

 
ᨨ泐
�矄

犨ᨨ �  
ᨨ矄��ᨨ 矄
A
  B
�
 
��
ᨨ矄��ᨨ

�鼀
�

犨 �ᆠ
�矄
ᨨ ᨨ矄矄�


Topic-2 : Non-uniform Motion

x҈⏌
b
t
㦨

t ᨨ x
㦨
҈⏌

�  �� �
犨





�鼀

犨
�

҈


㡰

ᨨ





dx b ´
t =
dt








�


�鼀





v


u

㦨

aS 盐

a =

�矄 ᆠ

 ®

C

Þ
v

㦨
҈ 

R O �矄ᆠ鼀 

B




쟼

和

æFö \ v
= u
-
ç ÷ S èmø

҈ ҈ 鼀⏌矄 =

䰘鼀⏌矄
Þ v =



b㡰 x
㦨
at
졔
bt


ct

dx d = at + bt
+ ct  㡰 dt dt 㦨
a
졔

bt

ct


ᨨᆠ�泐

v =

꾀ᆠᆠ�ᨨ



® vf

Dv ␔

F m

é

� ´ ҈
-
ù

Þ v
= ҈㡰
- 
㡰 ê ú - ëê

´҈
ûú ҈






p 㡰
�矄ᆠ鼀 
㦨
矄 ᨨ矄
�矄 ᆠ


pr
㦨 pR A
b㡰 꾀�
ᆠᆠ�ᨨ

uur ur uur ur r r v f - v� v f + -v� 㡰 D v a= = = t t t

҈ æ ҈ ö ҈ � 
� 䰘
鼀 + = çè + ҈÷ø ´
+ ´ ´

=

    b㡰 ᨨ鼀
 
 �
⼄盐�ᨨ
ᨨ
鼀ᨨ�ᨨ

=


r ҈ =
´ =
鼀⏌矄

Q r
㦨
҈鼀
錐
t
㦨
҈矄㡰 t ҈

㦨 ACB =

�
矄㡰

҈ ҈ =   ҈ � SD =
- =   �矄 ᆠ
ᆠᨨ
泐
 
ᨨ泐
㦨

ᨨ
v㜠t
� 㦨

OABS㡰
졔

SCD㡰

 �矄ᆠ鼀   b㡰 
꾀�
ᨨᆠ�泐

㦨
�鼀

㦨
AB
㦨AOB
㦨

R �矄 ᆠ
㦨

 �

t

OS =
+

b
t ᨨ v
㦨

b㡰 ෌ 矄
�
ᨨ�ᨨ
ᆠ㡰
ᨨ矄��ᨨ 㜠 �矄 ᆠ �鼀
 
ᨨ�ᨨ
㡰
ᨨᆠ�泐㜠 ᨨ矄��ᨨ

ᆠᨨ矄ᨨ � �
ᨨ ᨨᆠ�泐㜠�鼀
�
ᨨ�ᨨ
㡰
  �矄
�矄 ᆠ㜠�鼀
�
�矄
矄
��

ᨨ犨

 ᆠ
�矄 ᆠ㜠�鼀
� ᨨ�ᨨ
㡰
�矄
� ᆠᨨᆠ


㦨




t
㦨
t㡰


A B v鼀⏌矄㡰
S D O
҈















�




C

æb ö ÷ è c ø

ᨨ



㦨

b

c
맔

t
\ t = ç

vf
㦨
졔�
鼀⏌矄


ö
æ æb ö æ b ö = a+b  c ç ÷ ÷ ç ÷ c ø è è c ø è c ø

 
v
㦨
a +
b ç

㜠v� 㦨
졔�
鼀⏌矄

r  Dv = v
f + v�
= �
+ �
= �
鼀⏌矄 r r  Dv  �
҈ = = a =
鼀⏌矄

�
ᨨ 㜠犨矄
�ᆠ�ᨨ
Dt ҈



dv d a 졔
bt 㜠 ct
㡰 㦨 dt dt





㡰 ᨨ
ᆠᨨ 矄 
ᆠᆠ�ᨨ

 
�矄
矄�� 
�  ®

ᨨ
㜠 �矄
ᨨ
a - t ® ® ᔼ ᆠ� 
矄�� 
� 
ᨨ
v - t

 
ᨨ
ᨨ
x - t


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EBD_8341 PS-12

�


Physics

ᆠ㡰 
�鼀
├
泐
꾀
ᨨ
ᆠ
� �矄 � �
ᨨ� 
�矄

\
�鼀
├
泐

ᨨ
ᆠ
� �矄 � �
ᨨ� 
㦨


졔


ᨨ
ἘtἘ
ᨨ

鼀� �鼀盐鼀


x

盐
㦨



҈é ҈ x ê- + v êë
d
+ x



꾀 㦨
҈


҈ ҈  = 꾀 = ҈ 

= 

+ 㡰


Þ ҈ 
= 
( 
+  ) Þ
Þ
ᨨ
㦨


(

)

҈  



(

҈ + 


҈  


)

 



  

㦨
҈
 + 
  


ᨨ

 = ҈




ᆠ㡰 矄� �
⼄盐�ᨨ

a
㦨


v u ҈

 
S
㦨
ut
졔
at t


�矄 ᆠ

泐
ᆠ
�
҈�
矄ᆠ
㦨


҈ 
�㡰
҈�㡰

҈�


䰘� 鼀
�矄 ᆠ

泐
矄ᆠᨨᨨ
�
҈�
矄ᆠᨨ 矄
㦨


맔
҈�
㦨

�
Q
�矄 ᆠ
㦨
矄
맔
�鼀㡰 �  ᆠ
犨
�矄 ᆠ

泐
ᆠ
 
矄ᆠᨨᨨ �
҈�
矄ᆠ


�


䰘
�
㦨
҈҈

�
鼀 
ᆠ
ᆠᆠ 矄
矄ᆠᨨᨨ
�
�鼀
錐

䰘


QM - x t҈ =




�㡰
v

�鼀
├
ᨨ

 
�矄 ᆠ
ἘRP Ἐ
�
 
�

t
=

d
+ x
v







��㡰

ᨨ
�鼀
矄
ᨨ
鼀ᨨ
 
ᆠ
ᨨ鼀
Q
ᨨ
P

t = t҈ + t
=

QM - x d
+ x
+ v
v

턼



㝐

d =
ᨨ
x = 

- ҈ ᆠ㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

ᨨᆠ
�矄
鼀ᨨ� �
犨� ᆠᨨ 矄 
��
ᆠᆠ�ᨨ
�




㦨

ᆠᨨ 矄 
ᔴ㡰

黨



├ 
= - ᔴ + ├





= +

ᔴ ᔴ  ᆠ

�
㡰
矄 矄
ᆠᨨᆠ泐
b㡰 �矄 ᆠ
ᨨ �

� 
�



�矄ᆠ鼀 
s㡰 㦨
t
Q s µ t  �� 㡰 泐
ᨨ盐
�  ��ᨨ
ᨨ
�矄ᆠ鼀 

犨
� ᆠᆠ�ᨨ


ds dt  dv d t
= =
t = = t

 
a = dt dt dt dt a
㦨

t
ᨨ
a µ t  ᆠ
�
㡰
�矄
ᆠᨨᆠ
҈

㡰 
⼄盐�ᨨ
ᨨ
 
��
v㜠x
�
�矄 v v
㦨

x + v




�㡰 x


V=

v dv =-
dx x


㦨



䰘� +
�t  ҈�㡰 = 
t
䰘
�
졔

�


䰘�
㦨

 Þ ҈�
㦨
䰘
� Þ 
㦨



�
矄ᆠ 㡰 
 
ᆠ
盐
ᨨ
 
�� 犨泐


�矄 ᆠ
ἘxἘ
ᨨ鼀  
ᨨ� 
㝐

和ᨨ

㝐
㦨
x 꾀 
�
矄
ᨨ
ᆠ
�
�
�矄
v

 
�鼀
├
泐
 
ᆠ ᨨ
ᆠᨨ
 
�矄 ᆠ
QR
㦨
QM

x ᨨ
 
�� 犨泐




 = -ᔴ  
㦨

ᔴ




 = (   
) =





҈  

= 


盐� �
 �矄
盐
ᨨ


�
⼄盐�ᨨ

�㡰

ù ú=
úû



d


 㦨



졔
㡰 꾀


㦨
 Þ 
㦨
҈







졔
㡰
㦨
҈


㡰






�㡰

dt =
dx

\



a
㦨 v Þ

ù v é v dv v =  ´ v = 
ê 
x + v
ú
ᨨ鼀
�㡰 dx x
x
ë x
û











v
v
a
㦨

x x x






��㡰

졠
ᆠᨨ鼀� �
 
⼄盐�ᨨ
��㡰
犨�
⼄盐�ᨨ
ᨨ

矄��  �  y
㦨
mx
졔
c

v



犨
�
m
㦨

= + 
x



�



q 㦨
졔


�



q �矄
ᆠ盐



v 꾀矄ᨨ
c
㦨
-



x

�



 
y㜠� ᆠ
�矄
��  ᆠ
�
㡰
ᆠᨨᆠ泐
�ᆠ矄
ᆠᨨ矄ᨨ � �
a㜠x
�
҈҈
b㡰
ᔴ  �
�
ᨨᆠ�泐
㦨

盐 
 
ᆠᆠ�ᨨ 㜠�鼀 � ҈

㦨
´ ҈
´ ҈҈ = �� 鼀 ⏌ 矄
和� ᆠ

� ��
ᨨᆠ�泐
�矄
ᨨ

� 
ᨨᆠ�泐
�



鼀 �鼀盐鼀 ᨨᆠ�泐
�矄
��
鼀⏌矄


WWW.JEEBOOKS.IN

PS-13

Motion in a Straight Line ҈


㡰 A







鼀矄 ҈ a
㦨


鼀矄 
x B




鼀矄 ҈




鼀

턼





鼀

ᨨ

A

u҈ =

 鼀矄 -҈

a҈ = -
鼀矄 -


s҈
㦨
x

t҈
㦨
t ҈
a҈t҈
x
㦨


t 
t


s҈ = u҈t҈ +

\




�㡰

ᨨ

B u

㦨




鼀矄҈

a

㦨

鼀矄


s

㦨



x

t

㦨
t

҈
s

㦨
u
t

졔






x
㦨




t 졔
t
꾀� �
⼄

�㡰
 
��㡰

҈


t
㦨





��㡰



㦨

矄


�

u



㡰

A




B t � �矄 ᆠ

㦨
꾀
ᨨ
矄㜠�鼀
� O

҈ ´ � ´
=

鼀
҈

㡰 �矄 ᆠ

�矄
犨�
�鼀

矄
x

㦨
at

졔

bt
졔
c























=

Þ
x

和 ҈



ᆠ  �
�
ᨨᆠ�泐
= =

鼀⏌矄
�鼀 ҈
ᨨ

QS
�矄
�ᨨ
㦨


鼀⏌矄
ᨨ

OP

a
ᆠᆠ�ᨨ 㡰
>


鼀⏌矄
ᨨ

PS
ᆠᆠ�ᨨ 㡰




鼀⏌矄
ᨨ

OR

ᆠᆠ�ᨨ
a



鼀⏌矄
ᨨ

RS

�ᨨ
a >


鼀⏌矄
 ᆠ
a
³



矄ᨨ鼀
ᨨ� 
ᨨ
ᨨ� 矄
�
�矄

҈

�㡰

t҈

 
�鼀
├
泐
 
ᆠ
ᨨ
�
 
鼀 �鼀盐鼀 ᨨᆠ�泐
vm
犨 �
�
�矄
ᆠᆠ�ᨨ
矄� �

㦨
u
졔
at v vm =

+ at҈ ᨨ
t҈ = m




�㡰 a 和� ᆠ
 
ᨨ
�鼀
矄
�矄
t

 
ᆠ
ᆠᨨ矄
ᨨ �鼀
t

㦨
t

t҈㡰
ᨨ
ᆠᨨ鼀
泐
矄

a
㦨
b
 
v
㦨

矄� 
v
㦨
u
졔
at
v
= vm - bt - t҈ 㡰
ᨨ
t҈ = t + m




��㡰 b 矄� �
�㡰
�
��㡰

犨
� =

\



㦨


�
\

졠 ᨨ

OQ

ᆠᆠ�ᨨ
a㡰

vm v æ ҈ ҈ö = t - m
ᨨ
t = vm ç + ÷ a b èa bø t ab ᨨ vm =  a + b㡰

dx dx at + b㡰 dx =
at +
b Þ x = at + b Þ = dt dt x dt






���㡰




Þx

d
x æ dx ö +ç ÷ =a dt
è dt ø


æ dx ö æ at + b ö a -ç ÷ a -ç ÷
d x è dt ø = è x ø Þ = x x dt


=

ax
- ( at + b ) 

2

=




ac - b


x x  Þ 
µ
 ᆠ


㦨
 ҈�


ᆠ

㡰 꾀
t
㦨


 
t
㦨
҈
ᨨ泐
�矄

矄 ᔼ ��泐
a
�矄
ᨨ矄��
矄ᨨ
 
 
ᨨ泐
ᆠ⼄盐�矄
矄ᨨ鼀 ᨨᆠ�泐


a 矄 ᨨ盐

��
矄ᨨ
 
 
ᨨ泐 ᆠᨨ鼀矄
ᨨ
矄

 ᆠ
a ᆠ

ᨨ
鼀�
ᨨ矄��
ᨨ

�鼀 �
 
� 

£ t £ ҈


ᨨ盐 泐
�矄
�ᆠ
�
 
ᨨᨨ犨� �
v㜠t
�


ᨨ
�鼀
ᨨ
ᨨ盐 泐
㦨
҈
矄ᆠ

ᨨ
�矄ᆠ鼀 
㦨
҈
鼀
㦨
꾀
盐 
v㜠t㡰
� v鼀 =

��㡰 ᨨ
�矄 ᆠ

㦨

ᨨ
D
ABC ҈ ҈














㦨
´ 矄 ´ �盐
㦨
´ t ´ v鼀






s
´҈ = =
鼀⏌矄 t ҈

҈ æ ab ö
ab ҈  t
㦨
ç ´t ´
è a + bø÷ a+b
Topic-3 : Relative Velocity in One Dimension















㦨


҈


㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

�
A
 
B

盐

� �
ᨨ


ᆠ├矄
�
 
ᨨᨨ矄�
�ᆠ�ᨨ


҈


├鼀⏌



├鼀⏌ A

WWW.JEEBOOKS.IN

EBD_8341 PS-14

Physics 꾀矄
 
ᨨᆠ�泐
ᆠᨨ鼀ᨨ  
ᨨ
A
 
B
ᨨ �
 
矄鼀
�  OA


VA = 
├鼀⏌
㦨
҈

鼀⏌矄

䰘

├鼀⏌

VB = -䰘
├鼀⏌
㦨



鼀⏌矄





VMA
㦨
҈


├鼀⏌
㦨


�
鼀⏌矄 V鼀

B
㦨
V鼀

A
졔
VA, B











㦨
V鼀

A
졔
VA  VB
㦨


�
졔
҈




㡰











㦨



�
졔


㦨

黨
�
鼀⏌矄
㡰  턼  �矄 ᆠ

턼
㦨

맔
 ᨨ

�矄 ᆠ
㦨
矄
맔
�鼀㡰 �矄 ᆠ

턼
㦨

 

ᆠᨨ矄

° =

\

t0
=

vA ҈

 = =



鼀矄-҈ ᆠᨨ矄
° 


�矄ᆠ鼀  �

�


= = = �矄 ᨨᆠ�泐 vB 矄� 
°


´ ҈


æö ç  ÷
쟼ᆠ
矄ᨨ
K

L

M

N
鼀ᨨ矄
犨�

盐 � ᨨ鼀
矄 è ø v
矄盐ᆠ
 
K
犨泐矄
鼀ᨨ
�ᆠ泐
ᨨ犨矄
L

L
�ᆠ泐 ᨨ犨矄
M

M
�ᆠ泐
ᨨ犨矄
N
 
N
�ᆠ泐
ᨨ犨矄 K

ᨨ
 
矄�矄
ᨨ
矄泐鼀鼀泐
犨
ᆠ
矄泐
 
K

L

M
  N
犨�
鼀

 
ᆠ 
ᨨ
 
矄⼄盐
ἘOἘ
꾀
 泐
� 矄 
ᨨᆠ�泐
ᆠᨨ鼀ᨨ  
ᨨ �
KO
㦨
v
ᆠᨨ矄

�

\








҈ ³矄ᆠᨨ³

矄ᆠᨨ  和 � �
 
矄ᆠᨨ
犨� ᨨ盐
犨├� �
 
矄
�矄 矄ᨨ 矄
矄
犨├� �
ᨨ 泐
�矄


和ᨨ

 
矄ᨨ 矄
矄
�矄


҈ ҈ ҈� ³矄ᆠᨨ³ + =




҈

矄ᆠᨨ 

和ᨨ

 
�鼀
ᨨ
�ᨨ
盐
 
矄ᆠᨨ
 = �㡰

҈


㦨



矄ᆠᨨ 
�

꾀
㦨
҈





꾀


º




 矄�




�


  
º

º

ᆠ鼀







 ᆠᨨ矄



졠


�
ᨨᆠ�泐
ᨨ
B
犨�
矄ᆠ
ᨨ
A
�矄
 �ᆠ盐 ᨨ
 
� 
ᨨ
鼀ᨨ�ᨨ
ᨨ
A
�



OA


黨


㝐

O



d



⧤




�  



�矄 ᆠ

�鼀
├
ᨨ
鼀

ἘOἘ
= ᨨᆠ�泐

ᨨ
t = 䰘




v

�矄 ᆠ
KO
㦨
d
ᆠᨨ矄

�
㦨


҈ ³矄ᆠᨨ³

矄ᆠᨨ  ␔├� �
犨�
 
矄ᆠᨨ
�ᨨ� �

 
矄



�


vB =

盐矄

 = ҈

矄ᆠᨨ  ᆠ㡰

Þ

㦨




鼀⏌矄ᆠ






鼀 ෌�

盐ᔴ
㦨
盐
㦨



ᔴ
㦨


鼀⏌矄



㦨


鼀⏌矄
 ᆠ
�
ᆠᆠ�ᨨ

ᔴ
㦨


鼀⏌矄ᆠ
҈ ᨨ犨

犨
├ ᨨ犨

矄 = 盐 + 

҈


= ´

Q 盐 =

 ᆠ

 
ᆠ
犨�
ᆠᆠ
盐
犨�
 
盐矄
 
�鼀





vB
ᆠᨨ矄


㦨
vA
㦨
҈






졠矄㡰



鼀⏌矄ᆠ
ᔴ

ᆠ㡰





턼 턼

҈  ´   = Þ  =





\

B

d




v




=

d v

 矄㡰
꾀矄
ᨨ 
�
�矄
鼀ᨨ� �
�
 
�ᆠ�ᨨ
ᨨ
 矄 ᨨ�ᨨ
 
ᨨ 

�
 
ᨨᨨ矄�
�ᆠ�ᨨ

和ᨨ

矄矄盐  
犨�

�  
b㡰 ᨨ

鼀ᨨ� �
ᨨ矄

  ҈


泐
ᨨᆠ矄

ᨨ
鼀ᨨ �
 
ᨨᨨ矄�
�ᆠ�ᨨ

 �
矄

�矄
�矄
ᆠ盐矄
   �盐
矄
ᨨ
 

泐 ᨨᆠ
犨


ᨨ矄
�矄
�

꾀矄 
 
ᨨᆠ
鼀ᨨ矄
犨泐
   �盐
ᨨᆠ�泐
ᆠ矄矄
    ᨨ
�矄
矄
矄鼀矄
ᨨ
 q
矄矄


�矄 
ᨨᆠ
鼀ᨨ矄 q҈ 鼀� 矄
矄�ᨨ 泐


Q q
> q҈ \w
> w҈ ᨨ鼀
 
ᆠᨨ ᆠ
ᨨ
�
ᨨᆠ�泐 r r r 
҈ 㦨 
෌  ҈෌ 犨 
෌
�矄
 
ᨨᨨ泐
鼀
㡰 㡰
ᨨ
 

B
ᨨ
�
 
ᨨ泐㜠A
�
ᨨᆠ�泐 ᨨ

犨



A

vBA 矄 ᨨ盐

ᨨ �
OA

和ᨨ

vBA 犨�
鼀├


 �

�
犨�
졔㡰
X㜠 �矄
B v BG 㡰 
q
㦨

� vTG
q
´
� = =

°
� \ f
㦨
҈�   犨�
졔
X㜠 �矄

� A ҈� ᔼ
DOMA
f q 㦨

�
q 㦨

�
Þ ÐOAM
㦨

�








O

M

WWW.JEEBOOKS.IN

PS-15

Motion in a Straight Line \ ÐOAB
㦨
҈� 꾀矄ᨨ
ÐBOA
㦨





�
㦨
҈� 矄� �
矄� 
犨
�
DOBA

ᨨ

�


vT 矄� ҈�°   - ҈㡰 ´

䰘҈ = ;
鼀⏌矄 矄� ҈�°


�

 ᆠ
矄
ᨨ

犨


�ᨨ盐 

vB
㦨


鼀⏌矄

ᨨ犨

v
㦨
u
졔
at Þ





҈ s = ut + at



Þ -h =


gh t -

҈
҈ gt Þ gt
-
gh t - h =





gh ±
gh +
´ ᨨ

t =


´

g ´h


ædö


䰘


矄� �

S = ut +

꾀矄
⼄盐�ᨨ
�矄
⼄盐�ᆠ

矄ᨨ

v㜠h
�
犨�


ᨨ v

Þ 

㦨



t h=d




d ҈

ᔼ ��泐
ᨨᆠ�泐
�矄
ᨨ犨 犨矄
㡰
 
 
  ᆠᨨ�矄�ᨨ
�
矄矄
�矄
�ᆠ�ᨨ
犨�
矄矄
鼀� �盐

�

ᨨᆠ�泐
�矄
盐犨矄
졔㡰
ᨨ
 
�鼀
t
㦨


ᆠᨨ矄ᨨ 矄
ᨨ
 
ᨨ� 
ᨨ
 
� 犨 
h
㦨
d
 
�鼀
ᆠᨨ�矄�ᨨ
├矄
ᆠ


㡰 ᨨ

ᨨ泐
 ᨨ犨
�ᆠ泐
盐犨矄
ᆠᆠ�ᨨ

鼀� 矄
ᆠᨨ 矄 
a
㦨

g㡰
 
ᨨᆠ�泐

 泐�鼀
t
�矄 ��
泐
V
㦨
u

gt 盐� �
�矄
ᨨᆠ�泐
ᆠ矄矄
� 泐
 
盐� �
 ᨨᆠ�泐
� ᆠ矄矄
� 泐
 
�ᆠ�ᨨ
�矄
ᨨᨨ矄�
ᨨ ᆠ
ᨨ 
 ᆠ
�
㡰
ᆠᨨᆠ泐
�ᆠ矄
ᨨᆠ�泐
矄盐矄
�鼀
b㡰 泐҈
㦨
҈


�

錐
泐

㦨




�
ᨨ 泐҈
㦨




鼀


㦨

矄 \ 泐


泐҈
㦨


ᨨ




矄






㦨
gt


Þt=

҈ ®

V
� ᆠ矄矄
ᨨ犨 犨矄 h
® ᨨᆠ�泐
ᆠ  �矄
�矄
�ᆠ�ᨨ


® 
V
ᆠ矄矄
盐犨矄

҈
҈ gt Þ S =
´ t + gt


᲼Q
S = ҈

m桌




g

…�㡰

æ ҈ö ҈ s

ᨨ泐
矄

�矄 ᆠ
ᨨ
҈黨
m

矄ᨨ
�
ç t  ÷
è
ø �矄 ᆠ

㦨

҈

ᔼ
矄


ᆠᨨ�矄�ᨨ
├矄
ᆠ





v æ vö çè ÷ø =

ᨨ
v
㦨
v Ἐ

Þ v Ἐ =
vἘ 꾀矄
 
�ᆠ�ᨨ
�矄
矄
 
矄
�矄
ᆠ矄
矄ᨨ �
㡰
ᆠᨨᆠ泐
�ᆠ矄
 矄
ᆠᨨ ��ᨨ 矄




㡰

 

├矄
�鼀

ᨨ
ᆠ
 
�ᨨ盐  \

ᨨ

t =

v
= u
±
gh

nu =

 ᨨ鼀
⼄盐矄�ᨨ 㡰 g g Þ
gH
㦨
nn

㡰u
㡰 ᨨ
盐 � ᨨ鼀泐
ᆠᆠ
ᨨ
ᆠ�ᨨ
鼀ᨨ�ᨨ

v
= u
±
gh
⼄盐�ᨨ
�矄
⼄盐�ᆠ
 ᆠ
v

h
� 犨�


ᨨ
ᔼ ��
ᨨᆠ�泐
ᨨ犨 犨矄
矄ᨨ
��
 
ᆠᨨ�矄�ᨨ

� 矄矄
�矄
�ᆠ�ᨨ
犨�
矄鼀
盐
 ᆠ
ᨨᆠ�泐 �矄
盐犨矄
盐 

v Ἐ
=
g ´ ç ÷ = gd 錐 è ø

g


h
gh
h ҈ +
㡰 Þ t = ҈ +
㡰
ᨨ

t = 

g g g ᆠ㡰 ᨨ
盐 � ᨨ鼀泐
ᆠᆠ⏌
ᆠᆠ
鼀ᨨ�ᨨ



u
g

u + u
+
gH

Þt =


gh

ᨨ犨

ᆠ├
犨�
矄
鼀ᨨ� �
盐 
��泐

ἘtἘ

 
�鼀
├
泐
 
ᨨᨨ
ᆠ├
ᨨ
ᆠ
  �ᨨ盐 


u
+
gh = -u + gt

�鼀
├
ᨨ
ᆠ
�� 矄
ᨨ� 
�矄
t =

ᆠ㡰 ᨨ
盐犨
鼀ᨨ�ᨨ
ᨨ
�ᆠᨨ


v
= u
+
gh Þ v
=
+
gh Þ v =

盐 

Topic-4 : Motion Under Gravity




ᆠ㡰

v
㦨
u
+
gh

vB =





҈
�
和
ᨨ
ᆠ � �
�ᨨ盐 

泐


泐҈
㦨











vB vT = 矄� ҈�° 矄� ҈�°

҈



>

矄





ᨨ犨






҈ æ ҈ö æ ҈ö g ç t  ÷ =
҈

\ g ç t  ÷ =
҈´

è
ø è
ø


҈´
æ ҈ö Þ çt  ÷ = g è
ø \

҈ ҈ = 



҈ ´
㡰
g

Þ

g =
҈

 黨
㡰

Þ

ᨨ鼀
⼄

�㡰

g =



\
�
㦨


m⏌s




T盐㡰
꾀矄
 

矄
ᆠ├
ᨨ
ᨨ� 
ᨨ
ᨨᆠ�ᨨ

矄ᨨ
s
㦨

\ v


u

㦨


맔
g
맔
s 㦨

\ v
㦨
u

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EBD_8341 PS-16

Physics

Motion in a Plane

3





Topic-1 : Vectors ҈


㡰 矄� �
R

㦨
A҈

졔
A


졔

A҈A
ᆠᨨ矄
q �

㦨


졔
�

졔


맔

맔
�
ᆠᨨ矄
q ᨨ ᆠᨨ矄
q
㦨



 ®ö æ ® çè
A҈ +  A
÷ø

�ˆ

®ö æ ®
ç  A҈ -
A
÷
㦨

A
맔
A ҈ ҈ è ø

҈


㦨

A҈

졔
黨A҈A

ᆠᨨ矄
q


A҈A
ᆠᨨ矄
q


A

㦨

A҈



A


졔
�A㜠҈A

ᆠᨨ矄
q 㦨


맔





맔
�

졔
�
맔

맔
�



㡰
㦨

҈҈

�




ᆠ㡰 ᨨ鼀
��盐
 r ˆ  ˆ r  ෌ = � + ├ Þ


 = ˆ + ├ˆ





䰘


( )

 ö  r r æ  ö æ \   ෌ = ç ˆ + ├ˆ ÷  ç ˆ� + ├ˆ ÷ = ˆ  ˆ� è

ø è

ø
r r 㡰 
鼀� �盐
ᨨ
犨ᨨ
ᆠᨨ矄
꾀  

㦨
 r r  꾀 +   = 
+ 
+

ᆠᨨ矄 q   r r  꾀    = 
+ 


éëᆠᨨ矄 (҈

°  q ) ùû





ᨨ



Þ

r r

 꾀






(҈+ ҈ 
ᆠᨨ矄 q)


Þ

(҈+ ᆠᨨ矄 q ) =
(҈ ᆠᨨ矄 q)

\

r \ 㝐� �盐
ᨨ
ᆠᨨ 
ᆠᨨ

ᔴ =

�



+
ᆠᨨ矄 wt = 
-
ᆠᨨ矄 wt ҈ + ᆠᨨ矄 wt = ҈ - ᆠᨨ矄 wt 㡰 p ҈ \ ᆠᨨ矄 wt =
ᆠᨨ矄 wt =
ᨨ
wt =
 p p ´t= \ t =




矄ᆠᨨ 矄


Topic-2 : Motion in a Plane with Constant Acceleration

盐矄� �
ᆠᨨ鼀ᨨ  ᨨ
 
�� ᨨ
 ᨨ鼀

犨
�





r

r r r r r
P 졔 Q R = P Þ P+Q = P
P P

졔
Q

졔

PQ

ᆠᨨ矄q
㦨
P
Þ Q
졔

P
ᆠᨨ矄q 㦨

q a Q Q Þ ᆠᨨ矄 q = 

�㡰
P
P 矄� q  a = = ¥ Q
 ᆠᨨ矄 q + Q =
㡰 Q +
P ᆠᨨ矄 q Þ a
㦨
黨
r r r r 



㡰 ෌�


A + B =  A - B

Þ

æ
҈ ö q = ᆠᨨ矄 ҈ ç
÷ r r r r r è +҈ ø 㡰 ᔼ
ᔴ = �ˆ + ˆ
 
꾀
ᔴ = 꾀
 
졔

㦨
҈





�㡰 rr r r 
ᔴ = 꾀




㦨
҈





��㡰 和ᨨ� �
⼄盐�ᨨ
�㡰
 
��㡰
犨
� ҈
=
=  



㝐� �盐
ᨨ
ᨨ⼄盐

 t  = ҈黨�
黨
㡰
෌�


\


+ 
+

ᆠᨨ矄 q =

+ 


ᆠᨨ矄 q


(҈+ ҈ +
ᆠᨨ矄 q)

kˆ

\




a �ˆ + a ᆠᨨ矄 wt �ˆ + a 矄� wt ˆ =  a �ˆ - a ᆠᨨ矄 wt �ˆ - a 矄� wt ˆ Þ




҈ + ᆠᨨ矄 wt 㡰�ˆ + 矄� w t ˆ =  ҈ - ᆠᨨ矄 w t 㡰�ˆ - 矄� w t ˆ




㦨

+ 


ᆠᨨ矄 q  
ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ


r r  꾀 +  =

ˆ

t =  ҈ -
= 䰘�ˆ - ҈҈ ˆ + �kˆ

졔
A
㡰
A҈㡰
ᆠᨨ矄
q


A҈㡰
A
㡰
ᆠᨨ矄
q

A

맔

A






҈黨�㡰 r ෌�


F = �ˆ +
ˆ + kˆ㡰
 r 꾀 

r = ᲼
�ˆ +  ˆ - kˆ㡰 - �ˆ +
ˆ + kˆ㡰桌 = �ˆ + ˆ -
kˆ r r

ᨨ⼄盐

t = r ´ F = �ˆ + ˆ -
kˆ㡰 ´ �ˆ +
ˆ + kˆ㡰

҈


㡰 ᨨ鼀
��盐⏌
��ᨨ ᨨ鼀泐
h҈ h
= 
�° \ h҈ = d d
h҈ h҈ + h
꾀 

=  
° d +




d A


�

d

Þ h҈ + h
㡰 ´  = 


d

҈
+ = 黨 黨

r r r S-P r r r 㡰 
P + bR = S




Þ R = b r r r 꾀矄ᨨ
R = Q - P r r r r r r S-P r r = Q - P

Þ
S - P = bQ - bP \
b r r r \ S = bQ + ҈ - b㡰 P

Þ h҈ + h
㡰 =

� 黨






d 

Þ d + h
=

\ h
= d r 㡰 ෌�


u = � ˆ 鼀⏌矄 r 꾀ᆠᆠ�ᨨ

a = ҈
�ˆ +
ˆ
  � 
ᆠᨨᨨ� 




y
㡰
�
�鼀
t
҈ S x = ux t + ax t




B




d C




d 

᲼Q ux =
桌

WWW.JEEBOOKS.IN

PS-17

Motion in a Plane

Þ

=
+








�


҈ ´ ҈
´ t
Þ t =
矄


҈ S y = u y ´ t + ay t

҈ y
= � ´
+ ´
´

= ҈
鼀
҈
ᆠ㡰 矄� �
S = ut + at
҈ y = u y t + a y t
ᨨ � y 꾀 �矄㡰
҈ ҈ Þ 
=
´ t + 
㡰t
Þ ´
´ t
= 
Þ
t
㦨


s

҈ S x = u xt + a xt
꾀ᨨ �
x 꾀 �矄㡰
҈ Þ x =  ´
+ ´
´

=


㡰 ෌�

ᨨ矄��ᨨ
ᆠᨨ
r r = ᆠᨨ矄 wt�ˆ + 矄� wt ˆ r r dr = w  矄� wt�ˆ + ᆠᨨ矄 wt ˆ 㡰 ᨨᆠ�泐

v = dt 꾀ᆠᆠ�ᨨ
r r dv a= = - w
ᆠᨨ矄 wt�ˆ + 矄� wt ˆ 㡰 dt r r
a = -w
r r \ a
�矄
 �
ᨨ
rr r r r r 꾀矄ᨨ
v
r =
\v ^ r

盐矄

 
�ᆠ
�矄
 ᨨ鼀� �
盐 � ᨨ鼀
ᆠ�ᆠ盐
鼀ᨨ�ᨨ
® 㡰 r = ҈�t
�ˆ + 
-

t
㡰 ˆ ®

v =

®

d r = 
t�ˆ -

tˆ dt

®

®

꾀ᆠᆠ�ᨨ

a = d v = 
�ˆ -

ˆ

dt








䰘





\ a = 
+

= �
鼀⏌矄
r r ҈r b㡰 꾀矄
和 = 盐 + 

r ˆ + ҈ 
�ˆ+
ˆ㡰
和 = ��ˆ +
㡰
= ҈
�ˆ +
ˆ +
�ˆ+
ˆ
r r  - � = ҈
�ˆ + ҈
ˆ r r r ᲼矄 矄 = ᆠ  � � ᨨ矄��ᨨ =  - � 桌 r  =

�ˆ +

ˆ r   =


ᆠ㡰 ᨨ鼀
��
⼄盐�ᨨ
r  = ⧤ 泐�ˆ + ˆ

(

)

 泐 = ├泐
 
= ├  





泐  = = 泐
Þ 泐泐
㦨
 ᨨ犨
 泐   ᔼ �� �
ᨨ
矄� 泐

㦨


졔
ᆠ �

㡰 ᨨ
��ᆠ
�A XA
㦨
t

졔

t
졔
҈
r VA = 

t 㡰�ˆ r aA = 
�ˆ

ᨨ
�ᆠ
�B YB
㦨
�


t

r VB = 

t
ˆ r aB = -

tˆ

꾀
t
㦨
҈
矄ᆠ

r r VA = 

t 㡰�ˆ =
�ˆ   vB = 

ˆ r r r \ V B ⏌ A =  v A + vB = 
�ˆ 

ˆ \
和
ᨨ
B
犨



A


v =

+




=
+ �䰘
= �



黨


\ v
㦨
�


鼀⏌矄㡰 

b

ᆠ

㡰
꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

⼄盐�ᨨ

泐=





泐=

꾀

㦨





=

泐 =
ü ý ��

盐 =҈ þ





㦨








㦨
҈
鼀⏌矄

泐=
泐 ҈   = ×
= Þ 泐
㦨
 
  ᨨ犨
�  ᆠ�
犨
�鼀


泐 


泐 =

=


æ  ö +ç ÷ 
è  ø




 ×  +  

 泐 = 
+  㡰
ᔼ

㦨
҈
 
�ᆠ
�矄

ᨨ���

㦨



泐
㦨

㡰 泐
㦨

泐
㦨
҈

㦨
҈
鼀⏌矄
㡰 
㦨

泐
㦨


졔

Þ
泐
㦨

ᔼ


㦨




㦨
ᆠᨨ 矄 
㦨
҈ Þ
泐
㦨
ᔼ

㦨
҈ ᆠ㡰 꾀

㦨




㦨





泐
㦨
 矄
㦨҈錐









泐
㦨


Þ

㦨
҈ 㡰 
㦨


�鼀�矄
 


㦨
҈矄 泐
㦨


졔

Þ
泐㦨
 Þ
泐
㦨

ᔼ

㦨


Þ


㦨
ᆠᨨ 矄 
� ��泐


㦨
҈㡰
Þ
泐
㦨
҈

㦨
҈ 꾀

㦨
҈
矄ᆠ 泐

㦨


졔
泐
맔

㦨
҈
맔
҈
㦨
҈ 泐  q = =

q
®
 �
犨�

 �矄㡰  \
 q =

泐

҈ = = ҈ Þ q =
�°  ҈

WWW.JEEBOOKS.IN

EBD_8341 PS-18

҈



Physics x
㦨
a
ᆠᨨ矄
 Þ ᆠᨨ矄
㡰
㦨




b

ᆠ㡰

x a

y b 和⼄盐� �
 
� �
⼄

�㡰
 
��㡰

犨
�
y
㦨
b
矄�

Þ 矄�
㡰
㦨


x


+





�㡰

R 㦨
vxT

=





��㡰

y


=҈ a
b
 ᆠ

ᨨ
 
�ᆠ �矄

�矄
ᨨ鼀
 
��
⼄盐�ᨨ 矄 犨
ᆠ
� 


Þ R
=



u
h ´ g


 u

g

ᆠ㡰 ෌�

y
㦨

x

黨x
졠
ᆠᨨ鼀� �
犨�


y = x  q -

gx



u
ᆠᨨ矄
q




␔


q
㦨


ᨨ
ᆠᨨ矄
q
㦨


҈




d x dx
= v x = - ap 矄� pt 錐

= ax = - ap ᆠᨨ矄 pt dt dt dy = v y 㦨

ᆠᨨ矄
pt dt  

d
y dt




= a y = -bp
矄� pt



꾀
�鼀
t
㦨


p p
ᨨ
pt
㦨




p =
㡰

졠 泐
vx
 
ay

 

ᨨ
犨
ᆠ
矄泐
 
ᨨᆠ�泐
�矄
ᨨ �

��
x㜠 �矄
 
ᆠᆠ�ᨨ
ᨨ �
��
y㜠 �矄
p

ᨨᆠ�泐
 
ᆠᆠ�ᨨ
ᨨ
 
�ᆠ  ᆠ


t
㦨

p 
ᨨ鼀
ᨨ
ᆠ
ᨨ 
꾀
t
㦨
t

ᨨ矄��ᨨ
ᨨ
 
�ᆠ r r t 㡰 = x�$ + y $
= a ᆠᨨ矄 pt $� + b 矄� pt ˆ a x
 
v y
ᆠᨨ鼀
ᨨ
ᆠ盐矄
ᆠᨨ矄





=黨 = 黨
ᨨ


u ҈ ⏌ �㡰

u ᆠᨨ矄
q \ u
㦨
�⏌
鼀⏌矄 㡰 R
犨�

矄鼀
ᨨ
q
 
黨


q.

�鼀
ᨨ
�� 矄

u 矄� q t҈
㦨

  g
u 矄� 黨
° - q㡰
u ᆠᨨ矄 q = t

㦨
g g



æ u
矄�
q ö
R æ
u 矄� q ö æ
u ᆠᨨ矄 q ö ÷= ᨨ犨

t҈t

㦨
ç
㦨
g ç g g è g ÷ø çè g ÷ø è ø b㡰 ᨨ
矄鼀
 �

 
 �
ᨨ
ᨨᆠ�ᨨ 矄


q
 
黨


q.
和ᨨ
h҈
㦨


u
矄�
q
 
g

h

㦨


u
矄�
黨
° - q㡰 u
ᆠᨨ矄
q =
g
g

 
ᆠᆠ�ᨨ
ᨨ
 
�ᆠ

u
矄�

q 꾀矄ᨨ

R
㦨
g

 ᨨ

ᆠᆠ�ᨨ
ᨨ
 
�ᆠ
�矄
犨泐矄
�ᆠ ᨨ犨矄
ᨨ���
p
�ᆠ
�矄



꾀
t
㦨



�ᆠ
�矄

a


㡰
 

t
㦨

p b㡰

 ᨨ

 
�矄 ᆠ
ᆠᨨ
�矄
ᨨ 
ᨨ盐
ᨨ
  ��ᆠ

 
ᨨ
a


h҈
h

㦨


r at 㡰 = ax $� + a y $ = - p
᲼a ᆠᨨ矄 pt $� + b 矄� pt $ 桌 r 㦨

p

᲼ x�$ + y $ 桌 = - p
r t 㡰

Topic-3 : Projectile Motion ҈


g

 





㡰 矄� �
� ᆠ�
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀 ᨨ
ᨨ�ᨨ 
鼀ᨨ�ᨨ

犨

mvx
㦨
mu
졔
mu
ᆠᨨ矄




u
ᨨ
�ᆠ
鼀ᨨ�ᨨ

vx =


h ҈
gT
Þ T = g

R
�矄
 
ᨨ�ᨨ 
�矄 ᆠ

泐
 
ᨨ泐
҈ R = v xT + 
㡰T 㡰
 ᨨ ᨨ�ᨨ  鼀ᨨ�ᨨ 㡰


�

҈


u
矄�
q u
ᆠᨨ矄
q
맔

g
g

u
u

矄� q ᆠᨨ矄 q㡰
R
㦨

҈
g ҈
ᨨ

R

㦨
҈

h҈
h
㡰 졠

� ᆠ� 
 

�鼀
ᨨ
�� 
 㡰
�矄
��
泐
u 矄� q T= g ᆠᨨ矄 a 和盐矄�盐� �
 
盐矄

犨
� 
㡰
矄� ҈�°㡰
矄� ҈�° T= = g ᆠᨨ矄 
° ҈
ᆠᨨ矄
° ҈
�矄 ᆠ

和
= 
ᆠᨨ矄҈�°㡰T - g 矄� 
°T 㡰
㦨


�





h =
+

泐



鼀⏌矄 q 㦨҈�


�矄�

� �ᆠᨨ矄


a 㦨



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PS-19

Motion in a Plane

矄� ҈�° æ҈ ö ҈
矄� ҈�° = 
ᆠᨨ矄҈�°㡰 - ç ´ ҈
矄� 
°÷ ø ҈

ᆠᨨ矄

° ҈
҈
ᆠᨨ矄 
° è


黨


҈
 - ҈
;

҈黨�
鼀 ;

ᆠ鼀

b㡰 ҈

鼀⏌矄

q

� ᨨ

҈



)

泐
㦨
 �  ҈
 鼀 ⏌ 矄 = ҈
 � 


ᔴ �

ᆠᆠ�ᨨ

㦨



� +҈

+ 䰘� ҈






�㡰 

҈
ᆠᨨ矄 q ᨨ鼀
��盐
盐矄� �
�㡰㡰

=

䰘


=

҈
 �  =
  Þq= ҈�° �

(

҈


  ҈
ᆠᨨ矄҈�

҈҈


) =


鼀

㡰 꾀矄
犨
├ ᨨ犨

 �

=

盐
矄�
q �

 


꾀
㦨
p
\

꾀
µ


ᨨ

꾀
µ 盐



�ᆠ
ᨨᆠ�泐
� ��㡰

�

㦨
uy t
졔
Þ uy
맔


졔


\

 q =

҈
gt


҈
҈
㡰
맔



ᨨ
�

㦨

uy




ᨨ
uy
㦨


uy ux

=

� 䰘 =




Þ 꾀 �
q
㦨
 ҈


䰘


q

u
⏌a t҈

u҈

q

u
⏌a
t


u


q

u
⏌a m

t u

q tn un

ᨨ
�矄ᆠ鼀  꾀�
ᨨᆠ�泐
= ᨨ
�鼀
├


u 矄� q

�鼀
ᨨ
�� 
T = g

\







鼀

ᨨ
盐 = ҈

= ҈
鼀矄 -҈ �� 
ᨨ
犨
꾀 ҈ �졠꾀㡰
㦨
졠꾀  q
盐
ᆠᨨ矄
q ҈ ҈
´ ҈
=
- ´ =


鼀 ҈ ҈
҈
´ ҈
´ ´

(4) 
u҈

u


u








 
ᨨ�ᨨ 
ᨨᆠ�泐
ᨨ
  ᨨᆠ�矄
 
t҈
t

t







 
�鼀
├
矄
矄 ᨨ犨
�

��盐
u


꾀҈ 盐҈
é ҈ ù ҈ = = = 꾀
盐

êë
úû ҈
ᆠ㡰 ᨨ鼀
⼄盐矄�ᨨ
ᨨ�ᨨ 
ᨨᆠ�泐
� ��㡰


ux = =

鼀⏌矄


꾀

盐
矄�
q
�  ´
�

� ´
´ ҈
\

盐
= = = ҈



矄� q æ ҈ ö ç ÷ è
ø

꾀 

㦨
҈
矄ᆠ
ᨨ�ᨨ 
ᆠᨨ鼀ᨨ  
ᨨ
ᨨᆠ�泐
x
㦨
�
鼀⏌矄 �ᆠ
ᆠᨨ鼀ᨨ  
ᨨ
ᨨᆠ�泐

 q=

犨

㝐 �鼀盐鼀
�� 


㦨


v泐
㦨
҈
ᆠᨨ矄


㦨
�  鼀⏌矄


+ 
泐


�

鼀

꾀矄

�矄
ᨨᆠ


 �

�
ᨨ
 
ᨨ�ᨨ    ᨨ

 �
㦨

 ҈
=

� 鼀 ᨨ ҈

㦨


Þ
 =
Q
 �
㦨


鼀
졔


鼀
㦨
҈
鼀㡰

ᨨ�ᨨ 
ᆠᨨ鼀ᨨ  
ᨨ
ᨨᆠ�泐 x
㦨
҈
ᆠᨨ矄



㦨
�
鼀⏌矄 �ᆠ
ᆠᨨ鼀ᨨ  
ᨨ
ᨨᆠ�泐

Þ =

b㡰 졠

 � ҈
- � 㡰

(

…
��㡰



�

�ᆠᨨ矄q





…
�㡰

҈ y =
t - ҈
t
㡰
ᨨ鼀
�㡰
 
��㡰

y
㦨

x

�x


=





r

b㡰 ᨨ鼀
⼄盐�ᨨ

v = �ˆ +
ˆ Þ

x
㦨
t

䰘
= �鼀 ⏌ 矄


ᨨ
��
盐
ᨨ
q

盐
ᨨ
T
犨�
ᆠ  �
犨�
 
盐 ᨨ
u
\ ᨨ
�鼀
├
㦨
t҈
졔
t

졔
t
졔



t t t ta = t҈ + ҈ + ҈ +




= ҈ = ҈
҈ a a a - ҈㡰 ҈a

ᨨ
�矄ᆠ鼀 
㦨
盐҈҈
졔
盐


졔
盐
졔







u҈ t҈ u҈ t҈ ut u t a

+


+




= ҈ ҈ = ҈
҈ ҈ a a a a ҈ -
a - ҈㡰 a ᨨ
�矄ᆠ鼀  \ 꾀�
ᨨᆠ�泐
= ᨨ
�鼀 u҈t҈a
a - ҈㡰 u҈a ´ =






















=
a +҈ t҈a a - ҈㡰 = u҈t҈ +

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EBD_8341 PS-20

Physics 犨 
a
㦨
҈
�
鼀⏌矄
ᨨ u
ᆠᨨ矄 q 㡰t
㦨

졔


䰘�
t




��㡰 ���㡰 ᨨ�ᨨ 
ᆠᨨ鼀ᨨ  
ᨨ
ᨨᆠ�泐
ᨨ
矄ᨨ 
㦨
�ᆠ ᆠᨨ鼀ᨨ  
矄
ᨨᆠ�泐
ᆠᨨ
�矄
� ᆠ� 


�
犨� ᨨ�ᨨ 
\ u
ᆠᨨ矄 q 㡰
㦨
gt

u
矄� q 㡰



���㡰  
�� 
 
矄�
�矄
犨�
gt

u
矄� q
ᆠ盐矄
  矄ᨨ 
�矄
�
�矄
ᨨ犨 犨
鼀ᨨ�ᨨ

\
gt
>
u
矄� q

ᔼ
盐犨 鼀ᨨ�ᨨ
盐矄� q >
gt㡰
㝐盐�泐� �
⼄盐�ᨨ
���㡰
犨�
t u
ᆠᨨ矄 q 㡰
t
졔
u
矄� q 㡰
t
㦨
҈
t




�㡰 ᨨ犨

�㡰

��㡰

�㡰
��矄



�
t





�
㦨


ᨨ
t
㦨
҈矄 和盐矄�盐� �
t
㦨
҈矄
�
�㡰
 
��㡰

犨
�
u
矄� q
㦨



�
鼀⏌矄
ᨨ
uy
㦨



�
鼀⏌矄  
u
ᆠᨨ矄 q
㦨

䰘�
鼀⏌矄
r ᨨ ux
㦨

䰘�
鼀⏌矄
\

u = u x $� + u y $

꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

�
ᨨᆠ�泐
㦨




҈ u҈a ᨨ



҈
=





�㡰 a +҈

u҈ ´ t҈ = u҈





��㡰 t҈ ᨨ鼀
⼄盐�ᨨ
�㡰
 
��㡰 a
㦨




a
졔





ᨨ

a





a
㦨






=
ᨨ



a
㦨





ᨨ

a =


҈

�㡰 

u
㦨
҈

鼀⏌矄

q
㦨


盐泐 盐
㦨
҈

鼀⏌矄 盐
矄�
q 㦨
�

 鼀⏌矄

V҈ =

r



盐 盐
ᆠᨨ矄

q 㦨
�
鼀⏌矄 \ ux
㦨
u
ᆠᨨ矄
q
㦨
҈

맔
ᆠᨨ矄



㦨
�
鼀⏌矄   uy
㦨
u
矄�
q
㦨
҈

맔
矄�



㦨
� 
鼀⏌矄 졠


´�  = = 矄 g ҈

a

 
ᆠᆠ�ᨨ
ᨨ
�
҈







҈
҈� = u xt - at
҈ Þ ҈
҈� = � ´  - ´ a ´  㡰

a Þ = �  - ҈
҈� =


� - ҈
҈� = 䰘
�

\ a = 䰘
� ´ = � 鼀⏌矄
 T盐㡰

쟼

㦨

쟼

졔
⧤
쟼

㦨
ᔴᨨ 矄  꾀
 
ᨨ

⧤
쟼

�矄
鼀� �鼀盐鼀
 

쟼

�矄
鼀 �鼀盐鼀

和� ᆠ ⧤
쟼

�矄
鼀� �鼀盐鼀
矄ᨨ
矄
�矄
矄ᨨ
鼀� �鼀盐鼀

ἘtἘ

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�鼀
 
犨 �ᆠ
 
矄ᨨ 
�矄
 
ᨨᆠ
  r q

 
 �
犨 �ᆠ
 
ᨨᆠ�泐
ᆠᨨ
u
鼀├矄
犨� ᨨ�ᨨ 
\

҈
҈



҈�


t=


u y

y

v x 㦨
u ᆠᨨ矄 q

u u矄� q

q

졠 u ᆠᨨ矄 q

gtu 矄� q 㦨᲼vy 桌 ҈

�鼀


� v

x

꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

犨

ᨨᨨ犨� �
 
ᆠᨨ ��ᨨ 矄
�㡰 �ᆠ
�矄ᆠ鼀 
ᨨ
矄ᨨ 
�矄
҈

�
鼀
҈ \ ҈

�
㦨
 u
矄� q 㡰
t

gt

犨 
�
㦨
҈

鼀⏌矄




�㡰 ᨨ u
矄� q 㡰
t

㦨
҈

�
졔
� t
��㡰 ᨨ�ᨨ 
�矄ᆠ鼀 
ᨨ
矄ᨨ  㦨

졔
�矄ᆠ鼀 
ᨨ
ᨨᆠ
A
҈
\ u
ᆠᨨ矄 q 㡰t㦨

졔
at


ᨨ u = 
䰘� $� +


� $ 㡰 鼀⏌矄 꾀 
 
�鼀
 
犨 �ᆠ
 
矄ᨨ 
�矄
 
ᨨᆠ

t
㦨
҈矄 㡰 
�
ᨨᆠ�泐
ᨨ
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�ᆠ
犨�
矄ᆠ
ᨨ  
ᨨ
�矄
u

ᨨ犨
 
�
ᨨᆠ�泐
�
x
 
y㜠�ᆠ�ᨨ

ux
 
uy
矄ᆠ�泐
和� ᆠ

 
�矄
ᨨ
ᨨᆠ�泐
ᨨ
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ᨨ
�
 
y㜠�ᆠ�ᨨ
  ᨨ
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�矄
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�ᆠ
ᨨᆠ�泐
ᨨ
 
�ᆠ
犨� 矄ᆠ
ᨨ
�ᨨ盐 
矄ᨨ


├� �
�
鼀矄
犨



ᨨ
Y㜠�ᆠ�ᨨ
鼀ᨨ�ᨨ

uy
㦨
졔
u
矄�
a ay
㦨

g
ᆠᨨ矄
q sy
㦨


ᆠ��泐
�矄
├
�
 
�鼀
 
�ᆠ
ᆠᨨ鼀矄 ᆠ├
ᨨ
 
ᨨ
㡰 ty
㦨
t ҈ ҈
sy
㦨
u y t + a y t Þ

㦨
u
矄�
a㡰
t

g ᆠᨨ矄 q ´ t



u 矄� a Þ t
㦨


ᨨ
t
㦨
g ᆠᨨ矄 q

├� �
�
鼀矄
犨



ᨨ
X
㜠
�ᆠ�ᨨ
鼀ᨨ�ᨨ

ux
㦨
졔
u
ᆠᨨ矄
a ; ax
㦨



tx
㦨
t

sx
㦨
sx

҈



sx
㦨
u xt + a xt
Þ sx㦨
u
ᆠᨨ矄
a
맔



u 矄� a u
矄�
a = g ᆠᨨ矄 q g ᆠᨨ矄 q

æ
u 矄� a ö b㡰 
�矄 ᆠ

泐
 
ᨨ
�
�鼀
ç ÷ è g ᆠᨨ矄 q ø 矄 ᨨ盐

⼄盐
ᨨ
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 �
ᨨ
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�ᆠ
ᨨ
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ᨨ矄 ᨨ
�ᨨ盐 
ᨨ
矄
ᨨ�ᨨ 
�矄ᆠ鼀 
ᨨ

ᨨ

  
矄
ᨨ
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ᨨ

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�鼀
ᨨ
ᨨᆠ�ᨨ
ᨨ
�ᆠ
 U


ᨨ
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鼀ᨨ�ᨨ
ᨨ

ᨨ
犨�
矄ᆠ
ᨨ
�ᨨ盐 
ux
㦨

U 錐
ax㦨

g
矄� q 錐
t y
㦨



u 矄� a -u
矄�
a 錐 sx
㦨
g ᆠᨨ矄 q g ᆠᨨ矄 q

҈
sx
㦨
u xt + axt


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PS-21

Motion in a Plane

æ
u 矄� a ö ҈ -u
矄�
a æ
u 矄� a ö = -U ç - g 矄� q ç g ᆠᨨ矄 q è g ᆠᨨ矄 q ø÷
è g ᆠᨨ矄 q ø÷

ᨨ
鼀矄矄

鼀
hᨨ犨
ᨨ鼀
C ᨨ
ᨨ�ᨨ 
鼀ᨨ�ᨨ QM
㦨
᲼҈

ᆠᨨ矄


桌
t QM
㦨
�
t




ᨨ
�ᆠ
鼀ᨨ�ᨨ

vy
㦨
҈

矄�



㦨
�  錐
ay
㦨
g sy
㦨
y
졔
҈
錐
ty 㦨
t

u ᆠᨨ矄 a + q㡰 ᆠᨨ矄 q

�
 
盐矄
ᆠᨨ�
�
�

ᨨ� 
P
 泐
ᨨᨨ犨
�   
ACP
 
ABP.  t҈
㦨
�鼀
├
泐
�矄
盐
�
ᆠ � �
P
t

㦨
�鼀
├
泐
矄ᆠᨨ 
盐
�
ᆠ � �
P
y u B ᨨ

ABP  �

R
㦨
u
ᆠᨨ矄


㡰t


u  �

R
㦨
u҈㡰 A C ᨨ

ACP h 和ᨨ� �
⼄

�㡰
犨
�

U
㦨


҈



\

ut
= ut҈


ᨨ t

㦨

t҈




�㡰 �� 
ᨨ

ABP

ᨨ犨

vy
㦨
�  졔
gt ҈  
sy 㡰
㦨
uy t
졔
ay t

҈
Þ y
졔
҈

㦨
�  t + gt
҈
ᨨ犨
盐� �
盐
ᨨ
y = gt
�
⼄

�㡰

҈
҈ 矄ᆠ gt + ҈
= �  t + gt
Þ t
㦨


 �矄 ᆠ
犨
ᨨ
ᨨ犨矄

P

҈

鼀





㡰

ᨨ ᨨ

-�  ´ t


\


㦨
҈䰘


鼀  ��㡰 꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
盐� � ᆠᨨ�矄�ᨨ
ᨨ
 
鼀矄矄矄

M㡰
�
 
ᨨ�ᨨ 
�ᆠ�ᨨ


ᨨᆠ�泐
ᨨ
ᆠᨨ鼀� 
鼀矄矄
ᨨ �
ᨨ�ᨨ 
㦨
v x m
맔
҈




m
҈

ᆠᨨ矄



㦨

m
맔 vx Þ ҈

m

҈

m
㦨

m
맔
vx
Þ vx
㦨

和� ᆠ

 
ᨨ�ᨨ 
鼀ᨨ鼀 盐鼀
ᆠᨨ鼀矄
ᨨ盐
ᨨ

ᨨ

  ᆠᨨ鼀� �ᨨ
ᨨ
鼀矄矄矄
犨�
ᨨ
�ᆠ泐
ᨨ犨 犨矄
  

E

\ 쟼
㦨
㝐
㦨
҈

t
㦨
҈

맔
㦨
҈҈
�
䰘
鼀 

-u ´ t
҈
҈
+ gt
= gt҈




҈
t҈
҈
t҈
=


-҈�t
+ ҈
t

= ҈

҈
ᨨ

- t
+
t

=
t҈
+

- ´
t҈ 㡰 +

t҈ 㡰
=
t҈
ᨨ -
t҈ + 
´
㡰t҈
=
t҈
ᨨ

t҈

=
t҈
ᨨ t҈
㦨
҈
矄ᆠ \ t

㦨

t҈
㦨


矄ᆠ ᨨ犨

R
㦨
ut҈
ᨨ

ACP






��㡰




���㡰

u
矄�
q 
黨㡰
矄�
´
�° =
㦨


�
鼀
g 黨

�矄 ᆠ
犨
 
ᨨ� 矄
ᨨ
 ᨨ犨
ᨨ
�ᆠ矄
�矄
矄ᨨ

�
鼀

�矄
鼀 矄
 
 
�ᆠ矄
矄 ᨨ盐
ᆠᨨ�

鼀� 犨泐
犨
A
 
B

 �
鼀 �鼀盐鼀
�� 
H
㦨






�㡰

u
矄�
q 
黨㡰
矄�
�°㡰

㦨

㦨

҈

�
鼀
g
´ 黨



ᨨ

ᨨ

R = �  ´҈ = �  鼀

҈



҈
gt҈
ᨨ

ACP
҈






= ´҈
´ ҈㡰 = � 鼀


҈䰘


u ᆠᨨ矄
�° =
黨

y㜠ᆠᨨᨨ� 
ᨨ
P
㦨
҈


h
㦨
҈


�
㦨
�
鼀




㡰

�鼀
� 
犨
 
�� �矄
㦨
t


t҈㡰








㦨



҈㡰
㦨
҈
矄ᆠᨨ 

҈
y
㦨
gt

















��㡰
ᨨᆠ�泐
�
�ᆠ
�ᆠ�ᨨ

vy
㦨
uy
졔
ay t
㦨
gt






���㡰


鼀⏌矄

꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀

 
ᨨ� 
ᨨ ᆠᨨ�矄�ᨨ

m҈ u ᆠᨨ矄
�°㡰 - m
u ᆠᨨ矄
�°㡰 = m
v - m҈ u ᆠᨨ矄
�°㡰

��㡰 ᔴᨨᨨ� 矄
ᨨ
ᨨ� 

P
㦨
R

y㡰
= �  鼀

�
鼀㡰 �㡰 
t

 
�鼀
├
ᨨ
ᆠᨨ�矄�ᨨ


�
 �
ᨨ� 
ᨨ
ᆠᨨ�矄�ᨨ
ᨨᆠ矄
 ᨨ犨
ᨨ鼀
꾀
 
ᔴ 盐
ᨨ �
ᨨ�ᆠ
 矄
ᨨ
鼀矄矄
m
hᨨ犨
hᨨ�ᨨ y
ᨨ鼀
A
ᨨ
ᨨ�ᨨ 
鼀ᨨ�ᨨ

㝐
㦨
VA
맔
t
㦨
҈

t













�㡰 ᨨ
�ᆠ
鼀ᨨ�ᨨ

uy
㦨

錐
sy
㦨
y錐
ay
㦨
g錐
ty
㦨
t \


 �
R
㦨


쟼ᆠ
�ᆠ
矄
 
矄鼀
矄
ᨨ

黨
鼀⏌矄
  ᨨ�ᨨ 
ᨨᆠ�泐
ᨨ
ᆠ
�ᆠ
�矄
矄鼀
�




꾀��
h =

\ �㡰





�㡰

BD
㦨
PM
졔
MQ
㦨
҈

t
졔
�
t
㦨
҈�
t
㦨
҈�
맔


҈ h = -u 矄�

°㡰t
+ gt


҈
h = gt҈
�� 
ᨨ

ACP





㡰




x

R





�㡰




맔

黨
맔


҈ ҈ ҈ 

´
黨 ´ +

V
㦨


´
黨 ´




\ V
㦨

�ᆠ
Q
�矄

矄
 
ᆠᨨ�矄�ᨨ
 
犨�

�ᆠ泐 ᨨ犨 犨矄

 
鼀�犨泐


�鼀
├
㦨



H
´
҈

�
㦨

㦨

�
矄 g 黨



Topic-4 : Relative Velocity in Two Dimensions & Uniform Circular Motion ҈


㡰 

R
㦨


҈
鼀

w=


p
p = =

҈
�

⏌矄 T



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EBD_8341 PS-22

Physics

꾀ᆠᆠ�ᨨ
ᨨ
 
�
ᨨ
 
ᆠᨨᆠ├
矄ᆠᨨ Ἐ矄
 


a = w
R = 

҈
�㡰


҈㡰 =



҈҈ = ҈
҈ ´ ҈
-
鼀⏌矄









㡰

 ᆠ

�
ᆠᆠ�ᨨ
�矄
ᨨ
 
ᨨ
ᨨ
҈





㡰 
��
矄�盐�ᨨ
�矄
矄 ᨨ犨
�
 
��鼀

 vr 
 
ᨨᆠ�泐
ᨨ
�
ᨨ


҈

q


vr

vr

p - q㡰

vr






�

- ҈


vᆠ 㦨

b 졔
҈㡰v


vᆠ 㦨
v

v

ᔴ  �
�
ᨨᆠ�泐


 D  = ҈
+ 

+
҈ 
ᆠᨨ矄 ( p  q )

␔ 
ᆠ
�矄
鼀ᨨ� �
犨�
矄
v
v 

° = r


�㡰 v ␔ 
ᆠ
�矄
鼀ᨨ� �
犨�
矄
҈ + b㡰v


vr b + ҈㡰v ��� �
�㡰
泐
��㡰
犨
�

�° =

㦨

矄�

ᆠ㡰
矄� q =




��㡰 䰘


b㡰

u
҈
= = v



 q 盐



ˆᨨ 㡰

B rBA

҈


㦨
҈

鼀⏌矄 ᆠ㡰 和

V
㦨
ᆠᨨ 矄 
 ᨨ鼀
⼄盐矄�ᨨ 㡰 ᔴ �
ᆠᆠ�ᨨ


V
r ra 㦨
ᆠᨨ 矄   ᆠ
�
ᆠ㡰
ᆠᨨᆠ泐
矄ᆠ�矄
�ᨨ
犨

ᆠᆠ�ᨨ
 
�盐矄
r b㡰 和 ᨨ矄
ᨨ盐
ᆠᨨ矄ᨨ 矄
ᨨ
vb
 �ᆠ盐
ᨨ �
ᨨ犨 ® vr d d \t= =
vb ® - vr
vbr ® vb d v ᨨ

t =

ᨨ



r r 

�ˆ + ҈�
ˆ 㡰 -

�ˆ - �
ˆ 㡰 rBA )( · vBA ) ( t鼀� �鼀盐鼀
=
=
҈

҈㡰
( vrBA )

҈
䰘 \t= = =


矄
҈

҈ ´ ҈

҈
҈ p p = ᆠ㡰 ᨨ鼀

q
㦨
w
㦨

w
w
和ᨨ

ᨨ

ᆠᨨ鼀
⼄盐
ᆠ�ᆠ

d ҈├鼀 = ҈ vb ҈


҈ =


҈
� - vr



Þ vr = 
├鼀⏌ 黨


҈
䰘



㡰 �鼀
├
ᨨ
ᆠᨨ矄矄
 
�
t
㦨


vs
ᆠᨨ矄
q

vs w ҈ 꾀


�
ᨨᆠ�泐


( )

w




d vs ᆠᨨ矄 q

 d ␔

q




 꾀    = w ҈ �ˆ - w
( � )

= w (
 ҈ ) �

r

br

�ˆ 쟼矄㡰

A r ˆ vA = 
� + �
ˆ ├鼀⏌  r vB = -҈
�ˆ㡰 ├鼀⏌  rBA = 

�ˆ + ҈�
ˆ 㡰 ├鼀 r r r vBA = vB - v A = -҈
�ˆ - 
�ˆ - �
�ˆ =

�ˆ - �
ˆ

�


r

(Q ҈  =  
) = 

a 㦨


ᨨ
q
㦨

犨�
矄ᆠ
ᨨ
ᨨ犨
㦨
黨

졔


㦨
҈





q


㦨


맔
҈
㡰
맔
矄� 
㡰
㦨


맔
҈

맔

v = b + ҈㡰v Þ b =  - ҈ =

䰘




҈

vr

쟼 和

ᨨ
�鼀
ᨨ

鼀� �鼀盐鼀

ᆠᨨ矄
q
㦨
鼀 �鼀盐鼀
�



q
㦨

 ᆠ
矄犨�鼀鼀
矄 ᨨ盐
矄犨�鼀
盐
ᨨ


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PS-23

Laws of Motion

Laws of Motion

4

Topic-1 : Ist, IInd & IIIrd Laws of Motion ҈


㡰 ෌�

r F = k  v y �ˆ + vx ˆ 㡰 \ Fx = kv y �ˆ
Fy = kv x ˆ mdvx dv k = kv y Þ x = v y dt dt m 和�鼀�泐



dv y








dv y

dt

=

\ = �


꾀

㦨




㦨
鼀盐
 

h =

k v m x



䰘


dp d mv㡰 æ dm ö F= = = vç


�㡰 ÷ dt dt è dt ø dM t 㡰 = bv
t 㡰 ␔

��




��㡰 dt

盐矄
ᨨ
 
矄�
æ dm ö
 F = -v ç ÷ = -v bv 㡰 = -bv
᲼矄� �
�㡰
 
��㡰桌 dt è ø -bv Þ F = M t 㡰 a = -bv Þ a = M t 㡰





H

g졔kv 㡰
㦨
a
ᆠᆠ�ᨨ 㡰

u

r F = mkv
- mg



Q mg
 
mkv

ᆠ








 


mV mV
= 鼀 

 L 
nr 㡰 L 
nr V 鼀 㡰 ᨨ鼀
⼄盐矄�ᨨ
㝐矄矄
ᨨ
ᨨ泐

m
㦨
�
├� ᨨᆠ�泐

t
㦨


u
㦨

�ˆ -
ˆ㡰
鼀⏌矄 ᨨᆠ�泐

t
㦨
҈
矄
v
㦨
졔

ˆ
鼀⏌矄 ᨨᆠ

F
㦨
䁸

꾀ᆠᆠ�ᨨ

a
㦨


ᨨᨨ矄�
ᨨ
ᆠ
ᨨ 㡰

r r F a= = -᲼kv
+ g 桌 m dv ö æ dv Þ v× = -᲼kv
+ g 桌 çèQ a = v ÷ø dh dh h
v × dv
҈ Þò
= ò dh  éëkv
+ g ùû = - h Þ u
k u kv + g

҈ é ku + g ù  ê Þ ú= h
k ë g û 㡰 
ᆠᆠ�ᨨ

 㦨

㦨

�
졔
g
㡰  
�鼀

⼄盐�
ᨨ
�矄
ᨨ
�矄
 �

㦨

㡰
矄ᨨ

 - = ò 
ò ᲼ ᨨ
鼀


㦨

桌
 � + g




é 
ù ├
  = ├  Þ  = ├ [ ] ê ú
Þ
 = Þ =
ò ò
êë
úû

├  㡰 㦨 㡰 ᨨ鼀
㦨
㡰 鼀  
   = ò ᔼ �� �
ᨨ
矄�矄
ò  鼀
҈ \   µ ᔼ
 =  鼀 b㡰 和ᆠ
犨
 
矄盐ᨨ矄
ᨨ
鼀ᨨ�ᨨ
ᨨ
矄
�矄 
 꾀�
ᨨᆠ
 � ᆠ
泐
ᆠ
矄盐ᨨ


F=

黨





u

P =
f

盐犨
�ᆠ�ᨨ
�矄
ᨨ矄��
 
ᨨ犨 犨
�ᆠ�ᨨ
�矄

��
b㡰 ᨨ鼀
犨ᨨ 矄
矄ᆠᨨ 
犨  = = ├  ᔼ �� �
ᨨ
矄�矄
犨
�


b㡰 ᨨ鼀
 
犨ᨨ Ἐ矄
矄ᆠᨨ 
犨


v
㦨



㡰 

㦨
盐



� ᨨ



㦨
u
-
gh
㝐ᨨ鼀 盐鼀


㦨
鼀
= m u -
gh

v = x Þ ò v y dv y = ò v x dvx dvx v y v
y = v x
+ C v
y - v x
= ᆠᨨ 矄  k r r v ´ a = v x �ˆ + v y ˆ 㡰 ´ v y �ˆ + vx ˆ 㡰 m k



k







= vx kˆ - v y kˆ㡰 = vx - v y 㡰 kˆ = ᆠᨨ 矄  m m

㡰

æ g 
ö  -҈ ç ç � ÷÷ g� è ø

҈

҈



v - u
ˆ - 
�ˆ -
ˆ 㡰 -�ˆ +
ˆ
㦨

鼀⏌矄
㦨
҈
� t

ᨨᆠ

F
㦨
ma  -�ˆ +
ˆ 㡰 㦨
� ´ =  -�ˆ +
ˆ 㡰 N � ur 㡰 ෌�


鼀ᨨ鼀 盐鼀
p t 㡰 = A ᲼�$ ᆠᨨ矄 kt 㡰 - $ 矄� kt 㡰桌

ur ur d p = Ak ᲼-$� 矄� kt 㡰 - $ ᆠᨨ矄 kt 㡰桌 꾀 

ᨨᆠ

F = dt ur ur r r 

F
P =


盐

F
p = Fp ᆠᨨ矄 q \ ᆠᨨ矄 q
㦨





Þ
q 㦨
黨

 ᆠ

 �
犨
 
ᨨᆠ
鼀ᨨ鼀 盐鼀

q
㦨
黨
҈҈




�㡰
꾀
盐 
 
F

t
�
��矄
 
�鼀盐矄 �鼀
ᨨ
 
ᨨ泐



鼀� �盐
ᨨ
ᨨ
�鼀盐矄
ᨨ
ᨨᆠ
ᨨ
 
ᨨ泐
ᨨ鼀 t
㦨


Ҙ矄
ᨨ
t
㦨
҈

Ҙ矄 㦨

BCDFEB㡰 㦨

ᨨ

BCFEB
졔

CDFC

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EBD_8341 PS-24

Physics F �
㡰



C













A

B

҈



҈


҈



҈�






D

F

E








҈
҈
҈
҈
T

҈ ҈ -
-
㦨



+


㡰 ´
´ ҈
졔
´ ҈
´


´ ҈


ᆠ�ᨨ

㦨





҈
졔







㦨




�
矄 矄



ᨨ犨
鼀ᨨ�ᨨ
ᨨ
ᨨᆠ├ �矄
盐
ᨨ
 
 盐矄
�矄矄

 ᨨ犨

ᆠ├犨
ᨨ
盐
ᨨ
矄盐ᨨ盐 � �
� 盐矄 � �
ᆠ├犨矄
ᨨᆠ├ 
 盐矄
�矄矄
 ᨨ犨

ᆠ├犨矄
�矄
ᆠ�ᨨ
 
ᨨᆠ├ 鼀ᨨ� �
ᨨ犨
�矄
ᆠ�ᨨ
쟼 盐矄
�矄矄

�矄
  ᨨ鼀 ᨨ
├矄
ᆠ
�
  꾀ᆠ�ᨨ

矄 ᆠ
ᨨ
�
矄
犨
b

㡰

쟼
�矄

ᆠᆠ
鼀
 
 ᆠ

ᆠ

ᨨ
 
� �
鼀
쟼
�矄
ᨨ� �
ᨨ盐 
 
矄盐
 
�矄
ᨨ� �
ᨨ盐
�矄 ᨨ犨
 �矄
b㡰 ᔼ
�矄
矄�
ᨨ
盐

泐
ᨨᆠ
 
ᨨ
盐矄
�
ᨨ
 
�ᨨ盐 

�矄
�矄
ᆠ盐矄
 
ᨨ鼀
ᆠ�ᨨ
�
 
ᆠ矄 ᨨ
盐� �
�矄
矄矄
矄
ᆠᨨ鼀
泐
盐矄 � �


㦨
m 㡰
 ᨨ  
�ᆠ�ᨨ 
ᨨᆠ
�矄
矄鼀
�
ᆠ矄
ᨨ
盐� �



鼀� �盐
ᨨ
�ᆠ�ᨨ 
ᨨᆠ
 矄
ᨨ
 
盐
ᨨ  
犨ᨨ
矄盐 ᆠ矄
�
ᆠᨨ ᆠ

盐
�矄
ᨨ
 
ᆠ盐ᆠ   �ᨨ
ᨨ
矄鼀 㜠҈
b㡰 ᔴᨨ
ᆠ

盐
ᨨ盐
犨� ᨨ盐
�矄ᨨ�� �
 
�矄 矄 ᨨ鼀
 

ᆠ盐矄
ᨨ
� �
犨
ᨨ
� �
�矄
 
犨ᨨ Ἐ矄
�矄
犨
ᨨ
鼀ᨨ�ᨨ
ᨨ
泐
ᆠ�ᨨ
 
�矄

⼄盐
 
ᨨᨨ矄�
ᆠ�ᨨ


�矄
�矄
犨ᨨ Ἐ矄
 �
犨
ᨨ
鼀ᨨ�ᨨ
Topic-2 : Motion of Connected Bodies, Pulley & Equilibrium of Forces

҈


�





= ҈
鼀矄 -


+ ҈
+


 矄�ᨨ

 
鼀�
ᨨ�  㝐矄矄
ᨨ
ᨨ ö æ

= ç 㝐
+ ÷
è ø 㦨
҈

졔

㡰
맔
҈

㦨
҈黨

 =








\

㦨
҈




҈




T a T ma

T

a

T矄� q T矄� q

q Tᆠᨨ矄q

x

x

m







�㡰

矄ᨨ
F
㦨

T
矄� q










Þ T
㦨
盐� �
 �矄
盐
ᨨ

�
⼄

�㡰

F
矄� q

æ F ö ᆠᨨ矄 q ÷ è
矄� q ø m

a
㦨
ç

䰘


졠
��� �
�㡰
泐
��㡰
犨
�





b㡰 ᨨ鼀
��盐

ᆠᆠ�ᨨ

ᨨ
鼀矄矄
m
�矄
盐
ᨨ
 
ᨨᆠ
T ᆠᨨ矄 q T
ᆠᨨ矄 q 㦨
ma T ᆠᨨ矄 q Þ a
㦨
m




��㡰

T ᆠᨨ矄
�° ҈

=


Þ F
㦨
҈


 T 矄�
�° F 㡰 
ᨨᆠ
ᨨ
 
ᨨ
㦨
ᨨᆠ
ᨨ
ᨨ 졔
�ᆠ�ᨨ 
ᨨᆠ Þ F
㦨
҈

m
졔
M㡰g
졔
f 犨 

m
㦨
鼀矄矄
ᨨ
矄ᨨ

M
㦨
鼀矄矄
ᨨ
ᨨ
f
㦨
�ᆠ�ᨨ 
ᨨᆠ Þ F
㦨
҈

맔



졔
黨

㡰
맔
黨


졔





Þ F
㦨






N Þ P
㦨
FV
㦨






맔

㦨






W 㡰 꾀
⼄盐���盐鼀
ᨨ
� 鼀� ҈

= 

�
㦨
ᨨ
�

F

\

㡰 ᨨ鼀
 

ᨨ泐
��鼀








T
ᆠᨨ矄
�
㦨
҈


N


�㡰 T
矄�
�
㦨
F

b㡰 ᨨ鼀
��盐
꾀ᆠᆠ�ᨨ
a
㦨
Ra …�㡰 a  
mg

T = ma …��㡰 ᨨ鼀
⼄盐�ᨨ
�㡰
 
��㡰 æ aö

T
맔
R
㦨
mR
a
㦨
mR
ç ÷ è Rø ᨨ T
㦨
ma

Þ mg

ma
㦨
ma 鼀  g Þ a= 鼀�
㡰 꾀ᆠᆠ�ᨨ
ᨨ盐ᆠ
�
盐犨
�ᆠ�ᨨ

= 㝐҈ + 㝐
+ 㝐矄矄
ᨨ
鼀
ᨨ





é F F x a
- x
ù êQ  q ú
=
m  q =
m x ê ú a
- x
ë û ᆠ㡰
 ᨨ
 
矄� �
�矄
ᆠ盐
 
 矄�ᨨ
T
矄
ᨨ
ᨨ ᨨ
 
鼀矄矄矄

m
 
m
  ᨨ
T
㦨
mg ␔ 
 
矄� �
�矄
ᆠ盐

 
鼀矄矄
m
�矄

泐
� �
ᨨ泐
  �矄
ᆠᆠ�ᨨ
㦨
ᆠᆠ�ᨨ
盐
ᨨ
��泐
㦨
g
ᨨ
鼀矄矄

m

盐矄
 
 
矄� �
�矄
ᆠ盐
鼀� T
鼀� 矄
mg
ᆠ盐矄
ᨨ
 
  矄�ᨨ

ᨨ
矄� �

鼀 鼀 g \ mg


mg
㦨

m
맔
a Þ

= a




鼀�
鼀� 㡰 꾀
⼄盐���盐鼀
T
㦨
Mg



ᨨ
盐泐 T

mg

T T

Mg

T㦨㝐�














F

F҈
㦨
m
졔
M㡰
g


矄盐 
ᨨᆠ
ᨨ
盐泐

F҈㦨m졔M㡰g



F
㦨
F҈
+ T

㦨
᲼ m + M 㡰 + M 桌 g

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PS-25

Laws of Motion 黨


ᆠ㡰 
 矄�ᨨ
�
ᨨ
矄� �矄
犨�

矄鼀
盐
ᨨ 矄泐鼀鼀泐


T ᆠᨨ矄
q T ᆠᨨ矄
q

ᨨ鼀
��盐



㦨
鼀�

q q

T T T 矄�
q A

T T 矄�
q

B

T C

Ö
mg mg

mg

ᨨ
⼄盐���盐鼀


mg = T ᆠᨨ矄 q + T ᆠᨨ矄 q =
T ᆠᨨ矄 q \

҈




mg =
mg 㡰 ᆠᨨ矄 q ҈ Þ q =
�° \ ᆠᨨ矄 q =
矄



ᨨ鼀


矄 ᨨ犨
�
ᆠ矄
㡰
ᨨ
鼀矄矄
m T

mg
㦨 ma



�㡰 ᨨ
鼀矄矄

m ᔴ矄
㡰

a


mg

T
㦨

ma



��㡰 ᨨ鼀
�㡰
 
��㡰 a
㦨
g⏌



���㡰 ᔴ矄
b㡰 T

mg
㦨
maἘ Þ
mg

mg
㦨
maἘ ᲼Q
T
㦨

mg桌 \ aἘ
㦨
g


�㡰  ᆠ

ᨨ鼀
⼄
���㡰


�㡰
a


aἘ ҈҈


㡰 꾀ᆠᆠᨨ� �
ᨨ ᆠᨨ 矄� 
�ᨨ

ᨨ鼀
��盐



m

T a

mg

҈


a T

T

U Q

҈
鼀

T=
mg

m


a' mg

q


q


�㡰 x

졔
z

㦨
l �  �� �









O M ⼄盐�ᨨ
�㡰
犨


ᨨ
t dz dl =
l

졔

z

Þ zvM
㦨 lU dt dt zö æ U U l = Þ vM
㦨
U = çèQ ᆠᨨ矄 q = ÷ø l z z ⏌ l ᆠᨨ矄 q l
㦨
㝐矄矄
ᨨ
盐 �
 �
ᨨ
犨�





├�⏌鼀
P U

mg

a҈ kx

T

x
҈ æ ҈
㝐� ö
㝐� = ç ÷=

è ├ ø ├
� ├
㝐�
� ´ = ᨨ鼀
⼄

�㡰
҈
㦨
䰘 ҈
㝐 x 䰘 x

�ᆠ
�矄

鼀
ᨨ矄��ᨨ
 
�矄
ᨨᆠ�泐
㦨
꾀 꾀w 꾀

㦨


㝐� ├ x
├ = 㦨

҈
㝐 ├ ҈
㝐 ҈

b㡰 
ᨨ鼀
��盐

AN
㦨
x
㦨
ᆠᨨ 矄 
矄
盐泐
A
  dz B

� 㡰

NO
㦨
z


ᨨᆠ�泐
ᨨ
鼀矄矄
m
㦨


꾀矄ᨨ
dt dl A x N B =U 
OA
㦨 l
 
dt Z ᨨ鼀
DANO l ᨨ x
㦨


F㦨
mg

T

l

҈
鼀


T


T 
+ 
Þ


졔

㦨

҈ T T a
Þ 

졔

㦨
҈
졔
҈ a Mg Þ ҈


㦨



҈
M Þ 졠�ᨨ
㡰
�矄
ᆠᨨᆠ 
�x

 
  矄�ᨨ
ᨨ
 
矄� �


ᆠ�
� 矄 


├
㦨

㝐҈
㝐�


㦨

㝐  㝐�


㦨
㝐
҈ 졠
矄ᨨ� �
犨
�
├



㝐
�
├x ҈
㦨



�㡰 䰘 ҈
㝐
㝐� ├ ├
\ w
㦨
\ w
㦨
҈
㝐 ҈
㝐
㝐�
├x +


��㡰  
㦨
䰘 䰘 ᨨ
҈
㦨


㝐 �鼀盐鼀
  矄�ᨨ
ᨨ
矄� �㡰
犨

ᨨ鼀
�㡰



� ├x
㦨

䰘 ҈
㝐 ├x
㝐�

\
�
㦨
\ x
㦨



㝐 ├ 㝐   ҈
㝐�
\ x

㦨

x
㦨

㝐 ├

m҈

m҈g 졔 l l
�

҈
㦨


鼀
҈ 㦨
҈
黨
├� 鼀

㦨


黨
├�

�㡰
 矄�ᨨ
T

鼀�ᨨ� 
ᨨ
ᨨ犨
犨�

l
㦨
 㜠 �
㦨


�
鼀 \ T

鼀҈
졔
ll㡰�
㦨
鼀҈
졔
ll㡰a T
㦨
鼀҈
졔
ll㡰a
졔
g㡰 㦨
᲼҈
黨
졔




맔


�㡰桌



졔
黨

㡰
㦨


맔
҈

㦨




��㡰  矄�ᨨ
T Ἐ

鼀�㜠ᨨ� 
ᨨ
盐
犨�

\
T Ἐ
㦨
᲼鼀҈
졔
t
맔

l㡰
졔
m
桌a
졔
᲼m
g
졔
l
맔

l�
졔
m҈g桌 ᨨ
T Ἐ
㦨
᲼m҈
졔
l
맔

l㡰
졔
m
桌
a
졔
g㡰 㦨
᲼҈
黨
졔




맔
҈㡰
졔


黨桌᲼



졔
黨

桌
㦨
�
맔
҈

㦨
�



Topic-3 : Friction ҈


㡰 ᨨ
 ᆠ� �

mg 矄� q = f = mmg ᆠᨨ矄 q

Þ  q = m = Rᆠᨨ矄q q h

 =

䰘�
f = mgᆠᨨ矄q

R h q mg矄� q

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EBD_8341 PS-26

Physics

Rᆠᨨ矄q

3 + 50
㦨

졔



R

q

�



q

h

�







æ
ö R h = R - R ᆠᨨ矄 q = R - R ç ÷ = è �ø � R \ h = =


鼀
᲼Q
�盐矄

R
㦨
҈鼀桌 � ᆠ㡰 꾀


N
㦨
��
졔



矄� 
҈ 㦨
�

졔



맔

㦨




f 꾀ᆠᆠ�ᨨ
a҈
㦨


N


ᆠᨨ矄

�
�

F- f

ᆠᨨ矄 
° - ҘN = m �




矄�



é ù  -


´

ú ê

´
ú
㦨
҈



鼀⏌矄


㦨
ê f � ê ú êë úû








矄�



N




ᆠᨨ矄



䰘





\

㦨
҈



㦨


 b㡰 ෌�


鼀҈
㦨
�├�錐
鼀

㦨
҈
├�錐
m
㦨


҈� 
ᨨ
鼀҈

鼀҈�


㦨
鼀҈ Þ

�



㦨
�
맔
  鼀  






҈�
鼀
졔
҈
㡰�

鼀
㦨
҈

졔
鼀㡰 m鼀졔鼀
㡰� 鼀졔鼀
㡰� ᨨ
矄
a
㦨

ᨨ

�

㦨


҈�
鼀
졔
҈
㡰
҈
 鼀 + ҈
㡰 Þ �=





鼀҈ 鼀҈�
㦨
�


҈

= 鼀 + ҈

\

鼀
㦨


├�錐 ᆠᨨ矄
ᨨ
ᨨ�ᨨ
㡰  b㡰 
ᆠᨨ �ᆠ� 矄
ᨨ
├� �ᆠ
�ᆠ�ᨨ
犨
 
ᨨᆠ  
 
� ᆠ�  ҈ ҈ö æ m =  q ç҈ -
÷ Þ m = ҈ -






Q

q
㦨

� 㡰 è ø

b㡰 
m

 
鼀� �鼀盐鼀
ᆠᨨ �ᆠ� 
ᨨ
�ᆠ�ᨨ

m鼀�ᆠᨨ矄q

�
�




N
㦨
��




矄�



҈

㦨






㦨
�





맔





m�矄� q

F - f é

ᆠᨨ矄 
° -


´

ù =ê a

㦨
ú
㦨
҈



鼀⏌矄
m � ë û
ᨨ犨
a


a҈
㦨
҈




҈



㦨




鼀⏌矄 b㡰 ├� �
꾀
졔
㡰
矄
矄泐矄鼀 F

mM
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m㡰g
㦨
M
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m㡰a F  m  M + m㡰 g Þ
a
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�

꾀
⼄盐���盐鼀

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ᨨ矄
ᨨ
鼀ᨨ
矄ᨨ
鼀�
矄� q
㦨
m鼀�
ᆠᨨ矄q

\ μ 鼀� =  θ





F - 


㡰
´҈
æ F -
ö =ç ÷


�㡰
è
ø 盐

a鼀
㦨
m�
㦨




맔
҈

㦨



㡰







f҈



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꾀矄矄盐鼀� �
ᨨ
 
ᨨᆠ├矄

矄�ᨨ 泐 
㦨
f҈
㦨



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㦨
҈


졔



㦨
҈



F -
\ =

Þ
㦨
҈

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├�



꾀

a=

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µ
鼀�ᆠᨨ矄q

㡰





黨


ᔴᨨ 矄�� �
 
犨ᨨ
ᨨᆠ├矄 矄
ᨨ 
矄泐矄鼀
 
盐
ᨨ ⼄盐���盐鼀
f
㦨
҈

 㡰 f
㦨
ҘM
졔
m㡰
g

f Ҙ M + m㡰 g = = Ҙg M +m  M + m㡰 㦨



�
맔
҈

㦨


�
鼀矄




҈



a=


졔
鼀�矄� �q









f


f҈


�



鼀�
矄�

�

㦨


҈




㦨
�



᲼Q鼀 = ҈
├�
� = 黨

鼀 矄 -
桌 m鼀�
ᆠᨨ矄q
㦨




맔

鼀�
맔


҈



졔
鼀�
矄� q 㦨

졔
m鼀�
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=


´ �



ᔼ �� 鼀ᨨ鼀 盐鼀


�V V
= = M + m㡰 ҈


�




m
㦨
�
� n

M
㦨
҈

├� 


v


u

㦨

as


u

㦨

as u

㦨

as


æ


�v ö ç ÷ =
´

� ´
è ҈


� ø

WWW.JEEBOOKS.IN

PS-27

Laws of Motion 和ᨨ� �
犨
�

v
=

҈
졠ᆠ
� �
ᨨ鼀
�� 
H


V = ҈


҈



mg

x



Þ

҈





mg
ᆠᨨ矄q

P


mg
矄� q

q

mg
ᆠᨨ矄q

f mg

m q

y

x
㦨
᠀҈

҈



x ҈ = m

㡰 ᨨᆠ
ᨨ
�ᆠ�ᨨ

ᨨ� 

҈ �ᆠ�ᨨ = 㝐 矄� q  ҈ 㦨
㝐 矄� 黨
° ᲼

q
㦨
黨
桌  㝐 㦨
 ᆠ㡰 㝐� �鼀盐鼀
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A 㦨
�ᆠ�ᨨ 
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犨
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矄盐 ᆠ矄 㦨
҈

N 꾀

N

q

q

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y =

҈҈


q

mg
矄� q

dy x
=  q
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⼄盐矄�ᨨ 㡰
dx Q ᔴᨨ �ᆠ� 
ᨨ
�ᆠ�ᨨ
m
㦨


�

� =

f

P


gH



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=
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H
㦨



鼀
㦨





├鼀 㡰 꾀
�鼀�� �
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m
㦨

q

\

N



├� �
├�


和鼀ᨨᨨ
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␔ 
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㦨
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f
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盐犨矄
ᨨ �
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 
 ᆠ
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ᨨ �
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� ᆠ� 



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㦨
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q Þ f = P  mg 矄� q ᨨ犨
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f
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 �
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㦨
 �
ᨨ
ᨨ矄
졠 ҈�
ᆠ q

�



 -҈ m
=  -҈  =

°
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㝐

鼀

ᆠ鼀



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m= 

犨

q

� ᆠ鼀
PM ҈
⏌
= = = OM ҈� ⏌
䰘
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q






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ᨨ
鼀ᨨ� �
ᨨ犨
ᨨ
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F
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F \ F
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ᨨ ᲼F
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D POM

 q =

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맔
a鼀










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矄
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´  ´ ҈
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㦨

㦨
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° - m 矄�

° ҈ ҈  ҈


҈



m =

 = 
�° ҈�


\

㦨

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ᨨ
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q
>
m

矄ᨨ
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ᨨᆠ
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� 矄ᆠ

mg
 
N


犨ᨨ
ᆠᨨ鼀ᨨ  矄
ᨨ
鼀�
 mg
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a
 ᆠ矄
N
m㦨҈⏌ a mg
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a
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 ᆠ f N 泐
 


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ᨨᆠ
\ N
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mg
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a mgᆠᨨ矄a a f
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mg
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a











mg mg矄� a 盐
f
㦨
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mg
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a ҈ \ Ҙ
mg
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a
㦨
mg
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Þ ᆠᨨ
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㦨
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a
㦨
 Ҙ m 㦨


� b㡰 �鼀�� �
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ᨨᆠ

fl
㦨
Ҙ矄N
㦨


�
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�
N
f � ᨨ
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N


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҈
맔
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㦨


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㦨
␔

WWW.JEEBOOKS.IN

EBD_8341 PS-28






Physics

㡰 
ᨨᆠ├
�矄

矄
ᨨ
⼄盐���盐鼀

�ᆠ�ᨨ 
ᨨᆠ


N

和盐矄�盐� �

g
㦨
҈

鼀⏌矄
㡰 Þ � + � m =
� - � m㡰 Þ � + � m =

-

m Þ
� m = ҈�

f

f
㦨
鼀�
矄�
q
㦨
鼀�
矄�



=
´ 黨

´
҈


mg
矄�



҈ = 黨




mg

 

=



= � ҈




I 和ᨨ

= ҈


҈


Þm=

mg
ᆠᨨ矄






�㡰 ᨨᆠ├
鼀ᨨ� �
盐犨 矄�ᨨ 泐









ᨨᆠ├


盐矄


鼀� 矄






f
㦨
mN F҈ q mg
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q mg
矄�
q f mg


�
F
mg
矄�
q


�










ᨨ
盐犨
鼀ᨨ� �
ᨨ
ᨨᆠ├

盐矄 � �
ᨨᆠ
F҈㦨
mg
矄�
q
졔 f
\ F҈
㦨
mg
矄�
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m
mg
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㦨
mg
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矄�� �
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ᨨ ᨨᆠ├
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矄�ᨨ 泐
F

㦨
mg
矄�
q

m
㦨
mg
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m
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\ 矄�
q
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q
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m
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m
㦨
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m㡰 ᲼Q
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m
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㦨
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m
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맔


�
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├�
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�
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f

 ᨨ

�ᆠ�ᨨ 
ᨨᆠ


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㦨
m
맔
a
㦨
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㦨
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N
鼀� 

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鼀ᨨ� �
盐

� ᆠ� 
 


a҈ = g 矄� q + mg ᆠᨨ矄 q Þ a҈ = g 矄� 
° + mg ᆠᨨ矄
°

g mg  +


矄� �
v - u
=
a s 㡰 =



�㡰

Þ v

-

=
a҈  s㡰

Þ

v



v

a҈

Q q 㦨

ᨨ㡰

Q 盐
㦨

㡰




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= g 矄� q - mg ᆠᨨ矄 q
Þ a
= g 矄� 
° - mg ᆠᨨ矄 
°

g m  g


���㡰

矄� �
��
v
- u
=
as
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鼀ᨨ�ᨨ

Þ a
=




v
æv ö Þ ç
÷ =
a
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a
è
ø

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ᨨ
P
矄
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q

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v

和ᨨ

tP


tQ

䰘
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-
a҈  s 㡰 =


Þs=






矄


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� 턼  
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cos q ␔ 
q
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q
>

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矄� q
>
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Q
␔ 
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>
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P
㡰 꾀ᆠᆠᨨ� �
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
A
 
ᨨ�ᨨ 
矄矄
ᨨ Q q ᨨ
 
鼀矄矄矄
�矄
 
矄鼀
A B v 꾀矄
ᨨ
ᨨᆠ
�矄
ᆠ� �
�
  Nᆠᨨ矄q N ᨨ�ᨨ 
�ᆠ�ᨨ
  q N矄� q ᨨᆠ�泐
ᨨ
Q
鼀� 矄
  P 矄鼀
�
ᨨ�ᨨ 
O mg




�㡰

쟼⼄盐� �
⼄盐�ᨨ
��㡰
 
�㡰

æg m ö g mg  v

v
=
ç =
Þ a҈ =
a
Þ + ÷ ç



÷ø a҈
a
è

ᨨᆠ├
犨�
ᨨ

矄�
ᨨ
犨�


矄
� m҈
졔
m
㡰g 矄�
q
£
mm

g ᆠᨨ矄
q

 q £

mm
g ( m҈ + m
) g

mm
Þ  q £ m҈ + m


f
㦨
mN


N
q 
m҈ 졔
m
㡰
g ᆠᨨ矄
q


m҈ 졔
m
㡰
矄�
q



 ´
҈ m҈
졔 m

㡰g Þ  q £ £ q ҈+
� Þ 
q
£




�



q £
҈҈
� �



ᔼ
 
 �
q


҈҈
�
 
�ᆠ�ᨨ 
ᨨᆠ
�矄
矄矄
 

mN


㦨
mm
� 㦨



맔


맔
g
㦨




�  
�矄
⼄盐
ᨨ
m҈
졔
m
㡰g
矄�
q ᨨᆠ├矄
犨�
ᨨ
矄�
ᨨ
 
� ᆠ� 
 
 
�ᆠ�ᨨ
�矄 矄�ᆠ
꾀
q
>
҈҈
�
 
ᨨ�矄
矄
鼀ᨨ� �
ᨨ
 
� ᆠ� 
   
�ᆠ�ᨨ
�矄
├� �ᆠ
 
⼄盐
ᨨ
mm
g
ᆠᨨ矄
q


ᨨ鼀
ᆠ�ᨨ

N҈
㦨
mma
矄�
q
 
N

㦨
mg 꾀泐� �
矄盐ᨨ
ᨨᆠ
ma
 
矄ᨨ� �
�
F 
㦨
mar ma
ᆠᨨ矄 q 
f҈
졔
f
㡰
㦨
mar ma
ᆠᨨ矄 q 
ҘN҈
ҘN

㦨
mar

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PS-29

Laws of Motion ma
ᆠᨨ矄 q

Ҙma
矄� q

Ҙ
mg
㦨
mar Þ ar
㦨
a
ᆠᨨ矄 q

Ҙa
矄� q 
Ҙg




- ´
� ´ - ´ ҈
= ҈
鼀 ⏌ 矄
� � � � 꾀ᆠᆠ�ᨨ
ᨨ
ᨨᆠ├
ᨨ犨
 
  mN = f K mg 矄� q - m k mg ᆠᨨ矄 q a
㦨
m

矄
ᆠᨨ ��ᨨ 矄
犨�

矄�矄 �
犨 
T


҈�
N
矄泐
T
㦨
҈
 
f҈
㦨
f

㦨
҈

N


 ᨨ
 
ᆠᨨ ��ᨨ
f҈
㦨

f

犨�
ᨨ

矄�矄 �


盐矄
m҈
 
m

ᨨ
ᆠ Ἐ
鼀�
矄�ᨨ 泐
ᔼ
 
矄ᆠᨨ 
ᆠ矄

犨 
m҈
 
m

ᨨ
鼀ᨨ
f

㦨
f
㡰
鼀

㦨
҈�
N \
f҈
㦨

f

㦨


N 和� ᆠ
f҈


f҈㡰鼀

 
�矄
ᨨ
�
鼀ᨨ�ᨨ
犨
m҈  
M

�.e.


 
鼀矄矄矄
鼀ᨨ
犨�
矄鼀
ᆠᆠ�ᨨ
矄泐
ἘaἘ

ᨨ泐
��鼀矄
 
⼄盐�ᨨ 矄
ᨨ
鼀ᨨ�ᨨ

矄
ᨨᨨ犨矄


㦨

� ´

\

aA
㦨
g
矄� q 
Ҙk,A
g
ᆠᨨ矄 q



q

mg矄� q

mgᆠᨨ矄q

㦨
�
矄�

�

Ҙ
ᆠᨨ矄

�

N = mg ᆠᨨ矄 q

mg

æ ҈ ö æ ҈ ö = ҈
ç - 


㡰҈
㡰 ç =

è
÷ø è
÷ø 꾀  aB
㦨
g
矄� q 
Ҙk,B
g
ᆠᨨ矄 q 㦨
�
矄�

�
m├
�
ᆠᨨ矄

� æ ҈ ö æ ҈ ö = 
�
鼀⏌矄
= ҈
ç - 

㡰҈
㡰 ç ÷ è
ø è
÷ø 
aAB
�矄
�
ᆠᆠ�ᨨ
ᨨ
A
犨



B



aAB
㦨
aA
 aB


�
�矄 ᆠ
犨
A
 
B

L
҈

L
㦨
a AB t


㡰
L
L = = ᨨ t

㦨
a AB a A - aB

- 
�


(

) (

)

Þ
t

㦨


ᨨ
t
㦨

s
�矄 ᆠ
鼀ᨨ
泐
A
盐� �
 
�鼀
�矄
��
泐 ҈ ҈
SA
㦨
a
t

㦨
´

�
´
=

鼀



҈


和�鼀�泐
ᨨ
B 㦨
䰘
鼀
 ᆠ
ᨨ
 
ᨨᆠ├矄
犨�
ᆠᨨ鼀
�
� 
 
A
矄
 
�矄 ᆠ


鼀
ᨨ犨
 
  㡰 
ᨨ泐
��鼀
ᨨ
鼀矄矄
M T T

T

m mg



�㡰 mg ᆠᨨ矄 q + m 矄� q �  �� �
 
ᨨ
⼄盐�ᨨ
犨


q

犨
� dF m mg = ᲼- 矄� q + m ᆠᨨ矄 q桌 =
d q ᆠᨨ矄 q + m 矄� q 㡰


f҈

F

T

㡰 
鼀 �鼀盐鼀
盐
ᨨ
ᨨᆠ
ᨨ
�ᆠ�ᨨ
犨
鼀҈  
㝐 f҈㡰鼀
㦨


㡰


㡰
҈
㡰
㦨



N


�㡰


鼀 �鼀盐鼀
盐
ᨨ
ᨨᆠ
ᨨ
�ᆠ�ᨨ
犨
鼀
 
㝐 f
㡰鼀
㦨


㡰
�㡰
҈
㡰
㦨
҈�
N


��㡰 ᨨᆠ矄
ᨨ
m҈
 
m

�
ᨨ�ᨨ 
�ᆠ�ᨨ

矄
ᨨᨨ犨矄


m҈

f҈

f


m


F
㦨


\ q 㦨
 ҈Ҙ

�矄
�矄
 
 �
ᨨ
鼀� �鼀盐鼀
ᨨᆠ


ᨨ
� 
 
鼀� �鼀盐鼀
ᨨᆠ
矄盐矄�盐� �
 矄
盐矄
�

⼄盐�ᨨ
�㡰 Ҙ ҈ 矄� q 㦨




ᆠᨨ矄 q 㦨






Ҙ +҈ Ҙ +҈

N

T

T

M

F f҈㦨
 f
㦨҈� ᨨ
鼀矄矄

m ҈





T
㦨



a



���㡰 ᨨ
鼀矄矄

m



T

҈�
㦨
�
a



�㡰 ᨨ 鼀矄矄
㝐


F




㦨
�

a



㡰 꾀� �
⼄

���㡰


�㡰

犨
�

ᆠᆠ�ᨨ
a
㦨
鼀 ⏌ 矄
� �´ ᨨ鼀
⼄

�㡰


҈�
㦨

Þ


㦨
҈

 �  ᨨ鼀
⼄

㡰




㦨
�
´ Þ


㦨



 � 
ᨨᆠ


�
ᨨ
鼀ᨨ
 
ᨨ泐


 �
q
ᨨ  
ᨨ�ᨨ 
F矄� q


ᨨ泐
犨�
鼀ᨨ
犨 



F N F
ᆠᨨ矄 q
㦨
Ҙ
N q ᨨ鼀
��盐

ᨨ鼀
ᆠ�ᨨ ҘN Fᆠᨨ矄q N
㦨
mg

F
矄� q
ᆠᨨ矄q

㦨
Ҙ鼀�


矄� q㡰 Þ

N҈ Mg

m


m҈

a

F
㦨


Ҙ mg ҈ Ҙ
+ ҈

T

+




ᨨ 泐
犨ᨨ
ᨨ矄矄����矄
ᔴ矄
ᔼ
쟼� 
ᨨ
m҈
 
m

犨�
鼀�
矄�ᨨ 泐
犨


�ᨨ盐 㡰 ᔴ矄
ᔼᔼ
ᨨ
m҈
 
m

犨�
鼀ᨨ
犨



�ᨨ盐 㡰

�矄
ᆠ矄 �矄
ᨨ矄矄�
犨 

ᨨ T


f҈㡰鼀


ᨨ
T





N

  T


f
㡰鼀


ᨨ
T


҈�
N

m

+1





f҈㦨


a T

Þ F
㦨






Öm


黨


m Ҙ
+ ҈

Ҙ mg  Ҙ + ҈㡰


Ҙ +҈

=

´m

q

1

Ҙ mg

Ҙ
+ ҈

Þ F
㦨
mg
矄� q 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

鼀矄矄
M҈
鼀ᨨ矄
ᨨ犨 犨矄
犨� 
盐 � ᨨ鼀
ᨨᆠ�泐
�




ᆠᆠ�ᨨ
ᨨ
 
矄泐矄鼀
�矄 ᨨ

졠

盐� �
ᨨᆠ
ᨨ
 
矄泐矄鼀
�矄
ᨨ


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EBD_8341 PS-30

Physics

ᨨ
⼄盐���盐鼀
㡰 M҈g
㦨
M
g
矄�
䰘
졔
mM
g
ᆠᨨ矄䰘
졔
mMg ᨨ M҈
㦨
M

矄�
䰘
졔
mM
ᆠᨨ矄
䰘
졔
mM

F

æ ö æ
ö = 
㡰 ç ÷ + 


�㡰
㡰 ç ÷ + 


�㡰
㡰 =


├� è �ø è �ø 㡰 和� ᆠ

M
�矄
鼀ᨨ� �
犨�
盐 � ᨨ鼀
ᨨᆠ�泐


㦨
m҈m
g
㦨



�㡰
맔


맔
黨


㦨
黨


 

T

 
 矄�ᨨ
�
 
ᨨ

ᨨ� 
P
 
a

  ᆠᆠ�ᨨ
ᨨ盐ᆠ
�
 
ᨨ
F 㝐矄矄

盐 �
 �
ᨨ
 
ᨨ
�矄
m = L A P B F Ll l L ᨨ
 

AP錐 T
㦨
mL

㡰a





�㡰 ᨨ
 

錐 F

T
㦨
m⏌a





��㡰 é T ù F - T = ml ê ᲼矄� �
⼄

�㡰桌 ú ë m L - l 㡰 û Tl é l ù ÞT ê + l ú = F錐 L-l L l ë û é L ù æ L -lö Tê = F ÞT = Fç è L ÷ø ë L - l úû l ᨨ

T = F é҈ - ù ê Lú ë û 

꾀矄
ᆠ盐矄
ᨨ
ᨨ
矄�
ᨨ犨

 
 矄
 ᆠ
 泐
 矄 a 矄鼀
ᆠᆠ�ᨨ
鼀҈ 矄盐ᨨ
ᨨᆠ㡰 ᔴᨨ 矄�
 

ᨨ
  a ᆠ盐矄
ᨨ �
� ᆠ�  鼀҈ �


졔
鼀҈
ᆠᨨ矄a
㦨
m҈g
矄�
a
®



�㡰


졔
鼀
�
矄� b
㦨
m
g
ᆠᨨ矄
b
®
��㡰 鼀

ᆠᨨ矄
b 鼀
쟼⼄
�㡰

쟼⼄
��㡰 b 鼀
鼀҈
ᆠᨨ矄a
졔
鼀

ᆠᨨ矄b㡰f b 㦨
鼀҈矄� a
졔
鼀
矄� b㡰� Þ

�


T

f
= m m A + mB 㡰 g

f҈ = m mA + mB + mC 㡰 g 꾀矄
C
�矄
鼀ᨨ� �
犨�
ᆠᨨ 矄 
矄 F
㦨
f҈
졔
f

졔
T






�㡰



ᨨ
鼀矄矄
B
�矄 mm A g = f m m A + mB 㡰 g = f


B

T

꾀矄
B
�矄
鼀ᨨ� �
犨�
ᆠᨨ 矄 
矄
f

졔 f
㦨
T







��㡰 和盐ᆠ� �
⼄

��㡰
ᨨ鼀
�㡰 F

f

졔 f㡰
㦨
f҈
졔
f

졔
T

T
㦨
f҈
졔
f
Þ F 㦨
f҈
졔

f

졔
f
㦨
Ҙ
mA
졔
mB
졔 mC㡰
g
졔
Ҙ
mA
졔
mB㡰
g
졔
Ҙ
mA
g F
㦨
Ҙ


mA
졔

mB
졔
mC㡰
g ෌�

鼀꾀
㦨
├�

鼀
㦨

├�

㝐
㦨
�
├�

 
m
㦨



�㡰
㦨



�
᲼

맔

졔

맔


졔
�桌
맔
黨


㦨
䰘҈

�
N  ᆠ

ᨨᆠ
ᆠ矄矄泐
ᨨ
�


㦨
䰘҈

�
N

Topic-4 : Circular Motion & Banking of Road ҈


F -T =

鼀҈矄� a + m
矄� b㡰 f = g 鼀҈ ᆠᨨ矄 a + 鼀
ᆠᨨ矄 b㡰

C

mv
= mrw
r w
㦨

pn
㦨

p
맔

�
㦨
䰘p
⏌矄ᆠ �盐矄

r
㦨
҈

�
ᆠ鼀
㦨
҈

�
맔
҈


鼀 ᔴᨨ �ᆠ� 
ᨨ
�ᆠ�ᨨ

Ҙ
㦨
䁸 m rw 㡰
mmg =
Q

v
㦨
rw㡰 r Ҙmg
㦨
mrw


㡰 矄� �

mmg =



ö æ ҈

� ´ ҈
-
´ ç 䰘 ´ ÷ è rw 䰘ø = Þ m= g ҈


O

�矄ᆠ









鼀҈�
矄�
a

鼀
�
矄�
b

鼀
�

é m 矄� a + m
矄� b ù ᨨ鼀
⼄
�㡰
= 鼀҈ g 矄� a㡰 - 鼀҈ᆠᨨ矄 a㡰 ê ҈ úg 
ë m҈ ᆠᨨ矄 a + m
ᆠᨨ矄 b û

é m҈
ᆠᨨ矄 a 矄� a + m҈m
ᆠᨨ矄 b 矄� a - m҈
ù ê ú ᆠᨨ矄 a 矄� a - m҈m
ᆠᨨ矄 a 矄� b ú ᨨ

= g ê ê ú 鼀҈ᆠᨨ矄 a + 鼀
ᆠᨨ矄 b㡰 ê ú êë úû m m [ ᆠᨨ矄 b 矄� a - 矄� b ᆠᨨ矄 a ] g ᨨ

T = ҈
鼀҈ᆠᨨ矄 a + 鼀
ᆠᨨ矄 b 㡰 ␔ 
ᨨᆠ
F
�矄
�
ᨨ
ᨨᆠ├
C
犨�
鼀ᨨ
ᨨ犨矄
   
 
ᨨᆠ├
ἘἘ
犨�
鼀ᨨ
ᨨ犨矄
�� 
盐
ᨨ
ᆠ�ᨨ
ᨨ C
ᨨ
B

犨 �
ᨨᆠ├
A
犨泐矄
鼀� 矄

矄
=






ᨨ
鼀矄矄
C �矄

҈

�
ᆠ鼀

=

҈

� ´ ҈
-
´


=


҈


㡰 ෌�

q
㦨

�


㦨




鼀

�
㦨
҈

鼀⏌矄
鼀







�㡰

矄� q = 

ᆠᨨ矄 q = 鼀�






��㡰 ᨨ鼀
⼄盐�ᨨ
�㡰


��㡰
犨



 q =



q


�



㦨
�
Q

q
㦨

�  ᆠ

矄
ᨨ
 
 盐盐鼀
�
�矄
ᆠ�ᆠ盐



 = � =


´ ҈

㦨


鼀⏌矄 ᆠ㡰 㡰 
ᨨ�ᨨ 
ᆠᨨ鼀ᨨ  
ᨨ
 矄�ᨨ
ᨨ�矄
 

ᆠ矄矄泐
ᆠ �
ᨨᆠ
\
矄�
q =

T´

鼀

㦨
鼀 w



q L

R R = mRw
᲼Q
矄�
q
㦨
桌 L L



T = = \w=

� ´

� mL

=

T
ᆠᨨ矄
q T
q

T
矄� q R

mg



҈
= 
⏌矄



=

�

�

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PS-31

Laws of Motion �


㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

 
矄
犨�
犨 �ᆠ
  ᨨᆠ├
 矄
 
ᆠ├
�矄
 
矄鼀
�

 
ᆠ├矄
 
  ᨨᆠ├
�矄矄
ᨨ
 
矄鼀
�� 
矄ᨨ
ᨨ鼀
犨
ᨨ
ᆠᨨ 矄�ᨨ

ᨨ
 �泐

矄
ᨨ
 
ᨨᆠ├

�� 矄
ᨨ� 
犨�

矄鼀 �

ᨨ盐
ᆠ矄矄

 
ᨨᆠ�泐

 
�� 矄
ᨨ� 

v





b

ᆠ㡰
꾀
ᨨ �
�盐矄

ᨨᆠ
㦨
ᆠ �
ᨨᆠ

æ Mv
ö ç ÷ ç l ÷
è ø 꾀 
ᨨ �
 � 

ᨨᆠ 㦨
鼀
矄
 
鼀ᨨ�ᨨ
ᨨ

 盐盐鼀
�矄  

ᨨ
ᆠ�ᆠ盐
鼀ᨨ�ᨨ





\ 
졔
鼀�㡰
ᨨ�矄
 
ᆠ �
ᨨᆠ


mv


犨  l l
�矄
 
 �
ᨨ
 
 盐盐鼀
 
v
�矄
 
ᨨᆠ�泐
ᨨ
  ᨨ
ᨨ
⼄盐���盐鼀
ᨨ
 
ᨨ

T

mg
ᆠᨨ矄



=

\
mg ᆠᨨ矄 q - N =

mv R

黨


q Mg Mgᆠᨨ矄 q

g

gᆠᨨ矄q

꾀 

mat
㦨
mg 矄� q
Þ
at
㦨
g
矄� q �盐矄
ᨨ
ᆠ�ᆠ
㦨
ᔼ
DOCP

OC
㦨
CP
㦨
R \ ÐCOP
㦨
ÐCPO
㦨



Þ ÐOCP
㦨


\ DOCP
�矄

⼄盐�
� �
Þ OP
㦨
R

C

R R





ᨨ

mg
ᆠᨨ矄




A h

N v q

mg
矄� q

mg

P mg

O

R R = =x




ᨨᆠ矄
ᆠ� �

矄 ᨨ犨
�
 
��盐
�㡰 ᨨ鼀

ᨨ
 
� � ᨨᆠ
�
 
 � �
�ᆠ�ᨨ

㦨
F
ᆠᨨ矄


졔
mg
ᆠᨨ矄

㦨
᲼kx
졔
mg桌
ᆠᨨ矄

\

쟼  矄�ᨨ
ᨨ
矄� �
㦨
R -

N B

N = mg ᆠᨨ矄 q -
mg +
mg ᆠᨨ矄 q N = mg ᆠᨨ矄 q -
mg N = mg ( ᆠᨨ矄 q -
)
ᔴ泐
犨 
ᆠᨨ矄 q > N
�矄
ᨨ矄��
ᆠ矄
�泐
ᨨ盐犨矄 
 

�
ᆠᨨ矄 q <
N
ᆠ矄
�泐
� 犨矄 

N F㦨kx

M

mg

mv
Þ
N = mg ᆠᨨ矄 q




�㡰 R mg ᆠᨨ矄q 꾀 
泐
 �泐
ᆠᨨ 矄�ᨨ
q ҈
mv = mg [ R - R ᆠᨨ矄 q ]
v
=
g (҈ - ᆠᨨ矄 q ) \
R v
盐� �
 �矄
盐
ᨨ

�
⼄

�㡰 R N = mg ᆠᨨ矄 q - m [
g -
g ᆠᨨ矄 q] Þ Þ Þ

Mg矄� q

q g矄� q

T

꾀矄
 

�矄
鼀ᨨ� �
꾀
ᨨ

�



�
 
ᆠ�ᆠ盐






q

l

 ᆠ

ᆠ泐
T
>
mg
ᆠᨨ矄



㡰

T







mv
T
㦨
mg
ᆠᨨ矄



졔
l

mv

�矄
犨泐矄

ᨨ矄��
⼄盐 �泐

l 䰘


Mv T

Mg
ᆠᨨ矄 q = l

鼀

ᨨ
 
ᨨ泐

mv
N + mg = R mv
N= - mg ᨨ R
 
�盐矄
ᨨ
ᆠ盐盐㡰
�
�矄
ᆠ矄
�矄
鼀� �鼀盐鼀
 ᆠ

ᨨ鼀
ᆠ�ᨨ

犨�

鼀 �鼀盐鼀
�
�矄
ᆠ矄
矄



 �盐
鼀�盐
ᨨ
 
 盐盐鼀
�矄



��
犨 
�矄
 �盐
�矄ᆠ鼀 
�矄



 

\

l

v








졔
鼀�





q

F矄� 
ᨨ



ᨨ

F 졠


ᨨ

鼀�矄� 
ᨨ 
ᨨ

Fᆠᨨ矄
ᨨ 鼀� 鼀�ᆠᨨ矄
ᨨ

 � �

� mg �  




\
Ft
㦨
mat Þ at
㦨
�

꾀矄ᨨ

犨 
 
� �
�矄
盐矄
矄 N
졔
F
矄�


㦨
mg
矄�

mg ö ҈ mg æ Þ N
㦨
mg

F㡰
矄�



㦨
ç mg ÷´ = è
ø

Ft

㦨


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EBD_8341 PS-32

Physics

Work, Energy & Power

5

ᔼ �� �
ᨨ
矄�矄


Topic-1 : Work Done by Constant & Variable Force ҈


W =

㡰 
��

矄�盐�ᨨ
ᆠ
 F 犨
� �ᆠ泐
矄 B A 矄 ᨨ犨
�
��盐



 ␔ᨨ├
ᨨ 
㦨
꾀 盐 
F㜠x
� E ҈

 㦨
꾀
ᨨ
ᆠ � ABCD
졔
꾀
ᨨ C F D �盐鼀
BCFE ҈�鼀 
鼀 ҈ W = 


´ ҈�㡰 + ҈

+


㡰 ´ ҈� = 


+

�



䰘


ᆠ㡰 ᔼ
AC
㦨
l
 
ᆠᆠᨨ� � ᨨ
⼄盐矄�ᨨ

BC
㦨

l
 
AB 㦨
l
ᨨ盐� 

犨ᨨ├
ᨨ 
泐

  m ᨨᆠ矄
�矄
ᨨ
q W �ᆠ�ᨨ
졔
Wmg
㦨

A

和鼀ᨨᨨ

mg l 㡰矄� q - mmg ᆠᨨ矄 ql 㡰 =


\k =  






r uur

ᆠ㡰 ␔ᨨ├
ᨨ 

W = ò F × ds

(

)(

= - x�$ + yˆ × d ´ �$ + dyˆ


)

黨


҈

҈ ҈ Þ W =  ò xdx + ò ydy = æç
+ ö÷ + = ҈J è
ø
҈




�






鼀� 鼀� Þ㦨 㡰 

  鼀� 㦨
鼀 㦨



㦨
ᨨ鼀
ᆠ�ᨨ

ᨨ犨

犨ᨨ├
ᨨ 
泐
ᨨ鼀
ᆠ�ᨨ
�
ᨨ

r r æ 鼀� ö æ ҈
ö ᨨᆠ├
�
�鼀


␔㦨和㦨 çè ÷ø çè �⏌
 ÷ø

鼀�

ᨨ

␔ 㦨

矄 㡰 ᨨ鼀
⼄盐矄�ᨨ

 =  矄 = 

  ᨨ

矄 =  Þ 和 =

= 鼀´
鼀


҈

␔ᨨ├
ᨨ 
= ´ = 鼀 


ᆠ㡰 ␔ᨨ├
ᨨ 
�
矄ᆠ � �
 
盐㜠 
泐
 �矄 ᆠ
dx
�矄 dW
㦨
F dx
㦨
ax
졔
bx
㡰dx




㡰 �盐矄
ᨨ
ᆠ�ᆠ盐

㦨



⼄盐�ᨨ
ᨨ
ᆠ�ᆠ
�矄


졔
泐

㦨

෌�


ᨨᆠ

泐

l矄� q

Þ mmg ᆠᨨ矄 ql = mgl 矄� q
Þ m =  q = k  q




aL
bL +


é �ˆ r 泐ˆ ù = ⧤ ê
⏌
+
 ⏌
ú 


㡰  㡰 ûú ëê  㡰

B

C

L


ò axdx + ò bx dx =

é ù r �ˆ 泐ˆ + = ⧤ê ú

⏌

+ 泐
㡰 ⏌
ûú ëê  + 泐 㡰

Þ W = �
�




L

矄 

泐㡰

ˆ r ⧤ 泐 =  éë �ˆ + 泐ˆùû  

㡰 �ˆ


ᨨᆠ
ᆠ矄
�泐
ᨨ盐犨矄
矄
矄 ᨨ犨
�
 
��盐  
 
�矄ᆠ鼀 
�矄
 � �
ᨨ
 
ᆠ�ᆠ盐


 
 �
犨
 
ᨨᆠ
犨 �ᆠ
ᆠ矄
�泐 ᨨ盐犨矄
 
�矄ᆠ鼀 
犨 �ᆠ
�矄
 � �
ᨨ
  ᆠ�ᆠ盐

�矄
黨
\ ␔ᨨ├
ᨨ 


␔
㦨
和
ᆠᨨ矄
q
㦨

b㡰 ᔼ

ᆠᨨ 矄�
�
犨ᨨ├
ᨨ 
ᨨ矄
ᨨ
  ᨨ
 

�.e.,

�   


���ᨨ 
� �矄

ᆠᨨ 矄�
�
\ W҈
㦨
W

㦨
W r r r ᆠ㡰 dW
㦨
F
dS   F = - K  y�ˆ + xˆ㡰
��

r dS = dx �ˆ + dy ˆ + dz kˆ dW =  K  y �ˆ + x ˆ 㡰
dx �ˆ + dy ˆ + dz kˆ㡰









㦨

Kydx
졔
xdy㡰
㦨

K ᲼dxy㡰桌 \

a
a

a
a

ò


dW
㦨

K ò


d  xy㡰 a
a

҈



ᨨ W = 
K ᲼ xy 桌



ᨨ W
㦨

Ka

 ᆠ
ᨨ
犨ᨨ├
ᨨ 
泐
 
ᨨᆠ
ᨨ
 
�ᆠ
W
㦨

Ka
㡰 
犨ᨨ├
ᨨ 
�
� �� �
 
鼀矄矄
盐
犨�

⼄盐 ᨨ
 
ᆠ  �
�
ᨨ �
 �泐
ᨨ
 
鼀矄矄
h = ⏌
㝐g 㝐矄矄
㦨
醨  ᔴ
㦨
ᔴ 
ᨨ
鼀矄矄

C

�.e.
W
㦨
ᔴ  �
�
ᨨ �
 �泐 M L MgL ´g´ =




㦨
mgh
㦨

҈
҈҈

㡰 ␔ 
 
矄泐矄鼀
�矄
矄
T  mg = ma


�㡰 Mg  T = Ma


��㡰 ᨨ鼀
⼄

�㡰


��㡰





 M - m㡰 g = g ⏌ a= M +m   T
㦨



mg⏌

T a

m 㦨





├� T mg M a Mg

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PS-33

Work, Energy & Power ᨨ
bᨨᆠ├
鼀
=




├� u
㦨



a
㦨
g⏌
t
㦨
҈

s
㦨
䁸 ҈
҈ g s
㦨
ut
졔
at
㦨


졔
´ ´ ҈
= g ⏌



m
㦨


䰘

├�㡰

 \ ␔ᨨ├
ᨨ 
泐
 
矄� �
ᨨ
m
´


´ ҈
´҈
mg g ´ ´҈ = =
 T s
ᆠᨨ矄


㦨


´
҈



䰘�㡰 r ෌�


ᨨᆠ

F 㦨
ay�ˆ +
axˆ㡰   a
㦨

҈
鼀҈ r r ␔
├ ᨨ犨
 
␔ = ×  = a泐 $� +
a $ ×  $� + 泐 $ \
␔
㦨
a泐
졔

a 泐 Wᨨ├
ᨨ  ᨨ鼀
꾀
®

泐
㦨



矄
泐
㦨
҈

(

҈

ò

)(





҈ ´



=
 Þ 
= 鼀⏌矄
ᔼ ��
ᆠᨨ鼀矄矄�ᨨ
ᨨ
矄� � ҈

�
맔
҈




㦨


Þ


»

盐矄� �
犨ᨨ├
 �泐
 ᨨ鼀
␔�
졔
␔矄
㦨
D⧤쟼





҈

ò

�





ᨨ鼀

®
ᔴ

㦨



矄

㦨
҈

�

\
␔

=

ò

҈



�


a 泐 =

ò
a泐 =
a (-

�) = -a ҈

ᨨ鼀
ᔴ
®

泐
㦨



矄
泐
㦨


�

�

\
␔
=

a

ò a ´

� = -






҈

ᨨ鼀

®
E

㦨



矄

㦨


�


\
␔

=

a

ò
a ´

�泐 = -




�

ᨨ鼀
E
®

泐
㦨



矄
泐
㦨

\
␔�
= ò a ´
´  =
ᨨ鼀

®
꾀

㦨


矄

㦨



ò

\
␔

=
a ´
 =
\
ᨨ
犨ᨨ├
ᨨ 
␔
㦨
␔
졔
␔
졔
␔
졔
␔
졔
␔
졔
␔ ҈






a a a - ( -҈) =
㦨


䰘� 㦨
a

a

- =








�




Topic-2 : Energy ҈


䰘


ᆠ㡰 ␔
├ ᨨ犨

盐 
㜠
�
��矄
 
犨ᨨ├
ᨨ  泐
 
ᨨ泐

҈ ´  +
㡰 ´  -
㡰 +
´

㦨


�
졔


㦨


�

矄� �
犨ᨨ├
 �泐
 ᨨ鼀
D ⧤
쟼
㦨
犨ᨨ├
ᨨ  \
D ⧤
쟼
㦨


�
 nl l   l
= ᆠ㡰 l҈ + l
= l   l҈ = nl

\
l҈ = n +҈ n +҈ ҈ 꾀矄
k
µ

l k҈ l ⏌ n + ҈㡰 ҈ \ = = k
nl 㡰 ⏌ n + ҈㡰 n b㡰 ᨨᆠ�泐
ᨨ
҈
├�
ᨨᆠ├
盐矄
 ᨨ
�
ᆠᨨ�矄
犨�

├�

\W=








ᨨᆠ├
=
� =



鼀⏌矄



├�



҈ ҈
´ ҈

� ´ ҈



-
㡰 =
- ´
´ 

矄ᨨ� �

»


ᆠ鼀 㡰 ␔
㦨
盐

盐� æ mg L ö MgL
=
-ç´ = è n
n ÷ø
n
ᆠ㡰 ␔
├ ᨨ犨

盐 
㜠
�
��矄
 
犨ᨨ├
ᨨ  泐
 
ᨨ泐 ҈ \ W = ´  +
㡰 ´  -
㡰 +
´












㦨


�
졔


㦨


�
 矄� �
犨ᨨ├
 �泐
 ᨨ鼀
D ⧤
쟼
㦨
犨ᨨ├
ᨨ  \

D ⧤
쟼
㦨


�
 㡰 㝐 �鼀盐鼀
矄
�矄

鼀
ᨨ矄��ᨨ
ᨨ
⼄盐���盐鼀 꾀
⼄盐���盐鼀
ᨨ矄��ᨨ


㦨
├
Þ = ├ ᨨ鼀
犨ᨨ├㜠 �泐
 ᨨ鼀
␔ + ␔矄 = D⧤쟼 ҈
҈
 㡰

├ = 鼀 -




҈ æ ö ҈ æ ö ç ÷ - ├ ç ÷ = 鼀

Þ ҈ = ҈ 鼀
è ├ø
è ├ø

⧤
ᨨ

 鼀 = 鼀├ ⧤ ¶盐 ˆ =  ˆ ᆠ㡰 = ¶  和� ᆠ
�ᆠ
�矄
鼀ᨨ� �
�
ᆠ�ᆠ盐
 鼀
⧤ ⧤ = = Þ 鼀
=    
҈ ⧤ \ ⧤
쟼
= 鼀
=




ᨨ
 �泐
㦨

쟼

졔
⧤
쟼
⧤ ⧤ ⧤ = -
+
= ᨨ





Q 
쟼
= -
�� 㡰


 㡰 
Vf
�矄
 
� 
矄
ᨨ
 
ᨨ泐
ᨨ鼀
⼄盐矄�ᨨ 矄
V
҈ ҈ mV f
= mV


= � m⏌ s Þ Vf =


dV æ dV ö
F = mç 㦨
kV
÷ = -kV


\ ҈

㡰 dt è dt ø Þ

´ +

)

\
␔҈
= a泐  = a  = a


矄� �
� ᆠ�
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
盐矄  ᨨ
 
盐矄
 
ᆠᨨ�矄�ᨨ

犨
�





�

ò

dV

҈
V




҈


= -҈

K ò dt


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EBD_8341 PS-34

黨


҈



Physics

҈ ҈ - = ҈

K ҈
㡰







ᨨ
K
㦨
҈


kgm҈ � ҈
㡰 ⧤
쟼
㡰Ἐ
㦨
�
奠
ᨨ
⧤
쟼

 
�
�


 ҈ �
҈ 鼀 Ἐ
= ´ 鼀

Þ
 Ἐ =
҈



҈ ᔴᨨ �ᆠ� 
ᨨ
矄�盐�ᨨ
=
ᨨ犨

ᨨ
�矄 ᆠ

泐
ᨨᆠ
�矄 ҈ö æ æ ҈ + 
ö ç ҈+
÷  = ç ÷ = ç ÷ =  ç ҈ - 
÷ ҈ è ø çç ҈ - ÷÷ è
ø dV ᆠ㡰 矄� �


㦨
鼀
㦨
m dt dV
t = ҈



























᲼Q m
㦨
҈
├�
�� 桌 dt

ar  ar  �  + = ar 

҈

b㡰 꾀泐� �
鼀ᨨ鼀 盐鼀
ᆠᨨ 矄�ᨨ

m҈u҈
졔
m
u

㦨
m҈v҈
졔
m
v


҈u
졔
m
㡰
㦨


҈
㡰
졔
m㡰

҈u
㦨
m 㦨


҈ ҈

҈u
= m㡰


和ᨨ� �
犨
�

u
㦨



҈
҈ æ xö ҈ kx = K ç ÷ + 

҈㡰


è
ø

kx =

黨 Þ  ´ ҈ kx
=

黨 Þ


҈
\ Kx = 0.6  (ᨨ
� ��
 �泐
ᨨ
 
矄� �)


҈

ét
ù ҈ ò dV = ò
t dt

V =
êë
úû =  鼀矄

᲼Q
t
㦨
҈
矄ᆠ
�� 桌

ᨨ鼀
犨ᨨ├㜠 �泐
 ᨨ鼀
v

(

҈҈


ᨨ
 �泐
㦨
⧤
쟼

졔

쟼


҈�




盐 - ҈㡰
ᨨ

 = 盐 - =



ᔼ


 
� �
ᨨᆠ
�
泐
ᆠ
 ├

 

盐矄� �
犨ᨨ├

 �泐
 ᨨ鼀


)

҈ ҈

␔
㦨
D⧤쟼
㦨
m V - u 㦨
´ ҈ ´ 黨 =

� 

㡰 ␔ᨨ├
ᨨ 
泐
�ᆠ�ᨨ

턼
㦨
Ҙ鼀�x

\

҈
㦨


Þ
PQ
㦨

m
PQ ␔ᨨ├
ᨨ 
泐
�ᆠ�ᨨ

턼
㦨
Ҙ鼀�
맔
ᔴᨨ矄


맔



҈ ҈ ҈ ҈ ( - ҈)
㦨
鼀盐


鼀

㦨
鼀盐


鼀盐









ù é ҈
҈ - ( - ҈) ú 㦨
鼀盐 ê
ê ú


ë û ҈ æ
- ҈ ö = 鼀盐
ç ÷
è
ø 


 
盐鼀
ᨨ
 ├矄
⼄盐�
ᨨ
矄ᨨ
 
盐


ᨨ
�矄 ᆠ

泐
 
盐
 ᨨ
ᆠᨨ鼀� �
ᨨ 矄
㦨
 矄� �
犨ᨨ├㜠 �泐
 ᨨ鼀
��


 ´
㦨

 Ҙ鼀�
和� ᆠ
犨ᨨ├
ᨨ 
泐
�ᆠ�ᨨ
ᨨ
矄
PQ
 
QR

⼄盐
Ҙ鼀�
x
㦨

 盐鼀�

㦨
Ҙ鼀�
맔


Þ x
㦨

 @

�m 矄� �
犨ᨨ├
 �泐
 ᨨ鼀
鼀�
矄�


맔


㦨

 Ҙ鼀�
졔
Ҙ鼀�x

҈


Þ

㦨

 Ҙ Þ Ҙ
㦨
0.29 ␔ 鼀� ´ ҈


҈
´ 黨

´ ҈ ´ ҈


= = b㡰
㦨 � 盐 � 盐 � 盐 黨



ᔼ 盐
㦨
㦨

黨
맔
҈





黨 ´ ҈


㦨
҈


黨
맔
҈
├�

盐矄
㦨


´ ҈
䰘 b㡰 꾀矄
犨
├ ᨨ犨


㦨

 r

U =

ò ar





dr =

ar  




�㡰

mv
꾀矄

= ar
r m
v

㦨
mar ҈  ᨨ


鼀⧤쟼㡰
㦨
ar


��㡰


ᨨ
 �泐
㦨
ᨨ �
 �泐
졔
├� �ᆠ
 �泐 ᨨ犨

ᨨ鼀
⼄
�㡰
 
��㡰

盐


㦨





ᔼ
� �

矄�


㦨


҈



㡰 
盐

 
� ��
ᨨᆠ�泐
ᨨ
 
盐
ᨨ
鼀矄矄
鼀
꾀 
矄矄� �
 ᨨ盐�

 ├
ᨨ
犨�


�矄
ᨨᆠ�泐 ᆠ矄矄
ᨨ



(  ) =

҈ 鼀盐
-




é҈
(
- ҈) ù ҈ = 鼀盐
\
 =

ú ᨨ

 ( ) =  ê
鼀盐


û

- ҈ ë

҈



㡰 ෌�


├ 꾀 = 

 ⏌ 鼀

├  =


 ⏌ 鼀 
犨 
 
ᆠᨨ鼀� �ᨨ
ᨨ
矄� �矄
�矄
ᆠᨨ鼀矄矄
泐 ᨨᆠ


和� �
꾀
�矄
ᆠᨨ鼀矄矄
泐


 ᨨ
ᆠᨨ鼀矄矄�ᨨ

�
矄� �


 = 

䰘� - 㡰 ᆠ鼀 = 

´ =




䰘� - 㡰 和ᨨ� �
犨
�


㦨
�
ᆠ鼀  =

䰘� - � = 
䰘�ᆠ鼀 ҈ ├  㡰
쟼꾀
꾀 꾀 

´ �㡰

= = =
҈ 쟼  ├    㡰



´ 
䰘�㡰


WWW.JEEBOOKS.IN

PS-35

Work, Energy & Power ҈䰘


ᆠ㡰 꾀
盐 
F  t �
��矄
 
� 
� 
鼀ᨨ鼀 盐鼀 ᨨ
 
ᨨ泐


Ux
㦨



x
㦨


 


x
㦨


Ft㡰

꾀
ᨨ
D
AOB
㦨


҈ ´  ´
=
矄 A
N


\ \

OA CD 

�
s = 졠 C t s OB CB
CD D = Þ CD
㦨

 ҈
� é҈ ù 꾀
ᨨ
Dᔴ
= - ê ´ ҈
� ´
ú = -҈
� 矄 ë
û


� 
� 
鼀ᨨ鼀 盐鼀
㦨



҈
�
㦨


�
矄

p


�㡰
= = �



m
´
҈

ᆠ㡰 矄
犨 
 
ᨨᆠ├
B
�矄
�矄ᆠ
ᨨ犨矄
犨
҈
ᨨ 泐
矄� �
S҈
�矄
ᆠᨨ鼀矄矄
 
S

�矄
�
�矄
盐
矄 矄
 
ᨨ 
 
ᨨ
S

�矄

҈

 ᨨ
 
 �泐
矄ᨨ
�
 
矄泐矄鼀
㦨
k҈
x



␔ 
 
ᨨᆠ├
�矄
矄

�
犨�
ᆠᨨ鼀
ᆠ├
ᨨ
  ⼄盐���盐鼀
ᨨ矄��ᨨ

��
鼀ᨨ鼀 盐鼀

ᨨ矄 ᨨᨨ
ᨨ ⼄盐���盐鼀
ᨨ矄��ᨨ
 
鼀ᨨ
ᨨ犨矄
犨



꾀矄
 �矄  矄

 
矄� �
S҈
ᆠᨨ鼀矄
ᨨ
�矄
盐
 �
 
S
�矄
ᆠᨨ鼀矄矄



쟼

矄ᨨ
�
 
矄� �
S҈
�矄 矄ᨨ
矄
 

쟼

ᨨ
矄� �
S

犨 
 
ᨨᆠ├
B
ᆠ 矄
�矄  鼀
ᨨ矄��ᨨ
 
ᆠᨨ鼀矄矄� �
S

泐
y
ᔼ
�矄
ᆠ盐矄
ᨨ
�ᆠ�ᨨ
 泐犨 
和ᨨ

 �泐
�矄
ᆠᨨ 矄 ҈ ҈




҈ ҈ \
k҈

x

㦨 k

y




Þ 맔 kx 㦨 맔


ky




y ҈ Þ x

㦨

y








\ = x
҈黨
b㡰 ᨨ
ᆠᨨ 矄�
ᨨᆠ矄
DU
㦨

W \

⧤� �ᆠ
 �泐
ᨨ
 
ᨨᆠ├
=

x

x

DU
㦨
- ò Fdx
ᨨ
DU
㦨
- ò k xdx


Þ Ux㡰

U
㡰
㦨
-

Q U
㡰
㦨

㡰

kx
-
U x Þ x

㦨

k ᔼ
矄 矄

ᨨ
ᨨ犨
x

 �矄
矄泐鼀鼀�ᆠ b㡰 
x

 
鼀 �鼀盐鼀
  矄�ᨨ
ᨨ
 
矄� �

 鼀ᆠ  �ᆠ
 �泐
�矄
ᆠᨨ 矄

矄ᨨ
ᆠ矄
�
  ���ᨨ 
ᨨ �
 �泐
ᨨ
矄� �
鼀矄矄
矄泐矄鼀
Mgx㡰 \












kx


Ux㡰
㦨
-

æ҈ è


ö ø

㦨
��
�
矄� �
矄�ᆠ
ᨨ �
 �泐
ç kx
÷

҈

Mg kx Þ x = k
du 㡰 ᨨ鼀

F
㦨

dx dUx㡰
㦨

Fdx

Mgx =


҈


x

Ux
㦨
-ò - kx + ax 㡰dx \


x



kx
ax
Ux
㦨
- ò Fdx
=





k a ␔

ᨨ �
 �泐
ᨨ
犨�ᆠ
ᨨ盐
ᨨ
犨 �ᆠ
ᨨ 
�矄
 ᨨ���
꾀矄ᨨ

犨 
犨
盐
x 㦨


�
 
��
盐 ᆠ�ᨨ
dU 犨
�
F
㦨



盐
F
㦨
dx dU =

�


 
矄ᨨ
ᨨ
 
�
矄 ᨨ盐 Þ 꾀
x
㦨

錐
dx 
ᨨ


矄
ᆠ ᆠ�矄�ᆠ矄

矄 
泐
�
㡰



b㡰 ␔ 
矄� �
�矄
ᆠ盐
� ᨨ
�ᆠ矄
 
 
 �
ᨨ 꾀  Ux
㦨
��
ᨨ
x
>


꾀矄ᨨ
Þ


k 錐 a


L  
 
ᨨ��� 
 �
ᨨ
矄� �

L
 
矄� �
ᆠᨨ 矄  �矄
K
�� 㡰 ᨨ

矄� �

K
맔
L
㦨
ᆠᨨ 矄  
K Ἐ

 
矄� �
ᆠᨨ矄  
ᨨ
ᨨ �
�ᆠ
ᨨ
矄� �
L  \ K
맔
L
㦨
맔
K Ἐ

Þ K Ἐ
㦨
K 

p

ᆠ㡰 ⧤
쟼

㦨
錐
᲼⧤
쟼

㦨
⧤� �ᆠ
 �泐錐

㦨
鼀ᨨ鼀 盐鼀桌
m p҈
p

= ⧤쟼҈
㦨
⧤쟼
\ m҈ m
ᨨ �
�ᆠ
ᨨ
矄� �
㦨


\


�㡰

p҈


pl m҈ ml ҈ ҈ = = =
Þ

p m m


␔ᨨ├
ᨨ 
㦨
ᔼ ᆠ矄
�

쟼

졔
��
�
⧤
쟼
F
맔
d
㦨
mgh
졔
��
�
⧤
쟼
҈

맔
�
㦨
҈
맔
҈

맔


졔
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⧤
쟼
෌�
�
⧤
쟼

㦨
�


㦨
҈


n
㦨
�

p



=

\ \
�

㡰 

ᨨ矄矄
�
⧤
쟼

ᨨ
 
ᨨᆠ├
㦨
��
�

쟼

ᨨ
 
矄� � 졔
犨ᨨ├
ᨨ 
�� 矄
�ᆠ�ᨨ

҈
҈

mmgx + kx
mv
㦨
kx + mmg
x



Þ

v
㦨



m
´

҈ ´

҈
´ ҈
´


+
´


´


v
㦨


҈

 \
v
㦨

㦨





















\

㦨

҈
҈



҈�



㡰

ᨨ鼀
犨ᨨ├
 �泐
 ᨨ鼀


W = F × s = DKE =

҈
mv



V
=
gh

\F ×s = F ´


҈ ҈� = ´ ´
´ ҈
´

҈

҈



\ F = ҈�



WWW.JEEBOOKS.IN

EBD_8341 PS-36


䰘







Physics

҈



㡰


⧤� �ᆠ
 �泐
㦨
ᆠ  �
�
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⧤쟼
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m
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├�
Dh
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҈
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҈
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KE
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J 

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Ӹ
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ᨨ�
泐
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犨�� 
鼀�

\ 鼀� 矄� 
° =

\ 
=

҈
҈ 鼀
= 鼀� + 鼀




鼀�
矄�



\ 

㦨




�




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\ ᨨ鼀
⼄

�㡰










































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t







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



v

L

B L

u

A

盐� �
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ᨨ
v
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⼄

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r
 D v  㦨

u - g l㡰 

ᆠ

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鼀ᨨ�ᨨ



ᨨᆠ
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ᆠᨨ 矄 
�
鼀� �盐

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矄 ᨨ犨
 
矄
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ᨨ
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 
 ᆠ
├� �ᆠ
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犨�
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ᆠᨨ 矄 
v

mV
gt
=



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v

F F

v

v


ᨨᆠ�泐
 
ᆠᆠ�ᨨ
ᆠ  �矄
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盐 ᨨ
ᆠ  �
�
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�ᆠ�ᨨ

꾀
®


⼄



錐


®
⼄

矄錐

ᔴ
®


⼄



矄錐


®




 ᨨ
꾀

dU  d é U
æ æ x ö ù -
U
ê ç҈ - ç ÷ ú = =  x - a㡰 x  x + a㡰
㦨
 dx dx ê
ç è a ø ú a è ë û
㦨

 x
㦨



x
㦨
a

x
㦨

a   u
㦨

 x
㦨

a

x
㦨
a U

dV dé g ù g =m ê ´ t ú 㦨
m = gm
㦨
ᆠᨨ 矄  ᨨ犨 F
㦨
ma
㦨

m dt dt ë m û m 

b㡰
ᨨ鼀
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��
�
⼄盐矄�ᨨ
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ᨨ
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

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꾀
㦨
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㦨

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꾀

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쟼
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⧤

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ᔴ

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ᨨ
 
犨
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
 �泐 쟼꾀
㦨
쟼
㦨
쟼ᔴ



�㡰 쟼꾀
㦨
쟼
Þ
쟼ᔴ

>
⧤ᔴ


��㡰 쟼꾀
㦨
쟼ᔴ ⧤꾀
졔
鼀� 꾀
㦨
⧤
졔
鼀� ᔴ ⧤ - ⧤꾀 ᨨ
꾀 - ᔴ = ᔴ


���㡰 鼀�

U U0





g 矄 g g  ´ t

�




µ 
錐
V = =  Þ 矄=  鼀 鼀
m 和ᨨ
V
�矄
ᨨᨨ�ᨨ 
ᨨ
�
 
�矄 ᆠ
ᆠ

ᨨ
 ᨨᨨ�ᨨ 
ᨨ
ἘἘ


\
V =




�㡰






� = 

-
�
 ᆠ
ᨨ�ᨨ
㡰
�矄
ᆠᨨᆠ
꾀
ᨨ� 

 

ᨨ
ᆠ�ᆠ盐


 
ᨨ� 矄


矄鼀
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矄矄
 


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
ᨨ� 
泐
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和ᨨ
⼄盐�
ᆠ �
ᨨᆠ
= �矄
鼀 �鼀盐鼀

ᨨ� 矄 
 

d æ҈ dk
ö ҈ 

b

㡰

G
㦨
gt

  k = mV
\
ç mV ÷
㦨
gt dt
dt è ø
m dV dV ´
V
㦨
gt \
mV
㦨
gt Þ
dt dt V

҈
҈ mv + mgL = mu
졔





r r

㦨
 v - u  = v + u

�











































��㡰


m ò VdV = g ò tdt

ᨨ


Þ v
㦨
u -
gL





�㡰 r ᔴ  �
�
ᨨᆠ�泐
 D v 

鼀


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泐
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ᨨᨨ鼀
ᨨ�   
ᨨ� 
Ӹ 泐


黨


҈


Þ 꾀
>
ᔴ錐
⧤ᔴ


⧤꾀 ᨨ鼀
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�㡰
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 
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犨
�
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ᔴ \ ⧤
>
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E
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E
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E
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a

졔a

x

x U

U

졔a

ᨨ




Fx =

x

a

졔a

x

 dU æ xö = -U
ç ÷ dx èaø

 dU e- x ⏌ a = U
x x - a 㡰 x + a 㡰 dx a U dU ᨨ


Fx = = -
᲼ x - a㡰 x + a㡰桌 dx
a





ᨨ
ᔴ


Fx =

WWW.JEEBOOKS.IN

PS-37

Work, Energy & Power 

b㡰

N

mg
ᆠᨨ矄
q
犨�
ᨨ�
⼄盐�
ᆠ �
ᨨᆠ








mv R mv
N
㦨
mg
ᆠᨨ矄
q
졔
R

N

mg
ᆠᨨ矄
q
㦨


҈ ҈´ ҈
㡰
+
㦨
䰘
�
 



�
b㡰 ␔ᨨ├
ᨨ 
�� 矄


矄�

�ᆠ�ᨨ
㦨
ᔼ ��
鼀ᆠ  �ᆠ

鼀  �泐

� 
鼀ᆠ  �ᆠ   �泐
턼 ҈ 鼀�
矄�


҈�

㦨
mgh


鼀



鼀� ҈ 鼀�
ᆠᨨ矄


Þ ҈�

㦨
mg R
矄�




鼀

Q h
㦨
R
矄�

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㦨
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맔
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맔



맔



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v


\ v
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鼀⏌矄 

ᆠ㡰 ᔼ
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ᨨ
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�.e.

ᆠ矄
�

҈ 鼀ᆠ  �ᆠ
 �泐
㦨

mv


␔ �
�矄
矄ᆠᨨ 
ᆠ矄



ᨨ
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맔
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맔


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ᨨ
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ᨨᆠ├

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ᆠ矄
�
鼀ᆠ  �ᆠ
 �泐

҈
mv

 mgh


 ᨨ
矄鼀 
҈
�矄
ᆠᨨᆠ
和鼀 


�矄
犨ᨨ �


ᆠᨨ �ᆠ� 
ᨨ
�ᆠ�ᨨ
犨

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ᨨᆠ├
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䰘
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鼀矄矄
�
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m

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� 矄 
㦨
v
ᆠᨨ矄
q
㦨





v
� 盐 
M
犨�
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m
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盐
泐

҈
鼀  �

҈㡰
鼀
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鼀 ᆠᨨ 矄  �ᨨ
졠 㝐 q ᆠ矄
�

쟼
ᨨ
M
㦨
� ᆠ矄 ҈
鼀 h v
ᆠᨨ矄
q �

쟼

ᨨ
m
졔 �
鼀 v � ᆠ矄
�
⧤
쟼
q ᨨ
ᨨ
  h' 

�

҈㡰
鼀 v
㦨
v
ᆠᨨ矄
q ᨨᆠ├矄
鼀 ҈
҈
Mgh
㦨
mgh'
졔
mv + m ᆠᨨ矄 q㡰

҈
ᨨ

㡰
黨

㡰
҈㡰
㦨


�㡰
黨

㡰
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´ v

æ
v ö ҈ 졔
´

� ´ ç ÷
è �ø 和ᨨ� �
犨
�

v
㦨


黨
鼀⏌矄



⧤
쟼

ᨨ
ᨨᆠ├
㦨
犨ᨨ├
�� 矄
�ᆠ�ᨨ
졔

쟼

ᨨ
矄� � ҈
҈ mv = m k mg  BD + x 㡰 + kx


҈
҈
mv
㦨
Ҙk
mg


҈

졔
x㡰
졔
kx

҈ ҈
맔


�
맔


㦨




맔


�
맔
黨




҈

졔
x㡰
졔



맔
x




҈

졔
x
졔
x

㦨



� \ x

졔
x



҈҈
㦨

҈ =

҈ 졠
矄ᨨ� �

犨
�
x
㦨
҈



矄� �
�矄
ᆠᨨ鼀矄矄
泐


҈
鼀 矄ᨨ� �
ᨨᆠ

Y
㦨
kx
㦨


맔


҈
㦨




N �ᆠ�ᨨ 
ᨨᆠ

Y
㦨
Ҙs
mg
㦨





맔


�
맔
黨


㦨
҈

䰘

N 和� ᆠ
�ᆠ�ᨨ 
ᨨᆠ
>
矄ᨨ� �
ᨨᆠ

 
ᨨ泐
犨�
矄ᨨ
 �.e.

 
ᆠᨨ鼀矄矄� �
 
矄� �
泐
x \ 
ᨨ
�矄 ᆠ
 㦨
AB
졔
BD
졔
x
㦨


졔


҈

졔


҈
㦨





鼀


A

B
鼀

D

C



҈
鼀

x

Y

黨



 

ᨨᆠ
ᆠ� � v N ᨨ
A
 
B �矄
ᨨ

和� ᆠ T  
ᨨᆠ├矄
꾀
 

 B 鼀ᨨ� �
犨�
ᆠᨨ 矄  C T f ᨨᆠ�泐
mBg 
 
  矄�ᨨ
ᨨ
  T v 矄� �

x
A


犨�

ᨨ
�ᆠ�ᨨ

mg ᨨᆠ
犨
ᨨᆠ├
A  
C Q f
㦨
ҘN


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�矄
ᨨ
ᨨ鼀
ᆠ�ᨨ
ᨨ
A ᆠ盐矄
A
�矄
ᨨ
盐矄 � �
C
矄ᨨ
�ᆠ�ᨨ 
ᨨᆠ
犨

ᨨᆠ├
꾀
 
ᔴ
f

㦨
m
㦨

꾀泐� �
F 
㦨
ma
ᨨ
A
mAg

T
㦨
mA
맔

\ T
㦨
mAg



�㡰 꾀泐� �
F 
㦨
ma
ᨨ
B
T

f
㦨
mB
맔

\ T
㦨
f
㦨
ҘN


㦨
ҘmBg



��㡰 ᨨ鼀
�㡰
 
��㡰 A

mA
= = ҈
├� m



 
ᨨᆠ
ᆠ� �
ᨨ
 
矄� �
�矄
 
 矄�ᨨ
⼄盐
ᨨ 矄ᨨ� �
ᨨᆠ ma g T ҈黨

\ T
㦨
kx Þ x
㦨
\ x
㦨

㦨
k k k 쟼 �泐
矄ᨨ
�
矄� � ҈ ҈ ҈黨

҈黨

҈黨

´ ҈黨

=


黨
 U = kx
= ´ k ´ ´ 㦨


k k
´ ҈黨

ҘmBg
㦨
mAg

Þ
mB
㦨


WWW.JEEBOOKS.IN

EBD_8341 PS-38






Physics



⧤
쟼


C
㦨
ᨨ矄矄
�

쟼


犨ᨨ├
ᨨ 
泐
�ᆠ�ᨨ

ᔼ
ᨨ
ᆠ矄矄
犨ᨨ├
ᨨ 
泐
�ᆠ�ᨨ
㦨
mmgx




ᆠ =

A

҈
mvc = mg y - mmgx


















= gy - mgx

ᨨ


q(

\ vc
㦨

g  y - mx 㡰 B ᨨ犨

⧤
쟼


F
㦨
ᨨ矄矄
�

쟼


犨ᨨ├
ᨨ 
泐
�ᆠ�ᨨ

҈
D mv F = mgy - mmgx








a( G M ҈
vF = gy - mgx Þ
\

C





�.e.
vC
㦨
vF
㦨

gy -
mgx

E

F

N


҈


ᨨ


犨 


�矄
 ᨨ犨
盐
 
���ᨨ 
盐
ᨨ 

�矄
ᆠ� �
ᨨ
 

犨 �ᆠ
�矄
矄ᨨ 矄�
ᨨ
  ᆠ  �
�
鼀ᨨ鼀 盐鼀







ᨨ

矄� �

矄� �
ᆠᨨ 矄 㡰
k
맔
 � 㡰
l
㦨
ᔴᨨ 矄  l ᔼ
 �
�矄
鼀
ᨨ 
 �

�.e.


 
矄� �
ᆠᨨ 矄   ᆠᨨ鼀矄
 
�鼀矄
�.e


⧤
矄� �
⧤� �ᆠ
 �泐
⧤
쟼
㡰
 
鼀ᨨ鼀 盐鼀
�ᨨ






⧤
쟼

㦨


҈ p

ᨨ
⼄盐
盐
ᨨ
p,

⧤
쟼

µ
鼀矄矄
m

�
䰘






\ P = mv

dv v dt

dv Þ P dt = mv dv dt t

ᔼ �� �
ᨨ
矄�矄


黨





ò

�矄 ᆠ

s = v dt =









ᨨ


\

ᆠ㡰

w E ᨨ犨

P = = = ᆠᨨ 矄 t t

t


P ҈⏌
t dt = m

ò


Þ

V
㦨
krt

\

v
= ᆠᨨ 矄  k 㡰 \ t

҈
mv
= ᆠᨨ 矄 t

v
㦨
kt҈⏌


 




ᨨ
ds = kt҈⏌
dt 泐
� �� �

犨
�


P t ⏌
× m ⏌



P ⏌
× t
Þ 矄 µ ⏌
黨m 和ᨨ

�
㡰
�矄
ᆠᨨᆠ
b㡰 ᨨ
ᨨᆠ
⼄盐�
ᨨ
� 
鼀 �鼀盐鼀
ᨨ
ᆠᆠ�泐 �� 矄
�ᆠ�ᨨ 
ᨨᆠ
㦨




N Fᨨ
=
Mg
졔
�ᆠ�ᨨ

㦨





맔
҈

졔




㦨







졔





㦨






N 矄� �
ᨨ犨

P
㦨
F
맔
v


×
䰘


㦨






맔
v Þ
v
㦨
҈



鼀⏌矄
»
҈
黨
m/s  ᆠ
矄
ᨨ
 
ᨨ

盐
ᨨ
�矄
ᆠᨨ矄
ᨨ
҈
黨
鼀矄҈

Þs=

d d æ dm ö ÷ = v r´ ᨨ盐鼀㡰 = vr ᨨ盐鼀㡰 dt dt è dt ø 㦨
vr
맔
Av㡰
㦨
Arv
ᨨ犨
㦨
ᨨᆠ
맔
ᨨᆠ�泐
㦨
Arv

맔
v
㦨
Arv  ᆠ


ᨨ犨 P
µ
v b㡰 
ᆠ �
ᆠᆠ�ᨨ
ac
㦨
k

r
t
ᆠ㡰 F
㦨
v ç

ᨨ鼀
犨ᨨ├㜠 �泐
 ᨨ鼀

ᨨ
犨ᨨ├
ᨨ 
㦨
ᆠ  �
�

├� �ᆠ
 �泐


æ
P ö ҈⏌
҈
÷÷ t 


= mv
Þ v = çç
è m ø t

\
和
ᨨ
ᆠ
犨 �
鼀ᨨ� �
ᨨ犨 �

㦨
҈�
鼀⏌矄
㡰

b㡰

v

ò P dt = m ò v dv

q ᨨ矄 ҈ 犨ᆠ  q = ҈


犨 æ 犨 犨 ö = ç - ÷
è ҈


ø v Þ v = ҈� 鼀⏌矄



v
= k
rt
r dv
㦨
kr at
㦨
dt ᨨ犨
P
㦨
Ftv
㦨
matv \ ᨨ犨
㦨
mk
r
t

b㡰 ␔
├ ᨨ犨
  ᨨ犨

P = Fv 盐
F = mav = m


㦨






㦨
  ᆠ 㦨 㦨  ᨨ犨
㦨
鼀
㦨
鼀


㦨
㝐


ᆠ㡰 ␔ �
鼀ᨨ� �
ᨨ犨 �
ᨨ犨

 =
 =


Topic-3 : Power ҈



=



wö æ P = ç w 矄� q + ÷҈


è ø 犨 æ犨 犨ö  = ç + ÷҈
=
è ҈


ø

b(

vF
㦨

g  y - mx 㡰

b㡰 ᔴ �
ᆠᆠ�ᨨ
ᆠ
㦨




Þ s
㦨
҈




kt  ⏌
+ C

Þ
s
µ
t⏌


�.e.

�矄 ᆠ
鼀ᨨ
S
µ
t⏌
b㡰 ᔼ
鼀ᆠ � 
�矄
盐�ᆠ
犨�
ᨨ�
�ᆠ�ᨨ
�矄
盐ᆠ
졠盐盐 犨ᨨ├ 㝐ᆠ  �ᆠ
 �ᆠ� ᆠ泐
㦨
ᔼ 盐  �泐 盐
ᨨ
矄矄
�ᆠ�ᨨ
ᨨ盐盐
犨ᨨ├
犨�
� ᆠ矄

盐矄
  鼀ᆠ  �ᆠ
 �ᆠ� ᆠ泐
� ᆠ矄矄
㝐ᆠ  �ᆠ
 �

㝐
꾀

㦨


҈҈


ds = kt҈⏌
dt

ᨨ  ᨨ

�㡰 矄� �

犨ᨨ├

 �泐
 ᨨ鼀

D
⧤
쟼

㦨
␔
㦨

맔
 ҈ 鼀
=  ´ 
\ =


 = 鼀


´

� ´ � = �鼀矄 -҈




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PS-39

Work, Energy & Power ҈




 
� 
ᨨᆠ�泐

vἘ

盐矄� �
鼀ᨨ鼀 盐鼀
ᆠᨨ 矄�ᨨ

҈
㡰



෌�

㝐矄矄
ᨨ
 
ᨨ泐

m
㦨


├� ᨨ犨
�
泐
 �� 

P
㦨
҈
⏌矄

�鼀

t
㦨
黨
矄ᆠᨨ 矄 ᨨ犨

P
㦨
Fv

᲼Q F = ma桌 dv ö æ çèQ a = ÷ø dt

Þ P = mav Þm

v ҈
҈ ᨨ犨
� ��
 �泐
E� = mv


v ҈ ҈ mv
� 
 �泐


E f = ´ ҈
m ´ æç ö÷ =

҈
è ҈
ø mv = ҈
mv Ἐ Þ v Ἐ =

dv v=P dt

҈
҈ v
mv - m

҈
҈
é ҈ù ҈
é҈� ù Þ mv ê҈ - ú Þ mv ê ú

ë ҈
û ë҈
û

쟼 �泐
ᨨ矄矄


E� - E f =

P Þ v dv = dt m ᔼ �� �
ᨨ
矄�矄
犨
� v

Þ ò v dv =


Þ

dx = dt x

Þ ò dx =


t

P dt m ò


Þ

v
Pt æ
Pt ö = Þv= ç è m ÷ø m


҈⏌



ᨨ
 �泐
ᨨ矄矄
�矄
P奠
ᨨ
 
ᨨ��� 
 �泐
쟼 �泐
ᨨ矄矄 ´ ҈

졠��� 
 �泐 ҈
é҈� ù mv ê ú
ë҈
û ´ ҈

= 黨
䰘�奠 =










҈
mv
\ 奠P =

dx ö æ çèQ v = ÷ dt ø


P ҈⏌
t m t


P ҈⏌
t dt m ò



P t ⏌

P
⏌
= ´ t \
�矄 ᆠ

x = m ⏌
m 
´ ҈
⏌

Þx= ´ ´ 黨 = ´
䰘 = ҈


 




Topic-4 : Collisions ҈


㡰  ᨨ
ᆠᨨ�矄�ᨨ
ᨨᆠ�泐
ᨨ
�ᆠ
A

u҈ =  �ˆ + ˆ 㡰 鼀⏌矄 ᨨᆠ�泐
ᨨ
�ᆠ
B

u

㦨

꾀 
ᆠᨨ�矄�ᨨ
ᨨᆠ�泐
ᨨ
�ᆠ
A

v҈ = �ˆ +  ˆ㡰 ᨨᆠ�泐
ᨨ
�ᆠ
B

v

㦨

矄� �
� ᆠ�
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
 �盐
鼀ᨨ鼀 盐鼀

r r r r m҈u҈ + m
u
= m҈v҈ + m
v
r Þ
m
 �ˆ + ˆ 㡰 + m
´
=
m
�ˆ +  ˆ 㡰 + m
´ v
r Þ
�ˆ +
ˆ =
�ˆ +
 ˆ + v
r r Þ v
=   - ҈㡰�ˆ -   - ҈㡰 ˆ Þ v҈ = �ˆ +  ˆ r r ᨨ
 �
犨
v҈
 
v

r r v ×v
  - ҈㡰҈ - 㡰 ҈ -  ᆠᨨ矄 q = ҈r r
= = v҈v

´

  - ҈㡰

Þ q = ҈
�° r r 꾀 �
犨
v҈
 
v

�矄
҈
�





 ᆠ

盐
ᨨ
P
�矄
ᆠᨨ矄
ᨨ
黨

b㡰 ෌�
㝐矄矄
ᨨ
ᨨᆠ├

m҈
㦨
҈
黨
├� 㝐矄矄
ᨨ
盐

m

㦨


҈
├� ᨨᆠ�泐
ᨨ
盐

v

㦨



鼀⏌矄 
v

 
ᨨᆠ�泐
ᨨ
 
ᆠᨨ鼀� 
矄泐矄鼀

ᔼ
�矄


� 矄�ᆠ
ᆠᨨ�矄�ᨨ
矄� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀

m҈ ´
+ m
´ v
= m҈ + m
㡰v Þ

҈ ´

= 

҈ + ҈
黨㡰 ´ v Þ v = ҈ 鼀⏌矄 矄� �
犨ᨨ├
 �泐
 ᨨ鼀 ␔ᨨ├
ᨨ 
㦨
ᔴ  �
�
⧤� �ᆠ
 �泐 
K

 
⧤� �ᆠ
 �泐
ᨨ
ᆠᨨ鼀� 
矄泐矄鼀
鼀҈
졔
鼀

㡰� ҈ = ⧤ -
鼀҈
졔
鼀

㡰

҈ Þ
´ g ´҈ = K - ´
´ ҈
Þ K =
҈ 
ᆠ㡰 ᨨ鼀
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀 ur mu�ˆ +
= mvˆ + mv Ἐ ur u v v Ἐ = �ˆ - ˆ  

b㡰 꾀ᆠᆠᨨ� �
ᨨ
 
⼄盐矄�ᨨ


ᆠᨨ�矄�ᨨ 矄

 ᆠ泐 � 矄�ᆠ

矄ᨨ
 
 
� 
ᆠᨨ�矄�ᨨ


ᨨᆠ├矄

鼀ᨨ� � ᨨ� 
m

ᨨ鼀
├� �ᆠ
 �泐
ᆠᨨ 矄�ᨨ
æ æ u ö
æ v ö
ö ҈ ҈ ҈ mu
= mv
+ m㡰 ç ç ÷ + ç ÷ ÷ çè  ø è  ø ÷


è ø

ᨨ

mu = mv +

\v =
m

m v'

 ᨨ ᆠᨨ�矄�ᨨ

v
=   - ҈㡰$� -   - ҈㡰 $

m

m

x




�

v

m

m u

v҈ = $� +  $





v

y


m

꾀  ᆠᨨ�矄�ᨨ

mu
mv
+  

u



m

WWW.JEEBOOKS.IN

EBD_8341 PS-40

�


Physics

b㡰

盐⏌


黨


泐

ᆠ㡰





├� ҈


鼀

鼀

盐

鼀

盐⏌


҈


鼀⏌矄



졠






├�

 x㜠�ᆠ�ᨨ

mu u mu + =
mvἘx Þ VxἘ =

mu u Ἐ =
mv y Þ v Ἐy = 泐㜠�ᆠ�ᨨ

+



ù é ҈ ҈ æ uö æ uö K
E
� = m u
+ m ê ç ÷ + ç ÷ ú è ø è
ø ú

êë
û

�鼀
├
ᨨ
 
�ᆠ矄
ᨨ
ᆠᨨ�


 ҈

= =҈矄ᆠ  ҈

和
ᨨ
犨ᨨᨨ
盐矄
 ᨨ
ᆠᨨ�矄�ᨨ
㦨
�
㦨
҈

鼀⏌矄
  矄
ᨨ
盐
盐矄
 ᨨ
ᆠᨨ�矄�ᨨ
㦨


� 㦨
҈



҈

㦨
黨

鼀⏌矄 ҈ 和
㦨
҈


맔
҈


맔
҈

맔
҈
㦨
黨�
鼀
ᨨ犨

盐矄� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
盐矄
 ᨨ  
 
 
ᆠᨨ�矄�ᨨ





㡰
҈
㡰
졔




㡰
黨
㡰
㦨



�㡰v Þ
҈�

㦨
�v \

v = 

鼀⏌矄 㝐

�� 
ᆠ 
泐
ᨨ泐 
㦨


҈
mu
mu
mu + =




҈ ҈ K
쟼
f = (
m) vxἘ + (
m) v Ἐy

éæ u ö
æ u ö
ù � ҈




=

m êçè
÷ø + çè
÷ø ú =
mu êë úû \
ᨨ矄矄
�
KE
㦨
KEf

KE� =

( )

( )



㦨
=

 ᨨ



├� ҈

鼀⏌矄


æ
� ö mu = mu
ç - ÷ = è

ø









㡰 꾀泐
ᆠᨨ ᆠ�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
�

 
Ӹ �ᆠ�ᨨ
ᨨ
 
矄泐矄鼀
 

㝐
҈
ᆠᨨ矄
㡰
졔

㝐
�ᆠᨨ矄
� 㡰
㦨

㝐
҈ᆠᨨ矄
㡰 졔
㝐
ᆠᨨ矄
� 㡰

�  +�
=

 ҈ +









├�

҈







҈㡰




꾀矄ᨨ
㝐�矄�
� 㡰

㝐҈
矄� 
㡰
㦨

㝐҈矄� 


㝐
矄�
�

�
- � = ҈ -









㡰
和ᨨ� �
⼄盐�ᨨ
҈㡰
 

㡰  + ҈ ҈ = �  + ҈

- � Þ ҈ =

�鼀⏌矄

(

)



㦨



鼀⏌矄 䰘


b㡰 m҈v҈
졔
m
v

㦨
m҈v

졔
m
v҈ ᨨ





m҈v҈
졔


�m҈㡰v

㦨
m҈

�v҈
졔


�m҈㡰v


졠
矄ᨨ� �

v҈
㦨
v


v
㡰 矄� �
ᆠᨨ 矄�ᨨ
ᨨ
鼀ᨨ鼀 盐鼀
mv

㦨
鼀


鼀҈

҈
҈ 鼀҈ =


´ 鼀



Þ ҈ =






ᆠᨨ�矄�ᨨ
�矄
矄�ᆠ

和ᨨ


a

҈҈


҈

㝐


\ 㝐
㦨

鼀

Þ
鼀
= 鼀㝐

黨

鼀⏌矄




�
├�

\
�� 
ᨨ
ᨨ犨
㦨



鼀 ᆠ㡰 ⧤� �ᆠ
 �泐
ᨨ
ᨨᆠ├
꾀 ҈ ├҈ = 鼀


\
ᨨ鼀
� ᆠ�
ᨨ
� 
鼀ᨨ鼀 盐鼀
ᆠᨨ 矄�ᨨ

鼀
鼀
= (
鼀 + 㝐 )  Þ  =
鼀 + 㝐 � 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

ᨨ

 
� ��
├� �ᆠ
 �泐
�矄
ᨨ矄
�
犨 ᨨ
ᨨᆠ矄矄
҈ 鼀

├�
=
\ =

Þ
├ ҈ æ 鼀
ö (
鼀 + 㝐 ) çè
鼀 + 㝐 ÷ø

鼀 + 㝐 㝐 Þ =
\ =
鼀 鼀 꾀 
ᔴᨨ�矄�ᨨ

b㡰  ᨨ
ᔴᨨ�矄�ᨨ

鼀

꾀 
ᆠᨨ�矄�ᨨ

a

꾀  





㝐




҈ ҈ 㝐

=



´ 鼀

᲼\ 㝐
㦨
鼀矄矄
ᨨ
盐ᆠ盐矄桌

鼀 Þ 
= ´




鼀
= 鼀㝐 ´



 鼀 ´



㝐

Þ҈

鼀 =


鼀㝐


´ 
=
� 鼀
´ ҈


鼀

Þ

҈ 鼀


鼀

和�ᨨ 泐

҈
҈
æ҈ ö 鼀҈ + 鼀
= ç 鼀

÷


è
ø  Þ ҈
+ 

= 





�㡰
ᨨ鼀
鼀ᨨ鼀 盐鼀
ᆠᨨ 矄�ᨨ

鼀

㦨
鼀҈
졔

㡰



��㡰 和⼄盐� �
ᨨ
矄�矄
҈
졔

㡰

㦨


Þ



҈

졔



졔

҈

㦨




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PS-41

Work, Energy & Power



҈
= -

҈ - 
㡰
= 
҈ + 

-
҈
=

⧤
쟼

ᨨ矄
�矄
盐矄
ᨨ
盐鼀
ᨨ鼀
҈矄
ᨨ�
ᨨ


ᨨ� D⧤
쟼

㦨
҈


 㝐� �鼀盐鼀
⧤
쟼

ᨨ
盐ᨨ
ᨨ
� 矄�ᆠ
ᆠᨨ�矄�ᨨ

҈ 鼀
㦨
 ҈


㦨









+




和ᨨ� �
犨
�
�
ᨨᆠ�泐
犨
 
犨ᨨ
�ᆠ矄

҈ - 
= ҈



Ӹ 
㦨

鼀 





㡰 ᔴ  �
�
鼀ᨨ鼀 盐鼀

^  ˆ  ˆ  ˆ  ˆ + + �� D =



 ^

 ˆ   = ᔼ 鼀ᨨᆠ盐 D = ^
�
�

�  ^�
 ˆ   ^
 Þ ᔼ犨 =

 矄矄盐




Q = ᨨ
ᨨ �ᆠ矄㡰 㦨 = 꾀 꾀
´ 

´ ҈
-
䰘 ´ ҈
 ´҈

 =
㦨

�
맔
҈
⏌鼀

´ ҈
-
҈
㡰 ᨨ
ᆠᨨ�矄�ᨨ
ᨨ
盐ᨨ
犨�
盐�盐鼀
 ҈ 
鼀 鼀
鼀
鼀 꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
鼀ᨨ鼀 盐鼀

鼀
졔


㦨
鼀҈
졔

鼀





�㡰 


҈
㦨





��㡰 Q

ᔴᨨ�矄�ᨨ
�矄
矄�ᆠ


㦨
҈  ᨨ鼀
⼄
�㡰
 
⼄
��㡰
҈ = 

҈�


㡰


鼀

ᔼ ��
鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀

^�

�
㦨
᲼鼀
㡰
´
鼀
㡰
桌 㦨

鼀 ´
 � 
鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀
㦨
鼀 泐
 
犨
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
鼀ᨨ鼀 盐鼀



鼀v = 鼀
Þ

҈





v = ᆠᨨ鼀�  

ᨨ矄矄
�
 �泐 ҈ ҈ ҈
D쟼 = 鼀҈҈
+ 鼀


- 鼀҈ + 鼀
㡰ᆠᨨ鼀� 



� D쟼 = 鼀
- 鼀
= 鼀

㦨
��
��奠   ᆠ �
ᨨ矄矄
�
 �泐
盐� �
 
ᆠᨨ�矄�ᨨ
;
�
奠 㡰 ᨨ鼀
 
犨
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
鼀ᨨ鼀 盐鼀
犨
├ ᨨ犨  
鼀҈盐҈
졔
鼀
盐

졔





㦨
鼀҈҈
졔
鼀


졔




෌�
鼀҈
㦨
鼀

鼀

㦨

鼀
 
鼀
㦨
鼀  
盐҈
㦨
盐

盐

㦨

盐
 
盐
㦨
盐

Ӹ
鼀


盐




鼀҈

鼀҈ 졔 鼀
㡰


矄�


 鼀

盐








�

 �
ᆠᨨ矄


㡰
矄�


鼀

盐

ˆ
졔

鼀
맔

盐
-ˆ� ᆠᨨ矄

° - ˆ矄�

°㡰 鼀
맔
盐
�㡰 ® 졔

鼀
맔
盐 -ˆ� ᆠᨨ矄

° + ˆ矄�

°㡰
㦨
鼀
졔

鼀
졔
鼀㡰


҈ ҈ æ ҈҈ ö 鼀
- 鼀 ç  ÷



è ҈ ø = »



ᆠ = ҈
҈
黨 鼀


�ˆ


® æ  ˆö æ  ö �ˆ  ÷ - 鼀盐

+ 鼀盐 ç ˆ ÷ =
鼀  ÷ ç ÷
è
ø è
ø

Þ

鼀盐�ˆ -
鼀盐 -
鼀盐 ç ç

®

  ˆ 鼀盐�ˆ - 鼀盐�ˆ 鼀盐 =
鼀 

® ҈  ˆ 鼀盐 =
鼀  Þ - 鼀盐�ˆ

® 盐 ˆ - � - ˆ Þ  = ҈
҈䰘
b㡰 꾀矄


�矄

�矄
ᨨ

矄ᨨ
� ��
ᨨᆠ�泐
u
㦨

҈
҈ ҈


K.E. 㦨
mv
㦨
m [ u + at ] 㦨
m [
+ gt ]


҈
\
K
E = mgt \ K
E µ t



�㡰
Þ



҈  㦨 ҈ 졔 ҈㡰


® 㡰  ᨨ   㡰㡰 ® 꾀 

҈ ҈
æö
҈ 鼀 
鼀㡰 ᨨ矄矄
�
⧤
쟼

㦨
ç ÷ = 鼀



è ø 㦨




 
ᨨᆠ�泐
犨 
 泐
矄�ᆠ├
㦨




ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ


҈҈  ҈

㡰 ᨨ
� 矄�ᆠ
ᆠᨨ�矄�ᨨ
Ἐ 㦨


�

v

ᨨ犨

ᨨ
ᆠᨨ�矄�ᨨ
ᨨ
盐ᨨ
犨�
ᆠᨨ
盐ᆠ盐矄  ҈ 
鼀 鼀 ҈
鼀 ҈
鼀 꾀泐� �
ᔴᨨ 矄�ᨨ
ᨨ
鼀ᨨ鼀 盐鼀 mv
졔


㦨
mv1졔
҈
mv2



���㡰 
㦨



҈



�㡰 ᨨ鼀
⼄
���㡰
 
⼄
�㡰

҈



�

®

҈ ҈ 鼀
- 鼀҈



= =


黨  = ҈
黨 鼀


҈ = -

鼀
v

(

)

WWW.JEEBOOKS.IN

EBD_8341 PS-42

Physics

�.e.

�ᨨ
犨
k
 
t
�矄
ᨨ�ᆠ
�矄
 
├� �ᆠ
 �泐
犨�
� ᆠ矄
矄

⼄
�㡰

꾀矄
  矄
ᨨ盐ᆠ 矄
 
�ᨨ盐 
�
矄矄
 ᨨ鼀� �
 
ᨨ矄矄
�矄 ⧤
쟼


犨 
 
 ᨨ鼀�ᨨ
�矄
鼀 �鼀盐鼀
⧤
쟼

㦨



꾀矄
  
鼀ᨨ矄
盐
�
ᨨ矄矄
⧤
쟼

 
��
���ᨨ 
ᨨ �  �泐
�
 
矄鼀
�鼀
� 

矄
ᆠ ᆠ�矄�ᆠ矄
 矄
�盐矄
泐
kt
�
矄 ᨨ犨
�
㡰







ᆠ㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

犨
ᆠᨨ�矄�ᨨ

  �ᆠ矄
鼀ᨨ
犨�
ᆠᨨ 矄 
矄
꾀
�矄
ᆠᨨ�矄�ᨨ
ᨨ 
�ᆠ
� �
矄

v
犨�
ᨨ

맔
҈


㦨




犨 �
ᨨ 
�ᆠ
� �
矄
v
犨� ᨨ
҈



 ᆠ

�矄
ᆠᨨ�矄�ᨨ
├矄
ᆠ



꾀
�矄 ᆠᨨ�矄�ᨨ

 泐
犨�
 ᆠ  �
 �
ᨨᆠ��矄
矄
 
ᆠᨨ�矄�ᨨ

�矄
矄�ᆠ
 
 
�ᆠ矄

⼄盐
鼀矄矄矄

꾀��
矄ᆠᨨ  ᆠᨨ�矄�ᨨ
├矄
ᆠ

ᔴ


盐

矄�
a
g 盐矄� �



盐

㦨

� é u
矄�
a ù v҈
- u

=
- g 㡰 ê
ú ëê
g ûú

҈

㡰

Þ Þ

泐

a 鼀


҈


盐


鼀盐ᨨ
ᆠᨨ矄a

鼀

꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
�

Ӹ㜠�ᆠ�ᨨ


鼀
矄�
q
㦨
鼀҈
㦨
鼀盐
ᆠᨨ矄a


�㡰 꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
�

㜠�ᆠ�ᨨ


鼀
ᆠᨨ矄q
㦨
鼀盐

ᆠᨨ矄a


��㡰 ��� �
�㡰
 
��㡰
犨
� p 
q
㦨
҈










\
q
㦨

�
㦨

㡰 Fᨨr vert�cal mᨨt�ᨨn ᨨf bullet ᨨr ball 盐
㦨





㦨
�鼀





㦨
䁸




�
㦨
҈
鼀⏌矄
 鼀
㦨



҈
├�

�
鼀



ᔴ

ᆠᨨ�矄�ᨨ

r

�
㦨
�


鼀⏌矄


҈  ᆠ
� ��
ᨨᆠ�泐
ᨨ
 
盐


㦨
�


鼀⏌矄

r

r

r

Dp
㦨
F 
Dt
㦨
( m҈v҈ Ἐ+ m
v
Ἐ) - ( m҈v҈ + m
v
) 㦨
m҈
졔
m
㡰
g
맔

t




㡰  ᆠ

盐矄
 
ᆠᨨ�矄�ᨨ
犨�
A
ᨨᆠ�泐
ᨨ
B  m - m A 㡰u B
m A u A
+
m ´ 黨 vB = B + = =
鼀⏌矄 mB + mA m A + mB m +
m \

m


m

m

A

B

C


ᆠᨨ�矄�ᨨ
犨
B
 
C
�矄
ᆠᨨ鼀泐
� 矄�ᆠ \ mB vB
㦨
mB
졔
mc㡰
v


´
m =
鼀⏌矄

m + m

�㡰 盐
ᨨ
矄�ᆠ

ᨨ
ᆠᨨ�矄�ᨨ
ᨨ
⼄盐
鼀矄矄
m
ᨨ ᨨ
ᨨᆠ�泐

 
�� 矄
ᨨ� 
ᨨ
ᨨ
�
ᨨ
矄� �
l҈
�矄 ᆠ⼄盐�
泐
 
ᨨ
�
ᨨ
矄� �
l

l \
�l҈ = ��l
Þ ҈ =� l



҈

㡰
m v=

v


盐

҈ ҈ h = ut + gt

Þ
� = ´ ҈
´ t




















Þ 
㦨
҈
矄ᆠ ᨨ
hᨨ�ᨨ 
鼀ᨨ�ᨨ
ᨨ
b 
㦨


Þ


㦨

맔
҈
Þ

㦨



鼀⏌矄 ᨨ
hᨨ�ᨨ 
鼀ᨨ�ᨨ
ᨨ
b盐 盐
㦨
盐
맔

Þ ҈


㦨
盐
맔
҈
Þ
盐
㦨
҈


鼀⏌矄 꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
盐� � ᆠᨨ�矄�ᨨ
鼀
㦨
鼀盐
졔
㝐


҈

㦨



҈
맔
҈


졔




맔


\

㦨


҈




v ᨨ犨

矄
矄 ᨨ犨
�
��盐

 
犨ᨨ
ᆠᨨ�矄�ᨨ 矄
 泐
犨�
��

ᆠ

ᨨ� 
꾀
㡰 F 
㦨
m҈
졔
m
㡰g
 
Dt
㦨

t
Dp F 
㦨
Dt

Þ

㝐
㦨




├�




鼀 ҈


鼀

v

v


鼀

盐
矄�
a =

�

盐


҈



҈




v

҈矄
ᆠᨨ�矄�ᨨ

q
鼀
ᆠᨨ矄q



҈黨



v

v҈
= u

҈ - 矄�
a㡰 = u

ᆠᨨ矄
a ҈
㦨
盐

ᆠᨨ矄
a
鼀
矄� q

꾀

v

�� 

h
㦨


q q


鼀 v
⏌


v


m


�


㝐ᨨ鼀 盐鼀
ᆠᨨ 矄�ᨨ
ᨨ �
x
�ᆠ�ᨨ
v ҈
mv
ᆠᨨ矄 q =
m
Þ ᆠᨨ矄 q = ᨨ
q =

°

 ᆠ
 �
犨
 
� ��
ᨨᆠ��矄
ᨨ
 
犨ᨨ
ᨨ�矄 = q + q =

° +

° = ҈

°
҈



㡰 ᨨ鼀
鼀ᨨ鼀 盐鼀
ᆠᨨ 矄�ᨨ
�
 �ᆠ盐
�ᆠ�ᨨ

ᨨ
� ��
鼀ᨨ�ᨨ
mu҈ 矄� q҈ = ҈
mv҈ 矄� q



�㡰 ᔼ
�矄
��
 
 �泐
ᨨ
m
盐ᆠ
泐


ᔼ
u҈

ᨨᆠ�泐 ᨨ
m  
ᆠᨨ�矄�ᨨ

 

WWW.JEEBOOKS.IN

PS-43

Work, Energy & Power

   - = =    ␔ 

�矄
 
�矄 ᆠ

泐
 
�矄ᨨ
�
�鼀




鼀� 盐矄
矄��
� �ᆠ矄
ᆠ矄
�
�
犨�
�鼀


u æ ҈
ö ҈ ҈
Þ u҈ = çè mu ÷ø = mu҈



ᔼ
v ҈

 
ᨨᆠ�泐
ᨨ
鼀矄矄
҈

鼀
 
ᆠᨨ�矄�ᨨ

 

҈ ҈ u
u ´ ҈
m ´ v҈
= m Þ v҈ =




ᨨ鼀
⼄盐�ᨨ
�㡰

犨


\

 


矄� q҈ = ҈
矄� q





 ᨨ
ᆠᨨ�矄�ᨨ

v

v




v




ᨨ

 =








=


犨

   = 

⏌
҈ \ ⧤
쟼 
⏌
= 鼀

㡰

⧤
쟼 
⏌
҈ \ =
\ ⧤
쟼 
= 鼀

⧤
쟼 


黨



b


㡰 犨 
 =

졔





盐




盐

 ᨨ
ᆠᨨ�矄�ᨨ



҈









盐矄
 
ᆠᨨ�矄�ᨨ

v -v v + v
҈= ҈ ҈= v +u uv \
v҈
㦨
u
졔

v \
v

㦨
u


v \
Dv҈
㦨

u
졔

v  
Dv

㦨

u


v dp҈ = rAu + v 㡰
u +
v 㡰 ᨨ犨 F҈ = dt dp  
F
=
= rA盐 - 㡰
盐 -
㡰 dt \ F҈
㦨

rAu
졔
v㡰

 
F

㦨

rAu

v㡰
DF
�矄
 

ᨨᆠ
盐
ᨨ
 
�
鼀ᨨᆠ盐矄
ᨨ
 



꾀 
ᆠᨨ�矄�ᨨ

v

졔

v



  =-ò  

  \ l

= - l





҈


㡰

ᨨ
矄�ᆠ
ᆠᨨ�矄�ᨨ
KE�
㦨
KEf

҈ ҈ ҈ ҈ m ´
� + ´ m ´ 黨 = m ´ 
+ mvB





= 
+ VB
Þ VB =
҈ ҈ ҈ KEB = mvB
= ´

҈ ´
=

҈J = J

҈
\
x
㦨
҈
䰘




㡰

㝐 �鼀盐鼀
�� 
æ҈ö u
ç ÷ u
矄�
q è
ø 
㦨
Þ
҈


㦨

g
g \ u

㦨




� ᨨ
�� �
 
�ᨨ盐 

�
ᨨ矄矄

ᨨ
�矄
├� �ᆠ
 �泐 ҈
\ ⧤
쟼� ��
㦨
mu
㦨




鼀�
҈ ⧤
쟼 � 
㦨




鼀�㡰
㦨
҈


鼀�
҈
mv
㦨
҈


鼀� \


㦨




� \
꾀 
 
ᨨ盐 ᆠ

 
鼀 �鼀盐鼀
�� 
 

ᆠ 矄 æ҈ö


g ´ ç ÷ v
矄�
q è
ø \ ¢
㦨

㦨

㦨


鼀
g
g




㡰

ò

\

�矄ᨨ







␔ 
 
矄鼀
�ᆠ
鼀ᨨ� �
犨�
ᨨᆠ�泐
v
盐 �ᨨ矄

矄�ᆠ
ᆠᨨ�矄�ᨨ
犨�
 
泐
鼀ᨨ �矄ᨨ
鼀ᨨ� �
犨�
ᨨᆠ�泐
v

�
ᆠ⼄盐�矄

犨
ᨨᆠ�泐
v
졔

v

和ᨨ

 
� ᆠ矄
�
ᨨᆠ�泐
 
泐
ᆠᨨ�矄�ᨨ
�矄

v


�鼀
�ᨨ
ᨨ
ᆠᨨ�矄�ᨨ
犨 
 
�矄ᨨ
�矄



�矄 ᆠ �
ᨨ鼀
 
ᆠᨨ矄
  �矄 ᆠ


= = 矄  ␔ 
v
�矄
 
矄
ᨨ
 
�ᆠ

 
�鼀
\
⼄盐 ᆠ泐
ᨨ


犨 �ᆠ
 
�ᆠ
矄�├矄
 
�矄ᨨ

 㦨





ᨨ
ᆠ  �
ᨨ
矄
ᨨ
 
�ᆠ  =
㦨
 ⼄盐 ᆠ泐㡰
맔

  
  \  =






DF P= =
ruv ҈


DF = F҈ - F
=
rAuv

\
A



ᨨᆠ
Fnet
㦨
F

DF
㦨
ma \
F


rAuv
㦨
ma 

ᆠ㡰 꾀ᆠᆠᨨ� �
ᨨ
犨
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀 m҈u҈
졔
m
u

㦨
m҈v҈
졔
m
v
҈
맔
u҈
졔
�
맔


㦨
҈

㡰
졔
�
v
㡰 Þ
u҈
㦨



졔
�
v



�㡰


ᆠᨨ �ᆠ� 
ᨨ
矄�盐��ᨨ

v
- v҈ v
-  -
㡰
Þ ҈= u u҈ - u
҈
Þ
u҈
㦨
v

졔




��㡰 ᨨ鼀
⼄
�㡰


��㡰
u҈
㦨

鼀⏌矄
 
v

㦨
҈
鼀⏌矄  ᆠ
ᨨ
鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀
㦨

├�
鼀⏌矄
  ⧤
쟼
ᆠ鼀
㦨


䰘�
 e=

WWW.JEEBOOKS.IN

EBD_8341 PS-44

Physics

҈


㡰 ᨨ鼀
犨
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀


� ��
� 
鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀

p�ˆ - p�ˆ =
\
� 
� 
鼀ᨨ鼀 盐鼀
矄 ᨨ盐
矄ᨨ

ᨨ
�.e.

p҈¢
졔
p
¢
㦨

졠�ᨨ


p҈¢
졔
p
¢ = a҈ + a
㡰�ˆ + b҈ + b
㡰 ˆ + c҈kˆ
㦨
�  鼀ᨨ鼀 盐鼀
ᔼ
�矄
��
 
҈

҈

ᆠ҈






 
ᆠ


ᨨ 㜠ᨨ
盐矄
ᔼ
҈
㦨

 


㦨


꾀矄ᨨ
�
҈
㦨
泐
 


㦨
泐
 
 

ˆ�  
ˆ
ᆠᨨ鼀ᨨ  矄
ᆠᨨ鼀
ᨨ

盐
 
 �
鼀
� � ├ˆ
ᆠᨨ鼀ᨨ  
�矄
ᨨ 㜠ᨨ

�矄
��矄

 � �
�  鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀
犨 �ᆠ
�ᨨ矄
ᆠᨨ 矄�ᨨ
ᨨ � 
鼀ᨨ鼀 盐鼀

矄ᨨ
 �矄
�矄

犨ᨨ �
ᨨ�ᨨ
졠�ᨨ

p҈¢
졔
p
¢
㦨
a҈ + a
㡰�ˆ +
b҈ ˆ
¹


ᆠ盐矄
b҈
¹

ᨨᨨ犨� �
 
矄鼀
矄ᨨ � �
矄
ᨨ
 
ᨨ�ᨨ

�矄 矄ᨨ
犨ᨨ �


b

㡰
盐矄
 
 
ᆠᨨ�矄�ᨨ
ᨨ
C
犨�
A

C
矄ᨨ矄
  A
ᆠ⼄盐�矄

ᨨᆠ�泐
v
ᆠ盐矄
ᨨ
㜠ᨨ
矄�ᆠ
ᆠᨨ�矄�ᨨ

犨
� �ᆠ
鼀矄矄矄


␔ 
A
矄矄
鼀ᨨ� �
ᨨ犨矄
�� 

 
矄� �
矄盐 
 ᆠᨨ鼀矄矄�ᨨ
盐
ᨨ
犨 �ᆠ
B
矄ᨨ
矄矄
鼀ᨨ� �
ᨨ犨矄 �� 


ᆠᨨ鼀矄矄�ᨨ
ᨨ
 
矄� �
ᆠᨨ � 盐矄
�
 
�矄 �
ᨨᆠ�泐
犨
A

 
B

␔ 
 �矄
� ᨨᆠ�泐
ᆠᨨ鼀矄
ᨨ

ᨨ
A
 
B
鼀ᨨ
犨�
 
矄鼀 ᨨᆠ�泐
vἘ
 
 
矄� �
�矄
�

矄
ᨨ
鼀 �鼀盐鼀 ᆠᨨ鼀矄矄�ᨨ
矄泐
x
㦨
ᔴ

 

Ἐ

Ἐ

꾀



㝐
ᆠᨨ鼀矄矄�ᨨ


ᨨ鼀
� ᆠ�
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀 v mv
㦨
mvἘ
졔
mvἘ

Þ vἘ
㦨

\ ⧤
쟼

ᨨ
A

B
矄泐矄鼀

鼀 �鼀盐鼀
ᆠᨨ鼀矄矄�ᨨ





҈ ҈ mv
æQ v Ἐ = v ö mv Ἐ
+ mv Ἐ
= mv Ἐ
= çè ÷
ø


꾀泐� �
 �泐
ᆠᨨ 矄�ᨨ

҈
҈ ҈ ҈ mv = mv Ἐ
+ mv Ἐ
+ Kx




҈
҈
҈
Þ mv = mv + Kx


m ҈
҈
Þ x
㦨
v Kx = mv

\
K

㡰 
v¢

 
矄
ᨨ
ᨨ 
�ᆠ
ᨨ
矄 
 
  ᆠᨨ�矄�ᨨ
꾀矄
ᨨ 
�ᆠ
ᆠ矄
�矄


 
矄
ᨨ
 �矄
�ᆠ
盐矄  
 ᨨ矄�ᨨ
矄 ᨨ盐

v
ᆠᨨ矄
q  v

v矄� q q




h

D



ᔴ 꾀

꾀

A



㦨


ᔴ

꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀

 
�� 矄 ᨨ�  m m m
v
ᆠᨨ矄 q 㡰
㦨
´ v¢ - ´ v ᆠᨨ矄 q

\
v¢
㦨
v
ᆠᨨ矄
q 

ᆠ
㡰


ᔼ
� 矄�ᆠ
ᆠᨨ�矄�ᨨ
ᨨ 泐
鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀 鼀泐
鼀�
ᆠᨨ 矄
㡰 �矄
� ᆠᨨᆠ
ᆠ盐矄
 
鼀ᨨ鼀 盐鼀
ᨨ

ᆠ  �矄 �
鼀� �盐
矄
犨
矄
�ᆠ�ᨨ
㡰 �矄
� ᆠᨨᆠ
ᆠ盐矄
ᨨ
ᆠᨨ�矄�ᨨ

矄ᨨ鼀
鼀ᆠ  �ᆠ  �泐
�矄
ᆠᨨ 
� ᨨ


矄ᨨ盐 
 �泐
ᆠ㡰 �矄
ᆠᨨᆠ
ᆠ盐矄
ᨨ

졔

矄泐矄鼀
 
�鼀ᆠ ᨨᆠ
�矄

�  
ᨨᆠ
㡰 �矄
ᆠᨨᆠ

ᨨ
 �泐
ᨨ
 

 
 

�矄 ᆠᨨ 矄
�
b㡰 꾀矄
 
� ᆠ� 
 
�矄
�ᆠ�ᨨ 矄矄



⧤

쟼




㦨

쟼


A ҈

mv = mgh

Þ
v =
gh


졔 m





B

v
ᆠᨨ矄



v
矄�

 =
�


鼀

쟼

ᔼ
D
ADB





㦨
\ h
㦨

鼀

C

 鼀



\
v
㦨

� =

鼀 ⏌ 矄

�矄
�矄
 
ᨨᆠ�泐
ᨨ
 
ᨨᆠ├
盐矄
 ᨨ
ᆠᨨ�矄�ᨨ


�矄
ᨨᆠ�泐
鼀├矄

 �
ᨨ


犨�
 
�ᆠ
꾀矄ᨨ
�
�� 
 �
� �
BEC

ÐEBC㦨





 ᨨ
 鼀├矄

 �
ᨨ


犨�

 
矄ᆠᨨ  � ᆠ� 
 
BC


ᆠᨨ鼀ᨨ  
ᨨ

ᨨ �
BC
�矄
v ᆠᨨ矄


ᔼ
�矄
��
 
 
ᆠᨨ�矄�ᨨ

B
�矄
 ᆠ泐
� 矄�ᆠ   ᨨ
 
�鼀ᆠ
ᨨᆠ矄
ᆠ
ᨨ鼀
ᨨ
 
  矄盐ᆠ
 
 
�ᆠ
ᆠᨨ鼀ᨨ  
ᨨ
ᨨᆠ�泐
ᆠᨨ鼀矄 ᨨ


ᆠᨨ鼀ᨨ  
ᨨ
ᨨᆠ�泐
ᨨ �
 
� ᆠ�  BC
鼀� 矄
盐 ᆠ  �
 
�矄
⼄盐
ᨨ
v
ᆠᨨ矄



 ҈

= =
� 鼀 ⏌ 矄

҈ BE BE Þ = Þ BE = 鼀 

b㡰 ᔼ
D BCE

 

= CE    ᨨ鼀
鼀ᆠ  �ᆠ
 �泐
ᆠᨨ 矄�ᨨ
� ᆠ�
㝐ᆠ  �ᆠ
 �泐

B
㦨
鼀ᆠ  �ᆠ
 �泐

C
҈ ҈ M
� + M ´ ҈
´  = Mvc


㦨


´

( )

\
vc
㦨
҈
� 鼀 ⏌ 矄 䰘
ᆠ㡰 
ᨨᆠ�泐
ᨨ
 
ᨨᆠ├
ᨨ �
BC
盐矄
 ᨨ ᆠᨨ�矄�ᨨ
�矄

ᆠᨨ矄




�鼀ᆠ
ᨨᆠ矄
ᆠ
 �ᆠ盐 ᨨ
 
矄盐 ᆠ
矄ᨨ
 
ᆠᨨ鼀ᨨ  
ᨨ
ᨨᆠ�泐
ᨨ �
  � ᆠ� 
鼀� 矄
盐 ᆠ  �

�
졔



㦨
vc


vᆠᨨ矄q

vᆠᨨ矄q

v' m


m


vᆠᨨ矄q

WWW.JEEBOOKS.IN

PS-45

Work, Energy & Power

v
矄�

v

B


v
ᆠᨨ 矄






C

盐矄
 
ᆠᨨ�矄�ᨨ

꾀矄ᨨ
矄� ᆠ
 
ᆠᨨ�矄�ᨨ
�矄
矄�ᆠ

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�ᆠ
ᆠᨨ鼀ᨨ   ᨨ
ᨨᆠ�泐
v
矄�

㡰
 ᨨ
ᆠᨨ�矄�ᨨ
ᆠ  �矄
�
�ᆠ�ᨨ
 
鼀� �盐
鼀� � �
 
矄鼀
矄
矄 ᨨ犨
�
 
��盐
和ᨨ
 
ᆠ �盐
ᆠᨨ鼀ᨨ  矄
ᨨ
ᨨᆠ�泐
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ᆠᨨ�矄�ᨨ


矄
矄 ᨨ犨
�
 
��盐

�矄
鼀 矄
 
 
� 
ᨨᆠ�泐 ᨨ
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ᨨᆠ├
矄 ᨨ盐
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ᨨ�ᨨ 
鼀├� �
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 �

º
犨� BC

 ᨨ
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� 
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ᨨᆠ├
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ᨨ




㡰 㝐 �鼀盐鼀
 �泐
ᨨ矄矄
㦨





鼀
鼀 + 㝐㡰 é 
҈
ù = 鼀 ú êQ ⧤
쟼
=
鼀
ëê ûú


é 㝐 ù ҈ ì 㝐 ü = 鼀
í ý ê ú
鼀 ë 鼀 + 㝐㡰 û
î鼀 + 㝐 þ 和鼀 
ᔼᔼ
�矄

ᆠ矄
ᨨ
 ᆠ泐
� 矄�ᆠ
ᆠᨨ�矄�ᨨ
泐
ᆠᨨ鼀� �
 
⼄盐�ᨨ
��
�
矄鼀 
ᔼ
犨�
ᨨ ⼄盐�ᨨ

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�\ æ 㝐 ö æ 鼀 ö =ç
� 矄
ᨨ
÷ ç ÷ è 鼀 + 㝐ø è㝐+鼀ø  ᆠ
矄鼀 
ᔼ
�矄
犨ᨨ �
 
矄鼀 
ᔼᔼ
�矄
ᆠᨨᆠ
㡰 和鼀 
҈


ᨨ

矄�ᆠ
ᆠᨨ�矄�ᨨ

 
ᆠᨨ �ᆠ�  ᨨ
矄�盐�ᨨ
㦨
҈  v
- v҈  Þ 
v


v҈
㦨

u҈

u

 e
㦨
 u҈ - u
 Þ �
ᨨᆠ�泐
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ᆠᨨ�矄�ᨨ
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盐
�
 
矄鼀 
�
矄 �矄
��
和鼀 




� 
鼀ᨨ鼀 盐鼀
鼀� 矄
ᆠᨨ 矄
�


矄�ᆠ
ᆠᨨ�矄�ᨨ

�矄
矄鼀 
�矄
盐
ᨨ鼀
P
ᨨ
Q
෌�

u
㦨
҈

鼀⏌矄



 

m
㦨



�錐
s
㦨


鼀 é mg 矄� q + f ù é mg 矄� q + mmg ᆠᨨ矄 q ù = -ê a
㦨
- ê ú úû m m ë û ë =

黨







R

M uA



鼀

Q

N m mg
矄� q

q mg
ᆠᨨ矄q P mg f q

㦨

᲼g
矄� q
졔
Ҙg
ᆠᨨ矄 q 桌
㦨

g
᲼矄� q
졔
Ҙ
ᆠᨨ矄 q 桌 㦨

҈

᲼


�
졔



�
맔


黨黨桌
㦨



黨黨
鼀⏌矄
矄� �


v


u

㦨

as
Þ v

㦨
҈


졔





黨黨㡰
맔


Þ v
㦨


鼀⏌矄

 ᆠ
ᨨᆠ�泐
ᨨ
鼀矄矄
m
盐矄
 ᨨ
ᆠᨨ�矄�ᨨ
㦨


鼀⏌矄



ᨨᆠ�泐
ᨨ
鼀矄矄
M
盐矄
 ᨨ
ᆠᨨ�矄�ᨨ
㦨


鼀⏌矄
�� 㡰

v҈

 
ᨨᆠ�泐
ᨨ
鼀矄矄
m
 
ᆠᨨ�矄�ᨨ
 
v

  
ᨨᆠ�泐
ᨨ
鼀矄矄
M
 
ᆠᨨ�矄�ᨨ

ᨨy
ᨨ
鼀矄矄
M 鼀ᨨ� �
ᨨ鼀
Q
ᨨ
R
 
ᆠᨨ鼀� �
ᨨ
矄

꾀 
ᆠᨨ�矄�ᨨ
u
㦨
v
v
㦨

a
㦨



黨黨
鼀⏌矄
s
㦨


�


v


u

㦨

as
Þ 
㡰


v
㦨





黨黨㡰
맔


� Þ v

㦨
҈
䰘
鼀⏌矄 ᨨy
ᨨ
鼀矄矄
m
鼀ᨨ� �
ᨨ鼀
Q
ᨨ
P
 
ᆠᨨ�矄�ᨨ

h

S㦨 h
㦨


矄�
q
㦨


矄�



㦨


맔



� q P u
㦨
v҈ v
㦨
졔
҈鼀⏌矄 ⧤
쟼

졔

쟼
㡰� ��
㦨
⧤
쟼

졔

쟼
㡰 � 
졔
␔ �ᆠ�ᨨ

҈
҈ mv҈ + mgh = mv
+
+ m mgs

҈
҈ v҈ + ҈
´ 
´


�㡰 = ҈㡰
+


� ´ ҈
´


v҈
㦨

�
鼀⏌矄 \ ᔴᨨ �ᆠ� 
ᨨ
矄�盐�ᨨ

矄� q =

e
㦨


Q h

� ᨨᆠ�泐 ᨨ 矄�ᨨ

� ᨨᆠ�泐 ᨨ ᨨᆠ

-� - ҈
䰘 =




-
꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
 ᨨ
   
ᆠᨨ�矄�ᨨ
mv
졔
M
맔


㦨
m
맔
v҈
졔
m
맔
v


맔


졔
M
맔


㦨


맔

�㡰
졔
M
҈
䰘㡰

= ҈�


├� \ M
㦨
҈
䰘
҈
␔ 
 
矄ᨨ 
ᆠ 矄
 
ᨨ� 
B

 
ᆠᨨ鼀ᨨ  
ᨨ ᨨᆠ�泐
�
 
졔Z
�ᆠ�ᨨ
v
ᆠᨨ矄
q㡰
ᆠᨨ鼀矄
ᨨ
盐
ᨨ  
���ᨨ 
ᨨᆠ
�
 
Z
�ᆠ�ᨨ
㦨


Z

졔Z

졔Y v

vᆠᨨ矄q q v矄� q A vx 㦨

m t 㦨



L

B

v' Y

vx 㦨

鼀⏌矄

X t=t


矄ᨨ 
矄
犨ᨨ
ᨨᆠ��矄

B vx
�
 
졔X
�ᆠ�ᨨ


鼀⏌矄㡰 v
矄� q �
 
졔Y
�ᆠ�ᨨ


矄�


㦨

鼀⏌矄㡰 矄盐 
ᨨᆠ�泐
ᨨ
 
矄ᨨ 



vἘ
㦨
v x 㡰 + v 矄� q㡰
㦨

+  = �鼀 ⏌ 矄 �㡰 꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀

B
ᨨ ᆠᨨ�矄�ᨨ
犨�

鼀矄矄
ᨨ
⼄盐
鼀� �盐
和� ᆠ

 
ᆠᨨ�矄�ᨨ
�矄
ᆠᨨ鼀泐
� 矄�ᆠ
 
犨ᨨ
鼀矄矄矄 犨�
矄�ᆠ├
ᨨ� 

v
�矄
 
ᨨᆠ�泐
ᨨ
 
犨ᨨ
鼀矄矄矄
盐矄  
ᆠᨨ�矄�ᨨ
桌 m
맔
�
㦨

m
맔
v \ v
㦨


�
鼀⏌矄

WWW.JEEBOOKS.IN

EBD_8341 PS-46

Physics

��㡰 ␔ 
 
矄� �
�矄
盐 �ᨨ� �
ᆠ�ᆠ盐
鼀ᨨ�ᨨ




Ҙgd 㦨
-
Ҙgd �


��
矄��
矄 ᨨ犨矄
 
 
鼀矄矄
m
ᨨ盐 矄

ᔼ  
ᨨᨨ犨矄

ᨨᆠ�
鼀ᨨ�ᨨ
 
�矄
 矄
ᨨ�ᆠ
uy
㦨



sy
㦨
d

ay
㦨
g

ty
㦨
䁸 ᨨ
�ᆠ
鼀ᨨ�ᨨ




= - ´ �


mv
 泐
�泐
ᨨ矄��ᨨ

T


mg
ᆠᨨ矄
a
㦨
l 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

T
㦨


犨 
a
㦨
黨

mv
Þ v
㦨

l �.e.

�
 
ᨨ�ᨨ 
ᨨ矄��ᨨ

v
㦨

꾀泐� �
 �泐 ᆠᨨ 矄�ᨨ
ᨨ鼀
B
ᨨ
C
a \






㦨


҈
mv
=
mg l
















Þ l=


d ҈
҈ at
Þ d
㦨
gt


Þ t
㦨
g




ᨨ�ᨨ 
�矄 ᆠ

泐
鼀矄矄
m
盐� �
 �矄 �鼀
t ᨨ鼀

 犨矄

S
㦨
ut +

C

T

v


�㡰
= 㦨


҈

鼀
g
´ 黨





mg 矄� a

a



mg ᆠᨨ矄a


mg







d =
Ҙd

㦨

d µ g

ᔼ ��泐
犨 
 
ᨨ
ᨨ
 盐盐鼀
�矄

꾀

�矄

쟼

㦨
mgl ␔ 
 
ᨨ
矄
ᨨ鼀
꾀
 
矄�├矄
ᨨ
 
犨



쟼

ᆠ  �矄
ᨨ
⧤
쟼

 
犨 
�
�矄

ᨨ矄��ᨨ
�ᔴ
   �盐
鼀�盐
�矄



ᔼ
DOCM x
㦨

Ҙgd ´

B

졠ᆠ
ᨨ
鼀矄矄
m
ᆠᨨ�矄
犨�
ᨨᆠ├
B
ᨨ
鼀矄矄

m
和� ᆠ
 
ᆠᨨ�矄�ᨨ
�矄
矄�ᆠ
�
盐
泐� � ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀 mv
㦨

鼀㡰
u
졔
mv¢ 犨 
u
�矄
 
ᨨᆠ�泐
ᨨ
鼀矄矄

鼀
 
ᆠᨨ�矄�ᨨ
 
v¢
�矄  
ᨨᆠ�泐
ᨨ
鼀矄矄
m Þ v¢
㦨
v


u



�㡰 꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
├� �ᆠ
 �泐

҈
҈ ҈ mv = 
鼀㡰 u
+ mv Ἐ





m Þ v

㦨

u

졔
vἘ




��㡰 ᨨ鼀
⼄

�㡰


��㡰

v

҈
mvB


Þ


vB
㦨

gl

 
n
ᆠᨨ�矄�ᨨ 矄

 
 �盐
鼀�盐
�矄



犨 

 
ᨨ
��
鼀ᨨ矄
ᨨ犨矄
 
犨
ᨨ鼀
C

 
ᨨᆠ�泐 vἘB
 ᨨ
ᆠᨨ�矄�ᨨ

mg l =


m

A

꾀矄ᨨ


l OM Þ OM
㦨












l


ᨨᆠ�泐
ᨨ
ᨨ

B


ᆠᨨ矄



㦨


B


m


d

l ҈ = mv Ἐ
B


Þ

vἘB
㦨
gl



�矄
鼀 矄
 
 
ᨨᆠ�泐
ᨨ
 
ᨨ
矄 ᨨ盐
盐ᆠ


v v

㦨

u

졔
v


u㡰

Þ u
㦨
� ᨨᆠ├
B
矄矄
鼀ᨨ� �
盐
 
ᨨᆠ├
A
鼀� 矄

矄
꾀矄
 
�矄
ᨨ
�ᆠ�ᨨ
犨
A
 
B ᨨ
ᨨᆠ├
A
ᨨ
ᨨ

ᨨᆠ├
B
矄 ᨨ盐
鼀ᨨ

�矄 ᆠ

d

 
�ᨨ
ᨨ盐ᆠ
�
B
盐
ᨨ
�ᆠ�ᨨ
ᨨᆠ 犨
B
 
 


 F
㦨
mN Þ


m㡰a
㦨
m


mg㡰

Þ a
㦨
҈
�
Ҙg ᨨ
 
鼀ᨨ�ᨨ
ᨨ
B
u
㦨


mg

ᨨ鼀

gl
ᨨ



� 
ᨨᆠ�泐
 
n
ᆠᨨ�矄�ᨨ 矄
�矄
\

n


m҈
㦨
m

m

㦨

m

u҈
㦨
�
Ҙgd
u

㦨

Þ

҈ æ
ö = ᨨ� n
ᨨ�
ç

Þ n
㦨

҈ ÷ è �ø


v
=


m҈ - m
㡰 u҈
m
u
+ m҈ + m
m҈ + m


v҈
㦨


m -
m㡰�
Ҙgd +
m +
m

n

g l
Þ
æç
ö÷ = ҈ è �ø


├� �
ᨨ�
ᨨ
ᨨ
矄�矄
犨
�




v҈
㦨


gl

æ
ö çè ÷ø ´
g l = �


v
v
㦨



s
㦨



a
㦨

҈
�
Ҙg �

�
µgd
ᨨ
矄�ᆠ
ᆠᨨ�矄�ᨨ
犨
犨ᨨ
ᨨ�矄

en 
g l 㡰 =

gl

犨 
e
�矄
ᆠᨨ �ᆠ� 
ᨨ
矄�盐�ᨨ


æ
v ö ᨨ犨

v


u

㦨

as
Þ

㡰


ç ÷ =
 -҈
�Ҙg 㡰
d è �ø Þ

gl
盐
ᨨ
ᆠᨨ�矄�ᨨ 矄
犨�
犨矄







 ᆠ

盐鼀
ᨨ
ᆠᨨ�矄�ᨨ 矄
㦨



ᨨ� 
鼀矄矄
�
 
犨
犨�
 
矄
v




ᨨᆠ�泐
ᨨ
鼀矄矄
m

 �矄
� 矄 
㦨
v
ᆠᨨ矄
q
㦨





v
� 盐 
M
犨�


�矄 ᆠ
ᨨ
҈
鼀
犨 �
m
犨�
�矄
盐
泐  �

҈㡰
鼀
ᨨ鼀
 �泐
ᆠᨨ 矄�ᨨ

ᆠ矄
�

쟼

ᨨ
M
㦨
� ᆠ矄 �

쟼

ᨨ
m
졔
� ᆠ矄
�
⧤
쟼

ᨨ
ᨨ
 
ᨨᆠ├矄


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PS-47

Work, Energy & Power 꾀泐� �
鼀ᨨ鼀 盐鼀
ᆠᨨ 矄�ᨨ


҈
鼀

鼀 졠 ҈
鼀

㝐

mv

㦨
mv¢
졔

mv¢


\

v¢
㦨


v






��㡰

ᨨ鼀
⼄

�㡰
 
��㡰

q




h

v

-  ´

v
ᆠᨨ矄
q �
鼀

v q v
㦨
v
ᆠᨨ矄
q


䰘


h' 

�

҈㡰
鼀 鼀


mv
v

k

矄� �
ᆠᨨ 矄 㡰 = x



Þ
k
㦨
黨 m x



꾀
 
�� 矄
ᨨ� 
P

 
ᨨᆠ�泐
ᨨ
 
盐
㦨
u
ᆠᨨ矄
q ᔼ
D
AQR R �㡰
ᨨ鼀
��盐


QR � 矄�


㦨


QR
㦨
҈
⏌  

҈
҈
Mgh
㦨
mgh'
졔
mv + m ᆠᨨ矄 q㡰

҈ ᨨ

㡰
黨

㡰
҈㡰
㦨


�㡰
黨

㡰
 � - ҈㡰 + ´
´ v




졠

Y ҈
⏌

P

u
ᆠᨨ矄q

u

 R ,H 㡰






�


æ
v ö ҈ 졔
´

� ´ ç ÷
è �ø 和ᨨ� �
犨
�

v
㦨


黨
鼀⏌矄
ᔼ ��
ᨨ矄��ᨨ
ᨨ
ᔴ
㝐
m҈x҈ + m
x
҈

´
+ ҈

´ 黨
x҈
㦨
m + m
㦨

黨
鼀 㦨



҈
҈


�

㦨


ᨨ㡰 ᔼ ��
ᨨᆠ�泐
ᨨ
ᔴ
㝐

q O

Mu ᆠᨨ矄 q u ᆠᨨ矄 q =
M
ᨨ鼀
 �泐
ᆠᨨ 矄�ᨨ
� ᆠ�
K
E

ᨨ
 
盐㜠鼀矄矄 矄泐矄鼀

P
㦨
P
E

ᨨ
 
盐㜠鼀矄矄
矄泐矄鼀

R v
㦨


m҈u҈ + m
u
҈

´
黨 + ҈

´

㦨



�
鼀矄҈ m҈ + m

㦨



u҈ 꾀ᆠᆠ�ᨨ
ᨨ
ᔴ
㝐 ac
㦨
黨


鼀矄
ᔴ
㝐 �矄ᆠ鼀 
ᨨ
ᔴ
㝐
�矄 sc
㦨

黨
鼀 꾀泐� �
黨
鼀矄҈ ҈
S
㦨
ut
졔
at ҈


g
졠
ᨨ��� 㡰 
黨
㦨



�
t



黨
t
t


�t

҈

㦨

� ±
� +

� ±
� t
㦨

㦨

㦨


�
矄





ᆠᨨ�矄�ᨨ
犨
C
 
A
�矄
矄�ᆠ
 
 �
鼀矄矄矄 
⼄盐
矄ᨨ
 泐
犨�
 ᆠ  �
 �
ᨨᆠ��矄
A
犨矄 � ��泐

矄

  ᨨ
 
矄盐
ᨨ
ᆠᨨ�矄�ᨨ
犨�

  C
犨�
ᆠᨨ鼀
ᨨ
矄
 
A
犨�
� ��泐
矄
鼀ᨨ� �
犨�
 ᨨᆠ�泐
v


盐
矄� ᆠ
A
�矄
ᆠᨨ

ᆠ
ᨨ
B
犨�

矄� �
 
矄� �
犨�
�
ᆠᨨ鼀矄矄

v¢

 
ᆠᨨ鼀鼀ᨨ
ᨨᆠ�泐
ᨨ
A
 
B
m v

ᨨ

A m

B
m

㦨


C

A m

҈ 
M 㡰v
= 
M 㡰 gh


k
x
m

黨

´ � ´
´ ҈


Þ
q 㦨
䰘 �
´ �
Q
u
㦨
�

鼀⏌矄
�� 㡰 ��㡰 �ᆠ
ᆠᨨ鼀ᨨ  
ᆠᨨ矄
q 㦨


R u
矄�
q �
´ �
矄�
´ 䰘o = = = ҈



鼀

g
´ 黨

ᨨ�ᨨ 
ᆠᨨ鼀ᨨ  
H
㦨


u
矄�
q �
´ �
´ 矄� 䰘°㡰
= =

鼀
g
´ 黨








mV A

C

G

vἘ

҈
҈ ҈ ҈ mv
= mv Ἐ
+
mv Ἐ
+ kx





犨 
x

�矄
 
ᆠᨨ鼀矄矄�ᨨ
�
 
矄� �

t
㦨
t


v

= v Ἐ
+

҈ u
ᆠᨨ矄
q æ ҈
� ö 
M 㡰 =
Mg ´ ç + ÷ è  ø
҈


vἘ B
m

꾀
t
㦨
t


 
ᨨᆠ��矄
ᨨ
A
 
B
ᆠᨨ鼀
矄鼀
꾀泐� �
 �泐
ᆠᨨ 矄�ᨨ 錐

\





�㡰

X

ᨨ犨
ᨨ鼀

ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀

 
�� 矄 ᨨ� 
㜠
 M
u
ᆠᨨ矄
q㡰
졔
M
맔


㦨
M
졔
M㡰
v

uc
㦨


C

Q

A

B B

C

 ᨨ
ᆠᨨ�矄�ᨨ

mV

꾀 
ᆠᨨ�矄�ᨨ

 ᨨ
ᆠᨨ�矄�ᨨ
泐
矄泐鼀鼀泐

 

鼀ᨨ鼀 盐鼀
ᨨ
  矄泐矄鼀
�矄
ᨨ
꾀 
ᆠᨨ�矄�ᨨ

鼀ᨨ鼀 盐鼀
�矄
矄ᨨ
ᨨ
矄
ᨨ
矄 
ᨨᆠ �矄
ᆠ� �
ᨨ
 
矄泐矄鼀

꾀
ᆠᨨ鼀矄
ᨨ
矄
矄ᨨ

 
ᔴ
矄 ᨨ盐 
⼄盐
 
ᨨᨨ矄�
鼀ᨨ鼀 

矄ᨨ
ᨨᆠ�泐
ᨨ
C
矄 ᨨ盐 
矄鼀
�
鼀� �盐
�.e.

V
盐
�
ᨨᨨ矄�
�ᆠ�ᨨ
ᨨ ᨨᆠ�泐
ᨨ
B


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EBD_8341 PS-48


黨


Physics

꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀 ᨨ
盐
 

A

mv
㦨
mv҈
졔
m҈v



�㡰 ᨨ




v
㦨





v҈
졔
҈
맔
v


㦨


m
㦨





├�

v҈

v

Topic-5 : Miscellaneous (Mixed Concept) Problems

v


҈
v


ᨨ


꾀��
泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
ᨨ ᆠᨨ�矄�ᨨ

B

mv҈
㦨
m
졔
m
㡰v
ᨨ





v҈
㦨


黨

졔




㡰
v

㦨
v






v҈  ᨨ鼀
⼄

�㡰
 
��㡰 Þ v

㦨


t

\
s
㦨
v
ò e


v
= Þ v
㦨
v҈ Þ  v҈ 

t ⏌ t

R

奠
ᨨ矄矄
�
ᨨᆠ�泐



v
v
q = wt =
t R R $ vp
㦨
(  v
矄� q� + v
ᆠᨨ矄 qˆ )

pf
㦨
m
V
+ M
v


ᨨ

Mv ҈ Mv
Þ
a
㦨


mV mV 犨 �ᆠ
��矄
 
�ᆠ�ᨨ
ᨨ
 
鼀ᨨ鼀 盐鼀


y r pf a

æ è

vp
㦨
ç  v
矄�

x




éæ v
ö $ æ v
ö ù 㦨
m êç v
矄� t ÷ � + ç v
ᆠᨨ矄 t  v҈ ÷ ˆ ú ø û R ø è R ëè 㡰 ᨨ
ᆠ�ᆠ盐
鼀ᨨ�ᨨ
ᨨ
 


 
ᆠ矄矄泐 ᆠ �
ᨨᆠ
�矄
ᨨ�
泐
mg
ᆠᨨ矄 q 
N㡰
mg
ᆠᨨ矄 q 
N
㦨


Mvˆ

⧤
쟼 �

p
f m + M 㡰 ⧤
쟼 f ҈ 
㦨
҈ 
㦨
҈ ҈ ⧤
쟼� mV
+ Mv








㦨
҈ 





m V +M v

m + M 㡰mV
+ Mv
㡰

v
$ v ö t� + v
ᆠᨨ矄
tˆ ÷ R R ø

  vm
㦨
v҈ ˆ \ � 
鼀ᨨ鼀 盐鼀
ᨨ
�ᆠ
犨



鼀
矄

盐 ᆠ�ᨨ

ᨨ
�鼀 Lpm
㦨
mvp

vm㡰


a
㦨


mV�ˆ

x

t
㦨



w
㦨


� 
ᨨ泐㡰

��㡰


ds
㦨
v
et⏌t Þ
ds
㦨
v
e⏌t
dt dt

dt
㦨
v
t҈

e҈㡰
㦨


�
맔


맔




㦨




鼀

v - v҈ =


� v

⧤
쟼 �  ⧤
쟼 f

҈ J =
㦨


�
鼀矄҈ m




꾀 �盐
矄
ᨨ
�ᆠ
ᨨ盐
ᆠ 
ᨨ
 
ᆠ�ᆠ盐
 y v
v
q

҈ v - v҈
맔
҈


㦨
´ ҈

=
�奠
v �㡰
꾀ᆠᆠᨨ� �
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀 r r p f
㦨
p�
㦨
mV�ˆ + Mvˆ
㝐� �盐
ᨨ
 
鼀ᨨ鼀 盐鼀
ᨨ \



㦨









v҈  v  ҈ = Þ ҈ - ҈ = ҈ - = =


� v
v



\


㦨
鼀


Þ

꾀矄ᨨ
㡰
㦨
v
e⏌t \






��㡰





v҈




v
㦨





v҈
졔


�






㡰 ෌�

鼀矄矄
m
㦨




├�錐
�鼀盐矄

㦨
҈


矄
t㡰
㦨 v
et⏌t錐
t
㦨

矄
 
e҈
㦨


䰘
ᔼ鼀盐矄
㦨
ᔴ  �
�
� 
鼀ᨨ鼀 盐鼀

m

㦨


黨

├�

m ҈
㦨
҈
├�

Þ

m + M 㡰mV
+ Mv
㡰

DK
E
mM v
+ V
㡰
㦨
K
E� m + M 㡰mV
+ Mv
㡰

B

A

m
V
+ mMv
+ mMV
+ M
v
 m
V
 M
v


mv




�㡰




dö æ R + çè ÷
ø

B C A D

N
㦨
ᨨ鼀
ᆠ�ᨨ

 �
q 꾀ᆠᆠᨨ� �
ᨨ
 �泐 ᆠᨨ 矄�ᨨ

R

mgᆠᨨ矄q

q

NA

d⏌


q

V

mg矄� q

mg

dö ҈
æ mv = mg ç R + ÷ ҈ - ᆠᨨ矄 q㡰 è

ø dö æ Þ v

㦨

� ç R + ÷ ҈

ᆠᨨ矄
q㡰
ø è

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PS-49

Work, Energy & Power 盐� �
 �矄
盐
ᨨ
v

�
⼄

�㡰 N
㦨
mg

ᆠᨨ矄 q 

㡰 㡰 

犨�
ᨨ矄
ᆠᨨ ᆠ
犨�
 
�


矄 
犨 

N
㦨



æ
ö è ø 꾀 
 �矄
�
鼀├矄
ᆠᨨ ᆠ
犨�
ᨨ盐
矄 
 
ᨨ鼀 ᆠ�ᨨ
矄矄
ᆠ� �
ᨨ犨矄
 
ᆠ 


ᨨ


ᆠᨨ矄
q



㦨



ᨨ

q
㦨
ᆠᨨ矄҈
ç ÷

和ᨨ
ᨨ

æ
ö è ø

q £
ᆠᨨ矄҈
ç ÷

\ ux
㦨
҈


ᆠᨨ矄


졔
� 
㦨
��  \ 꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
ᨨ
  ᆠ

ᨨ
㜠ᨨ泐
矄泐矄鼀
�
ᨨ�ᨨ 
�ᆠ�ᨨ
mux
졔
M
맔


㦨
m
졔
M㡰
vx

mu x 犨 
m
㦨
鼀矄矄
ᨨ
ᆠ

ᨨ

m+M 

M
㦨
鼀矄矄
ᨨ
ᨨ泐

vx
㦨
ᨨᆠ�泐
ᨨ
 
ᆠ

ᨨ
㜠 ᨨ泐
矄泐矄鼀㡰

\

vx
㦨


҈´ ��  = �
�  鼀矄 ҈ ҈+ 黨 ᨨ�ᨨ 
�矄 ᆠ
ᆠᨨ
泐
 
ᆠ

\

NB
㦨

NA
㦨
mg

ᆠᨨ矄
q


㡰

 

ᨨ�ᨨ 
ᆠᨨ鼀ᨨ  
ᨨ
ᨨᆠ�泐
ᨨ
 
ᆠ

ᨨ
 鼀� 矄
 
矄鼀

æ
ö   ᨨ q ÷ è ø NA
㦨

  NB
㦨
mg




ᆠᨨ矄
q㡰


NA

ᆠᨨ矄
q
 
NB

ᆠᨨ矄
q
� 矄

矄
ᨨᨨ犨矄
³
ᆠᨨ矄҈ ç

NB �
鼀�

NA


vx 㦨


P 㦨
҈

맔
�  =

 鼀
Q
和ᆠᨨ 

犨矄 ᨨᆠ
 
҈

矄ᆠᨨ 
㡰 和� ᆠ
 
矄ᆠᨨ 

矄ᨨ
矄盐ᆠ├
 
ᨨ泐
\
�
�鼀
҈

矄ᆠᨨ 矄

 
ᨨ泐
ᆠᨨ矄

�矄 ᆠ
ᨨ



 鼀
ᨨ
ᨨ泐
 
҈

矄ᆠᨨ 矄錐

u = �  鼀⏌矄



v
㦨
䁸
t
㦨
҈

矄 m

mg

s
㦨


 鼀


鼀�

҈






⏌

ᆠᨨ矄
q

졔҈

҈


⏌ 졔҈

s
㦨
ut +

ᆠᨨ矄
q

ᔴᨨ 矄�
 
�ᆠ
鼀ᨨ�ᨨ
ᨨ
 
ᆠ

ᨨ
 uy
㦨

졔
҈


矄�


�ᆠ
ᆠᨨ鼀ᨨ  
ᨨ
ᨨᆠ�泐㡰 sy
㦨

҈


鼀 �ᆠ
�矄ᆠ鼀 㡰
ay
㦨

҈

鼀⏌矄

A

m 㦨
黨├� B

u 㦨

T

t 㦨
t

 �鼀
ᨨ
�矄
�鼀ᆠ
ᨨ
ᆠ��㡰 ҈





sy
㦨
uy t + a y t


\

҈


㦨
�

t


� t


Þ \ \

t


҈

t





㦨



-҈
㡰 ± ҈

-
҈㡰 -

㡰 㦨
҈

ᨨ



᲼ᨨ
�桌 t

㦨


t

㦨
҈

矄ᆠ


҈
at


҈

 㦨
�
�  맔
҈

졔

맔
a
맔
҈




\


a
㦨
  ⏌ ҈
鼀矄

\
v
㦨
u
졔
at
㦨
� 
鼀⏌矄


æ ö v
㦨
u
졔
at

㦨
�
�   çç ÷÷ ´҈
è ҈
ø

ux

C

ux

Þ

졔 P uy u 
ᨨ 힘

҈

m

M

v
㦨


�  鼀矄 ҈

ᨨ
� 
 
� 
ᨨᆠ�泐
vf
ᨨ
 
ᆠ��
 
 
矄ᆠᨨ  �鼀ᆠ
꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
�
  ᨨ�ᨨ 
�ᆠ�ᨨ

mux
졔
M
졔
m㡰
vx㦨
M
졔

m㡰
vf \

҈
맔
�� 
졔
黨
졔
҈㡰


� 
㦨
黨
졔

㡰
vf vf
㦨
҈�
䰘�
鼀⏌矄

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EBD_8341 PS-50

Physics

System of Particles and Rotational Motion

6

㝐矄矄
ᨨ
鼀� � �

Topic-1 : Centre of Mass, Centre of Gravity & Principle of Moments ҈




pR r  p҈㡰 r   ᔴ 
ᨨ
鼀矄矄
ᨨ
鼀� � �

M r + M
r
X ᔴ졠㝐 = ҈ ҈ M҈ + M
M  鼀� � �㡰 =

b㡰 ෌�


æxö ÷ èLø

� 
鼀矄矄
 矄�泐

r x 㡰 = a + b ç

X CM =

ò xdm ò dm




é
 ù é
ù  êë  pR rúû
+ êë  p҈㡰  r㡰 úû ᲼ R ҈桌 Þ 
 R 㡰 =

pR r + p҈㡰 r㡰  

L

ò dm = ò r x㡰dx
Lé

æxö = ò êa + b ç ÷ èLø ê
ë





Þ


ù

bL ú dx = aL +  úû L Læ æ aL
bL
ö bx  ö xdm = ax + dx = + ç ÷ ç ÷ ò òç ç

÷ø L
÷ø è

è æ aL
bL
ö + çç ÷

÷ø è \ X ᔴ㝐 = bL aL +  L æ
a + b ö Þ X ᔴ㝐 = ç ÷
è a + b ø b㡰

 R ҈㡰 

 R ҈㡰

=
 R

Þ

 R ҈㡰  R ҈㡰 R
+ R + ҈㡰

=
R

Þ
R

졔
R
졔
҈㡰



R㡰
㦨
҈



㡰

Xcm =

X cm

m҈ x҈ + m
x
+ m x m҈ + m
+ m ҈ ´
+ ҈
� ´  +

� ´
҈
� ´  = = =

黨cm ҈ + ҈
� +

� �

m y + m
y
+ m y Ycm = ҈ ҈ m҈ + m
+ m ҈ ´
+ ҈
� ´
+

� ´


� ´
Ycm = = =
cm ҈ + ҈
� +

� �  ᆠ

ᆠ 
ᨨ
鼀矄矄
ᨨ
矄泐矄鼀
�矄

ᨨ� 


黨


㡰 ᨨ
��
鼀�  x y m҈
㦨
҈

C҈
㦨
҈
�



�㡰 m

㦨

C

㦨


�

҈
�㡰 m x + m
x
҈
� + ҈
� = =

䰘� X cm = ҈ ҈ m҈ + m



m҈ y҈ + m
y


� +

� = = ҈
䰘� m҈ + m

\
ᔴᨨᨨ� 
ᨨ
ᆠ 
ᨨ
鼀矄矄
ᨨ
�
矄 
鼀�  

䰘�

҈
䰘�㡰 㡰 㝐矄矄
ᨨ
矄 
㦨
ᨨ盐鼀
ᨨ
矄 

 矄�泐
ᨨ 矄 

�


=


 p r 

㝐矄矄
ᨨ
ᆠ�泐
M ᆠ�泐 =


p҈㡰 r 


m㡰aˆ + m ´ a�ˆ + ma -�ˆ㡰 +
m ´ a- ˆ 㡰
m + m +
m + m
a�ˆ -
aˆ a ˆ ˆ = = � -  㡰 ҈
� 㡰 ␔�
矄ᆠ
ᨨ
ᨨ� 
q

 
ᔴ㝐
ᨨ
 
ᆠ盐㜠ᨨ
ᨨ�ᨨ

æa bö ç
÷

è

ø MX - mx 矄� �

xᔴ㝐 = M -m M a M ´
- ´

=- a = M ҈
M
=

Ycm =




㡰 꾀ᆠᆠ�ᨨ
ᨨ
ᆠ 
ᨨ
鼀矄矄
aᆠ鼀㡰
�矄
��
泐 r r m a + m
a
+







r \ aᆠ鼀 = ҈ ҈ m҈ + m
+













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PS-51

System of Particles and Rotational Motion

b ҈
和ᨨ
ᔴ㝐
ᆠᨨᨨ� 矄
ᨨ    yCM = -

҈



a a �a - =
҈
҈
b b �b =  
y
=
҈
҈
ᆠ㡰 ᨨ
ᨨ盐ᆠ
鼀 �鼀盐鼀
鼀ᨨ鼀 
ᨨ
ᨨᆠ
� 
ᨨ
ᆠ�ᨨ

ᨨ
ᨨᆠ
鼀盐矄

 �ᆠ盐
ᨨ
� 
꾀


ᆠ㡰 ᔴ 
ᨨ
鼀矄矄
xᆠ鼀

x
=

䰘


꾀 q





黨


҈҈


h D

B

O

C

h

C

h

h'

B

P

B D = h' - h = OB - OP = OC - OP ҈ æ ö =
h - h = h
- ҈ = h -҈ ç æ pö ÷ ç ᆠᨨ矄 çè ÷ø ÷ è ø n

(

)

 ᆠ
ᨨ
 泐
盐
ᨨ
n
D = h

҈ æ ö -҈ ç æ pö ÷ ç ᆠᨨ矄 çè ÷ø ÷ è ø n

꾀

ydy

h

y




�㡰

a

dy h

r 

R

ᔴ

h

é y
ù ê ú  ë
û
ycm
㦨


㦨
h
㦨
 
h h ҈

b㡰 ᔴ 
ᨨ
鼀矄矄
ᨨ
 
ᨨ
�矄
��
泐
 ò  y dy

ò ax +


L

bx
㡰 dx L bx

ò a + L 㡰dx



A O

h


ò r
㦨



ò ydm ò dm

҈
҈
pR hr R h   ᨨ

ᆠᨨ 

犨
├ ᨨ犨
  r y y =



\
r
㦨
R R n n

xcm =

B

D D

dyr

L

n
㦨
㡰 OD 㦨
h
 
矄ᨨ
O
�矄
 
ᆠ ᨨ� OA h Ἐ
= = Q OA = OB = h Ἐ㡰 \ OD h ҈ Þ h'
㦨

h æ ҈ ö æ ҈ ö D = h Ἐ- h =
h - h = h ç -҈ = h -҈ ç p ÷ è ҈ ⏌
÷ø çè ᆠᨨ矄 ÷  ø ᨨ犨
ᆠᨨ 矄�
矄⼄盐
n
㦨

㡰


h

㦨








h'

O

C

ò y pr

ᔴ

泐

㡰 
 矄�泐
ᨨ
ᆠᨨ 
㦨
r


h

O







㡰

q


҈ \  q = =

ᆠ㡰 꾀ᆠᆠᨨ� �
ᨨ
� ᆠ�
ᨨ
鼀ᨨ鼀 矄
犨 

矄泐矄鼀
�矄 矄
ᨨ
 ᆠ

 
 �㜠ᆠᨨᆠ├犨�矄
鼀ᨨ鼀 
�矄
⼄盐
ᨨ ᆠᨨᆠ├犨�矄
鼀ᨨ鼀 
�




ᨨ
맔
ᨨ
鼀
㦨
 ᨨ
맔
 ᨨ
鼀 ␔ 
�
鼀�
犨�� 
�矄
ᆠ

ᨨ
鼀
矄 � 矄
ᨨ
 
矄�
 ᆠ
 
鼀
ᆠᨨ鼀矄
矄 ᨨ
 
�� 
鼀
b㡰 ᔴᨨ 矄�
矄�鼀矄
ᨨ泐�ᨨ
�



� � A A C

꾀

Bc y ҈ +  \ = = ҈
䰘 AB x






鼀

æ x ö҈ rx 㡰 ç ÷ + ry y ⏌
è
ø
r x + y 㡰

҈ y y
+ =
x x


Þ

ᔴ 
ᨨ
鼀矄矄

ycm
㦨




鼀

x =


A

aL bL
L æ a + b ö + ç ÷  = è
ø =




bL b aL + a+

a b + 䰘L
 = ᨨ犨
b ҈
a+
졠
矄ᨨ� �
犨
�

b
㦨

a ҈
ᆠ㡰 ҈

㡰 ҈

㡰 
犨� �
��
�
 
��盐
�矄
鼀
盐
ᨨ
� ᨨ�矄
�



 
ᆠ�ᆠ矄
 
犨ᨨ
矄�� 
� 
ᨨ
盐 � ᨨ鼀 鼀矄矄
�矄�盐�ᨨ
ᨨ
犨
ᆠ
矄矄盐鼀
 
矄泐矄鼀
ᨨ

鼀 盐
ᨨ
�
ᨨ� 
鼀矄矄矄
犨 
 
鼀矄矄
ᨨ
ᆠ

�矄 ᆠᨨ 矄�


�矄
�ᨨ鼀�ᆠ
ᆠ 
Y a
a
㡰 m

m

Y

m 



㡰




a㡰 m

a
a
㡰 m

a
a
㡰 m

a
a
㡰

X



m





㡰

X

m 


a㡰

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EBD_8341 PS-52

҈�


҈



Physics


泐㜠ᆠᨨᨨ� 
ᨨ
 
ᆠ 
ᨨ
鼀矄矄
�矄 m y + m
y
+ m y + m
y
+ m� y� ycm = ҈ ҈ m҈ + m
+ m + m
+ m�
m ´
+ m ´
+ m ´ a + m ´ a + m  - a 㡰 \ ycm =
m + m + m + m + m ma a =







= ҈
m ҈
㡰 㝐盐盐
ᨨᆠ
ᨨ
ᆠ�ᨨ
犨�
犨 �ᆠ
犨ᨨ
�ᆠ矄
꾀  

鼀ᨨ
ᨨ犨矄
ᆠ
ᨨ 
�矄

�  
ᨨᆠ

 
ᨨ
  
ᨨᆠ矄
ᆠ� �
ᨨ
 
矄泐矄鼀
␔
├ ᨨ犨
F 
㦨
Maᆠ
鼀
aᆠ
鼀

㦨



Þ vᆠ
鼀

㦨
ᆠᨨ 矄 
和� ᆠ

� ��泐

vᆠ
鼀

㦨

\


� 
ᆠ鼀
㦨

㡰 ᔴ 
ᨨ
鼀矄矄
ᨨ
矄ᨨ�
鼀�矄 
ᨨ
�盐矄
R
�矄


R
ᨨ
 
ᆠ 
ᨨ

矄�
ᨨ
鼀�矄 

R  ´
\ hᆠ鼀 = = =  ᆠ鼀





㡰



s

 
鼀矄矄
 矄�泐
ᨨ
ᆠ�ᆠ盐
�矄ᆠ
졠��� 
鼀矄矄
ᨨ
 
�矄ᆠ

m
= pa
s 
�矄 ᆠ


҈䰘


a
s
æ a
ö æ
p - ҈ö 鼀� � �

鼀矄矄

m Ἐ = ç pa
- ÷ s = a
ç s è
÷ø
è ø Y 鼀ᨨ
鼀矄矄

m =

a
a


X ҈

犨
ᨨ矄��ᨨ
ᨨ
ᆠ 
ᨨ
鼀矄矄

a
a ´ m x - mx

X CM =

= m
- m a
pa

a -a  ⏌
-a -a = = = =



æ p - ҈ ö a


p - ҈㡰
p -
çè ÷
ø \ x =
  RA
㦨
ᨨ鼀
ᆠ�ᨨ

A RB
㦨
ᨨ鼀
ᆠ�ᨨ

B ᨨ
 
ᨨ
ᨨ

�
⼄盐���盐鼀
�
ᨨ�ᨨ 
ᨨ矄��ᨨ

  RB 鼀ᨨ鼀 
ᨨ

 
ᨨᆠ矄 R A x  d  x 㡰 ᨨ盐
A
ᨨ
ᨨ盐
B
矄 ᨨ盐 
ᨨ C B \ 㝐ᨨ鼀 
ᨨ
ᨨᆠ矄
ᨨ盐
A
A pa
´
-

҈



W 㦨
犨�� 
ᨨ
ᨨ㡰 xW RB = d

xW
㦨
RBd ᨨ

꾀��

鼀ᨨ鼀 
ᨨ
ᨨᆠ矄
ᨨ盐
B,

æ d - xö RA = ç W






��㡰 è d ÷ø xW æ d - xö \ ᨨ鼀
ᆠ�ᨨ
ᨨ
A = ç W
 
ᨨ
B =
è d ÷ø d ҈黨
矄


�矄
ᨨ
  
ᨨᆠ
�矄
ᆠ� �
ᨨ
 
犨ᨨ
�ᆠ 矄泐矄鼀 和ᨨ
ac.m 㦨


 
 ᆠ
Vc.m
㦨
ᔴᨨ 矄 



b


ᆠ㡰 
 
ᨨᆠ├
ᨨ
鼀矄矄
㝐
鼀ᨨ矄
泐
�矄 ᆠ
ᨨ犨矄
 
Mx
㦨
mR

x㡰 mR Þ x= ᨨ犨矄
  M +m mR \ x= M +m ᔼ

�矄
 
ᨨᆠ�泐
ᨨ
鼀矄矄
Ἐ鼀Ἐ
矄
�
矄
 
ᨨᆠ├
 
 �矄
 
ᨨᆠ�泐
ᨨ
ᨨᆠ├

 
� 矄 
 
ᆠᆠᨨ� �
ᨨ ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀 鼀

㦨
㝐 泐
 �泐
ᆠᨨ 矄�ᨨ

҈
҈
鼀�
㦨
mv + MV

d 
x㡰W = RAd

ᨨ

m
gR
gR
 


㦨
m m M ҈+ ҈+ M M
҈


ᆠ㡰
␔ 
� �
矄ᨨ
盐
 
ᆠ泐ᆠ
�矄
�
鼀ᨨ�ᨨ
  犨 矄
鼀ᨨ
�
 
�ᆠ�ᨨ
矄盐ᆠ
 
 
ᆠ 
ᨨ
鼀矄矄 ᨨ
 
犨 
鼀ᨨ
ᨨ犨

ᨨ� �
�ᆠ�ᨨ
犨�
ᆠ
�
  ᨨᨨ矄�
�ᆠ�ᨨ
ᨨ
 
�
鼀ᨨ�ᨨ
ᨨ
 
ᆠ 
ᨨ 鼀矄矄
ᨨ
 
ᨨ泐
犨�
矄ᆠ
ᨨ
�ᨨ盐 

 ᆠ
 
ᨨ� � �ᆠ�ᨨ
犨�
ᆠ
�
ᆠ├犨
�ᆠ�ᨨ
�
ᨨ
 
犨 矄
盐� �
� �
矄�� �
�ᆠ�ᨨ
犨�
ᆠ
�
 
ᨨ犨 �ᆠ�ᨨ
ᨨ

犨 



㡰 
�矄
ᨨ
  
ᨨ⼄盐
ᨨ

ᨨ泐
ᨨ盐
�矄
ᆠ 
ᨨ 鼀矄矄

矄ᨨ
ᨨ

�矄
ᆠᨨ 矄 

ᨨ
ᨨᆠ�泐
ᨨ
ᆠ 
ᨨ
鼀矄矄
ᨨ 鼀�
ᆠᨨ 矄 
 

ᨨᆠ
ᆠ� �
ᨨ

ᨨ泐
鼀盐矄

ᨨ



� 
鼀ᨨ鼀 盐鼀
ᨨ

�矄ᨨ
矄泐矄鼀
鼀� 矄
ᆠᨨ 矄 


�
� 
ᨨ
矄 
矄 矄
� ��
矄

ᨨ
� 
ᨨ 矄 
矄 矄
� 
矄
和ᨨ� �
犨
�



㦨


Y
R

4m O

LἘ㜠� R

L

m (L,O) (L',O) R

X

L'+�R

␔ 
矄鼀
矄 
M
ᆠ  �矄
�矄
ᨨ矄��ᨨ
ᨨ
ᨨ 
 鼀 ᨨ矄��ᨨ

 
�矄
ᨨ
  
ᨨᆠ
�
 
ᨨ�ᨨ  �ᆠ�ᨨ

 ᨨ
 
㜠ᆠᨨᨨ� 
ᨨ
ᆠ
鼀

犨�
ᨨ
ᆠ  �
᲼xᆠ
鼀
桌� ��
㦨
᲼xᆠ
鼀
桌 �  M҈ x҈ + M
x
M҈ x Ἐ҈ + M
x Ἐ
Þ = M҈ + M
M҈ + M



m ´ L + m ´ � R + L 㡰
m ´ L Ἐ+ m ´  L Ἐ- � R 㡰 =
m + m
m + m Þ �L
졔
�R
㦨
�LἘ
�R Þ �L
졔
҈
R
㦨
�LἘ Þ L
졔

R
㦨
LἘ Þ

W





�㡰

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PS-53

System of Particles and Rotational Motion






꾀矄
 
� ��盐
ᆠ 
ᨨ
鼀矄矄
ᨨ
 
犨ᨨ
矄 矄 矄

y
ᆠᨨ㜠ᨨ� 
ᨨ
�
�矄
� ��
矄
 
�矄
� 
矄
矄ᨨ

  y㜠ᆠᨨᨨ� 
ᨨ
ᆠ
鼀

ᨨ
 
犨ᨨ
矄 
矄泐矄鼀
犨�
鼀�
ᨨ
 ᆠ

 
ᆠᨨᨨ� 
ᨨ
ᆠ 
ᨨ
鼀矄矄
ᨨ
���
矄  L
졔

R


㡰
�㡰 
ᨨ
矄 矄
�ᆠ
ᨨ �
 泐
泐
 �矄

␔ � ᨨ� �

 
ᨨᆠ
ᆠ� �
ᨨ
 
ᨨ

�矄
犨�� 
 

ᨨ鼀
ᆠ�ᨨ

矄
ᨨᆠ矄

�ᆠ
ᨨᆠ矄
  ᆠ

ᨨ
ᆠ

ᨨ�ᨨ 
鼀ᨨ�ᨨ

 ᆠ
 
ᆠ 
ᨨ 鼀矄矄
鼀ᨨ矄
�ᆠ泐
ᨨ犨 犨矄

盐矄
 

ᨨ
  ᆠ 
ᨨ
鼀矄矄
ᨨ
 
ᨨ
�矄

矄�� 
� 
��㡰 ᔴᨨ 矄�

�泐
ᨨ� 
P
ᨨ
 
ᨨ
ᨨᆠ
 x

y㡰
 


�矄 ᆠ
r
ᨨ鼀
 
 
B


ᨨ
鼀├矄


 �
q
犨�
 
ᨨ�ᨨ 

 �矄
ᨨ矄��ᨨ
y ᔼ
D
BNP

矄� q =



�㡰 A Y r x + BN x + r ᆠᨨ矄 q ᆠᨨ矄 q = = C L⏌
L⏌
P
x
y㡰 x y r Þ ᆠᨨ矄 q = q




��㡰 L O N B X -r x


Q
矄� q + ᆠᨨ矄 q = ҈

y



�


m

r


+






㦨





㦨


s p R
- s p r
- 
҈㡰
´ 䰘




㡰 - 
҈㡰




=-


҈ ´
҈ ´ 䰘 = -黨 ᆠ鼀 䰘 ´
黨

�.e.

ᆠ 
ᨨ
鼀矄矄
�矄


�矄 ᆠ
ᨨ
黨
ᆠ鼀
ᨨ鼀
 
ᨨ���

ᨨ犨矄
 


Topic-2 : Angular Displacement, Velocity and Aceleration ҈


dy =  q =
Cx dx 꾀
P

 q =
Ca

㡰 y =
Cx
Þ

y

w




mw aᆠᨨ矄q N

R r

R 
r v O 졔Y

ᔴ矄
�㡰

졔X

r

v'

a x

2

ᔴ矄
��㡰

ᔴ矄
��㡰
ᆠ 
ᨨ
鼀矄矄
ᨨ
 
矄泐矄鼀
�
x㜠ᆠᨨᨨ�  M ´ x
+ m ´ x
= x
xἘC
㦨
M +m xC
㦨
xἘC
Q   
㦨


�
x㜠�ᆠ�ᨨ 㡰 mx҈ m R - r 㡰 = \ x

㦨
M +m M +m m R - r 㡰 �.e.

 
ᨨᆠ├
鼀ᨨ
x
=
犨 

M +m


ᆠ泐� 
ᆠ 矄
 
ᨨᨨ鼀
ᨨ� 
b
ᨨ
 
ᆠ├ 꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
P�
㦨
Pf

㦨
MV

mv MV \ v
㦨
m 꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
 �泐
҈
҈
Þ mg R

r㡰
㦨
MV + mv



MV 桌 m

s ´ p R

㡰 - s ´ p r
´ 䰘

졔Y x 㦨Rr㡰

M

᲼Q v =

m
Þ
mg
R

r㡰
㦨
MV

졔
M V m é Mù
ém + M ù
mg
R

r㡰
㦨
MV

ê҈ + ú = MV ê ú m ë û ë m û
g R - r 㡰
m
g  R - r 㡰 Þ = V


Þ V
㦨
m
M m + M 㡰 M m + M 㡰



s

 
鼀矄矄

盐 �

ᨨ
盐 � ᨨ鼀

\ 㝐矄矄
ᨨ
 
犨 ᨨ
�矄ᆠ
㦨
s
맔
p R
㝐矄矄
ᨨ
 
ᨨ�ᨨ
鼀ᨨ
㦨
s
맔
p r
R R
㦨



ᆠ鼀錐
r
㦨

҈
ᆠ鼀錐
OP
㦨
䰘
ᆠ鼀 ᨨ矄��ᨨ
ᨨ
ᆠ 
ᨨ
鼀矄矄 r m҈ x҈ - m
x

x
㦨















O P m҈ - m


1

r

M
V




x


=҈
æL ö çè - r ÷ø
 ᆠ
矄 
ᨨ
 

ᨨ
 �矄
ᨨ� 
�矄

�矄
ᔴ矄
�㡰
ᆠ 
ᨨ
鼀矄矄
ᨨ
 
矄泐矄鼀
ᨨ
犨ᨨ
ᨨ�矄
�

x㜠ᆠᨨᨨ�  M ´
+ mx҈ mx҈ = xC 㦨
M +m M +m

\

Þ
mg
R

r㡰
㦨
MV

졔
m

q Pa, b㡰 mg

q

mg矄� q




mw a x

mgᆠᨨ矄q

ᨨ
矄泐
ᆠ�ᆠ盐
鼀ᨨ�ᨨ

mw
a ᆠᨨ矄 q = mg 矄� q





Þw=

g  q a

\w =

g ´
aC =

gC a

ᆠ㡰 

rdr w
r = rgdh R

h







Þ w
ò rdr = g ò dh

w

dh dr

w
R
= gh
෌�
R
㦨
�
ᆠ鼀㡰 Þ

\h =

w
R

�w
=
g
g

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EBD_8341 PS-54




Physics

ᆠ㡰 
ᨨ泐
��鼀
�
 
鼀
ᨨ
�矄ᆠ kx








® mw  l
+ x 㡰 ¬¾¾ m ¾¾

 㦨 �

ᨨ
ᨨ� 


ᨨ�ᨨ 
矄
ᨨ
 
盐鼀

\ mw
l
+ x 㡰 = kx

Þ x=





ml
w

k  mw


x mw
= k l


b㡰 꾀
ᨨ �
ᨨ矄��ᨨ
x㡰
mv
F� = = mr w
r

\ kx = ml + x㡰w
Q r = l + x 㡰 ├
㦨
鼀lw

졔
鼀 w


mlw
k  mw
b㡰 N
矄�
q
㦨
鼀w

r⏌
㡰 \x=

�





�㡰

矄� q =

r⏌
҈ = Þ q = 
° r


 
N
ᆠᨨ矄
q
㦨
鼀�

w
r ᨨ 
q
㦨

g w
r w
r ҈ ᨨ 


㦨
ᨨ


㦨

g
g 
g \



ᆠ㡰

w

㦨


r 

犨�
 
ᨨ
矄 


Q
�矄
  
ᨨ���

�矄
ᨨᆠ�泐
犨�

泐 ᨨ �
턼
矄ᨨ
�
犨�
 
�
盐 矄  
҈҈


҈



 ᨨᆠ
㦨



Þ w µ +҈


p w

�鼀
�ᨨ


µ ᆠ㡰 ᨨ⼄盐

 �
q

=

p
p

和

҈

鼀
㡰



+҈




턼


ᨨᆠ�泐
v㡰
ᨨ
矄ᨨ
㦨
dx⏌dt  
 
 �盐
矄
w㡰
ᨨ
矄ᨨ
�� 

㦨
w,a

ᨨ鼀
D
SOP


x Þ h dx d f = h 矄ᆠ
f ᨨ
dt dt v
㦨

矄ᆠ


�
맔


҈ v
㦨

맔


맔


҈
㦨




鼀⏌矄


f
㦨
Q Q

\


ᆠ鼀

턼 

+҈


l t = Mg 矄� q

꾀矄ᨨ t
㦨
la l \ l a = Mg 矄� q


Ml
l
a = Mg 矄� q 
矄� q la =g Þ 




矄�



�

턼

҈

�鼀 �ᨨ =

䰘





��㡰




鼀w
= ᨨᆠ µ Þ w
µ

 � ҈
㦨 㦨 ҈

�


鼀 �鼀盐鼀
ᨨ�ᨨ 
矄
ᨨ
 
盐鼀
�
ᨨ盐�ᨨ 矄 
鼀� 盐

҈
ω鼀㡰 㦨 㦨
䰘

p ҈

� T 黨
㡰 꾀
t
㦨



t =
 
t = T  
�
ᨨᆠ�泐
犨�

ᨨ

T T 꾀
t
=
 
t =


 
�
ᨨᆠ�泐
犨�

鼀 �鼀盐鼀

�
鼀� �盐  ᆠ
�
㡰
ᆠᨨᆠ泐
�ᆠ矄
vr
矄盐矄
t
�
w ҈ ҈

ᆠ㡰 ᨨ

ᨨ
ᨨ�ᨨ
ᨨ
�矄ᆠ
R
Q p ҈
p O t= ´ =
w
w x㜠ᆠᨨᨨ� 
ᨨ
P
㦨
wRT P pR > R 矄�
�°





















= t
㦨




ᨨ
ᆠ
 
盐 矄 


�ᆠ㜠 P
矄
ᨨ

ᨨ�ᨨ 
 �
> 턼 R
矄�

�
;


䰘
졠 盐
�矄
 �
�矄

矄矄
 
⏌

矄ᨨ
� ω㦨




ᨨ
k >>
mw


Þ

㡰 ᨨ
盐矄
ᆠᨨ鼀
ᨨ�ᨨ

é Ml
ù êQ I rᨨd = ú  û ë  g 矄� q a=
l

\ \ \ ҈

㡰

x
㦨
h

f v
㦨
h
矄ᆠ

f㡰
w ᲼Q q + f = 黨
° 桌

ᨨᆠ
ᨨ
 
ᨨᆠ├
ᨨ �
矄ᨨ
㦨
鼀
w

㦨
鼀
㦨
鼀  

\

df dt

ò  = ᨨ



ò




w

Þ
 = w  -

 


 =





WWW.JEEBOOKS.IN

PS-55

System of Particles and Rotational Motion

\



= ò w 




졠
矄ᨨ� �
犨
�

ᨨ





\

=

w
㦨







w
-
 w  졔
 =




w (  + -w )


wk㡰

®

� 㦨
鼀
Ù ᨨ w 

y㡰






�㡰

w é +  -w ùû
ë


O ¬

� =
鼀

\ \

q

 ´
҈
=
⏌矄 � ´҈ � m = ҈
├� ᨨ犨
盐矄� �
 �泐
ᆠᨨ 矄�ᨨ

 
ᆠᨨ�矄�ᨨ

҈
l I w =
mg ҈ - ᆠᨨ矄 q㡰 + mgl ҈ - ᆠᨨ矄 q㡰

҈ æ �
ö 黨v
Þ ç ml ÷ =
mgl ҈ - ᆠᨨ矄 q 㡰 ø
�l

è  Þ mv
=
mgl ҈ - ᆠᨨ矄 q㡰 �´
 

䰘 ´ = ҈ - ᆠᨨ矄 q Þ ҈ = ᆠᨨ矄 q ҈

´ ҈
�

 ᨨ

ᆠᨨ矄 q = \q ;
°
�
㡰 
ᨨ⼄盐

t 
ᨨ盐
B
�矄
ᨨ

⼄盐���盐鼀 \ TA ´ ҈

- mg ´ �
-
mg ´
� =


Þ TA ´ ҈

= ҈

mg

vr

Þ TA = ҈ mg   矄�ᨨ
�
 
矄� �

A㡰

Ù w w  - -w ) w  (


TA

Ù 鼀 w
w � =  -  -w )  (


�

ᆠ鼀
�
ᆠ鼀

®

®

ᆠ�ᨨ =

TB B

A

Ù Ù 鼀 w w  -  -w ) + 鼀� ⧤ (



Topic-3 : Torque, Couple and Angular Momentum ҈


x®

 w -w
㦨 
㦨
éë w - w ùû



��㡰 
ᨨ鼀
�㡰
 
��㡰

\


®

M
㦨


├�

ᨨ

w =

® Ù Ù Ù æ ö æ Ù Ùö ᨨ = � +
鼀 ç ᨨ � ÷ ´ w ├+ 鼀 ç w ├´  � ÷ ´ w ├ è ø è ø Ù ® Ù Ù \ 鼀
w

� = � +
鼀 ᨨ wæç -  ö÷ + 鼀w
 � è ø

盐 
㦨


w

� v Þ mvl = ml
w Þ w =  �l

®

҈
b㡰

㡰 矄� �
ᆠᨨ 矄�ᨨ
ᨨ
 �盐
鼀ᨨ鼀 盐鼀

æ
ml
ö w mvl = ç ml
+  ÷ø è

ᨨ




 + 
-









ò

mg

mg





F

㡰

ᆠ㡰 꾀 �盐
鼀ᨨ鼀 盐鼀

L = I w R Ra 꾀 �矄

l

Mg

m

x

N f

a

ᨨ
矄
盐

F ´ R ³ Mg ´ x l⏌


m

m

l⏌


x=

R
-  R - a㡰

ᨨ鼀
��盐

F鼀�

Mg æ R-aö = ´ R
-  R - a㡰
= Mg ҈ - ç ÷ R è R ø




æ l ö I = m
㡰
+ m ç ´
+ m
l 㡰
è
÷ø

�





ᆠ㡰


ml
+
ml
= ml

꾀 �盐
鼀ᨨ鼀 盐鼀
L = I w = ml
w =

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EBD_8341 PS-56

Physics

꾀ᨨ盐
ᨨ� 
졠
 �盐
鼀ᨨ鼀 盐鼀 � ��
㦨
 � 

Þ

mV


´

҈ é
mL
mL
ù =ê + ú´w
ë ҈

û


V 
V = 䰘L 䰘
L b㡰 ෌�
 


㦨


졔

ᆠᨨ矄
w1 泐
㦨
泐

졔

矄�
w2 \w=





 =  

泐 = 泐
㦨
w2
ᆠᨨ矄w2㡰   泐  = 
㦨

w21
ᆠᨨ矄
w1㡰


㦨
泐
㦨

w22
矄�
w2㡰   꾀

㦨




㦨


졔


泐
㦨
泐
2
























㦨

w 1 ,
泐
㦨

r r r r r ᨨ犨

t =  ´ = 鼀´ 㡰 Þ 
㦨

w1
矄�
w1㡰


 

䰘


㦨
᲼
+ 㡰 �ˆ+ 泐
ˆ桌 ´ 鼀 -  w҈
ˆ�㡰 = + 鼀泐
 w҈
├ˆ r 㡰 ␔

��
r =
t�ˆ - t
ˆ

r r  t =
㡰 =
�ˆ - ҈
ˆ r r dr v = =
�ˆ -
tˆ ᨨᆠ�泐

dt r v  t =
㡰 =
�ˆ - ҈
ˆ r r r L = mvr 矄� qnˆ = mr ´ v 㡰 =

�ˆ - ҈
ˆ 㡰 ´ 
�ˆ - ҈
ˆ 㡰 = -

kˆ





㡰 꾀 �盐
ᆠᆠ�ᨨ
w - w

� ´
p -
a=
=
㦨
҈

p ⏌矄
t � t
㦨
ᔼa

æ�

ö æ�ö mR
÷ a
» ç ÷ � ´҈
- 㡰҈
-
㡰҈
p è
ø ø �
㦨




맔
҈
鼀 黨
㡰 ҈

ᆠ㡰 꾀ᆠᆠᨨ� �
ᨨ
犨ᨨ├㜠 �泐
 ᨨ鼀 Þ t
㦨
çè


鼀�
㦨

҈ ҈ 鼀
  鼀
꾀




�
㦨 
  
꾀


맔
҈

맔
҈

㦨    �


Þ
vB
㦨҈�
鼀⏌矄 꾀 �盐
鼀ᨨ鼀 盐鼀
ᨨ盐
O
LO
㦨 mvr 㦨


맔
҈

맔


LO
㦨

├�
鼀
⏌矄

r  ˆ ҈҈
㡰 ෌�

҈ =  -ˆ�㡰 + - 㡰

r ҈ =
�ˆ +
ˆ

ᨨ⼄盐
盐
ᨨ
҈
ᨨᆠ æ r r r  ˆ ö ˆ t ҈ = ҈ ´ ҈
=
ˆ ´ çç  -ˆ�㡰 +  - 㡰 ÷÷

=  ├㡰

è ø

ᨨ⼄盐
盐
ᨨ


ᨨᆠ r ˆ ´ ├ˆ =  �ˆ +
 - ˆ㡰 t
= 
�ˆ + 㡰 r r r ˆ t  = t ҈ + t

=  �ˆ +
 -ˆ㡰 +  ├㡰 ˆ



= �ˆ -
ˆ + ├㡰 r r r ҈

㡰 ᨨ⼄盐
ᨨ盐
 
ᨨ���
㦨

t =  ´ 㦨



矄�
q Þ

�
㦨
҈
맔
�
矄�
q ҈ p 矄� q=

� = Þ q=

҈
㡰 ᔴᨨ 矄�

矄�
ᨨ
�盐矄

 
 �ᆠ├ 矄矄



ᨨ⼄盐
盐
ᨨ
�ᆠ�ᨨ
ᨨ
 �矄
矄� 
ᨨ⼄盐
㦨
å ᨨ⼄盐
ᨨ
� �

ò t =



ò




m

p  p



Ҙ  ·

Ҙ t=  ҈

㡰 꾀泐� �
ᨨ⼄盐
⼄盐�ᨨ
ᨨ盐
ᨨ� 

t 㦨
ᔼ
a
㦨
᲼
㝐

l㡰

졔
�㝐
l
桌a Þ �㝐
�l 


㝐
�l
㦨
᲼
㝐

l㡰

졔
�
㝐
l
桌a Þ t=

Þ 㝐
�l 㦨
҈㝐
�l
㡰a

� ҈l ҈�
㡰 ෌�
 

 
ᨨ
�矄
ᨨ
盐 � ᨨ鼀
鼀矄矄
 矄�泐
  꾀
㦨
ᔴ 
鼀矄矄
ᨨ
ᨨ 
ᨨ
�矄
鼀
 ᆠ� �
ᨨ⼄盐
ᨨ盐
� �
ᨨ� 
鼀�
ᔴ҈㡰
㦨

鼀�
ᔴ
㡰 \ a 㦨


æL ö æL ö mg ç 矄� q ÷ = mg ç ᆠᨨ矄 q - L 矄� q ÷ è
ø è
ø  mgL Þ mgL 矄� q = ᆠᨨ矄 q

矄� q ҈ ҈ Þ
㦨 ᨨ



q
㦨 ᆠᨨ矄 q   ҈

㡰 ҈䰘
㡰 ␔
├ ᨨ犨
 

㦨
mvr^ 泐

 



 

 꾀

⏌


졠

ᔴ  

 










ᔼ
ᨨ 
ᨨ
 
ᆠ矄矄

 
 �ᆠ盐

æ R ö + a÷ �矄 ᆠ
^ �矄
ç è
ø

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PS-57

System of Particles and Rotational Motion ҈



㡰 泐
�ᆠ
⼄盐���盐鼀


 
ᆠᨨ矄




졔

矄�


㦨
҈


� 

g Þ

㦨

…
�㡰  泐
ᨨ�ᨨ 
⼄盐���盐鼀
㦨
ᆠᨨ矄



 ҈
 N= 㦨



ᨨ⼄盐
ᨨ盐
꾀








矄�




x

h

l⏌
҈

� 

҈

�
맔
꾀
㦨

맔
꾀 f  l 

g l ´ x

\ = x







…
�㡰 ҈


�
맔
ᆠᨨ矄

° =



h

x ᨨ鼀
⼄

�㡰
 
��㡰

\ x=

盐
ᆠᨨ矄


㦨


҈黨



�


h


…
��㡰 ᆠᨨ矄 
°




l h h   =

\
= ᆠᨨ矄 
°
l ҈
㡰 꾀 �盐
鼀ᨨ鼀 盐鼀
L

㦨
鼀
矄�
黨




㦨


맔




맔
҈

맔
҈
맔
҈ ᲼꾀矄
V
㦨
rw

和�
黨

㦨
҈桌 和ᨨ

L

㦨
҈



├�鼀
⏌矄

ᨨ


鼀

㡰 
ᨨᆠ├
�矄
�
⼄盐���盐鼀
ᨨ
 矄�ᨨ 
⼄盐���盐鼀 SFx
㦨


Þ
F
㦨
N SFy
㦨


Þ
f
㦨
mg

y

f
a




鼀

x

҈
鼀

a C

b

졠






b㡰 P ᆠ㡰 ᨨ

ᨨᆠ�ᨨ 矄
ᨨ
鼀  
 �盐
鼀ᨨ鼀 盐鼀
ᨨ
  P mass m
ᨨ盐
O �.e., Lᨨ
�矄
mr
w  
�矄
�ᆠ
ᨨ犨
졔 z �ᆠ�ᨨ


 �盐 鼀ᨨ鼀 盐鼀
ᨨ
鼀矄矄
m
ᨨ盐
P l �



Lp
�矄
mvl
 
�矄
�ᆠ ᨨ
 
��
ᨨᆠ�ᨨ
ᨨ
m
矄 矄 ᨨ犨
�
 
��盐

ᨨ O v �  
ᨨᆠ�ᨨ
ᨨ
m

  r �ᆠ�ᨨ
ᨨ
P
鼀� 矄 m r O ᆠ  �� �
b㡰 꾀矄
�ᆠ
�矄
ᨨ� �
�
ᆠ�ᆠ盐

 
� 
矄 
�矄
ᆠ矄� �

L
�矄
ᨨ
ᆠᨨ 矄
�
鼀� �盐

和� ᆠ  
�ᆠ
矄
犨ᨨ
ᆠᆠ�ᨨ 矄
ac
 
at
  ᨨ
 


ᆠᆠ�ᨨ
�矄
ur ᨨ
ᨨ犨矄
 
ᆠ 



�ᆠ�ᨨ
ᨨ
L
鼀� 矄
矄鼀

犨 
 
矄 ᆠ矄矄


F

N

ᆠ㡰 ᨨ⼄盐
犨ᨨ├� �
ᨨ
 
ᨨ
ᨨ
鼀矄矄
鼀
�矄

t
㦨
mg
맔
l
矄�

q

�ᆠ�ᨨ

ᨨ
 
ᨨ
ᨨ�ᨨ
ᨨ
�ᆠ㡰

mg

l 鼀


҈


q

ᨨ
ᨨ�ᨨ 
⼄盐���盐鼀 Stc
㦨



ᨨ⼄盐
ᆠ
泐
�ᆠ�ᨨ 
ᨨᆠ
 f 㡰
ᨨ盐
C
㦨
f
맔
a
�

ᆠᨨᆠ├犨�矄
�ᆠ�ᨨ



�矄
 ᨨ 
ᨨ⼄盐
犨 �ᆠ
矄 ᨨ盐
ᆠᨨ盐 
 �矄
ᨨ⼄盐



ᨨ鼀
ᆠ�ᨨ
N
ᨨ
 
ᨨᆠ├
犨�
ᆠ

ᨨ⼄盐 N
맔
b
�
 
 �ᆠᨨᆠ├犨�矄
�ᆠ�ᨨ
和盐ᆠ
  f
맔
a
㦨
N
맔
b  ᆠ
N
犨�
ᨨ盐ᆠ
ᨨ⼄盐


l 鼀�

r 꾀矄
t
�矄
 �ᆠ盐
ᨨ
L

�ᆠ�ᨨ
ᨨ
L
ᆠ  �矄
盐 鼀� �盐
鼀� 矄
矄鼀
ᆠ㡰 ෌�



鼀
㦨


҈


├� q 㦨



㦨
҈

鼀⏌矄 ® ® 꾀 �盐
鼀ᨨ鼀 盐鼀
 =  ´ 鼀  㦨

鼀
ᆠᨨ矄
q é 
矄�
q ù 
矄�
q  = ᆠᨨ矄 q ê ú 㦨

� ûú
� ëê

҈
2 ´ 矄�


°´ ᆠᨨ矄

°
´҈
㦨




├�
鼀
⏌矄


㡰 꾀 �盐
鼀ᨨ鼀 盐鼀

㦨
鼀

맔
㡰 æ  ö 㦨

├� ç ´ ÷ =
├�
  ´ �� -

ˆ㡰 è  ø 㦨


├�



 ├ˆ 㡰
㦨


├�
맔



맔

鼀

矄҈ ├ˆ 㦨


├ˆ


䰘







㦨



黨


b㡰 K
E
㡰ᨨ�ᨨ 
㦨


L


I




L
㦨
ᆠᨨ 矄  \ K
E
㡰ᨨ�ᨨ 
맔
I
㦨
ᆠᨨ 矄 
 ᆠ
犨 
I
�矄
ᨨ盐

K
E
ᨨ�ᨨ 
ᆠᨨ鼀矄

㡰 

ᨨᆠ
ᆠ� �
ᨨ

�ᆠ
盐 �ᨨ� �
盐 � ᨨ鼀 ᆠ�ᆠ盐
鼀ᨨ�ᨨ
�矄
ᆠ �
ᨨᆠ
ᆠᨨ犨矄
ᆠ 
ᨨ
  ᆠ�ᆠ


ᨨ⼄盐
盐
ᨨ
 �矄
ᨨᆠ
ᨨ盐
 
ᆠ 
�矄
ᨨ
ur   ᨨ

 �盐
鼀ᨨ鼀 盐鼀
L
ᨨ盐
ᆠ 
�矄
ᆠᨨ 矄
㡰 꾀泐� �
ᆠ  �
�
 �盐
鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀 㦨
 �盐
�鼀盐矄
��
ᨨ
 
矄泐矄鼀
ᨨ盐
 
ᆠ 
ᨨ 鼀矄矄
꾀 �盐
鼀ᨨ鼀 盐鼀㡰f


꾀 �盐
鼀ᨨ鼀 盐鼀㡰� L 㦨
Mv
맔 㦨
IC w



�㡰
 w �矄
 
 �盐
ᨨᆠ�泐
ᨨ
 
ᨨ
㝐ᨨ鼀 
ᨨ
ᔼ �
ᨨ
 
矄泐矄鼀
ᨨ盐
�矄
 �矄
ᨨ
ᨨ�ᨨ

᲼ᆠ 
ᨨ
鼀矄矄
ᨨ
 
矄泐矄鼀桌

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EBD_8341 PS-58

Physics

æ Lö





æ Lö





IC
㦨
M ç ÷ + M ç ÷ = è ø è ø


ML
ML
=



ᨨ鼀
ᆠᨨ 矄�ᨨ
ᨨ
 �盐
鼀ᨨ鼀 盐鼀

鼀㡰
㦨
Iw ҈

맔



�
맔
黨
맔



�
㦨
맔



�
맔


�㡰

맔
w
\
w
㦨



矄҈ 
㡰 ᨨ鼀
ᆠᨨ 矄�ᨨ
ᨨ
 �盐
鼀ᨨ鼀 盐鼀 I҈w҈
㦨
ᔼ
w


�

㡰

V MVL æ ML
ö =ç ÷ w Þ w
㦨
L

è ø 

b㡰 
�矄
ᨨ
  
ᨨ⼄盐

C  �盐
鼀ᨨ鼀 盐鼀
鼀� 矄 ᆠᨨ 矄

꾀矄
鼀ᨨ鼀 
ᨨ
� � � ��泐

犨 
 
ᨨᨨ�矄
鼀ᨨ矄 B ᨨ鼀
꾀
ᨨ

ᆠ矄矄
 
 

� ᆠ矄矄
犨 
ᨨᨨ�矄
鼀ᨨ矄 ᨨ鼀

ᨨ
ᔴ

矄ᨨ
w
犨�
� ᆠ矄 A � ��泐
 
 
ᆠ矄矄

和ᨨ
  ��ᨨ
ᨨ

w
�矄
ᨨ � 
҈
b㡰 盐
ᨨ
� ᆠ矄
�
 �ᆠ盐
�矄 ᆠ
ᨨ
 
 犨�
 
 �矄
ᨨ
ᨨ�ᨨ
M
I

ᨨ盐
 
 �矄
ᨨ
ᨨ�ᨨ
�矄

ᨨ
ᆠᨨ 矄 

和ᨨ
I
� ᆠ矄矄



�矄
ᨨ
  
ᨨ⼄盐
�矄
ᆠ� � dL \
t 
㦨

Þ
L
㦨
ᆠᨨ 矄 
Þ
Iw
㦨
ᆠᨨ 矄  dt 盐

I
� ᆠ矄矄
矄ᨨ
w
ᆠ矄矄


ᆠ㡰 ␔ 

�矄ᆠ
ᨨ矄

�
矄
犨ᨨ
泐矄
ᨨ
鼀ᨨ�ᨨ
�


 矄�ᨨ 
 
ᨨ�ᨨ 

 ᨨ
 

犨ᨨ
泐矄 ᨨ
 �盐
鼀ᨨ鼀 盐鼀
 
 
ᨨ
 �盐
鼀ᨨ鼀 盐鼀
�矄  
ᆠᨨ
矄盐鼀
ᨨ
 矄
犨ᨨ
y ur ur ur L = LT + L R LT
㦨
 �盐
鼀ᨨ鼀 盐鼀 w 盐
ᨨ
 矄�ᨨ 
鼀ᨨ�ᨨ
R LR
㦨
 �盐
鼀ᨨ鼀 盐鼀 v 盐
ᨨ
ᨨ�ᨨ 
鼀ᨨ�ᨨ

C ᨨ盐
C.M
L
㦨
MV
맔
R
졔
Iᆠ鼀w x O \





҈ 㝐
´ ω҈ I҈ω҈
\ ω
= = ᔼ
ì҈

ü í 㝐 +
᲼
鼀 桌ý î
þ ෌�

M
㦨
�

├�

R
㦨




m

w҈
㦨
҈

ᨨ⏌矄 m
㦨



�
├�
 
r
㦨




m

҈ ´ �
´


´


´ ҈



=



㦨 㦨
 ⏌ 矄 ҈ ´ �
´


´


+
᲼
´


� ´


´


桌
졔҈
䰘

㡰 
 
ᆠ 
ᨨ
鼀矄矄
ᨨ
 
� 泐
矄
矄泐矄鼀


  ᨨ���

ἘOἘ

�矄 ᆠ
犨
AO
㦨
x
 
OB
㦨
d

x㡰 졠 꾀  x d㜠x ᆠ
鼀 ҈҈
Ms



Ms ᨨ���

d


M s - x㡰 + ҈҈M s d - x 㡰
=


M s + ҈҈M s �d
ᨨ

� 泐
矄
矄泐矄鼀

 �盐
矄
w
ᨨ盐
 
ᆠ  ᨨ
鼀矄矄

Ἐ졠Ἐ
�矄
矄鼀
ᨨ
ᨨ
 
矄矄


Þ


㦨




Ms 
㡰
졔
҈҈Ms
d 
x㡰

Þ x =




Iᆠ鼀
㦨
M.I

ᨨ盐
ᆠ 
ᨨ
鼀矄矄
C
 
V
㦨
Rw

҈ 

㦨
M
R
w㡰
R
졔
MR w 㦨
MR w




\

Y

b㡰 꾀 �盐
鼀ᨨ鼀 盐鼀

ur ur L = r^ ´ p

v

M

w

꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

㝐


 
y
ᆠᨨ 矄 
 ᆠ
�矄  �盐
鼀ᨨ鼀 盐鼀
犨



ᨨ���
鼀� 矄
ᆠᨨ 矄 
ᆠ㡰 和� ᆠ
 
犨ᨨ
ᨨᆠ矄

ᆠ
� 泐
ᨨ
 
� �
  ᨨ
ᨨ
  
ᨨ⼄盐
�矄
ᆠ� �
ᨨ
 
矄泐矄鼀

 ᆠ  �盐
鼀ᨨ鼀 盐鼀
鼀� 矄
ᆠᨨ 矄 
w1 w2

m


r

m





I
㦨
Mr I
㦨Mr 졔mr \ I҈w҈
㦨
I

w
Þ Mr

맔
w҈
㦨
Mr

졔

mr
㡰
w
(Q w҈ = w ) w

㦨


Mw M +
m


㦨





㡰

M, L㡰

X

O

\




ædö ҈҈ M s ç ÷ ´ w è
ø

y

L
㦨
Mv
맔
y









LTᨨtal = LB




æ �d ö ædö


M s ç ÷ w + ҈҈ M s ç ÷ ´ w è
ø è
ø

m

v

 ᨨ
ᆠᨨ�矄�ᨨ 꾀 
ᆠᨨ�矄�ᨨ

矄� �
� ᆠ�
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
 �盐
鼀ᨨ鼀 盐鼀
犨
 r r L� = L f Þ mvL = I w

æ ML
ö Þ mvL = ç + mL
÷ w è  ø æ

黨 ´ ҈
ö Þ

҈ ´

´ ҈ = ç +

҈ ´ ҈
÷ w  è ø


æ  ҈ö Þ
= ç + ÷wÞ
= w è ҈
҈
ø ҈
Þ w =


⏌矄ᆠ


WWW.JEEBOOKS.IN

PS-59

System of Particles and Rotational Motion 黨


I҈w҈ = I
w


I҈ =  I M






␔ 
m
�矄
 
鼀矄矄
ᨨ
矄ᨨ   
w
�矄
 
 �盐
ᨨᆠ�泐 
 
� 矄 
ᨨ
�鼀
t
L 꾀矄ᨨ

L
㦨
Iw
Þ w = ҈ 盐� �
 �矄
盐
ᨨ
w
�矄
⼄
�㡰

黨


㡰 
M

㦨




├�

m
㦨



├� 矄� �
ᆠᨨ 矄�ᨨ
ᨨ
 �盐
鼀ᨨ鼀 盐鼀

L� = Lf

M


æ M R
ö + I m 㡰 = ç
+ mR
÷ ç
÷ è ø

m

҈ I
= M
R

 
w҈ = �
鼀
æ M R
ö � \w
= çç
+ mR
÷÷ ´
è
ø M
R

�R 

+ ҈

㡰 = 黨
鼀

=
´ ҈

R 
�



ᆠ鼀㡰 

҈

鼀�

dx d
x = - Aw 矄� wt

Þ = - Aw
ᆠᨨ矄 wt
dt dt \
㝐� �盐
ᨨ
 
鼀 �鼀盐鼀
ᆠᆠ�ᨨ
ᨨ
 
 ᨨ鼀 \ 
㝐
ᆠᆠ�ᨨ

㦨
Aw
␔ 
 ᨨ鼀
鼀ᨨ矄

ᨨ⼄盐
ᆠ矄
ᨨ
 
ᆠ泐� 
   

ᆠ泐� 

ᨨ矄
ᨨ盐
�矄
 �矄
f 꾀ᆠᆠ�ᨨ
ᨨ
ᆠ泐� 


a҈ = m

ᨨ⼄盐 t 㦨 f R 㦨 Ia fR fR = Þ a = I MR
⏌


҈

ᆠ鼀

ᔼ ��泐




㝐� 㝐� = 黨
黨

ᨨ⼄盐
ᨨ盐
ᆠ 
tᆠ 
㦨

�
�㝐� \ ҈�
㡰
㦨



㡰  = 㝐� = 黨
黨 �
㦨

 
졔

㦨
㝐�

 =

҈
㦨
m⧤

҈
㦨
m和

҈
㦨





҈
㦨





⧤

⧤


f
f




ᨨ
Ra 㦨
㦨
a
㡰 MR M \
ᨨ
ᨨ
鼀
� 
ᆠᆠ�ᨨ
f
f f + = a鼀 㦨 a҈ + a


㦨 M M M
f MAw ᨨ

Aw
㦨



Þ f


㦨 M 

盐矄

鼀 �鼀盐鼀
ᨨ⼄盐








⧤
㦨



















Þ



㦨





҈

㦨
�
    = 

㡰 Þ   = 
� ᨨ犨

㦨
�矄 ᆠ
犨 
�� 
� �
矄ᨨ矄
 
 
� � 矄矄
鼀ᨨ� � ᲼ ᨨ⼄盐
ᨨ盐
ᆠ 

ᆠ 
㦨

桌 
㦨
  㡰  
㦨
⧤


















Þ



㦨




҈

㦨




㡰 =   �


҈


҈

=
�

ᆠ鼀 �

mv
㦨

mrw
r

㦨

MAw
R ҈

㦨
MARw  


�矄
ᨨ
  
ᨨᆠ
 
 ᆠ
ᨨ
ᨨ⼄盐
�矄
�
 
 �盐
鼀ᨨ鼀 盐鼀
鼀� 矄
ᆠᨨ 矄  �.e.錐
l҈㦨
l
\ I҈w҈
㦨
I
w
ML
´ w
M w
Iw ҈
\ w

㦨
҈ ҈ = 㦨


M +
m I
ML æ Lö +
m ´ ç ÷ è
ø ҈



矄 ur I҈w҈
㦨
I
w
Q L = ᆠᨨ 矄  㡰 t鼀 㦨

f
맔
R

㦨


҈
I҈
㦨
MR


҈ ҈ M
æ
+ ҈ö � MR
+ R =ç MR
= MR
÷ è
ø



w҈
㦨
w ; w

㦨
䁸 ҈ MR
´ w I҈w҈


=+ = w= w w

㦨
� I

´� � MR



I

㦨




 泐
� 矄 
ᨨ
�鼀
t

 
�盐矄
ᨨ
 
ᨨ�ᨨ  矄盐 ᆠ
㦨
r
T
㦨


a

ᨨ




㦨





㦨



  ⧤ ᔼ

㦨
�矄 ᆠ
ᨨ
 
� �
ᨨ鼀
ᆠ 
犨 
�� 
� � 矄矄
鼀ᨨ� � t
㦨

㡰ᨨ盐
ᆠ 
Þ 
㦨
 

㡰

\ =

m

æ L
ö = ´ r Þ T = ç ÷ r - è mø mr
㡰
mL


Þ




黨

ᆠ鼀

T

æ L
ö = A ᆠᨨ 矄  ÷ çè ø m


ᔴᨨ 矄�� �
 
鼀ᨨ�ᨨ
ᨨ
 
 ᨨ鼀 x
㦨
A
ᆠᨨ矄
wt �� 㡰




ᆠ鼀

�


I


r

ᨨ


= Ar -
\
n 㦨






T
㦨


mrL


w





�㡰

WWW.JEEBOOKS.IN

EBD_8341 PS-60


�


Physics



ᆠ

㡰
ᨨ鼀
 �盐
鼀ᨨ鼀 盐鼀
ᆠᨨ 矄�ᨨ
ᨨ盐
  �ᨨ
ᨨ� 



䰘


꾀泐� �

F =

�ᨨ
ᨨ� 

æ 鼀
ö 鼀 = ç + 鼀
÷ w ç  ÷ è ø

b

ᆠ㡰
෌�


ᨨ �
 �泐

㡰
㦨
⧤
⏌


ᨨ
r
㦨
R

F
㦨
kR



dv d é kr
ù = ê ú = kr dr dr êë
ûú

mv

Q �ᆠ
�矄
鼀ᨨ� �
�
ᆠ�ᆠ盐
ᨨ�㡰 R mv
k = kR Þ v = ´R ᨨ

R m

꾀矄ᨨ
F =

᲼꾀矄
 
ᆠᨨ鼀� 
矄泐矄鼀 ᨨ矄
犨�
 �盐
矄 w ᨨ盐
 
�ᨨ桌




鼀  鼀  \w = =
鼀
 + 

+ 鼀   w=
 + 
 ᆠ
ᨨ�ᨨ
㡰
�矄
ᆠᨨᆠ
w =
ᨨ
鼀 �鼀盐鼀
 �盐
ᨨᆠ�泐



盐

æ k ö R÷ R = kmR
÷ m è ø

꾀 
 �盐
鼀ᨨ鼀 盐鼀

L = mvR = mçç




ᆠ㡰 ᔼ
 
ᨨᆠ
 㡰
�矄
�


 � �
 
 
t 鼀� 矄
ᆠᨨ 矄 
 
t 㦨

맔


ᔼ
ᨨᆠ
�矄
�
ᨨ鼀
ᨨ


 
矄
 
犨 矄
ᆠ�鼀矄
 
 
 �ᆠ盐
�矄 ᆠ
ᨨ
ᨨᆠ
ᨨ鼀
턼
犨�
�ᨨ ᨨ
ᆠ  �� �
� ��泐
 
 �ᆠ盐
�矄
턼㝐


� ᆠᨨ鼀矄
턼㝐Ἐ


ö  æ 
ç +  ÷ =
÷  çè ø

Þ


x






 

+ =
Þ x =

㝐¢



  和ᨨ
ᨨ�ᨨ
ᆠ㡰
�矄
ᆠᨨᆠ


턼

\ 鼀 =

w鼀 =






 



 
+ 


=




=



 


æ  ö 
+  ç ÷ è ø  ᆠ
ᨨ�ᨨ
㡰
�矄
ᆠᨨᆠ
ur r 

ᆠ㡰

෌�
F = at�ˆ + bˆ ᨨ F = t�ˆ + ˆ r md v \ = t �ˆ + ˆ dt r \ dv = t dt �ˆ + dt ˆ \

v

r


黨


Q a = 1NS -҈   b = 1N 㡰

ᔼ
 
ᨨᆠ
 㡰
�矄
�
ᨨ鼀
ᨨ
 
ᆠ�ᆠ盐鼀  ᆠ
 ᨨ� 

  t �矄
ᨨ
ᔼ
 
ᨨᆠ
 㡰
�矄
�
 � �泐

ᨨ� 
和
 

t = F ´ R
 
 
犨 
ᆠ�鼀矄



ᆠ

㡰 ᨨᆠ
ᆠ� �
ᨨ
ᆠᨨ鼀

㦨



\

㦨



 ᨨ
 
ᨨᆠ ᆠ� �
�
�ᆠ
�ᆠ�ᨨ
犨�
鼀ᨨ
 
鼀�
ᨨ� 
ᨨ
ᆠᨨ鼀 ᨨ
 


�ᆠ泐
ᨨ犨 犨矄
Ӹ 


㡰

꾀

᲼Q m = ҈桌

t

t

ò
dv = ò
t dt �ˆ + ò
dt ˆ

黨


q

r t
v = �ˆ + tˆ
҈ r ҈ -҈ 꾀
t = ҈s
v = �ˆ + ˆ = �ˆ +
ˆ 㡰鼀矄

r

r dr t ˆ ˆ = � + t \
drr = t dt �ˆ + t dt ˆ 꾀矄ᨨ

v 㦨
dt

ᨨ




r

r



㝐

t t


t

r

t

t


ò
d r = ò

dt �ˆ + ò
t dt ˆ

\
r =
�ˆ +
ˆ

r ҈ ҈ 꾀
t = ҈
r = �ˆ + ˆ



r \ r =

r r r æ҈ˆ ҈ t=r ´F =ç � + è



r ҈ ᨨ
t = - kˆ 

\

҈ ҈ ҈
+ = 




ˆ ö÷ ´ �ˆ + ˆ 㡰





t
㦨
҈矄㡰 ø

r ҈ t =
鼀 



泐㡰 

q

⏌




L 矄� q


q

⏌
鼀�

L ᆠᨨ矄 q




␔ 
 

鼀├矄

 �
q
 
�� 
ᨨ
�矄
ᆠᨨ鼀 L 㦨
ᆠᨨ矄 q
\ �矄ᆠ鼀 
ᨨ
�矄
鼀�
ᨨ� 
ᨨ鼀
 
� ��
ᨨ矄��ᨨ

L L L - ᆠᨨ矄 q 㦨
҈ - ᆠᨨ矄 q㡰


ᔼ 矄  ᨨ盐矄
ᨨ⼄盐
ᨨ盐
 
ᨨ� 
ᨨ
ᆠᨨ ᆠ
 L t = mg ´ 矄� q


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PS-61

System of Particles and Rotational Motion ᨨ犨
x =

�



é ma

m
a㡰
ù ҈䰘
LCM = I CM w = ê + ma w úw =


ëê ûú 㝐� �盐
ᨨ
 �盐
鼀ᨨ鼀 盐鼀
ᨨ
ᔴ㝐
ᨨ
矄泐矄鼀
ᨨ盐 ᨨ� 
O.

L 矄� q



y = L 矄� 黨
° - q㡰 = L ᆠᨨ矄 q





x
y
æ
x ö æ y ö \ç ÷ + ç ÷ = ҈
ᨨ

+
= ҈ è L ø è Lø L L

盐矄

ᨨ
꾀
�矄

�矄
ᨨ
ᨨ


b㡰 r ෌�

r = at �ˆ + bt
ˆ

黨w 黨
҈鼀w
҈

鼀w
´ = = � � � � 㝐� �盐
ᨨ
㜠ᆠᨨ鼀ᨨ  
ᨨ
 �盐
鼀ᨨ鼀 盐鼀
ᨨ 矄泐矄鼀
ᨨ盐
ᨨ� 


Lz = L'
ᆠᨨ矄q  Lᔴ㝐 矄� q ¢ = �鼀 ´

r ҈
r = t �ˆ + �t
ˆm  r r dr = ҈
t
�ˆ + ҈
tˆ ms ҈ \ v= dt r r dv a = =

t�ˆ + ҈
ˆms -
 
dt r ҈
꾀

=
҈矄
rt =҈ = �ˆ + � ˆm
錐  r r -
vt =҈ = ҈
�ˆ + ҈
ˆ ms ҈
 
at =҈ =

�ˆ + ҈
ˆ ms r pt =҈ = �ˆ + ˆ kgms ҈ ˆ kˆ �ˆ r r r L = r ´ p = ҈
�
= kˆ é ҈
- �ù = - � kˆ ├�鼀矄 -҈ êë  úû  


҈

mwa


҈䰘
҈ ´ - ma w ´ � � � � ҈黨

҈䰘 æ ö = ma
w ç è
� ҈
÷ø �

㡰 꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
 �盐
鼀盐鼀 盐鼀
ᨨ盐  
 �矄 =

MR


맔
w
㦨
MR

맔
和ᨨ� �
犨
� Þ �


҈ ҈
r r F = ma = 
�ˆ + ˆ 㡰 N ˆ kˆ �ˆ

�҈


P

Þ �



l

ᆠᨨ矄 q =

a

q

O

w

l
l �

҈ l +a




=



a


a +

a




=



�

éQ l = ë



a �� ùû

OA
㦨

l 㡰
+ 
a㡰
= 黨
a
+
a
= ҈
a 
ᆠᨨ鼀
矄泐矄鼀
ᨨ矄
ᨨ盐
z㜠 �矄
犨�

ᆠᨨ 矄   �盐
ᨨᆠ�泐
w¢

w'
p 
a㡰 w = Þ w' = � w
p҈
a㡰 㝐� �盐
ᨨ
 �盐
鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀
ᨨ盐
�矄 ᆠ 
ᨨ
鼀矄矄

\

㡰 ෌�


r

A


҈  q =

å

f҈
㦨 m҈
҈

N҈

m҈

f

㦨 m




鼀�
r Fext =
r d p矄泐矄鼀

r 㦨


Þ
p矄泐矄鼀
㦨
ᆠᨨ 矄  dt �



� 
鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀
ᨨ矄
ᨨ
ᆠ  �
�

�鼀
盐
ᨨ
�  
ᨨᆠ矄
ᆠ� �
�
 
矄泐矄鼀

 
├� �ᆠ
  ᨨ �
 �泐
鼀泐
ᆠ  �
犨�
�鼀
꾀矄ᨨ
ᨨ
  
ᨨᆠ
鼀泐
ᆠ

ᨨ⼄盐

盐矄
  ᨨ⼄盐
犨�
ᆠ  �
 
 �盐
鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀
�

�鼀
��


b

ᆠ㡰 ur uur æ r dL ö r ur ur dL ur ur = A´ L ෌�
t = A ´ L Þ çQ t = dt ÷ è ø dt uur dL ᨨ鼀
ᆠᨨ矄矄㜠ᨨ盐ᆠ
盐


�矄
 �ᆠ盐
ᨨ
ᨨ dt ur ur A
 
L
r ur ᨨ鼀
ᨨ
ᨨ盐ᆠ
盐

L
L = L
�  �� �
犨�
矄ᆠ
ᨨ
�鼀

å Fext =


a

黨a �

ᆠ

㡰
␔ 
m҈
¹


 
m

¹

ᨨ�ᨨ 
⼄盐���盐鼀

҈
㦨
m

�ᆠ
⼄盐���盐鼀

鼀�
㦨


졔
m1҈ 和ᨨ� �
 
ᨨ
⼄盐�ᨨ 矄
犨
�

l 鼀� ´ ᆠᨨ矄 q = ҈ ´ l 矄� q



m m


R �

鼀� N



m mg ҈ + m҈m
q
␔ 
m҈
㦨

錐
ᨨ⼄盐
ᨨ盐


m
㡰 +
ml 㡰
l = m +
m �

w'


㦨



=

r r r ҈
é҈
ù -

ˆ t=r ´F = �
= kˆ ê - ҈
ú = k 鼀   ë û
҈


ᆠ㡰
ᨨ矄��ᨨ
ᨨ
ᔴ㝐
ᨨ
 
 �矄
ᨨ
ᨨ
xᔴ㝐 =


w M 黨 R

w M

w
졔

맔

맔

졔
r
맔
黨
黨 黨

�

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EBD_8341 PS-62

Physics

uur uur uur ur dL ur dL ur dL dL dL =
L L× + L× =
L

Þ
L × dt dt dt dt dt ur dL L^ 和� ᆠ
dt uur r dL dL =
\
L × =


Þ
dt dt Þ L
㦨
ᆠᨨ 矄 

盐矄

r  
鼀� �盐
ᨨ
L
犨泐矄
鼀� 矄
ᆠᨨ 矄 
꾀矄
A
�矄

ᆠᨨ 矄 
ᆠᨨ
 
�
�矄
犨泐矄
 �ᆠ盐 r ᨨ
t
r r 꾀矄ᨨ

L
�矄
 �ᆠ盐
ᨨ
A r r rr Q
L ^ A




\
L
A =
r  ᆠ

�
ᆠ

ᆠᨨ ᆠ盐
 
ᆠᨨ鼀ᨨ  
ᨨ
L
ᨨ � r A
�矄
ᨨ
�



ᆠᨨ 矄 
�

b

㡰
꾀 �盐
鼀ᨨ鼀 盐鼀
L = r ^ ´ P 꾀 �盐
鼀ᨨ鼀 盐鼀
ᨨ盐
ᨨ� 
O mv ´h L
㦨




�㡰
v
ᆠᨨ矄
� h

v

꾀 �盐
鼀ᨨ鼀 盐鼀

L
㦨
m᲼Vy x  Vx y桌 ᨨ



év
æv v
ö v ù ö æ v
´ ç
- v
÷ - ç
-
÷
ú L
㦨
m
ê ø è
g
g ø
ú êë
g è
û

é v
 v v v ù -
-
+
ú L
㦨
m
ê
g
g

g ûú ëê
g - mv
 mv
 é ҈ ҈ ù
㦨
ú g êë


û

g ur r ur 꾀��

L = r ´ p
 ᆠ

 
�ᆠ�ᨨ
ᨨ
L
�矄
 �ᆠ盐
ᨨ
 
 
ᨨ 鼀ᨨ�ᨨ
 
�矄
�ᆠ
犨泐
ᨨ鼀
 

L
㦨


Topic-4 : Moment of Inertia and Rotational K.E. ҈


㡰 ᨨᨨ犨
�ᆠ㜠ᆠ鼀
ᆠᨨ 
ᆠ

矄矄盐鼀
矄
矄
矄 ᨨ
�矄ᆠ矄
� �
�  
�盐矄

矄ᨨ

I = ò dI = ò dmr
㡰




�㡰

R

v Ö


h


�

r

H

O

꾀矄ᨨ

h
㦨


v
矄�
q v
=
g
g

᲼Q q =
�°桌

q





��㡰

ᨨ鼀
⼄

�㡰
 
��㡰 L
㦨


m


ᨨ鼀
��鼀



gh 㡰 h = m
gh

r R R =  q =
ᨨ
r = h h H H

v
mv  ´ = 꾀矄ᨨ L
㦨


g

g mv

�䰘


㝐矄矄
ᨨ
鼀 

dm = rpr
㡰dh ᨨ鼀
⼄

�㡰

��㡰
 
���㡰


v 

꾀 
�鼀
t
㦨

x

y㡰

 
ᆠᨨᨨ� 矄
ᨨ
 

꾀
ᨨ
鼀 

dA =
prdl =
pr

g �ᆠ
 
vx

vy

 
ᆠᨨ鼀ᨨ  矄
ᨨ
ᨨᆠ��矄

  �鼀


ᨨᆠ�泐
�
ᨨ�ᨨ 
�ᆠ�ᨨ

v


o

vx
㦨

v
ᆠᨨ矄
� =





Ö


v


I = ò dI =

v


x

Ö


꾀泐� �
v
㦨
u
졔

�
 
�ᆠ
�ᆠ�ᨨ

H

H

H







æR ö

ò dmr 㡰 = ò rpr 㡰dh çè H × h÷ø

æ æR


ö ö ÷ dh ø




=

v v
´
=
ᔼ
ᨨ�ᨨ 
�ᆠ�ᨨ
x = vx ´ t
㦨

g
g ҈ 2 ᔼ
�ᆠ
�ᆠ�ᨨ
泐� �


S
㦨
ut
졔
at


和ᨨ� �
犨
�

I =

)




v


v v
v
҈ v
´
- g
=
-
y
㦨

g
g

g
g v









ò r çè p çè H × h÷ø

æv ö v v y = v
矄�
�o - g ç
÷ 㦨
v
- g ´ v

㦨

- v
è gø g



(

dh ᆠᨨ矄 q

R R
+ H
ᆠᨨ矄 q

 

r = h  q 㡰

졠




���㡰


Mh  dh

㝐矄矄
ᨨ
鼀 

dm =

泐


�




��㡰

MR

dm

b㡰 A

B x

dx 㝐矄矄
ᨨ
 
矄鼀
鼀 
ᨨ
 
ᨨ

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PS-63

System of Particles and Rotational Motion

y

dm = l × dx 㝐ᨨ鼀 
ᨨ
� �
ᨨ
矄鼀
鼀 
xö æ dI = dm × x
= l
ç҈ + ÷ × x
dx è Lø 㝐ᨨ鼀 
ᨨ
� �
ᨨ
 
ᆠᨨ鼀
ᨨ
ᆠ

ᨨ� 
泐 � ��ᨨ

҈


ᆠ鼀

O




= l



L

䰘l
L ҈
㝐矄矄
ᨨ
 
 �
ᨨ
ÞI =

L

L







æ M = ò l dx = ò l
ç҈ + è




IO Ἐ = IO + M �
㡰


éL L x x + = l
ê + ú 
L

û ë 

ù




�㡰



l
L xö ÷ø dx = L



M \ l
= L 䰘 æ
M ö  䰘 ML
\I = ç ÷L ÞI = ҈
è L ø ҈
㡰 泐
 �盐
鼀ᨨ鼀 盐鼀
ᆠᨨ 矄�ᨨ

Lc = L f w wI +  I ´
=
I w Ἐ Þ w Ἐ =
҈
w  KE 㡰� = I w
҈  KE 㡰 f = I + I 㡰w Ἐ



DKE =





dx

OἘ dmR dI = + dmx

n= - L ⏌
dmR
M Þ I = ò dI = ò + ò dx ´ x

L n= L ⏌



I I

҈
҈

Iw - Iw = Iw





Iw  DKE
= =
\
ᆠ�ᨨ 
ᨨ矄矄
�
⧤
쟼

= ҈

KEl� Iw
ᆠ㡰 
p 
 
 矄�泐
ᨨ
 
�矄ᆠ矄
 
t
�矄
 
 �ᆠ├ 矄矄 ᨨ
�矄ᆠ矄
㝐ᨨ鼀 
ᨨ
� �
ᨨ
�矄ᆠ
�矄
��
泐

MR
᲼rpR
㡰t 桌R
=

꾀矄
r
 
t

矄鼀㡰 I µ R


I=

�


I
æ R
ö ҈
= Þ = a
Þ a =
I҈ çè R҈ ÷ø ҈ b㡰 㝐ᨨ鼀 
ᨨ
� �
ᨨ
ᆠ �盐
矄 
ᨨ盐

 �矄 矄矄� �
 ᨨ盐�
O
M
M a + b
㡰 = ᲼

㡰
+ 

㡰
桌 IO = ҈
҈


MR
ML
+
҈
M V ML
MV ML
ÞI = ´ + ÞI= +
pL ҈

pL ҈
dI mV M ´
L =+ =

҈
dL
pL

Þ V = pL Þ pR
L = pL    L \ =
R b㡰 ᔼ ��
 �盐
鼀ᨨ鼀 盐鼀
= I҈w҈ + I
w

w

 �盐
矄
ᨨ
 
ᆠᨨ鼀� 
矄泐矄鼀
� 
 �盐
鼀ᨨ鼀 盐鼀
= I҈w + I
w 꾀ᆠᆠᨨ� �
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
 �盐
鼀ᨨ鼀 盐鼀  I҈ + I
㡰w = I҈w҈ + I
w
෌�


I =

I w
҈ æ wö ´ 
I 㡰 ´ ç ÷ = è
ø






M 


+


㡰 IO ҈ ҈
= = M IO Ἐ



+


㡰 + M �
㡰
҈
ᆠ㡰 
 


ᆠ泐� 
ᨨ
鼀矄矄
m
 �
L
 
�盐矄 R

ᨨ犨

├
鼀 泐
�矄ᆠ
ᨨ
�盐矄
R
 
 �ᆠ├ 矄矄
dx 

�矄 ᆠ
ᨨ
x
ᨨ鼀
 �矄
OOἘ
 
鼀ᨨ鼀 
ᨨ
� � ᨨ盐
OOἘ
ᨨ
 �矄
鼀 
O

R









=




ᆠ鼀

x


ᆠ鼀 ᨨ鼀
 

 �矄
 ᨨ鼀

鼀ᨨ鼀 
ᨨ
� �
ᨨ盐
O Ἐ


æ x ö I = l
ò ç x
+ ÷ dx Lø
è L



O'

䰘


I҈w҈ + I
w


҈ ´ ҈
+


´ �

= = I҈ + I


҈ +


 � 
ᨨ�ᨨ 
├� �ᆠ
 �泐 Þw=

҈ ҈ ҈ æ

ö I҈w
+ I
w
= 

҈ +


㡰 ´ ç ÷ è ø




Þ Kf =  




Kf =

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EBD_8341 PS-64





Physics

㡰 㝐ᨨ鼀 
ᨨ
� �




æd ö
m ç ÷ + m ( AO ) � è
ø d  
AO = 

I҈ =

㝐ᨨ鼀 
ᨨ
� �
ᨨ盐
�졠
é
æ d ö
æ d ö ù I
= I҈ =  ê m ç ÷ + m ç ÷ ú êë � è
ø è  ø úû

҈ Md
҈
é
æ d ö
ù
æ d ö
꾀 
I A =
ê M ç ÷ + Md
ú + M ç ÷ êë � è
ø úû � è
ø
 Md
Þ IA = ҈
҈ Md
IO ҈
҈ \ = =

 IA Md
҈
㡰 矄� �
� ᆠ�
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
 �泐 Þ I
=

黨


҈ ҈ m҈ + m
㡰v
+ I w


҈ ҈ m҈  m
㡰 gh = m҈ + m
㡰wR 㡰
+ I w


Q v = wR 㡰

m҈  m
㡰 gh = Þ

Þ

w
é m҈ + m
㡰 R
+ I ù û
ë
m҈  m
㡰 gh

m҈  m
㡰 gh =

Þ w=

m҈ + m
㡰 R
+ I

҈

㡰 ␔ 
 
ᨨ
ᆠᨨ

�矄 ᆠ
� 

҈
҈
mv + I w


҈ ҈ mr = 鼀w 㡰
+ ´ ´ w
Q v = wr nᨨ sl�pp�ng 㡰




Þ mgh = mw r

gh ҈
gh Þw= =  r
r ҈҈
ᆠ㡰 㝐ᨨ鼀 
� �
ᨨ
 
ᨨ
矄矄� �
 ᨨ盐�

ᨨ�  l 犨泐
ᨨ鼀
 
ᆠ 
ᨨ
 
ᨨ
I
=
Ig
졔
鼀l
æ I
ö 䰘 MI
MI
Þ I= + M ´ç ÷ = ç ҈
÷ ҈


è ø 矄� �
mgh =

䰘MI
矄� �
I = MK =
K
㦨
�盐矄
ᨨ
�泐�ᨨ 㡰




Þ K=

䰘 I



҈

㡰 ෌�
鼀矄矄

盐 �

ᨨ
ᆠ�ᆠ盐
�矄ᆠ

s
㦨
A
졔
Br 꾀
ᨨ
 
� �
㦨


pdr 㝐矄矄
ᨨ
 
� �

dm
㦨
s
prdr 㝐ᨨ鼀 
ᨨ
� �


I = ò dmr
= ò s
prdr
r
a é Aa
Ba� ù Þ I =
p ò  A + Br 㡰r  dr =
p ê + ú � úû êë

é A Ba ù Þ I =
pa
ê + ú ë
� û b

҈
ᆠ㡰 I
㦨
ò  dm㡰r




a b

æ s
ö

ps
 b  r a 㦨
ò ç r ´
pr dr ÷ r 㦨
ø  aè


ps
 b - a  㡰  㝐矄矄
ᨨ
 
�矄ᆠ
㦨


b

s
m
㦨
ò r ´
pr dr
㦨

ps

b

a㡰 a

�盐矄
ᨨ
�泐�ᨨ
k
㦨

I m


ps
⏌ 㡰b - a 㡰 a
+ b
+ ab 㦨
㦨

ps
b - a 㡰  ҈

㡰 꾀矄
ᨨ
  
ᨨ⼄盐
�矄
ᆠ� �
矄ᨨ
 �盐
鼀ᨨ鼀 盐鼀 矄 ᨨ盐

ᆠᨨ 矄 ᔼ҈
졔
ᔼ
㡰
w
㦨ᔼ҈w҈
졔
ᔼ
w

᲼wᆠ
㦨
ᆠᨨ鼀鼀ᨨ
 �盐
ᨨᆠ�泐
ᨨ  
矄泐矄鼀

犨 
�矄ᆠ矄

�
ᆠᨨ ᆠ桌 ᔼw ᔼ҈w҈ + ҈ ҈
æ � ´
öw wᆠ = ç ÷ ҈ ᔼ
ø ᔼ҈ + ҈ è
�w҈ wᆠ =
҈ ҈ ҈ 쟼 - 쟼� = ( ᔼ҈ + ᔼ
) wᆠ
- ᔼ҈w҈
- ᔼ
w




�w҈ 盐
ᔼ

㦨
ᔼ҈⏌

 
wᆠ
㦨
�w҈⏌

␔
�

ᔼ w
쟼  쟼� =  ҈ ҈

҈�
b㡰 꾀矄
ᨨ鼀
 
⼄盐矄�ᨨ
 矄�泐
s㡰
㦨
├


㝐矄矄
ᨨ
�矄ᆠ
㝐 =

ò ├


Þ├=




㡰
p =
p├


p├
=




㝐

p
\
㝐ᨨ鼀 
ᨨ
� �
ᨨ盐
 
 �矄
ᨨ
 
�矄ᆠ






�㡰

WWW.JEEBOOKS.IN

PS-65

System of Particles and Rotational Motion

 = ò  = ò ( 鼀 ) 
= ò s꾀
= ò ├ 㡰
p㡰


=




æ
㝐 ö p´ ç ´


÷ p ├
è p ø = ò
p├ � = = = 㝐
  
᲼盐� �
盐
ᨨ
├
ᨨ鼀
⼄




�㡰桌 ҈

b㡰


é

ù pr
R  ú R
= MR
� êë  � û






b㡰


҈


㡰 꾀ᆠᆠᨨ� �
ᨨ

 矄
 ᨨ鼀

ᔼ=





鼀
+ 鼀
�

 ᆠ
�
㡰
ᆠᨨᆠ泐
�ᆠ矄
ᔼ
矄

㡰 
鼀矄矄
ᨨ
 
�
� �
㦨
㝐 和�
ᨨ
�
� �
㦨
l 㝐ᨨ鼀 
ᨨ
� �
ᨨ
�
� �
㦨
鼀


㝐 㝐矄矄
ᨨ
矄鼀
� �
㦨

 �
ᨨ
矄鼀
� �
㦨


҈
æ 䰘M ö ö
҈ æ 䰘 I҈ = ç = ç ´
÷ MR
= 鼀
÷ 
R㡰
è ҈
ø
è
ø
ö æ MR

æ M ö R
æM ö
Þ = = I ç ÷ I
= ç ÷ r
ç ÷ � è
ø çè
÷ø

�è
ø ҈
 ù é
 ê  pr r =
 pR ´ r ú ê ú ëêÞ r = R ⏌
ûú I҈ ҈
´

= = ҈

I



㝐ᨨ鼀 
ᨨ
� �
ᨨ
鼀ᨨ
� �

㦨



㝐 æö ç ÷ ᔼ
\ 鼀ᨨ =

è ø
㝐 㡰 ᔼᨨ���  ᔼ ᔼ 鼀ᨨ =
҈






æ M w
ö \w f = ç ÷ è M +
m ø ҈黨
ᆠ㡰 ├� �

ᆠ�ᆠ盐
� �
ᨨ
�盐矄
r
 
 �ᆠ├ 矄矄
dr
矄
 鼀矄矄
鼀 

矄ᨨ
ᨨ
鼀矄矄
R

M = ò r
r ´
prdr =


R


pr
R  

IC = ò r
r ´
prdr ´ r
=



㦨

a

ᨨ鼀
犨ᨨ 矄
矄ᆠᨨ 
犨


졔

㦨
鼀
 a㡰

├� �
ᨨ⼄盐
ᨨ盐


犨
�


맔



맔

㦨
ᔼa


맔



맔

㦨
鼀

a




㦨
鼀
a 和ᨨ� �
⼄盐�ᨨ
�㡰
 
��㡰

a= = ҈
 ⏌ 矄
鼀










pr
R � �

矄� �

 �矄
 ᨨ鼀 � æ ҈ ҈ ö ҈
pr
R \ I = I C + MR
=
pr
R � ç + ÷ = ҈� è �ø











æ ML
ö æ Lö ö +
m ç ÷ ÷ w f ÷ w
= ç ÷ ç ҈
è
ø ÷ø ø è

㝐æö
çè
÷ø

ᔼ ҈�ᔼ
和ᨨ

ᔼ
㦨
ᔼ
-
= ҈
҈
b㡰

졠

҈
I w = kQ

æ
k ö ᨨ
w = çç ÷÷ Q è I ø dw
K æ dQ ö æ
k ö = ᨨ
a = ÷w ç ÷=ç p I è dt ø çè I ÷ø æ
k öæ
k ö =ç q
k q ÷ç ç I ÷ç ÷÷ = I I è øè ø ҈

ᆠ㡰 I�w�
㦨
Ifwf ҈䰘
㡰

æ ML
ᨨ
çç ҈
è

l







�㡰




��㡰

b㡰 㝐ᨨ鼀 
ᨨ
� �
ᨨ
�矄ᆠ
҈
ᨨ盐
졠졠¢
㦨
ᔼ҈
㦨


㝐



㝐
졠
ᔼ
ᨨ


ᨨ盐
졠졠¢

҈ æ 㝐
ö 㝐

+ 㝐
÷ + 㝐 = 㦨
ᔼ

㦨 ç
è
ø
㝐
졠
ᔼ
ᨨ

ᨨ盐
졠졠¢ ҈ æ 㝐
ö 㝐

+ 㝐
÷ + 㝐 = 㦨
ᔼ
㦨 ç
è
ø
矄ᨨ

矄盐 
㝐
졠
ᔼ
ᨨ盐
졠졠¢
�矄
ᔼ
㦨
ᔼ҈
졔
ᔼ

졔
ᔼ Þᔼ =

æ 㝐
ö 㝐
+
ç + 㝐
÷
è
ø










=

㝐
㝐

+ +
㝐

㦨

㝐



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EBD_8341 PS-66

Physics


ᨨ
鼀矄矄
ᨨ
 
�矄ᆠ 㦨
s
맔
p
R

㦨
黨M


鼀矄矄
ᨨ
 
ᆠ�ᆠ盐
�矄ᆠ
ᆠ盐

├
�
ᆠ㡰 꾀矄
犨
├ ᨨ犨


w = ᔼ

w=

é êëQ ᔼ

├ 鼀l




ᨨ =

҈
ù 鼀l ú  û

 矄�ᨨ
犨 
�
矄矄矄
 ᨨ盐�
 
鼀
ᨨ矄��ᨨ


= 鼀w
q







l ├ l = 鼀
q

= ├q
  鼀l l

҈ æ Rö 㦨
´ M ´ ç ÷ è ø




é҈ æ Rö æ
R ö ù ú 㦨
ê ´ M ´ ç ÷ + M ´ ç ÷ è ø è  ø ú êë
û ᲼矄� �
 �ᆠ盐
 �矄
 ᨨ鼀桌 \

ᨨ
M
I

ᨨ
 
矄泐矄鼀
ᨨ盐

 �矄
矄矄� �
 ᨨ盐� O
 
 �ᆠ盐
ᨨ
 
 
ᨨ
 
�矄ᆠ
�矄 I
㦨
I҈
졔
I


ᔼ ��
ᨨ矄��ᨨ

ᨨ






é҈ ҈ æ Rö æ
R ö ù
㦨
黨MR - ê ´ M ´ ç ÷ + M ´ ç ÷ ú è ø è  ø ú
êë
û

� 
ᨨ矄��ᨨ

泐
 
ᨨ犨
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
 �泐

쟼

ᨨ
ᨨ
㦨
ᨨ�ᨨ 
⧤
쟼
鼀�


l ҈ 和�
q 㦨

ᔼw2



l ҈ ҈ 鼀l 2 ҈ 鼀l


l w ω
Þ 鼀� ´ = 和�


㦨






 ᨨ
ᆠᨨ鼀
 �
ᨨ
ᨨ




ᨨ矄 ҈
㡰 矄� �

 矄
 ᨨ鼀

鼀ᨨ鼀 
ᨨ
� �
ᨨ盐
�O ᔼᨨ
㦨
ᔼᆠ鼀
졔
鼀
w = �
(
l ) =

=

䰘㝐
��㝐
+
㝐 ´ 
㡰
㡰 =









꾀��

鼀ᨨ鼀 
ᨨ
� �
ᨨ盐
ᨨ� 
P

Ip
㦨
Iᨨ
졔
md





��㝐
҈
҈ 㝐
+ 䰘㝐 㡰
=




㡰 
s

 
鼀矄矄

盐 �

=

R/  O'


R ⏌ O

黨MR
黨MR
黨 - ҈㡰MR
=
MR

㦨

҈


黨
ᆠ㡰 꾀矄
犨
├ ᨨ犨

鼀ᨨ鼀 
ᨨ
� �
ᨨ

�矄ᆠ
ᨨ盐

 �矄 矄矄� �
 ᨨ盐�
ᔴ
෌

 
 �ᆠ盐
ᨨ
�矄
 
㦨


Þ 鼀�


䰘





\ M
I

I
㡰
ᨨ
 
ᆠ盐
ᨨ盐
ᨨ�ᨨ
ᨨ盐

 �矄
矄矄� �  ᨨ盐�
O
 
 �ᆠ盐
ᨨ
 
 
ᨨ
�矄ᆠ

㡰 㦨
�

ᆠ鼀




p R
=M 黨 
盐矄
ᆠᨨ 矄�
 
ᨨ
矄泐矄鼀
矄

ᆠᨨ鼀
�矄ᆠ
ᨨ 鼀矄矄
黨M
 

��
鼀矄矄
M
矄盐
�鼀ᨨ矄
ᨨ
�
㝐ᨨ鼀 
ᨨ
� �
I ҈ 㡰
ᨨ
 
ᆠᨨ鼀
�矄ᆠ ҈
㦨
黨MR ᨨ盐

 �矄
矄矄� �
 ᨨ盐�
O
 
 �ᆠ盐
ᨨ
 
 
ᨨ
 
�矄ᆠ
M
I

ᨨ
 
ᆠ盐
ᨨ盐
ᨨ�ᨨ
ᨨ盐

 �矄
矄矄� �
 ᨨ盐� OἘ
 
 �ᆠ盐
ᨨ
 
 
ᨨ
�矄ᆠ

æ Rö 

㦨
s ´ p ç ÷

㦨
s ´ è ø

鼀

㝐ᨨ鼀 
ᨨ
� �
ᨨ

�矄ᆠ
ᨨ盐

 � �
 �矄  �ᆠ盐
ᨨ
�矄
ᨨ犨
 
ᔼ =







ᔼἘ =

 鼀



\

鼀
ᔼ ᔼ Ἐ =


졠





⼄

⼄

鼀
=҈






ᆠ㡰 꾀矄
犨
├ ᨨ犨

鼀ᨨ鼀 
ᨨ
� �
ᨨ

矄ᨨ�
ᆠ泐�  ᨨ盐

 �矄
犨 �ᆠ
�矄
 �ᆠ盐
�矄ᆠᨨ

mR
ml
+
҈
mé l
ù I = êR
+ ú
ë û m é V l
ù
㦨
ê + ú
ë pl  û I=

l Þ

dl m é -V
l ù = + =
dl
êë pl
 úû


pl  
  l l
pl Þ
=
ᨨ

= pR
l =
R  R

V
l =
 pl

ÞV =




WWW.JEEBOOKS.IN

PS-67

System of Particles and Rotational Motion ҈


b㡰 㝐ᨨ鼀 
ᨨ
ᔼ �
ᨨ
ᆠᨨ鼀
�矄ᆠ
ᨨ盐
Ἐ졠Ἐ
ᨨ�  㝐
ᔼᨨ =
�盐矄
ᨨ
鼀ᨨ
�矄ᆠ
㦨
⏌
\ 㝐矄矄
ᨨ
鼀ᨨ
�矄ᆠ
㦨
㝐⏌҈
᲼꾀矄
㝐
µ
桌 㝐
ᔼ
ᨨ
鼀ᨨ
�矄ᆠ
ᨨ盐
�矄
ᨨ犨
 �矄
졠Ἐ㡰

�


b㡰 A' M

A

B'




㝐
㝐 æ  ö ҈黨 㝐
+ ç ÷
= �҈
�҈
҈
è
ø 㝐
ᔼ
ᨨ
鼀� � �
�矄ᆠ
䰘 㝐
҈黨 㝐
㝐

= �҈

�҈

R 㡰 
a = 
 pR M ᨨ犨

=  M¢ a
 pR   =  = p



æ
ö R÷ ç è  ø
M M¢= p ᔼ鼀� � �
=



 R MR
= Mr


Þ r
㦨
҈� �


㡰 
s

 
鼀矄矄

盐 �

ᨨ
 
�矄ᆠ
\ ᨨ
鼀矄矄
ᨨ
 
�矄ᆠ \

R/ 

㦨
s
맔
p
R

㦨
黨M 㝐矄矄
ᨨ
 
ᆠ�ᆠ盐
�矄ᆠ

ᆠ盐

æ Rö 㦨
s ´ p ç ÷ è ø 㦨
s ´

M ¢a





M æ
ö R÷ ´ç
p è  ø =
MR =
黨 p
㡰 ᨨ

 �
盐 � ᨨ鼀 矄⼄盐
矄 

ma
I҈
㦨
I

㦨
I
㦨
҈


㝐w 㝐 +
鼀 和ᨨ� �
犨
�
ᨨ矄矄
�
⧤
쟼
㝐鼀 w

㦨
㝐 +
鼀㡰


R ⏌




O

p R
=M 黨




҈ æ Rö 㦨
´ M ´ ç ÷ è ø
矄� �
 �ᆠ盐
 �矄
 ᨨ鼀 \ M
I

I
㡰
ᨨ
 
ᆠ盐
ᨨ盐
ᨨ�ᨨ
ᨨ盐

 �矄
矄矄� �  ᨨ盐�
O
 
 �ᆠ盐
ᨨ
 
 
ᨨ
�矄ᆠ

é҈ æ Rö æ
R ö ù ú 㦨
ê ´ M ´ ç ÷ + M ´ ç ÷ è ø è  ø ú êë
û \

ᨨ
M
I

ᨨ
 
矄泐矄鼀
ᨨ盐

 �矄
矄矄� �
 ᨨ盐� O
 
 �ᆠ盐
ᨨ
 
 
ᨨ
 
�矄ᆠ I
㦨
I҈

I


I҈ I


҈


ᆠ㡰 ⧤� �ᆠ
 �泐
ᨨ�ᨨ 㡰
⧤
㦨
ᔼw
҈
⧤� �ᆠ
 �泐
 矄�ᨨ 㡰
⧤
㦨
㝐
㝐
ᔼ
� ��㡰
ᔼ� �
㦨
㝐
錐
w� ��
㦨
w 㝐
ᔼ
 犨㡰
ᔼ¢矄泐矄鼀㡰
㦨
㝐
+
鼀
w¢矄泐矄鼀㡰
㦨

O'

҈
㝐ᨨ鼀 
ᨨ
� �
I҈㡰
ᨨ
 
ᆠᨨ鼀
�矄ᆠ
㦨
黨 MR ᨨ盐

 �矄
矄矄� �
 ᨨ盐�
O
 
 �ᆠ盐
ᨨ
 
  ᨨ
 
�矄ᆠ
M
I

ᨨ
 
ᆠ盐
ᨨ盐
ᨨ�ᨨ
ᨨ盐

 �矄
矄矄� �
 ᨨ盐� OἘ
 
 �ᆠ盐
ᨨ
 
 
ᨨ
�矄ᆠ

㝐ᨨ鼀 
ᨨ
� �
ᨨ
 
ᆠ盐
ᨨ盐
 
��
 �矄





Y'

ᨨ
矄ᨨ�
矄 
鼀ᨨ鼀 
ᨨ
� �
ᨨ盐
�鼀

IAB
㦨
MR = I � ᨨ
矄ᨨ�
�矄ᆠ ҈ 


IA'BἘ
㦨
IYYἘ
졔
Mr

㦨
Mr + Mr = Mr

IAB
㦨
IA'BἘ �� 㡰

㝐
ᔼ
ᨨ
鼀ᨨ
�矄ᆠ
ᨨ盐
졠 ᔼ鼀ᨨ
�矄ᆠ
㦨
ᔼᆠ鼀
졔
鼀


I=

M

r








B

R

҈ 㝐æ ö 㝐
= çè ÷ø =
҈

�҈


=

Y

I


㦨
w㡰

䰘




é҈ ҈ æ Rö æ
R ö ù
ú 㦨
黨 MR - ê ´ M ´ ç ÷ + M ´ ç ÷ è ø è  ø ú
êë
û 黨MR
黨MR
黨 - ҈㡰MR
=
MR
ᨨ
I
㦨

㦨

҈

㡰 㝐矄矄
ᨨ
�矄ᆠ
㦨

㝐 ҈
ᔼ
�矄ᆠ
㦨
MR
\ 㝐ᨨ鼀 
ᨨ
� �
ᨨ
ᨨ 
⼄盐
ᨨ
�矄ᆠ ҈ é҈
ù ҈
㦨
ê 
M 㡰 R ú = MR
ë

û

WWW.JEEBOOKS.IN

EBD_8341 PS-68





Physics

IXXἘ
㦨


MR
 + MR
= MR





鼀矄矄
M
㦨
Lr
 
�盐矄
R
㦨
\ 黨


ᨨ鼀
⼄ 矄

�㡰
 
��㡰

㡰 꾀ᨨ盐
 
�鼀
ᨨ
 
ᆠ�ᆠ盐
ᨨᨨ
� �㡰 ҈
I
㦨
MR
矄� �

 �矄
 ᨨ鼀 㝐ᨨ鼀 
ᨨ
� �
ᨨ
 
ᨨᨨ
ᨨ盐
XXἘ
 �矄

L 錐
p




Lr æ Lö = ÷ è
p ø
p





IXXἘ
㦨
 Lr㡰 ç

㡰 ᨨ鼀
矄泐鼀鼀泐
IAB
㦨
IAἘBἘ
 
ICD
㦨
ICἘDἘ AἘBἘ
^ AB
 
CἘ
DἘ ^ CD A' ᨨ鼀
 ᨨ鼀
ᨨ C'

D

 �ᆠ盐
 矄
Izz
㦨
IAB
졔
IAἘBἘ
㦨
ICD
졔
ICἘDἘ

q

A

Þ
IAB
㦨

ICD \




B

O

IAB
㦨
ICD 㦨
I

D'

C

B'


㡰 ᨨ
矄ᨨ�
矄 
A

 矄�泐

R

æ rö rA r 㡰 = k ç ÷ è Rø

dx

æ rö R

s

x

꾀  PBr㡰
㦨
K ç ÷ è ø






ᔴᨨ 矄�

矄 �ᆠ
矄 
ᨨ
�盐矄
x
 
 �ᆠ├ 矄矄
dx
㝐矄矄
ᨨ
 
矄 

dm
㦨
 矄�泐
맔
ᨨ盐鼀

æ xö = ç k ÷ 
p x
dx 㡰 è Rø 和ᨨ

鼀ᨨ鼀 
ᨨ
� �
ᨨ
矄 
ᨨ盐
�矄
�鼀


dI =


dm㡰 x



æ xö

÷ 
p x dx 㡰 x  Rø

㦨
ç k è

æ
p k ö � =ç x dx è  R ÷ø R

\

㝐ᨨ鼀 
ᨨ
� �
ᨨ
 
矄 
A

I A = ò dI



pk �
pk é x x dx = = ê ú ò R R ëê
úû



I
= kml


G


鼀ᨨ鼀 
ᨨ
� �
ᨨ
DEF
㦨
I҈
ᨨ盐
G


D I ml
æ m öæ l ö
ᨨ
I҈ =
÷ç ÷ µ ҈
҈
è
øè
ø m⏌
B 
I ADE = I BDF = I EFC = I
I �I \  I
+ I҈ = I
Þ  I
+
= I
Þ I
=
҈
҈


和ᨨ

I҈ µ ç

�㡰

R

æ
pk ö é x҈
ù I B = � ò x dx = ç � ÷ ê ú è R ø ëê ҈
ûú R



IB =

A

I
µ ml


B

æ
pk ö � \ IA = ç R è ҈
÷ø 和�鼀�泐

ᨨ
矄 
B

\


㦨
é
é


ù
ù 㦨

ê MR + Ma ú +
ê MR ú ë� û ë� û 꾀  Ӹ
é ù




=M
 㡰 + a ú =
´ MR +
Ma
ê� R � ë û
é æ ù
�ö -
㦨



� ê ´ ç ÷ +
´
ú ´҈
ç ê � è
÷ø ú ë û 㦨



� [
+ ҈
] ´ ҈
-
= 黨 ´ ҈
-
㦨
N
맔
҈


├�
m
\ 
㦨
黨

҈҈㡰


鼀矄矄
ᨨ
� �盐
鼀� 
㦨
m

 
 �
ᨨ
矄� 㦨
l

 
鼀ᨨ鼀 
ᨨ
� �
ᨨ
鼀� 
ᨨ盐

 �矄
矄矄� �  ᨨ盐�
ᆠ ᨨ�
G
 �ᆠ盐
ᨨ
 
 



ùR

R


pk

I B ҈

n = = = \n=
I A 
҈
҈

҈
㡰 
s

 
矄盐 ᆠ
鼀矄矄
 矄�泐
㝐ᨨ鼀 
ᨨ
� �
ᨨ
 
鼀� 
ᨨ盐
 矄
矄矄� �  ᨨ盐�
Ἐ졠Ἐ ҈

IO

㦨
s᲼p
R 㡰 桌 ´ 
R㡰
é҈



ù êë
s p R 㡰 + s p R 㡰 ´ R úû ҈






㦨
psR
㝐ᨨ鼀 
ᨨ
� �
ᨨ
 
鼀� 
ᨨ盐
 矄
矄矄� �
 ᨨ盐�
ἘἘ

IP
㦨

p s

졔
sp
㡰

맔

㡰

é҈


æ

ö ù ê s pR 㡰 R + spR 㡰 ç 
R㡰 + R ÷ ú è ø ûú ëê

㦨


p s


�
�
s p

㦨
҈

�
p s
I P ҈

�psR
䰘 = = » \ ҈ IO psR
҈



黨㡰 
 
ᨨ盐
矄 矄

꾀



ᔴ



 m

㦨
m
㦨
mᔴ
㦨
m
㦨


�
├�   㝐ᨨ鼀 
ᨨ
� �
ᨨ
  ᔴ 矄泐矄鼀
ᨨ盐
 
��ᨨ 
ᨨ  
矄⼄盐 ᔼӸ 㦨
ᔼ҈
졔
ᔼ
졔
ᔼᔴ
졔
ᔼ
㦨


ᔼ꾀
졔

ᔼ

ᆠ鼀

R

黨


pk � R 


��㡰

A

l⏌
G m⏌


F

E

C

 ᆠ

鼀ᨨ鼀 
ᨨ
� �
ᨨ
DECB
�



 
鼀ᨨ
 ADE æ � I ö æ I ö ҈҈I
NI
=
I
+ I҈ =
ç
÷ + ç
÷ = = ҈
è ҈
ø è ҈
ø ҈


 ᨨ
盐
ᨨ
N
㦨
҈҈


WWW.JEEBOOKS.IN

PS-69

System of Particles and Rotational Motion




҈ MR



(20) 꾀矄
犨
├ ᨨ犨
鼀ᨨ鼀 
ᨨ
� �
�矄ᆠ

I �矄ᆠ = w҈

w
M

I҈ R҈㦨R

Iᨨ

㦨
I҈
졔
I

㦨
I
졔
I


ᨨ Iᨨ
㦨

I

㦨
2I \
I

㦨
I I҈
㦨
I

㦨
I
㦨
I


 ᨨ

Iᨨ ᆠ
 ᨨ� 
泐
� �
 泐
犨ᨨ �



Iᨨ

㦨
I҈
졔
I

㦨
I҈
졔
I 㦨
I҈
졔
I

㦨
I

졔
I




㦨
I

졔
I

㦨
I
졔
I


w M

R҈M㡰 R
M㡰

I
R
㦨R/2

矄� �
 �盐
鼀ᨨ鼀 盐鼀
ᆠᨨ 矄�ᨨ

I҈w҈ + I
w
=  I҈ + I
㡰 ´ w f æ MR
MR
ö + çç ÷wf
÷ø è
҈ ҈ æ MR
ᔼ ��
⧤
쟼


K � = I w
= çç

è
MR
´w+
=


҈


ö
MR
w
÷÷ w =
ø

MR
w
MR
w

� = ´ ҈

=

奠 = P奠

MR w X
F rF  ᆠ

盐
ᨨ
P
㦨




�㡰 a 㝐ᨨ鼀 
ᨨ
� �
ᨨ
  矄泐矄鼀
ᨨ盐
 �矄
XE


I = I E + I F + IG E

a

rG Þ I = mrE 㡰
+ mrF 㡰
+ mrG 㡰

�
�
æaö ma Þ I = m ´

+ m ç ÷ + ma
= ma
=


è
ø \ N =
�


a

G




҈�㡰 

 �
ᨨ



㦨
҈
鼀





 �

q
㦨



҈
Iw

l ҈ æ ml
ö
Þ 鼀� l ´ ҈ = ҈ æç ml ö÷ w
Þ 鼀�㡰 矄� 
° = ç ÷w


çè  ÷ø

çè  ÷ø


DPE = DKE ᨨ mgh =

Þ w = ҈� ⏌矄



O

D

C


㡰 K
E
ᨨ
 
矄 
㦨
 矄�ᨨ 
K
E
졔
ᨨ�ᨨ  K
E

҈
҈
mv + I w

␔ 

I
㦨
鼀ᨨ鼀 
ᨨ
� �
w
㦨
꾀 �盐

ᨨᆠ�泐
ᨨ
ᨨ�ᨨ

m
㦨
鼀矄矄
ᨨ
 
矄  v
㦨
� 
ᨨᆠ�泐
ᨨ
ᆠ 
ᨨ
鼀矄矄
ᨨ
矄 

Q
㝐ᨨ鼀 
ᨨ
� �
ᨨ
矄 
I = mR � ҈
҈


\
K
E = mv + ´ mR ´ w

�
v ö ҈ ҈
ævö æ Þ K
E = mv
+ ´ mR
´ ç ÷ çQw = ÷ Rø

� è Rø è
҈æ
öæ v ö Þ KE = ç mR
+ mR
÷ç ÷
è� øè R ø
҈ 䰘 v 䰘 ҈
� Þ KE = mR
´ ´ = ´ ´
� R
҈

҈

� Þ KE = ´ ҈

ᨨ盐
Þ
⧤쟼
㦨


䰘�
맔
҈


ᨨ盐 b㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

ᨨ
�矄
鼀ᨨ
�

ᆠ鼀
�


ᆠ 


㦨
�

ᆠ鼀


㦨
҈

ᆠ鼀
 
ᨨ
 


= ҈

㦨

�
ᆠ鼀 �盐矄
ᨨ
ᨨ


=

ᨨ
ᨨ
矄�� �

 
�ᨨ盐  ᆠ 
㦨
w \ ᨨᆠ�泐
ᨨ
矄ᆠ
㦨
ᆠ 
졔
w㡰 \ �矄 ᆠ
鼀ᨨ
泐
矄ᆠ
㦨
ᆠ 
졔
w㡰

꾀矄矄盐鼀� �
矄泐鼀鼀�ᆠ
鼀� 
ᨨ

�
xy
 

Ix
㦨
Iy 泐
矄泐鼀鼀泐 Ix
졔
Iy
㦨
Iz
 �ᆠ盐㜠 �矄
 ᨨ鼀㡰

 ö æ ö æ = ç ᆠ  + ᆠ  ÷  çQ w = ÷ è ø è ø

Iz B =


Ma
A Ix
㦨
Iy
㦨
a
a졔a=2a Q Iz
㦨
҈


Ma


�� 㡰







a ᨨ犨

ᨨ鼀
ᨨ
㜠 矄 X  ᨨ鼀
O
IAB



㦨
Ix
졔
M
a㡰 㦨




Ma

졔

Ma
㦨




Ma


b

ᆠ㡰 和� ᆠ

ABCD
�矄

矄⼄盐
鼀� 
 ᆠ
泐 矄泐鼀鼀泐I҈
㦨
I

 
I
㦨
I
ᨨ鼀
 �ᆠ盐
 矄
 ᨨ鼀
㝐ᨨ鼀 
ᨨ
� �
ᨨ盐

 �矄
 �ᆠ盐
ᨨ
矄⼄盐 
 
矄矄� �
ᨨ鼀
ᆠ 

O

=

Y






B

=

҈ æ MR
MR
ö ҈

MR
w
+ ç ÷ w =
÷ø
� �
çè
ᆠ �
ᨨ矄矄
�
├� �ᆠ
 �泐
奠
ᨨ矄矄


䰘


҈

Topic-5 : Rolling Motion
Þ wf = w �

� 
⧤
쟼


K f =


�



A

ᆠ   ×  = ´ �
= 䰘� ᆠ鼀

꾀

x 㦨



ᨨ

 ᨨ �
ᆠ鼀

꾀 

҈

ᆠ鼀




ᆠ㡰 ᨨ
矄 
҈
҈ mv
졔
ᔼw

㦨
mgh


䰘v
҈
҈ æ

öv ᨨ mv
졔

ç mR ÷
= mgh ᨨ
h
㦨
҈
g

è� øR

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EBD_8341 PS-70

Physics

ᨨ
ᆠ泐� 

䰘


҈
҈ æ mR
ö mv + ç ÷ = mgh Ἐ

çè
÷ø h 䰘v
⏌ ҈
g ҈
= = \ h Ἐ v
⏌
g ҈�



ᨨ

҈
҈ mvcm + I cm w



I cm ö
҈æ ҈
҈ æv ö = mvcm + I cm ç cm ÷ = ç m +
÷ vcm
R

è R ø è ø mR
ö
҈æ ᨨ
� �


鼀�
=
çç m +
÷÷ vcm R ø è
v \ h = cm
g mR
ö
҈æ ÷ vcm ᨨ
矄ᨨ�
ᆠ泐� 

mgh
=
çç m +
R
÷ø è v
\ h = cm
g ҈æ
mR
ö
m = + ç ÷ vcm ᨨ
矄 

mgh

ç � R
÷ø è 䰘v
\ h = cm ҈
g





 䰘
Þ


҈�
҈

҈


㡰




㦨
鼀

mR a


��㡰
ᨨ
盐
ᨨ� � 
㦨
a


���㡰 ᨨ鼀
҈㡰

㡰
 
㡰 mRa  F = maR
F = mRa


F a= mR ᆠ㡰 꾀矄
矄 ᨨ犨
�
 
��鼀

 
ᨨ鼀
ᆠ�ᨨ
ᨨ
꾀 ᨨ
ᨨ
犨�
矄 � 
ᨨ犨矄
졠


�矄
犨�

ᨨ
 � �
ᨨ
 
矄泐矄鼀
ᨨ
ᆠᨨ 矄
ᨨ
盐
 


졠

꾀

ᔴ

a



ᨨ

ᨨ

黨


ᆠ㡰

҈



ᨨ鼀
ᆠᨨ 矄�ᨨ
ᨨ
 �盐
鼀ᨨ鼀 盐鼀
ᨨ盐
 泐
� ᨨ� 
ᨨ
 
矄盐 ᆠ
mr
w

㦨

mr
w w
r


[Q v = r w ] Þ w
㦨
w
⏌

Þ

v =
㡰 꾀泐� �
 �泐
ᆠᨨ 矄�ᨨ
� ᆠ�
ᨨ


㦨
ᔼa
㦨








�㡰

mR F 㝐矄矄
㦨
m a = fR 졠
aù é mR
a = fR
êQ a = ú f Rû ë
R ma = f


��㡰
盐
 �矄
盐
�
⼄盐�ᨨ
�㡰
ma ma ma
㦨
F 
ᨨ
F =

㡰 꾀矄
犨
├ ᨨ犨
鼀� 矄� q 꾀ᆠᆠ�ᨨ

 = ᔼ 鼀+ 
㝐 ᆠ
�
矄� qᆠ 㝐 ᆠ
�
矄� qᆠ ᨨ
ᆠ泐� 

 ᆠ =

㦨
҈ 㝐ᆠ 㝐ᆠ
㝐ᆠ + 㝐 + ᆠ




ᨨ
ᆠ
㦨
� 矄� qᆠ  ᨨ
矄 
㝐 � 矄� q矄
㦨
㝐矄 � 矄� q矄 矄 = 矄 ᔼ
㝐
㝐矄 + 㝐 矄 + 矄
�
 �  矄 = � 矄� q矄 ᨨ
䰘 ��

ᆠ
㦨
矄 � � 矄� qᆠ 䰘 ҈�
� �


= = � 矄� qᆠ = � 矄� q矄 \ 矄� q矄
҈
 䰘 � 




�㡰






ᆠ㡰 ᨨ鼀
��盐
ma
㦨
F

f 꾀 

ᨨ⼄盐
t
㦨
Ia


ᨨ 盐矄㡰
mgh =

�ᨨ
ᨨ
�� 矄
҈
�


v
hἘ =
g

æ v
ö ҈
҈
mv + I w = mg ç ÷

è
g ø ҈
҈ v

mv + I
= mv

᲼Q
v

㦨

w桌 \

R

҈ v  ҈ ҈ ҈ I = mv
- mv
= mv
Þ I
㦨
mR
\
R






�矄
�矄
 
鼀ᨨ鼀 
ᨨ
� �
ᨨ
 
�矄ᆠ
 ᆠ
 
ᨨᆠ
�矄 �矄ᆠ
҈҈
b㡰 ᔼ
盐
ᨨ� �

 
ᨨ� 
ᨨ
ᆠᨨ ᆠ
�矄
  � 矄  ᨨ盐矄
ᆠ 
ᨨ
ᨨ�ᨨ
ᨨ

 
�ᆠ矄
ᨨ
  �矄ᆠ

졠
泐� �
v
㦨
rw ␔
� 
w
�矄
矄鼀
ᨨ

 
�ᆠ矄
 
v
µ
r
rQ  rC  rP \ rQ  rC  rP

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PS-71

System of Particles and Rotational Motion b㡰 ᔴ泐� 
ᨨ

鼀ᨨ� �
ᨨ

�ᆠ�ᨨ 矄矄
矄盐 ᆠ

ᔼ

ᨨ
 
ᆠ矄矄
ᨨ
盐
ᨨ
ᨨ犨
 
ᆠᆠ�ᨨ
ᨨ
  ᆠ 
ᨨ
鼀矄矄
ᨨ
 
ᆠ泐� 
�矄
�
矄�
q

꾀矄ᨨ
ᨨ
ᨨ⼄盐 ᨨ盐
 
ᆠ 
ᨨ
ᆠ泐� 
�矄
ᆠ� �
ᨨ
 
ᆠ泐� 
矄� ᆠ 犨
矄矄盐鼀
 
矄盐 ᆠ
ᨨ

�ᆠ�ᨨ 矄矄
 
 
ᨨᆠ矄 ᆠ� �
ᨨ
 
ᆠ泐� 
�矄
鼀�
 

犨 �ᆠ 矄矄
 ᨨ盐�
 
ᆠ 
ᨨ F ᆠ泐� 

 ᨨ
 
 g s�n
q ᨨ� 
ᨨ N 鼀ᨨ鼀 
ᨨ
 
ᨨ� 
ᨨ ᆠᨨ ᆠ ᆠᨨ ᆠ
�
ᨨ
 
ᆠ矄矄
�矄 g s�n
q �
 
ᨨ犨 犨
�ᆠ�ᨨ

 
 ᆠ
 
�ᆠ�ᨨ  mg s�n q ᨨᆠ
犨泐矄
ᆠ矄
�
  mg q 盐犨
�ᆠ�ᨨ
҈
ᆠ㡰 

盐
ᨨ
� F ᨨᆠ
ᨨ鼀
ᆠ�ᨨ
矄 � 矄 L N G ᨨ
ᨨ 
ᆠᨨ 

꾀
 �矄 矄�盐�ᨨ

 
ᆠ盐�ᆠ
ᨨᆠ├ L⏌
矄矄
ᨨ� �
ᨨ盐
ᨨ� 
O
O 꾀ᨨ盐
ᨨ� 
졠
ᨨ⼄盐 mg L mg F
맔
L
㦨
mg
맔

Þ F
㦨


 ᆠ
鼀� �鼀盐鼀
ᨨᆠ
⼄盐�
ᨨ
ᨨ
 
ᨨᆠ├
mg F=
҈

䰘㡰 K.ETᨨtal
ᨨ

ᨨ� �
�矄ᆠ
K.Etrans
졔
K.Erᨨt ҈




҈
҈
҈
҈ æ ҈ mR
ö æç v ç ÷ 㦨
鼀
졔
ᔼw
㦨 mv
졔


ø çè R




è

ö ÷ ÷ ø


mv
ᨨ
矄盐 ᆠ
AB k.E�
졔
ᨨ矄矄
�
���ᨨ 
ᨨ �
 �泐
㦨
K.Ef   m
㡰

졔
mg
㡰
㦨
m VB



�㡰

ᨨ
矄盐 ᆠ
CD   m
v
㡰

졔
mg

䰘㡰
㦨
m VD



��㡰

Q
VB
㦨
VD

\
ᨨ鼀
⼄

�㡰
 
��㡰   m
㡰

졔
mg
맔


㦨
m
v
㡰

졔
mg
맔

䰘

\
V

㦨
䰘
m/s ҈�

㡰


矄�ᆠ├
�矄

㡰
ᨨᆠ
矄ᨨ

ᨨ� 
ᨨ
ᆠᨨ ᆠ
졠
ᨨ
  � �
犨�
�ᨨ盐 
 矄
ᨨ
矄�

盐
 
�ᆠ�ᨨ 
ᨨᆠ
f
ᨨ矄
ᨨ
ᨨ犨
 �矄
 
ᆠ矄

ᨨ⼄盐
ᨨ盐
ἘᆠἘ
犨 �ᆠ 矄矄
ᨨ� �
 
� �

犨
 
� �


 
矄�ᆠ├

�ᆠ�ᨨ

ᨨᆠ
f҈
矄ᨨ
ᆠ矄
F

f

㦨
ma


f

㦨


맔




\ f

㦨
҈



C 꾀泐� �
t
㦨
Ia
ᨨ盐
C
 R R f҈ a  f


f҈㡰
R
㦨
Ia
㦨
I f
R 졠 ᲼Q
a
㦨
Ra桌



᲼Q
I
㦨
MR
桌 Þ
᲼҈



m
맔

桌
맔


�
㦨


맔


�㡰

맔


� P
\ m

㦨




㦨
㦨

\ P
㦨

҈
҈
K.ETᨨtal
㦨


҈

�
㡰 ᨨ
 
ᨨ
ᨨ

矄�
F
㦨
mmg
㦨




鼀� ᨨ
ᨨ
ᨨ� �

a æa ö F ç + b ÷ £ mg

è ø a æa ö Þ


mg ç + b ÷ £ mg Þ



a
졔




b
£


�
a

è ø b  Þ £ a
�


b
£


䰘�
a
盐
 �矄
�矄
ᨨ
ᨨ矄矄�
꾀矄
 
鼀 �鼀盐鼀
盐
ᨨ
b
ᆠ

⼄盐
ᨨ


�a
҈

b Þ = �
a ҈䰘


䰘�㡰


�鼀
├
ᨨ
ᆠ
 
�ᨨ盐 
ᨨ
矄ᆠ 

t=

҈
h æ K
ö ç҈ +
÷ 矄� q g çè R ÷ø h K
Q
=҈ g R

ᨨ
� �



t҈ =

҈
h
(҈ + ҈) = 矄�

° g 

ᨨ
�矄ᆠ


t
=

҈
h æ ҈ ö
h K
҈ + = = ҈ Q ç ÷ 矄�

° g è
ø g R



෌� t҈ - t
= \






-  ҈



h
-  h = g g ҈


Þ

h - h =  -













\ h »

䰘� 鼀 �� 
ᨨ
 
ᨨ
ᨨ
 
�
ᆠ� 
 㡰 ҈

矄

ᨨ
⧤� �ᆠ
 �泐
ᨨ
 
� �
E҈ 㦨
K.E.㡰 ᨨ�ᨨ
졔
K.E.㡰  矄�ᨨ  ҈
҈
㦨
I w + mvc

҈

҈
㦨
´ mr w + mr w 㡰 Q
I
㦨
mr

vc
㦨
rw)

Þ
h 


҈
䰘
㡰
㦨
҈
䰘


҈
�






Þ


h =






E҈ = mr w
ᨨ
├� �ᆠ
 �泐
ᨨ
 
ᆠ泐� 


E
㦨
K.E
㡰 ᨨ�ᨨ
졔
K.E
㡰  矄�ᨨ 

҈ ҈ I Ἐ w Ἐ
+ Mv Ἐc


҈æ҈ ö

҈
㦨
ç Mr ÷ w Ἐ + M r w Ἐ㡰 è ø





E

㦨
Mr w Ἐ
쟼⼄盐� �
E҈
㦨
E


mr
w

㦨
Mr w Ἐ


m


 wἘ = = ´ = ҈ Þ wἘ
㦨
w Þ
 M 


w  ᆠ
ᨨ
� �
 
ᆠ泐� 
犨�
ᆠ

 
矄鼀
�鼀


㦨


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EBD_8341 PS-72

҈黨







Physics

ᆠ㡰 ᔼ
 
ᨨᆠ
 㡰
�矄
�


 � �
 
 
t 鼀� 矄
ᆠᨨ 矄 
 
t 㦨

맔


ᔼ
ᨨᆠ
�矄
�
ᨨ鼀 ᨨ


 
矄
 
犨 矄 ᆠ�鼀矄

 
   �ᆠ盐
�矄 ᆠ 㝐¢ ᨨ
ᨨᆠ
ᨨ鼀
턼
犨�
�ᨨ 㝐 ᨨ
ᆠ  �� �
� ��泐
   �ᆠ盐
�矄
턼㝐
턼 
�
ᆠᨨ鼀矄
턼㝐Ἐ
ᔼ
 
ᨨᆠ
 㡰
�矄
�

ᨨ鼀
ᨨ
  ᆠ�ᆠ盐鼀  ᆠ

ᨨ� 
   t �矄
ᨨ
ᔼ
 
ᨨᆠ
 㡰
�矄
�
 � �泐

ᨨ� 
和
 

t = F ´ R
 
 
犨 
ᆠ�鼀矄
r 

b㡰 ᨨᆠ�泐

ᆠ 
Ἐ졠Ἐ \ vo = Rw iˆ R w ᆠᨨ矄
º kˆ


R w
R w 矄� 
º  -�ˆ 㡰


졠


҈



º

ur Rw Rw VP = Rw�ˆ 矄� 
°�ˆ + ᆠᨨ矄 
°kˆ

ur Rw ˆ ù  Rw ˆ é �ú+ k \ VP
㦨
êRw �ˆ
û
ë ur ҈҈  R kˆ ᨨ
VP = Rw �ˆ +
ur
w b

ᆠ㡰 

VA =
V C =
V
ur ur VB = V
ur ur VC =
V
ur ur VB = V
\ VC -VB ur ur = VB -
VA

ur






R w �ˆ

ur

ur

ur

F

m
f҈

f

mg
ᆠᨨ矄q

 ᆠ
�ᆠ�ᨨ 
ᨨᆠ
�矄
ᨨ�ᨨ
盐
� 矄  矄�ᨨ 
鼀ᨨ�ᨨ
꾀泐� �
F 
㦨
ma
ᨨ �
 
�ᆠ�ᨨ
ᨨ
� ᆠ� 
 
mg
矄� q 
f
㦨
mac
犨 
ac
㦨
ᆠᆠ�ᨨ
ᨨ
ᆠ 
ᨨ
鼀矄矄
ᨨ
 
ᆠ泐�  \ f
㦨
mg
矄� q 
mac …
�㡰 g 矄� q g 矄� q
= = g 矄� q 盐
ac 㦨

Ic  mR ⏌
҈+ ҈+
mR mR
mg 矄� q f
㦨
 ᔴ泐

�
q �矄
盐ᆠ

�ᆠ�ᨨ 
ᨨᆠ
�矄
盐ᆠ


a


m҈

a҈




和� ᆠ

 
�矄
ᨨ
矄�� �
 泐犨  \ a҈
㦨

a


VA =


   V C - V A  =
 V B - V C  N ᆠ

㡰

矄�� �
�ᆠ�ᨨ

ᆠ矄
�
 
ᨨᨨ矄� �ᆠ�ᨨ
ᨨ
鼀�
矄�
q
ᨨ ᨨᨨ矄
 
�
鼀ᨨ�ᨨ
q ᆠ盐矄
ᨨ
�ᆠ�ᨨ 
ᨨᆠ
 

q
矄� g mg m ᆠ泐� 
ᨨ矄
q



㡰 ᔼ
ᆠ矄
ᨨ
盐
ᨨ� �
ᨨ
� ᆠ� 
 
ᆠᆠ�ᨨ

ᨨ

ᨨ泐 � 矄� q = ᔼ ҈+ 㝐
ᔼ 㝐
ᨨ
ᨨᨨ犨
ᆠ泐� 
= =҈ 㝐
㝐
҈ 㝐
ᔼ ҈
ᨨ
矄ᨨ�
ᆠ泐� 
= =


㝐 㝐  ᆠ
ᆠᆠ�ᨨ

ᨨ
矄ᨨ�
ᆠ泐� 
�矄
鼀ᨨ
 
ᨨᨨ犨 ᆠ泐� 
 
  ᨨ
矄ᨨ�
ᆠ泐� 
犨�
ᆠ
 
ᨨᨨ鼀 ᨨ
 
� ᆠ� 
 
�矄
ᔼ
 
ᆠ矄
ᨨ
ᨨ� �
 
犨�

ᨨ
犨ᨨ├
ᨨ 
泐
�ᆠ�ᨨ


 ᨨ
ᨨ
鼀ᆠ  �ᆠ
 �泐
鼀� 矄
ᆠᨨ 矄
\ ⧤쟼㡰矄ᨨ�
㦨
⧤쟼㡰 ᨨᨨ犨
㦨
ᆠ矄
�
쟼
㦨
鼀�



鼀
�矄

ᨨᆠ

ᔼ矄
ᨨ�ᨨ 
ᆠᨨ鼀ᨨ  
F 盐矄 矄
 
 ├

 ᆠ
 
ᨨ� 
ᨨ
ᆠᨨ ᆠ
ᨨ
 
 ├ 犨�
 
ᆠ泐� 
犨�
 
ᨨ
鼀ᨨ
ᨨ犨矄
�� 
  �ᆠ�ᨨ 
ᨨᆠ
f҈
犨�
ᆠ
ᨨ犨矄
 
ᨨ
 
 ├
 f҈
㦨
�ᆠ�ᨨ
犨
 ├
 
ᆠ泐�  f

㦨
�ᆠ�ᨨ
犨
ᆠ泐� 
 
�ᨨ盐  a҈
㦨
ᆠᆠ�ᨨ
ᨨ
 ├ a

㦨
ᆠᆠ�ᨨ
ᨨ
ᆠ 
ᨨ
鼀矄矄
ᨨ
ᆠ泐�   


a
㦨  �盐
ᆠᆠ�ᨨ
ᨨ
ᆠ泐� 
ᨨ盐
�矄
ᔴ㝐
f҈

F - f҈ a҈
㦨
m
f҈ - f
a

㦨
m҈  f҈ - f
㡰 R  f҈ - f
㡰 R a= = ҈ I m R

҈
 f҈ - f
㡰 a= m҈R
 f҈ - f
㡰 a

㦨
Ra = m҈ 㡰 和ᨨ� �
⼄矄

�㡰
ᨨ
㡰

犨
�

f






�㡰



��㡰



���㡰





�㡰



㡰


F m҈ + m
꾀 
꾀ᆠᆠ�ᨨ
ᨨ
ᆠ 
ᨨ
鼀矄矄
ᨨ
ᆠ泐� 

F a
= m҈ +
m
m҈F 㡰 �ᆠ�ᨨ
犨
 ├
 
ᆠ泐� 
f҈ = m҈ +
m
m҈F �ᆠ�ᨨ
犨
ᆠ泐� 
 
�ᨨ盐 
f
= m҈ +
m
和� ᆠ


⼄盐 ��矄

ᨨ矄��

 泐

ᆠᨨᆠ泐 矄 ᨨ犨
�
��盐矄
꾀ᆠᆠ�ᨨ
ᨨ
 ├

a҈ =

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PS-73

System of Particles and Rotational Motion �


Topic-6 : Miscellaneous (Mixed Concepts) Problems ҈


㡰 
鼀 �鼀盐鼀
盐
ᨨ
⼄
�



⼄鼀

 
ᨨᨨ
�矄
ᨨ盐 ᨨ
ᨨ

 


㦨


 

 
ᨨᆠ矄
鼀�
 
҈㡰
鼀盐矄
矄矄  ᨨ盐�
ᆠᨨ ᆠ
ᨨ� 
ἘἘ 졠턼   =
ᨨ

矄�
q鼀
= \ ᆠᨨ矄黨


q鼀 㡰
= 졠 졠 q 鼀 턼  

ᨨᨨ 
鼀�





҈





鼀�

q

b㡰 꾀 �盐
鼀ᨨ鼀 盐鼀



 ᨨ
L 㦨
Iw
ᨨ盐
 �矄
ᨨ
ᨨ㡰 㝐ᨨ鼀 
ᨨ
� �
ᨨ
 
ᨨ㜠 � 矄ᆠ
矄泐矄鼀
I
㦨
Iᨨ
졔
mx

㦨
Iᨨ
졔
mv
t


m 㦨
鼀矄矄
ᨨ
� 矄ᆠ
\ L
㦨
Iᨨ
졔
mv
t
㡰w

q鼀

w

(

x
㦨
v t

dL = 
mv
t w㡰 ᨨ  t µ t dt �


 
�
�矄
矄�� 
� 
矄矄� �
 ᨨ盐�
ᨨ���
꾀 
�鼀
T

L 㦨
ᆠᨨ 矄  dL =
\  t ᨨ dt �



犨 
 
� 矄ᆠ
矄ᨨ矄
鼀ᨨ� �

L
ᨨ矄
ᨨ
ᆠ  �
    ᨨ
T
ᆠᨨ鼀矄
ᆠᨨ 矄 
㡰 ␔ 
犨

��� �

 �盐
ᆠᆠ�ᨨ
a㡰
ᨨ  
ᨨ

 

矄
� 矄  ᨨ盐矄
ᆠᆠ�ᨨ
ᆠ� 矄 㦨
La



矄

  ᆠ泐
ᨨ
鼀ᨨ
犨泐
ᨨ鼀
  ᆠ 

盐
盐
ᨨ
 
�ᆠ�ᨨ
犨
 

 
  ᨨ

 �矄
ᨨ矄
ᨨ


ᔼ
� 矄  ᨨ盐矄
 �盐 ᨨᆠ�泐
�矄
w
 



ᆠ矄矄泐
�ᆠ�ᨨ 
ᨨᆠ
�矄
ᨨ�
泐
�ᆠ�ᨨ  ᨨᆠ mLw

㦨
Ҙma㡰
Þ

mLw

㦨
Ҙ
mLa
Þ w

㦨
Ҙa

\

a
t

㦨
Ҙa
Þ t
㦨
a





㡰

a

v

Ҙ a

mv
T
矄� q 㦨
�矄
ᨨ盐ᆠ� � r

M

r w

q

T q

Tᆠᨨ矄q

T矄� q

mg

 
ᆠ矄矄泐
ᆠ �
ᨨᆠ ᨨ
ᆠ�ᆠ盐
鼀ᨨ�ᨨ





L

)

T
ᆠᨨ矄 q 㦨
mg

v
҈
´ ҈
= = ҈

ᨨ q 㦨

� rg ҈
´ ҈
㡰 
ᨨᆠ
ᆠ� �
ᨨ
 
鼀矄矄
ᨨ
�⼄盐�
dm
ᨨ
 � dx 

�矄 ᆠ
x
ᨨ鼀
 
 �矄
ᨨ
ᨨ�ᨨ
O
æ mv
ö ÷ dF
㦨
dm㡰
x
w



çQ F = r ø F è \

C

M

 q
㦨


x

O

O

a
a ᨨ


r =
泐
�ᨨ鼀泐


ᨨ⼄盐
ᨨ盐
ᨨ� 
O
�矄
ᨨ
 ᆠ

 �盐
鼀ᨨ鼀 盐鼀
L㡰
ᨨ盐
O
�矄
ᆠᨨ 矄
ᨨ
L�
㦨
Lf æ aö
MV ç ÷ = I
w =  Icm + Mr 㡰w è
ø ìï Ma
æ a
ö üï v

+ M ç ÷ ý w
= Ma w 㦨
w
㦨
í 
a è
ø þï îï
r

㦨


䰘


a

矄� �
w
㦨
w

졔
at
Þ w
㦨
at

dW =
dx ҈



x
d é
w \ x + ҈
䰘
- ҈
黨
x ù úû
dx êë

䰘
├� Þ
x

҈
黨

㦨

҈
黨
=

黨
鼀 \ x
㦨

ᆠ㡰 
ᨨ
�矄
盐 �ᨨ� �

ᆠ�ᆠ盐
鼀ᨨ�ᨨ
矄
 
ᆠ
�矄 鼀ᨨ� �
�
ᆠ�ᆠ盐
ᨨ�ᨨ 
ᆠ├
犨�

ᆠᨨ 矄 
矄



ᨨ
�矄
盐 
 
� 盐 ᆠ
ᨨ
犨ᨨ
ᨨᆠ矄
�㡰  矄�ᨨ

T
�
 
ᨨ v ��㡰 ␔�� 
ᨨ
 
ᨨ

mg ᨨ
犨ᨨ├
ᨨ 
ᨨ

鼀� �鼀盐鼀


ᨨ犨  t  =




ᆠ㡰 꾀矄
�
�矄
ᨨ
ᆠᨨ�矄�ᨨ
犨
犨ᨨ
� �ᆠ 矄 矄

 泐
犨�
 ᆠ  �
 �
� 
ᨨᆠ��矄

和� ᆠ  
矄 矄

矄鼀ᨨᨨ
 
犨�

ᨨ
�ᆠ�ᨨ
 ᨨ
ᨨ⼄盐㡰  
  ᨨ
 
犨�

ᨨ
 矄 
ᨨ
 �盐
鼀ᨨ鼀 盐鼀


盐矄
A

 
ᆠᨨ�矄�ᨨ
犨�
鼀�
犨�
�矄
� ��
 �盐 鼀ᨨ鼀 盐鼀

�


w A = w   wB =
ᆠ㡰 ᨨ
ᨨ� 
鼀矄矄
I
㦨
mr
㝐
ᔼ

ᨨ
 
矄泐矄鼀
ᨨ盐
 �矄
ᨨ
ᨨ�ᨨ
O I
㦨
I҈
졔
I

㦨


x

졔


䰘
҈



x㡰






㦨


x

졔


䰘
҈
黨

졔
x





x㡰







㦨
x

졔
҈
䰘


҈
黨
x


犨ᨨ├
ᨨ 
�
ᨨ� �
 
ᨨ
�矄
ᆠᨨ 
� ᨨ
�矄 ᨨ�ᨨ 
├� �ᆠ
 �泐


├� ҈
\ W
㦨
I w
x ҈

= ᲼ + ҈
䰘
҈
黨
桌 w x x


졠


M dx dF dx ´ xw

L 졠 M 犨 


�矄
� 
鼀矄矄
 矄�泐
ᨨ
�⼄盐�
L \ 
ᨨᆠ
ᆠ� �

 
ᨨ 
 
�矄
ᨨ
 
犨 ᨨ �⼄盐�
�
盐
LM
M
L w x dx = w ò x dx F
㦨
ò

L L

\

dF
㦨


L

ù MLw
M
é x
ù M
é L
w ê ú = w ê -
ú =



㦨
L
ëê
úû
L ëê
ûú

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EBD_8341 PS-74

Physics

҈ù é  êF ´ r ´ ú = I a
û ë ҈ ҈  ´

� ´

� ´ 㦨
´ ҈
� ´

� ´

� ´ a

Þ
a
㦨



矄҈ w
㦨
w

졔
at
Þ
w
㦨


졔


맔
҈
㦨



矄҈ ҈

␔ 
 
ᆠ盐
�� 矄
ᨨ
�
ᨨ盐
 
�
 
ᨨ鼀 ᆠ�ᨨ
N 犨�
矄矄
 ᨨ盐�
 
�
D ᨨ盐
犨 �ᆠ ᨨ�ᨨ
├矄
ᆠ


鼀ᨨ鼀 矄
盐
ᨨ
N
 
�ᆠ�ᨨ   ᨨᆠ
f
犨�

ᨨ


├� �
鼀ᨨ鼀 
ᨨ
ᨨᆠ B C
 
鼀�
ᨨ盐
D F 黨


҈



㡰


a a = mg ´ \ F
㦨
mg


a⏌


A f D
a mg  ᆠ

鼀� �鼀盐鼀
盐
ᨨ

F
㦨
mg
 t 矄
ᨨ⼄盐

t
㦨
Ia
\
a
㦨
I t
㦨 F ´ r ^ ᨨ⼄盐
�矄
矄鼀
�
ᨨ
 
ᆠ矄矄

盐
矄� ᆠ

I 犨�

�  
盐
ᨨ
�  
鼀矄矄
�矄�盐�ᨨ
ᨨ盐  
 �矄

矄ᨨ
a
犨�

�  


b

ᆠ㡰

ᨨ
�矄
矄
ᨨ鼀
矄
矄ᨨ
 
�
矄
泐 ᨨ� �
ᨨ盐
�矄
ᆠᨨ ᆠ
ᨨ� 
犨�
 
ᨨᨨ
犨� ᨨ盐 矄�� �


㡰 꾀矄
犨
├ ᨨ犨

ᆠᆠ�ᨨ
ᨨ
 
ᆠ 
ᨨ
鼀矄矄
ᨨ ᆠ泐� 
ᨨ� �
ᨨ犨

� ᆠ� 
 

g 矄� q I












҈+ MR
ᔼ
ᆠ矄
ᨨ
P
 
鼀矄矄
�矄
ᆠᨨ ᆠ 㜠 
犨泐
ᨨ鼀
 
 �矄
和ᨨ
IP
>
IQ \ aP


aQ Þ v


v턼

Þ
w


w턼 ҈



҈



ac

ᆠ㡰 
 �盐
�鼀盐矄
ᆠ
泐
 
�ᆠ�ᨨ 
ᨨᆠ 犨
 
� �
 
 

 矄
ᨨ
ᆠ矄
 
 �盐 矄
w
ᨨ
 
� �
ᨨ盐
졠
òt

F
맔


҈҈


C

ac =

 㦨
㜠矄 ᨨ 泐㦨҈矄

ᆠ

҈�


꾀 
 
ᆠᨨ�矄�ᨨ
w
ᆠ矄矄
盐
 
� �
鼀� 矄
ᨨ� �
�

 
 �ᆠᨨᆠ├犨�矄
�ᆠ�ᨨ

 ᆠ
 
�ᆠ�ᨨ
犨
  � �
 
 
�ᨨ盐 

 
ᨨ� 
ᨨ
ᆠᨨ ᆠ
�矄
ᨨ
 
 


ᆠ

㡰 꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
�



P�
㦨
Pf
m

v㡰
졔
m

v㡰
졔

m
맔


㦨

m
졔
m
졔

m㡰
vc
Þ vc
㦨

+ -


m

a

v a


a


v m

⏌
 ᆠᨨ矄









ar






at

ar
ᆠᨨ矄



졔
at ᆠᨨ矄



෌�
�
├� �ᆠ
 �泐
㦨
ᨨ矄矄
�
ᨨ �
 �泐

҈
l Iw
㦨
鼀� ҈

ᆠᨨ矄


㡰

l ml

� \ w
㦨
鼀� Þ w
㦨


l ᨨ犨

t
㦨
ᔼa ҈ l  � \ 鼀�
맔
矄�



㦨
鼀l
a Þ
a
㦨


l l  � 盐 

㦨
a
㦨


� l l � ´
㦨
꾀矄ᨨ 
㦨
w

㦨


l

ᨨ
�ᆠ
鼀ᨨ�ᨨ
ᨨ
ᆠ 
ᨨ
鼀矄矄 鼀�


㦨
鼀ᆠᨨ矄



졔

ᆠᨨ矄

㡰 \

é � ҈  � ù ´ ú 鼀�


㦨
鼀 ê ´ +

û ë



\

㝐� 
㦨
҈


҈



ᨨ犨

泐� �
ᆠᨨ 矄�ᨨ
ᨨ
 �盐
鼀ᨨ鼀 盐鼀
ᨨ盐 ᆠ 
ᨨ
鼀矄矄 �


L�
㦨
Lf
mv
맔
a
졔
m

v㡰
맔

a
㦨
Iw ҈ 


I
㦨

m㡰
맔

a㡰

졔

m
맔
a

졔
m
맔

a


㦨

ma
҈
v
\
mv ( a ) + m (
v ) ´
a = 
ma ´ w Þ w = �a ҈
I w \ 꾀 
ᆠᨨ�矄�ᨨ
 �泐


E
㦨



v mv ҈ 
= 㦨
´ 
ma ´
�
�a
r dr r r 㡰 
r t 㡰 = at�ˆ + btˆ
®
v
㦨
dt 㦨
a�ˆ + bˆ
㦨
ᆠᨨ 矄  r a
㦨

\ r r 盐 
P = mv
㦨
ᆠᨨ 矄 

҈

鼀 㦨
ᆠᨨ 矄 
r æ ¶U ˆ ¶U ˆ ö F = ç �+  ÷
㦨

¶y ø è ¶x

 
⧤
㦨


\ 
㦨
ᆠᨨ 矄  꾀矄ᨨ
쟼
㦨
⧤
졔

㦨
ᔴᨨ 矄  r \

dL r r r = t= r´F =
dt

Q
ar
�矄
ᆠᨨ 矄 㡰

r

\
L
㦨
ᆠᨨ 矄 

WWW.JEEBOOKS.IN

PS-75

System of Particles and Rotational Motion

r r dr r ˆ ˆ v = 턼
r t 㡰 = a ᆠᨨ矄 wt � + b 矄� wt  ® dt = -aw矄� wt 㡰 ˆ� +bw ᆠᨨ矄 wt 㡰 ˆ r r dv r a= = -w
᲼a ᆠᨨ矄 wt �ˆ + b 矄� wt ˆ桌 = -w
r \ dt r r r r r 꾀矄ᨨ
t = r ´ F =
\

㦨
ᆠᨨ 矄 㡰 \ r   F   ҈҈㡰

⼄㡰 � 
鼀ᨨ矄
盐

X
矄ᨨ
鼀ᨨ矄
盐
 
  ᨨ
  ���ᨨ 
 �泐
ᨨ
X
�矄


 ᆠᨨ � 盐ᨨ盐矄泐
� ᆠ矄� �
T
ᨨ
犨�� 
ᨨ
Y
ᨨ盐
P
矄
    �ᆠ盐
�矄 ᆠ
ᨨ
 
� 
ᨨ  ᆠ�ᨨ
ᨨ
ᨨᆠ
ᨨ鼀
 
ᨨ� 
P
�矄
ᨨ
Ӹ ᨨᆠ
 
泐
X
ᨨ
Y 㝐� 㦨 Mg + Mg =
mg 犨 
Mg
�矄
犨
 ᨨ
Y
 
Mg
�矄
 
ᨨᆠ
ᨨ
Y
盐
ᨨ
Z. 㡰 Mg

r r uur mw
r
DU = - ò F
dr = + ò mw
rdr =




\
U
µ
r




꾀矄ᨨ

= a
ᆠᨨ矄
wt + b
矄�
wt \
r
㦨
f t㡰 ᨨᆠ
�矄
ᆠ 
\
쟼
㦨 r dr r r ˆ ˆ
r t 㡰 = a t� + 矄� w t
®
v
㦨
dt ˆ ᲼ 矄�

t � ᆠᨨ矄 t ˆ 桌 a w w + w w
㦨
\ 
㦨
aw
㦨
ᆠᨨ 矄  r r
㦨
dv
㦨
aw
᲼ᆠᨨ矄
wt ˆ
졔
矄�
wt ˆ 桌 � a dt r a
㦨
 w
rr \ r r r t
㦨
r ´ F
㦨

\ 
㦨
ᆠᨨ 矄  ᨨᆠ
�矄
ᆠ 
�
盐
 ᆠ


 
⧤

矄ᨨ
ᆠᨨ 矄 
r dr b
r 和
r t 㡰 = aᆠᨨ矄 w t�ˆ + t ˆ ®
vr
㦨
㦨 at �ˆ + bt ˆ \

㦨
f t㡰 dt
r a
㦨

bˆ
㦨
ᆠᨨ 矄  r r F
㦨
ma 㦨
ᆠᨨ 矄 

 鼀 �


\ 
ᨨᆠ
ᨨ
Y
�矄
ᨨ
�


Mg = B + Fv B
졔
Fv

 
泐
X
ᨨ
Y



���ᨨ 
ᨨ �
 �泐
ᨨ
X
�矄 ᆠᨨ � ᨨ盐矄泐
� ᆠ矄� �
ᆠ盐矄
矄
Y
鼀ᨨ矄 ᨨ犨

 
ᆠ 
ᨨ
鼀矄矄
ᨨ
X
鼀ᨨ矄
盐


҈



N f

҈黨
q

q

mgᆠᨨ矄q mg

㦨
 
q + ҈ 㡰 mg ᆠᨨ矄 q = 矄ᆠ qmg ᆠᨨ矄 q = mg 㦨
犨�� 
ᨨ
Y
꾀��

盐
ᨨ
 
矄 ᆠ
ᨨ
�ᆠ�ᨨ 
ᨨᆠ
犨
Y  
X

 
鼀ᆠ  �ᆠ
 �泐
ᨨ
 
矄泐矄鼀
 X + Y 㡰 ᆠ矄矄
ᆠᨨ � ᨨ盐矄泐
矄
Y
矄�矄

ᨨ犨


B

Fv

Mg


鼀ᆠ  �ᆠ
 �泐
ᨨ
 
矄泐矄鼀
X
졔
Y㡰
�矄
ᆠᨨ � ᨨ盐矄泐 ᆠ矄� �
ᨨ
ᨨᆠᨨ鼀
 
�矄ᆠᨨ盐矄
ᨨᆠ矄


f
+ N
= m
N
+ N
= m
+ ҈N

mg矄� q

Mg

R =  Mg 㡰
+ m
+ M 㡰 g 桌
¹ Mg


鼀ᆠ  �ᆠ
 �泐
ᨨ
 
矄泐矄鼀
X 졔 Y㡰
�矄
ᆠᨨ � ᨨ盐矄泐 ᆠ矄� �
矄
 
矄泐矄鼀
�矄
ᆠᨨ鼀� �
ᨨ犨
 
�矄
ᨨ �  �泐
�矄
ᆠ矄� �

 
├� �ᆠ
 �泐
鼀� � �
 
矄鼀



ᨨ⼄盐
ᨨ
 
犨�� 
ᨨ
Y
ᨨ盐
P ¹
矄㡰 ᨨᆠ
ᨨ
Y
泐
X
�矄
㦨
犨

ᨨ
�⼄盐�
�矄ᆠ
犨 �ᆠ ᆠ

ᨨ

⼄盐
ᨨ
Mg 矄
 
 矄�泐
ᨨ
Y
>
 矄�泐
ᨨ
X 
Y
�矄
矄� ├� �㡰


���ᨨ 
ᨨ �
 �泐
ᨨ
X
� ᆠ矄矄 ᆠᨨ � ᨨ盐矄泐
ᆠ盐矄
矄
Y
鼀ᨨ矄
ᨨ犨

 
ᆠ 
ᨨ
鼀矄矄 ᨨ
X
鼀ᨨ矄
盐
㡰 和 
Y
�矄
鼀ᨨ� �
犨�
鼀� 
ᨨᆠ�泐



t

v

R 

ᔼ
 �矄
ᆠ矄
 
ᨨᆠ
 
泐
X
ᨨ
Y
㦨
ᨨᆠ
 
泐 Y
ᨨ
X


ᨨᆠ
ᨨ
X
盐
ᨨ
Y
�矄

r r  mb
t
D
㦨
 ò F × dr
㦨
 m ò bˆ × a�ˆ + b tˆ 㡰dt
㦨




҈
҈


⧤
㦨
mv = ma + b t 㡰

҈
꾀矄ᨨ

쟼
㦨
⧤
졔

㦨
ma
㦨
ᆠᨨ 矄 

҈䰘




꾀 ® 
㡰 錐
 ® ⼄
矄
㡰
錐
ᔴ ® 

㡰
 ® ⼄
㡰 꾀矄
 
ᨨᆠ�泐
�矄
ᆠᨨ 矄  f = mg 矄� q \ 盐
f = mN = mmg ᆠᨨ矄 q \ mmg ᆠᨨ矄 q = mg 矄� q
Þ
m =  q


ᨨᆠ
泐
X
ᨨ
Y
�矄
 
矄盐 
ᨨ
f
 
N

m
g

Ӹ





҈






æ R - rö ÷ R ø

ᨨ
├� �ᆠ
 �泐
ᨨ
 
� �
㦨
⧤� �ᆠ
ᨨ�ᨨ 
졔 ├� �ᆠ
 �泐
 矄�ᨨ 㡰 ҈ K
Etᨨtal = 
MR
㡰w
= M w

 R - r 㡰

㡰 mM w
鼀�  R - r 㡰 = Mg g \
w鼀� = m R - r 㡰 㡰 꾀 �矄
ᨨ
ᨨ�ᨨ
�矄

ᨨ
㜠 �矄

 ᆠ
ᨨ
ᨨ  
ᆠ矄矄

�
矄  ᨨ盐矄
 �矄
矄矄� �
 ᨨ盐�
�矄
�ᆠ
㡰 ᨨ
���
ᨨ泐
w
�矄
矄鼀
ᨨ
 泐
ᨨ� 
ᨨ
 
ᨨ泐
㡰
kx

f
㦨
Ma  

a
㦨
Ra a
kx - f é fR ù \
= Rê ú a ҈ M
ê MR ú ë
û
2kx é fR ù = Rê ú ҈
f ê MR ú ë
û ᆠ㡰 
w
R - r 㡰 = wR



\

w = wᨨ çè

WWW.JEEBOOKS.IN

EBD_8341 PS-76

Physics

和ᨨ� �
 �矄
⼄盐�ᨨ

犨
�
├
├ \  =
├ - =
├ =  

�矄
�矄
ᨨᨨ矄�
ᨨ
�矄ᆠ鼀 
kx f = 
kx \ Fnet = �ᆠ
ᨨ犨矄
 
⼄盐���盐鼀 

æ
kx ö ÷x è  ø
k F æ
k ö x = -w
x Þ w = \ a = net = - ç è M ÷ø M M



ᆠ㡰 㝐ᆠ  �ᆠ
 �泐
�矄
ᆠᨨ 矄
�
ᆠ矄
ᨨ
盐
ᨨ� � 鼀ᨨ�ᨨ


҈ é҈
ù
҈æ҈
ö æ v
ö Mv + MR \
ç ÷ ç ÷ =
ê kx 鼀 ú

è

R øè ø ë û M x鼀 = v
\
k
kx鼀
k M F鼀 = mMg = v
=  
k M v
= mg \ k
�
b㡰 ᨨ矄矄
�
├� �ᆠ
 �泐
㦨
K
E
㡰� ��

K
E
㡰 � 


㡰 Fnet = ҈ - ç


é҈ é҈ æ
ö ù
҈
ù 㦨
ê I 
w 㡰 + 
I 㡰 w ú - ê  I +
I 㡰 ç w ÷ ú è ø ú
ë
û êë
û







䰘








I w
㦨
 I w
- I w
=   㡰 ␔ 
�矄ᆠ
B
�矄
ᨨ盐� 
�
ᆠᨨ ᆠ
犨�
�矄ᆠ
A 
w  
ᆠᨨ鼀鼀ᨨ
ᨨᆠ�泐

ᨨ鼀
ᆠᨨ 矄�ᨨ
ᨨ
 �盐 鼀ᨨ鼀 盐鼀
ᨨ
 
犨ᨨ
�矄ᆠ
矄泐矄鼀

I

w㡰
졔

I
w㡰
㦨
I
졔

I㡰
wἘ

Þ wἘ
㦨
w 

ᨨ⼄盐
ᨨ
�矄ᆠ
A
I ´ w - I ´
w DLA L f - L� -
I w  = = = tA
㦨
t t t t 
��
矄��
� �ᆠ矄
 
 
ᨨ⼄盐
ᆠ矄
 �盐 �ᨨ
ᆠ㡰 ᨨ
�矄ᆠ
A 2w ҈
҈ kx҈ = I 
w 㡰

Þ kx҈
=
I w


A ᨨ
�矄ᆠ
B ҈
҈ kx
= ´
I w


Þ
kx

= I w
w

x k x҈

I w
B Þ ҈ =
= \


x kx
Iw

�矄

�
ᨨ�ᨨ 
ᨨ矄��ᨨ

\ 
ᨨ⼄盐
ᆠ� �
ᨨ
 

�矄
ᨨ
m b⏌


F

a

b⏌


mἘ

Mg

b⏌


b b =F´



�㡰

n 盐鼀
ᨨ
矄
ᆠ
ᨨ
鼀矄矄
m
矄�├� �
犨�
ᨨᆠ�泐
v ᨨ
 
矄 

ᨨ�ᨨ
ᨨ
 
 b dp F
㦨
n 맔
A
㦨
n
맔

mv㡰
맔
a
맔
Q
A
㦨
㡰
dt 盐� �
 �矄
盐
ᨨ

�
⼄

�㡰 b b b Mg
맔
㦨
n
맔

mv㡰
맔
a
맔

맔



´
Þ 
맔
҈

㦨
҈


맔


맔



҈
맔
v
맔
҈
맔

\ v
㦨
҈

鼀⏌矄 Y
黨
和� ᆠ
 
矄泐矄鼀
�矄
�
⼄盐���盐鼀
P Mg
맔


\

X

SFy
㦨



mg

Mg
졔
mg
졔
Mg

N

N 㦨



Þ Mg
졔
mg
졔
Mg
㦨
N
졔
N

N

Mg f

A

N

Mg f

B

m2

mg


M + m㡰 g Þ N
㦨




�㡰



 
�ᆠ�ᨨ 
ᨨᆠ

꾀
 


ᔴᆠ盐� �
ᨨ⼄盐矄 ᨨ盐
P Q P Mg
맔
PQ
졔
f
맔
PM
㦨
N
맔
OP O q Þ Mg
맔


L ᆠᨨ矄 q + f ´ L 矄� q
= NL ᆠᨨ矄 q

MgL NL ᆠᨨ矄 q ᆠᨨ矄 q
Þ f

㦨
L 矄� q

N A

l⏌
q f

Mg M

Mg ᆠᨨ q
éæ
N + m ö Mg ù ᆠᨨ q Þ f
㦨
ê ç ÷ø g è

úû ë gù é Þ f
㦨
ê( M + m) ú ᆠᨨ q
û ë 

꾀
盐
ᨨ
鼀矄矄
m
矄�├矄
 
犨ᨨᨨ
ᨨ�
ᨨ
鼀矄矄
M
   �
L
 
矄�ᆠ├矄
ᨨ
�
ur ur r r dL Þ t ´ dt = d L

ᨨ⼄盐
t = dt r r ␔ 
 �盐
�鼀盐矄
 t ´ dt 㡰
�矄
ᨨ

 
 �盐 鼀ᨨ鼀 盐鼀
�矄
ᆠᨨ 矄 

ᔼ
 �矄
ᆠ矄
ᨨ
 
犨ᨨᨨ

ᨨ�㜠盐
矄泐矄鼀

 
 �盐
�鼀盐矄
ᨨ盐
O
�矄
ᆠᨨ 矄 
\ ᲼ �盐
鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀桌� �� 㦨
᲼ �盐
鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀桌 �  Þ mv
맔
L
㦨
I

맔
w



�㡰 犨 
I

�矄
 
鼀ᨨ鼀 
ᨨ
� �
ᨨ
 
犨ᨨᨨ
ᨨ�㜠盐 矄泐矄鼀
 
ᆠᨨ�矄�ᨨ
ᨨ盐
O I

㦨
I犨ᨨᨨ
ᨨ�
졔
I盐 ҈

㦨
ML + ML



��㡰  ᨨ鼀
⼄

�㡰
 
��㡰 mv mv ´ L w
㦨

=  M + m㡰 L é҈

ù êë  ML + mL úû

= N ᆠᨨ q -

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PS-77

System of Particles and Rotational Motion ҈


㡰 
ᨨ� 
Ἐ졠Ἐ
 ᨨ
 
ᆠ 
ᨨ
鼀矄矄
ᔴ㝐㡰
ᨨ
  矄泐矄鼀 \

l 

OA
㦨
r =

  + w� = +  -҈㡰



���㡰 I AB ҈

+ 
x
ᨨ鼀
⼄

��㡰
 
���㡰  é ù - ҈ + ҈ú Q
w� = ҈
ᨨ⏌矄㡰

㦨


맔
x
ê
҈

+ 
x ë û  é ù ҈
㦨
�x
ê ú ë҈

+ 
x
û Þ ҈


졔


x

㦨
�x
᲼졔
桌
㦨
҈�x \ 
x


҈�x

҈


㦨

和ᨨ� �

犨
�
x
㦨


҈
ᨨ



盐

x
㦨




鼀
wf
ᆠᨨ鼀矄
ᨨ盐
ᨨ

��



�
ᨨ⏌矄㡰 犨 �ᆠ
�矄
ᨨ
ᆠᆠ

 ᨨ
㡰 x
㦨
�矄 ᆠ
ᨨ
ᔴ㝐
ᨨ鼀
� 
꾀
㦨


҈
鼀 㡰 和盐矄�盐� �
x
㦨


҈
鼀
�
⼄

��㡰
犨
�
wf
㦨
҈
ᨨ⏌矄
   �盐
ᨨᆠ�泐
犨�
犨 �ᆠ
矄 
ᆠᨨ鼀矄
ᆠ├
   
�矄
�鼀ᆠ
�矄
҈
⏌矄
ᆠ㡰 꾀矄
 
矄 
盐 矄
犨�
矄鼀
 �盐
ᨨᆠ�泐
ᨨ ҈
⏌矄

 
矄 
犨�

ᆠᨨ鼀
ᨨ
矄


\

泐
�ᨨ鼀泐㡰 Am㡰 y x

l

l

O CM F

 l


r

l

Bm㡰

C m㡰

F
㦨
ᆠ �
ᨨᆠ
ᨨ
F
㦨
m㡰
rw


æ ҈ ö
w F = m㡰 ç è  ÷ø ᨨ F = ml w
㡰 
a

 
 �盐
ᆠᆠ�ᨨ
ᨨ
矄泐矄鼀
ᨨ盐 ᨨ� 
A æ  ö  F 㡰 çç l÷
÷ø tA F è a= = =
IA
ml
ml ᨨ犨

ᆠᆠ�ᨨ
ᨨ
ᔴ㝐
ᨨ �
x㜠 �矄
�矄

ᨨ

æ l ö æ F ö ax = r a = ç ÷ ÷ç è  ø çè
ml ÷ø

F ᨨ ax =
m 
Fx

 
ᨨᆠ
�
泐
 
� �
ᨨ �
X㜠 �矄
\ Fx 졔 F = m㡰ax  F ö ᨨ Fx + F = m㡰 æç ᨨ Fx + F = F ÷ è
m ø
F

Fy

 
ᨨᆠ
�
泐
 
� �
ᨨ �
Y㜠 �矄 \ Fy
㦨
ᆠ �
ᨨᆠ ᨨ

Fx =








ᨨ Fy = ml w 
x

 
�矄 ᆠ
ᨨ
ᆠ 
ᨨ
鼀矄矄
ᨨ鼀
� 
AB
\ 㝐
ᔼ

ᨨ
鼀� 
矄 
ᨨ盐
AB IAB
㦨
Iᆠ
鼀

졔
mx
 ᨨ鼀

 矄
 ᨨ鼀㡰 IAB
㦨
҈


졔

x




�㡰




wf
㦨


ᔼ ��泐
 
�矄ᆠ
�矄
 �ᆠ
犨�
 
ᨨ� 
A

�矄 �� 矄
ᨨ矄��ᨨ

ᔼ
矄矄 ᨨ� �
犨 
ᨨ犨
ᨨ

 ᆠ
�矄
ᨨ �
 �泐 ᆠ  �矄
ᨨ
ᨨ�ᨨ 
├� �ᆠ  �泐


A m R C

꾀泐� �
 �泐
ᆠᨨ 矄�ᨨ

턼

P

ᆠ
鼀

B m
㦨

├�

IAB
wf

w�㡰
㦨


맔

�
㦨


R

  ᆠ 
ᨨ

쟼


R ö
R ö ҈ æ æ + mg ç R + = mgR + I w
÷ ÷ è è
ø
ø
␔ 
I
㦨
M
I

ᨨ
�矄ᆠ㜠鼀矄矄
矄泐矄鼀
ᨨ盐
PQ ҈
҈
R
R ҈ + mg = mgR + I w

Þ mgR
㦨
I w mg
맔





mgR I
I㡰PQ
㦨
I�矄ᆠ㡰PQ
졔
I鼀矄矄㡰PQ

Þ w
㦨






�㡰



é mR
æ Rö ù æ �R ö ê + M ú + m 㦨
çè ÷ø çè ÷ø
ûú
ëê


҈ MR
桌


mR
᲼
+ ҈ +
�桌 ҈�mR
= ҈

盐� �
 �矄
盐
ᨨ
I
�
⼄

�㡰 ᔼ
㦨



mgR ´


҈
g �R ҈�mR


� 
矄
ᨨ
 
�ᆠ
矄
�
ᆠ 矄
�矄
ᨨ犨矄 ᨨ矄��ᨨ

w
㦨


ᆠ盐矄
ᨨ
�鼀盐矄
 
 �盐
ᨨᆠ�泐
ᨨ
 
鼀�  矄 
犨�
ᆠ  �
 
泐
�鼀ᆠ
ᔼ鼀盐矄
㦨
ᆠ  �
�
� 
鼀ᨨ鼀 盐鼀

㦨


Vf 
V�㡰

㦨


맔
x
wf

w�㡰



��㡰 꾀矄ᨨ

ᆠ  �
�
 �盐
鼀ᨨ鼀 盐鼀
㦨
鼀ᨨ鼀 
ᨨ
�鼀盐矄 \ IABwf

IABw�
㦨
ᔼ鼀盐矄
맔
�矄 ᆠ

ᔴἘ

mg ç
R +

᲼Q
M
I

ᨨ
�矄ᆠ
ᨨ盐
�鼀
㦨
x

Q

R⏌


v

쟼 ��
 �泐
㦨
� 
 �泐

A

鼀

�

R/4

P




=

҈
g �R Rö æ ´ = � gR



\


v
㦨
÷ �R
è
ø 

㡰 
ᆠᨨ�矄�ᨨ
�矄
矄�ᆠ
 
 
矄 
�矄
� 

 ᆠ  
犨�
犨�
盐
ᆠ├
犨�
 
矄鼀
ᨨᆠ�泐 �
鼀� �盐㡰
v
㦨
w ç R +

WWW.JEEBOOKS.IN

EBD_8341 PS-78

Physics Z

Y

꾀矄ᨨ

X

\ 


F 




t fR
f
f = = = = ҈
f I mR
⏌
mR
´

҈ a
㦨
҈

f



��㡰



m

├�

├� z a=


ᨨᆠ
矄ᨨ 矄�
ᨨ
ᆠ  �
 
ᨨᆠ�泐
ᨨ
 
犨� �
X㜠�ᆠ�ᨨ
�矄
Fx
Fx
맔
Dt
㦨
mv



mv㡰 ᔼ鼀盐矄㡰
㦨
ᔴ  �
�
鼀ᨨ鼀 盐鼀㡰



҈鼀 O y


mv
mv
mv Þ
F
㦨

Þ
F
ᆠᨨ矄


㦨
Dt  Dt Dt Q Fx
㦨
F
ᆠᨨ矄


 
Fy
㦨
F
矄�

ᔼ
ᆠᨨ
鼀矄 ur F 㦨
Fx $� + Fy  - k$ 㡰 = F ᆠᨨ矄
°�$ + F 矄� 
° -k$ 㡰

\

Þ

f

ᨨ

$ ҈ � + F ´  - k$ 㡰




mv F + mg = + mg
Dt

�



mv Dt

ᨨ ᨨ ᨨ

 �
ᨨ
ᨨ
㦨



ᆠ鼀
㦨




鼀

\


a 

ᨨ f =

(

) ( )

r t҈ = -

҈ ˆ -

҈kˆ ´
�ˆ N - m

r t҈ =


kˆ -


ˆ r t҈ =


kˆ - ˆ 㡰 N - m





�㡰

ᨨ

(

)

t
=


- ˆ - kˆ 㜠鼀

r  t
 = 


㡰
+ 


㡰
=
´


=


� 㜠鼀 r r  t҈  =  t
 =


� 㜠鼀

�盐矄
ᨨ
ᆠ
�矄ᆠ
㦨
R
㦨
҈

ᆠ鼀
㦨


҈
鼀


a

㦨
� 
ᆠᆠ�ᨨ
ᨨ
ᆠ 
ᨨ
鼀矄矄
ᨨ
�矄ᆠ a
㦨
 �盐
ᆠᆠ�ᨨ
ᨨ盐
�矄
ᆠ 
ᨨ
鼀矄矄 f f a
= =



�㡰 m


ᨨ f =

r r r 和�鼀�泐
t
= r
´ f

෌�


㝐矄矄
ᨨ
ᆠ
�矄ᆠ
m
㦨


├�

�㡰

f =a



㦨



�
㜠鼀


�矄ᆠ㜠ᨨ
矄泐矄鼀
ᆠ
ᨨ
ᨨ
 
盐ᆠ├
犨� ᨨ盐
矄�� � 矄
 
�ᆠ�ᨨ
ᨨ
 
�矄ᆠ㜠ᨨ
矄泐矄鼀
犨�
 
ᨨᨨ
ᨨ
  鼀ᨨ� �
盐ᆠ├
�矄
�



 �矄
ᨨ
�矄ᆠ㜠ᨨ
ᨨᆠ
�矄
ᨨ�ᨨ 
 
 �ᆠ盐 ᨨ
 
�ᆠ�ᨨ
ᨨ
 
鼀ᨨ�ᨨ
ᨨ
盐ᆠ├

z㜠 �矄
�矄
 
�ᆠ泐 盐犨
�ᆠ�ᨨ


fö ÷ 졔


҈㡰҈

f 㡰
㦨

 ᨨ鼀
�㡰㡰
ø

r  t҈  = 


㡰
+ 


㡰
=
´


= ҈

҈
´




´h

꾀ᆠᆠ�ᨨ
ᨨ
盐ᆠ├
ᨨ �
x㜠 �矄
㦨
黨
鼀⏌矄


æ çè

��㡰 �ᆠ�ᨨ 
ᨨ⼄盐

t = r ´ f r 꾀  P
r҈ = 

҈ ˆ -

҈kˆ r 꾀  Q
r
=

҈ ˆ -

҈kˆ r r r \ t҈ = r҈ ´ f

æ
mv ö + mg ÷ k$ N
㦨
ç è D t ø

�矄
�矄
 
ᨨ鼀
ᨨᆠ
 
泐
 

ᨨ
 
犨� 盐� �
 
�鼀
D

b㡰 ᨨ⼄盐矄
ᨨ
犨�
ᨨ盐
 
ᆠ 
ᨨ
鼀矄矄
ᨨ
  犨� F
맔
h

ᨨ⼄盐
盐
ᨨ
N
졔
mg
맔


㦨


h
�矄
 
 �ᆠ盐
�矄 ᆠ
犨
 
ᆠ 
ᨨ 鼀矄矄
ᨨ
 
犨�
 
 
� 
ᨨ
ᆠ�ᨨ
ᨨ


ᨨ⼄盐
盐
ᨨ
N
㦨
F
맔
h
㦨


f


´ 黨

=
N 

�矄
ᨨᆠ
ᨨ
�ᆠ�ᨨ
f

�矄
ᆠ� �
�
ᨨ矄��
x㜠�ᆠ�ᨨ

�ˆ �矄
盐 �
ᆠᨨ
ᨨ �
x㜠 �矄
r \ ᔼ
ᆠᨨ
ᨨ鼀
f = 
�ˆ㡰
犨ᨨ





���㡰 ᨨ

쟼⼄盐���盐鼀
ᨨ
ᨨᆠ
�
Z㜠�ᆠ�ᨨ
ᆠ� �
ᨨ
犨�㡰 Fy
졔
mg
㦨
N

\

Q

和� ᆠ
 
�矄
ᨨ
矄�� �
犨
�矄ᆠ
 
盐ᆠ├
ᆠᆠ�ᨨ
ᨨ
P
㦨
ᆠᆠ�ᨨ
ᨨ
Q a

졔
R
a
㦨
a

ur F
mv   $� - k$ 㡰 F 㦨
  $� - k$ 㡰 =
Dt

Þ N
㦨




P

Fx
㦨



㦨
F
맔




҈鼀





㡰 
ᆠ泐� 
ᨨ
�盐矄

鼀矄矄
ἘmἘ
ᨨ矄
ᨨ
 
�
ᨨ  
ᆠ �盐
ᨨᆠ├
犨� ᨨ盐
矄�� �


ᨨ�ᨨ
ᨨ
  ᆠ泐� 
�矄
ᨨ盐
 
ᨨ� 
ᨨ
ᆠᨨ ᆠ
矄ᨨ
 �泐
ᨨ
  ᆠ泐� 
�矄
ᆠᨨ 矄

 
ᨨ��� 
ᨨ矄��ᨨ
ᨨ
ᆠ 
ᨨ
鼀矄矄
ᨨ
 
ᆠ泐�  
O

␔ �
ᨨ� �
ᨨ犨
ᨨ
 
�


 
ᆠ泐� 
 
矄盐ᆠ

ᨨ矄��ᨨ
 
�矄
ᆠ 
ᨨ
鼀矄矄
�矄


ᨨ矄��ᨨ
OἘ

ÐNPO

q

꾀矄
 
ᆠ泐� 
�矄
ᨨ� �

 
ᆠ
鼀

ᨨ矄

WWW.JEEBOOKS.IN

PS-79

System of Particles and Rotational Motion �

ᆠ�ᆠ盐



ᆠ �
ᨨᆠ
⼄盐�
ᨨ
  ᆠ�ᆠ盐
鼀ᨨ�ᨨ
N




҈ ҈ ҈
æ
g ö
mgR 㦨
´ mR ´ ç ÷ +
mv Ἐc 㡰 R

䰘 è ø

O

mvc
mg
ᆠᨨ矄 q

N
㦨









R


ᆠᨨ ��ᨨ
ᨨ
  ᆠ泐� 
� �
 
�
N
㦨



R

h M Rh

w q os

gc

m

q

OἘ

q R mg P

mg矄� q

vc
mvc

Þ ᆠᨨ矄 q 㦨




�㡰 Rg R 꾀泐� �
 �泐
ᆠᨨ 矄�ᨨ
ᨨ鼀
O
ᨨ
OἘ
ᨨ矄矄
ᨨ
ᨨ �
 �泐
ᨨ
ᆠ泐� 



㦨
��
�
 矄�ᨨ 
K
E

졔
��
�
ᨨ�ᨨ 
K
E


\

vc
gR ⏌ 䰘 = R R mgR
mgR Þ mgR


㦨
㦨
 矄�ᨨ 
K
E
䰘 䰘 ҈
mgR 꾀矄ᨨ

ᨨ�ᨨ 
K
E

㦨
Iw =
䰘

 矄�ᨨ  K
E
=
\ ᨨ�ᨨ  K
E
w
㦨


Q

vc

mg
ᆠᨨ矄 q 㦨


҈
҈
mgh
㦨
mvc + I w



��㡰

␔ 
I
�矄
 
鼀ᨨ鼀 
ᨨ
� �
ᨨ
 
ᆠ泐� 
ᨨ盐
OἘ
҈ I = mR
, w
�矄
 
 �盐
矄

Vc
�矄
 
ᨨᆠ�泐
ᨨ
ᆠ 
ᨨ
鼀矄矄


dL 䰘
和� ᆠ

t
㦨
 
t
㦨

dt

P


\ L
�矄
ᆠᨨ 矄 


꾀 �盐
鼀ᨨ鼀 盐鼀
 ᨨ
ᆠᨨ�矄�ᨨ
㦨
mv
맔

l
l 꾀 �盐
鼀ᨨ鼀 盐鼀
 
ᆠᨨ�矄�ᨨ
㦨
Iw I
㦨
Im
졔
I꾀
졔
I

vc R 盐� �
盐
ᨨ
I
 
w
�
⼄

��㡰

m 㦨



�
├�㡰

é ì æ l
ö æ l ö ï ìï æ l
ö æ l ö
üï ù
ï ê m 
l 㡰 + 㦨
í mA ç ÷ + ç ÷ ý + í mB ç ÷ + ç + l÷ ý ú è
ø ï ï è ҈
ø è
ø ï ú ҈
êë îï è ø þ î þû
ü

mgh 㦨


é æ l
l
ö æ l
黨l
ö ù
+ ÷ + mB ç + 㦨
ê
ml + m A ç ÷ú
ø úû êë è ҈

ø è ҈


é ë





���㡰


gr ҈ - ᆠᨨ矄 q㡰 Rg


䰘 b㡰 和
ᨨ
ᔴ
㝐

ᨨ
ᆠ泐� 
 ᨨ
� �
ᆠᨨ ᆠ
犨� �



gR
gR æ
ö
gR ᲼ ᨨ鼀
���㡰桌
Þ vc
㦨
ç҈ - ÷ = 䰘  è 䰘ø 䰘 ᆠ㡰  ᨨ
 
ᆠ 
ᨨ
鼀矄矄
ᨨ
 
ᆠ泐� 
ᆠ 矄
  ᨨ�ᨨ 
� 
ᨨ
 
�

�
矄
ᆠᨨ ᆠ
犨�
 
�
q = ᆠᨨ矄 -҈ = ��
҈�° 䰘  ᆠ
 
ᨨ�ᨨ 
K.E


犨 �ᆠ
 
ᆠ泐� 
�� 矄

  �鼀
ᨨ
� �
ᆠᨨ ᆠ
犨�
 
�
鼀� 矄
 
矄鼀
�
�矄 盐 
鼀ᨨ�ᨨ

 
 
ᆠ泐� 
�� 矄
 矄�ᨨ 
K.E
꾀��
泐� �
 �泐
ᆠᨨ 矄�ᨨ
ᨨ鼀
O
ᨨ
 
ᨨ�  犨 
ᆠ 
ᨨ
鼀矄矄
�矄
�
ᨨ�ᨨ 
� 
犨�
� ҈
҈
mgR
㦨
I w + mv Ἐc 㡰



䰘 ù m l
=


黨 ├� m
 B úû

Iw
㦨
mv
맔

l

mv ´
l


� ´ v ´
´


= 㦨



䰘
v


黨 I 꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
鼀ᆠ  �ᆠ
 �泐
 
ᆠᨨ�矄�ᨨ
ᨨ矄矄
ᨨ
K
E

㦨
��
�
P
E
Þ w
㦨


҈
æ lö æ l ö I w = mg 
l㡰 + m A ç ÷ g + mB g ç ÷ è
ø è
ø


Þ 
ᆠᨨ矄 q 㦨





ᆠᨨ矄 q

Þ
ᆠᨨ矄 q 㦨


vc
=

҈ 

㦨
ê
ml
+ m A l
+

ᨨ鼀
⼄

�㡰
 
���㡰

犨
�

ᆠᨨ矄 q =

mB B

v

꾀矄ᨨ
ᨨ
ᨨ� �

vc
㦨
wR
Þ w
㦨


v
҈
҈ ҈ mvc + ´ mR
´ c



R ҈
҈


gh
Þ gh
㦨
vc + vc = vc

Þ vc =


 R-h ᔼ
DO¢MP

ᆠᨨ矄 q = R Þ h
㦨
R
҈

ᆠᨨ矄 q 㡰
g
R ҈ - ᆠᨨ矄 q㡰 \ vc = 

mA A

҈ ´


黨 ´ 


䰘 v㡰

҈ ù é 㦨
ê


� ´
+


҈ ´ +



´ ú ´ 黨

´




û ë Y Þ v
㦨



鼀⏌矄 R T w 

�㡰 꾀ᆠᆠᨨ� �
ᨨ M ⼄盐矄�ᨨ

 
盐鼀
�矄 m T ��

� ��
 �盐 X ᨨᆠ�泐
矄盐ᆠ
 
  q ᨨᆠ├
X
矄矄
鼀ᨨ� �
盐 mg
矄� q mg mg ᆠᨨ矄q  
 
Þ

q=30 o

꾀矄
 
�鼀
矄矄矄
 
ᨨᆠ├
X
�矄矄
盐
 
ᨨᆠ�泐
ᨨ
  ᨨᆠ├
ᆠ矄矄


a

 
� 
�ᨨ
ᨨ
 
ᨨᆠ├
X. \ mg 矄� q - T = ma



�㡰

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EBD_8341 PS-80

Physics


� 
�ᨨ
ᨨ
 
ᨨᆠ├
a
 
 
 �盐 ᆠᆠ�ᨨ
ᨨ
 
盐鼀
a㡰


矄

a R 犨 
R
�矄
 
�盐矄
ᨨ
 
盐鼀



� �
ᨨ⼄盐
ᨨ
 
盐鼀
�矄
盐
ᨨ
 矄�ᨨ
T
�
  矄� �
t
㦨
T
맔
R
㦨
Ia
Q t
㦨
Ia㡰 a
㦨
a Þ a =

\

҈
T
맔
R
㦨
MR a


҈ é
ù êëQ I =
MR úû

a
T ҈ MR
Þ a
㦨

R M 和盐矄�盐� �
 �矄
盐
ᨨ

�
⼄

�㡰

Þ TR
㦨


mg
矄� q 
T
㦨
m
맔



T æ
m ö Þ mg
矄� q 㦨
ç҈ + ÷T è Mø M

mg 矄� q㡰 ´ M

� ´ 黨

´ 矄� 
° ´
= 㦨
҈


 M +
m
+
´

� ��㡰 �矄 ᆠ

泐
x

\

T
㦨



T
´ ҈

 = = ҈

 m ⏌ s
M
盐矄� �
V
㦨
Rw

Þ V 㦨




맔
҈

㦨


鼀⏌矄 Q R
㦨




m
 
w
㦨
҈

ᨨ⏌矄
�� 㡰 \ �矄 ᆠ

泐
ᨨᆠ├
x

a=


´
v
=
a
´ ҈

 \ �矄 ᆠ

泐
x
㦨
҈



鼀 㡰 꾀 �盐
鼀ᨨ鼀 盐鼀
ᨨ
 
矄泐矄鼀
ᨨ
졔
� 矄ᆠ㡰
ᨨ盐  
ᆠ 
ᨨ
鼀矄矄
Ἐ졠Ἐ
 ᨨ
ᆠᨨ�矄�ᨨ
㦨
 �盐
鼀ᨨ鼀 盐鼀 ᨨ
 
矄泐矄鼀
 
ᆠᨨ�矄�ᨨ
L M Mv
맔
= I w
v ␔ 
I
�矄
 
鼀ᨨ鼀  ᨨ
� �
ᨨ
 
矄泐矄鼀 盐矄
 
ᆠᨨ�矄�ᨨ
 
w A B O �矄
 
 �盐
ᨨᆠ�泐 L⏌
L⏌
盐矄
 
ᆠᨨ�矄�ᨨ

as
㦨



盐



Þ s =

黨


Þ

Þ

Mv


ù L é æ Lö ҈ = ê M ç ÷ + ML
ú w
êë è
ø ҈
úû

L ML
é ҈ ҈ ù ML
é  +
ù Mv ´ = + úw= ê

ë
û
êë ҈
úû


҈
v 䰘 ML
´ ´ w

Þ w
㦨
䰘 L
҈
b㡰 꾀 
ᆠᨨ�矄�ᨨ
 
�矄

 
鼀矄矄
M
ᨨ
 
� 矄ᆠ 犨 �ᆠ
ᆠ矄

ᨨ⼄盐
�
 
ᆠᨨᆠ├犨�矄
�ᆠ�ᨨ

犨 �ᆠ  矄
ᨨ
ᆠ
 �盐
ᆠᆠ�ᨨ
�
 
ᨨ

盐
  矄鼀
�矄
ᆠᨨ鼀 矄
泐
 
鼀ᨨ鼀 
ᨨ
� 矄ᆠ
ᨨ犨矄 B
盐
ᨨ
犨 �ᆠ
鼀ᨨ鼀 
ᨨ
� �
I
ᨨ
 
矄泐矄鼀
� ᆠ矄矄


 泐
� 矄 
ᨨ
�鼀
t
 
� 矄ᆠ



�矄 ᆠ
x





㦨


ᨨ鼀
 
ᆠ 
ᨨ
 
ᨨ
 
 
ᨨ
矄
盐 
 ᨨ盐� 
 �
q 㦨
wt㡰
犨


�矄
ᨨ��� 
ᨨ矄��ᨨ

dL d A =  I w㡰
㦨
dt dt M

dI
= w dt

㦨
w

A

q =w t

O x

d é҈ ù ML
+ Mx
ú ê dt ë҈
û

B N B

dx Mg dt

�矄
ᨨ⼄盐
�矄
 ᆠ
泐
 
ᨨ⼄盐
盐
ᨨ
犨�� 
ᨨ � 矄ᆠ
dx æ g ö ᆠᨨ矄 wt dt \
M
w
x 㦨
Mg
x
ᆠᨨ矄q㡰


Þ dx
㦨
ç è
w ÷ø dt 졠
� ��ᨨ

├� �
�鼀�矄 L⏌
g p ⏌
w ò L ⏌
dx =
w ò
ᆠᨨ矄 wt dt L 犨 

x =
wt =

L⏌
g p ⏌
w [ x] L ⏌
=
[ 矄� wt ]

w p L 犨 

x
㦨

wt =

g é p æ L Lö ù Þ ç - ÷= 矄� - 矄�
ú è

ø
w
êë
û

㦨


M
w
x

Þ

L g
g =

Þ
w
㦨
L

w


盐 w
㦨


҈
v ҈
v =
Þ 䰘 L 䰘 L


g 䰘
gL Þ v
㦨
L ҈


䰘
´ ҈
´ ҈

㦨

�
鼀矄҈ ҈



 ᨨ
ᆠᨨ�矄�ᨨ


⧤� �ᆠ
 �泐
ᨨ
 
矄泐矄鼀 ҈
҈
҈
⧤
쟼
� 㦨
m҈v҈ + m
v
+ MV


҈
҈




�㡰 㦨




´ ҈
+



´
+
= �




Þ v
㦨


M
㦨


҈

├� ҈

鼀⏌矄

A

m҈ 㦨





├�㡰

x =

�鼀 C x =

�鼀

m




鼀⏌矄 㦨





├�㡰

B

꾀泐� �
犨
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀
盐� � ᆠᨨ�矄�ᨨ

m҈
맔
v҈
졔
m

맔
v

㦨
M
졔
m҈
졔
m
㡰
Vc 犨 
Vc
�矄
 
ᨨᆠ�泐
ᨨ
ᆠ 
ᨨ
鼀矄矄
ᨨ
 

  �ᆠ矄
矄�ᆠ├
ᨨ
�
 
ᆠᨨ�矄�ᨨ






맔
҈

졔





맔


㦨


҈

졔





졔




㡰
Vc Þ Vc
㦨


鼀⏌矄

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PS-81

System of Particles and Rotational Motion \

⧤
쟼

 矄
 
ᆠᨨ�矄�ᨨ

҈
㦨
M + m҈ + m
㡰Vc =

�




��㡰
꾀泐� �
ᆠᨨ 矄�ᨨ
ᨨ
 �盐
鼀ᨨ鼀 盐鼀
ᨨ
 
  
犨ᨨ
�ᆠ
矄泐矄鼀
ᨨ盐
 
ᆠ 
ᨨ
 

TExternal
㦨

\ L
㦨
ᆠᨨ 矄  L � 㦨
m҈v҈
맔
x

m
v
x 㦨






맔
҈

맔


�






맔


맔


� 㦨










㦨


҈

├�
鼀
矄҈ ᔼ
ᆠᨨᆠ├犨�矄
�ᆠ�ᨨ 㡰 Lf

㦨
Iw

\


҈


҈
҈
⧤
쟼
ᨨ
㦨
I w = ´



´
=

҈




�㡰

K.E.f
㦨
 矄�ᨨ 
K
E

졔
ᨨ�ᨨ 
K
E
㦨


�

졔


҈

㦨


䰘

 D K
E

㦨
K
E
�

� 
K
E
f 㦨
�






䰘

㦨


䰘

 P
E


D
㦨
P
E


Q
졔
K
E
㡰Trans
졔
K
E
㡰Rᨨt ᨨ鼀
 �泐
ᆠᨨ 矄�ᨨ
� ᆠ� ҈
҈
Þ mg



㡰
㦨
mg
҈㡰
졔
mv + I w

҈
҈ æ

ö v
mv + ç mr ÷
ør

è� vö

æ çèQ I = mr
w = ÷ø � r

䰘v
ᨨ
g ´ ҈

=


Þ


v
㦨




鼀⏌矄 ҈
꾀 
ᨨ� 
Q

 

ᨨ
  D ᨨ泐
ᨨ�ᆠ
҈
Q



鼀⏌矄



鼀 h
㦨
gt


Þ


҈
㦨


黨 t
y
҈鼀 ҈


�矄ᆠ = \ ty
㦨
B C A

黨 �矄 ᆠ矄
BC
㦨
V
맔
ty
㦨




맔



�
㦨

鼀 盐� �
�矄
�� 
矄

ᨨᆠ�

 
矄 
ᆠᨨ � 盐矄
ᨨ ᨨ
ᆠ盐矄
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
 �盐
鼀ᨨ鼀 盐鼀

矄鼀
鼀矄矄
m
鼀ᨨ矄
ᨨ盐 

ᆠ�ᆠ盐

ᨨ
�盐矄
r

 
矄� �
鼀├矄

 �
q
犨�
 
�ᆠ
 
T   
 矄�ᨨ

T
ᆠᨨ矄 q 㦨
mg











�㡰 T
矄� q 㦨
mrw









��㡰 l \

 q 㦨


q

r w
g

ᨨ鼀
��盐


矄� q 㦨


Þ r
㦨
l 矄� q

T

r


l

r

q

T
ᆠᨨ矄
q

T
矄�
q mg

 q 㦨
l 矄� q


Þ
w =

w
g

 q
g ᨨ

w = l 矄� q

g l ᆠᨨ矄 q

g ҈



���㡰
Q w =
pn
p l ᆠᨨ矄 q ᨨ
M
ᨨ
鼀�
矄�ᨨ 泐

T
㦨
Mg  
T
ᆠᨨ矄
q
㦨
mg \ Mg
ᆠᨨ矄 q 㦨
mg m Þ ᆠᨨ矄 q 㦨
M 盐� �
 �矄
盐
ᨨ
ᆠᨨ矄
q
�矄
⼄
���㡰

犨
� Þ n
㦨


é M l
ù + m҈ x
+ m
x
ú w 㦨
ê ëê ҈
ûú é 

҈
㡰  㡰
ù +



´ 

�㡰
+ 



㡰 

�㡰
ú w 㦨
ê ҈
êë úû 㦨






w




w
㦨


҈

Þ w
㦨


⏌矄



���㡰

mg (


- ҈) =






\

n
㦨


҈
p

gM l m





O

OM ᔼ
DOPM

ᆠᨨ矄 q
=



l Þ OM
㦨
l ᆠᨨ矄 q
OM Ἐ ᔼ
DOPἘMἘ

ᆠᨨ矄 q = l Þ OMἘ
㦨
l ᆠᨨ矄q

q0

q

T

㝐 㝐Ἐ

P P'

  ᆠ mg
矄� q 
ᨨ

쟼


q mg 鼀�
ᆠᨨ矄
q

OMἘ

OM 㦨
l ᆠᨨ矄 q - ᆠᨨ矄 q
㡰 ᨨ矄矄
�
ᨨ �
 �泐
㦨
෌�
�
├� �ᆠ
 �泐 꾀ᆠ��泐P
ᨨ
PἘ㡰 ҈
Þ mg l ᆠᨨ矄 q - ᆠᨨ矄 q
㡰 = mv
Þ v

㦨

g l ᆠᨨ矄 q - ᆠᨨ矄 q
㡰



�㡰 T

mg
ᆠᨨ矄 q
㦨


mv
l

mv
+ mg ᆠᨨ矄 q \ T
㦨




��㡰 l ᨨ鼀
�

�㡰
 
��㡰 m T
㦨
´
g l ᆠᨨ矄 q - ᆠᨨ矄 q
㡰 + mg ᆠᨨ矄 q l \ T
㦨
mg
ᆠᨨ矄 q 


mg
ᆠᨨ矄 q
Þ T
㦨
mg

ᆠᨨ矄 q



ᆠᨨ矄 q
㡰 ᨨ鼀
⼄盐�ᨨ
��㡰
�
�矄
ᆠ
 
 
 矄�ᨨ
�矄
鼀 �鼀盐鼀 犨 
ᆠᨨ矄 q
㦨
҈
�



q 㦨

 ᆠ

T鼀
㦨


mv
+ mg l

ᨨ鼀
⼄

�㡰 v

㦨

g l ҈ - ᆠᨨ矄 q
㡰 ᨨ鼀
���㡰
 
�㡰 m T鼀
㦨
᲼
g l ҈ - ᆠᨨ矄 q
㡰桌 + mg l \ T鼀
㦨
mg


mg ᆠᨨ矄 q




㦨

맔






맔



ᆠᨨ矄 q






���㡰



�㡰

҈ Þ


ᆠᨨ矄 q
㦨



Þ ᆠᨨ矄 q
㦨


Þ q
= 
°


WWW.JEEBOOKS.IN

EBD_8341 PS-82

Physics

Gravitation

7

v


Topic-1 : Kepler’s Laws of Planetary Motion ҈






꾀  꾀 ҈
q ҈
҈
= 
꾀
㦨
 q Þ 㦨  w



꾀矄ᨨ


㦨
鼀
㦨
鼀 w 꾀 ҈  \ = 
鼀 ᆠ㡰 

ᨨ
�矄
ᆠ
㦨
�


 ᨨ ᆠ 졔 和ᆠ和㡰 꾀
ᨨ
和ᆠ和
㦨
졔

꾀
ᨨ

�矄
졔
꾀
ᨨ
� �㡰 ᆠ 




 = 꾀
ᨨ
和ᆠ和
㦨

꾀
ᨨ
和ᆠ和  ⏌
㦨
㦨 ⏌
꾀
ᨨ
和ᆠ和






҈ 


\

v҈
㦨


҈


RA RB

v
- v҈ 
҈
´ ҈

´ � ⏌҈

㦨
㦨

맔
҈


⏌矄 R
- R҈  ´ ҈

´ ҈


㡰 ෌�
l
㦨
꾀
졔

㡰


├� �
矄鼀
鼀 
鼀
ᨨ
 �



�矄 ᆠ
ᨨ鼀

㦨


㦨

矄ᨨ

鼀
㦨
l
 鼀
㦨
꾀
졔

㡰

R


mV

㦨
mRw

 
���ᨨ 
ᨨᆠ
ᨨ
ᆠ�ᨨ

R mRw

㦨
GMmR�⏌





 +  ෌鼀 꾀 + 
㡰








鼀

෌鼀鼀

Þ
= ò

R = ҈ T

R


R T
= ҈ Þ R
 = R҈ ´
T

R
 T҈


R



pR

p㡰 
´҈

㡰 = = p´ ҈

├鼀⏌ T



 =

MA

T҈


T҈


R҈

Topic-2 : Newton’s Universal Law of Gravitation




Þ mR ´
p = GMm
Þ T

µ
R䰘⏌
T
R� ⏌
�㡰 꾀ᆠᆠᨨ� �
ᨨ
⧤Ἐ矄
 �
犨
ᨨ
 泐
鼀ᨨ�ᨨ




µ


\

㦨




⏌






S҈ P

 ᆠ
 
矄
ᨨ
矄�
S

犨



S҈ 㦨
v


v҈
㦨
p
맔
҈




p
맔
҈


㦨

p
맔
҈


├鼀⏌ ��㡰 꾀 �盐
矄
ᨨ
S

犨



S҈

⏌
æ R
ö é R ⏌
ù = ҈
黨 泐矄 ÷
㦨

�
ê Þ T

㦨
T҈
ç ú ë R û è R҈ ø b㡰 
ᆠ �
ᨨᆠ
�矄
ᨨ�
泐
ᔴ �
ᨨᆠ


㦨


S


v҈


pR҈
p´҈

= =
p´ ҈

├鼀⏌ T҈ ҈ v
=


T҈
㦨

�
泐矄
錐


T

㦨
䁸錐
R҈
㦨
R


 
R

㦨


�


´

� 
矄
ᨨ
矄�
S



和

҈ 㦨 
ᨨ


㦨
 ҈

㡰 ᔼ ᆠ矄
ᨨ
� 泐
矄
矄泐矄鼀
 
���ᨨ 
ᨨᆠ
ᨨ
ᆠ�ᨨ

犨
 
矄矄
犨�
ᨨ�
  MB

ᆠ矄矄泐
ᆠ �
ᨨᆠ矄
和ᨨ
 �盐
ᨨᆠ�泐
w
ᨨ
ᨨ
矄矄 �矄
 
矄鼀

 ᨨ
�鼀
�ᨨ
T
p 㦨

鼀� 矄
 
矄鼀
w b㡰 꾀ᆠᆠᨨ� �
ᨨ
⧤Ἐ矄
犨
T

µ
R

( )

R
 = ҈






=

´ ҈
҈
҈
Þ R

㦨


맔
҈


├鼀
� 
矄
ᨨ
矄�
S҈

ᆠ㡰 꾀
ᨨᆠ�泐錐


㦨




Þ






+ é æ҈ ù ҈ ö é 꾀 ù = ෌鼀 ê 꾀 ç = ෌鼀 ê - +  ú ÷ +  ú ë û ë è   + ø û 㡰 꾀矄
犨
├ ᨨ犨

෌��ᨨ 
ᨨᆠ
ᨨ
ᆠ�ᨨ
෌㝐鼀 =
෌㝐  鼀 ෌㝐  㝐 矄 ҈ =  
=
҈ 



D ҈ =


෌㝐  鼀 ҈

D҈   D
=

෌㝐  㝐 矄 


D


æ 鼀 ö æ 
 ö æ D҈ ö D ҈ 鼀D҈ 
 =  =ç ÷ ÷ç ÷ç D
҈ 㝐矄 D
è 㝐矄 ø è ҈ ø è D
ø

矄� �
D҈
㦨
D

㦨


 錐
鼀
㦨


맔
҈

├�錐 㝐矄
㦨


맔
҈


├� ҈
㦨




맔
҈


├鼀
 


㦨
҈�

맔
҈


├鼀 

D ҈ æ
´ ҈


ö æ ҈�
´ ҈

ö =ç ÷ç ÷ ´҈ @
D
è
´ ҈

ø è


´ ҈

ø

WWW.JEEBOOKS.IN

PS-83

Gravitation 



F ᆠᨨ矄
�° + F ¢ =

㡰

␔ 
F =


F ¢ =

GM



R㡰 GM


R
´ GM


Þ

Ἐ

㝐





Mv

 ᨨ鼀
��盐㡰 R


 



 R + h㡰


G
㦨
���ᨨ
ᆠᨨ 矄  M
㦨
鼀矄矄
ᨨ



ᆠᆠ�ᨨ
盐
ᨨ
��泐


h �矄

ᨨ




+

GM


=

Mv
R

GM æ ç҈ R
è ෌�

g
㦨
g' gἘ =

㝐

㝐


 R
㡰

R
GM
é ҈ ҈ ù
+ ú = Mv R êë

û

Þ

\

GM æ
+
ö ҈ GM ҈ +

㡰 ç ÷= R çè

÷ø
R 㡰 ᨨ盐鼀
ᨨ
鼀ᨨ
矄  


æRö
æ ҈ö ÷ = pR ç ÷  è
ø  è
ø

V鼀ᨨ
㦨
p ç




æ҈ö æ䰘ö    è
ø è
ø

 ᨨ
鼀矄矄
ᨨ
矄 
ᆠ
 
鼀� � �
矄  ҈ 䰘 

矄ᆠ�泐
㝐
 
㝐




 ᨨ

���ᨨ 
ᨨᆠ
犨
 矄
犨ᨨ
矄 
䰘M ҈ GM m G
´
M
҈ ෌M
䰘 GM
; F
㦨

㦨
=




r

´ 黨 R
R 㡰


Topic-3 : Acceleration due to Gravity









�






 
g d =

GM  R - d 㡰 

R Rö æ çè R + ÷ø
GM GM  R - d 㡰
R - d 㡰 = Þ =
 黨 R R æ R ö çè ÷ø
Þ
R = 黨 R - 黨d Þ � R = 黨 d d � \ = R 黨 b㡰 盐
ᨨ
g

⼄盐ᨨ

g A = g × - Rw
盐
ᨨ
g

�� 
h
ᨨ
 
ᨨ


㡰 ␔�� 

ᨨ

w
㦨
mg
㦨
҈黨

N Þ m
㦨
҈黨


├� ␔�� 

⼄盐ᨨ

w
㦨
mg
㦨
mg

w
R㡰
é ù æ
p ö
p ö = ҈黨

ê҈
 ç ´



´ ҈
 ú 

æçQ w = ÷ ÷ è ø

´ 


è T ø êë úû 㦨
҈黨


᲼҈






桌
㦨
҈黨�

N We mge 黨 ge 黨 㡰 W = mg =

ᨨ

g =
p p p

GM ⏌ R


b㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

gh = g d = g҈

GM

GM æ hö ç҈ - ÷ R
è R ø

- R ± R
+
҈㡰 R

- R + �R  � - ҈㡰 = R








=



   V鼀�
㦨
pR - pR ç ÷


㦨
pR ç ÷

gh =

=

Þh=

ᨨ盐鼀
ᨨ
 
矄 
鼀� � �㡰

҈


GM  R + h㡰


hö ÷ Rø

\ R =  R + h㡰
 R - h㡰 =  R
+ h
+
hR 㡰 R - h㡰 Þ R  = R + h
R +
hR
- R
h - h -
h
R Þ h + h

R - R 㡰 - R
h =
Þ h + h
R - R
h =
Þ h
+ hR - R
=


\ v=





GM

g=

㝐






ᆠ㡰 
ᆠᆠ�ᨨ
盐
ᨨ
��泐


�� 
h
�矄
��

泐

ᨨ

h
㦨 R/

d

G M ⏌ 黨㡰 ⏌

\ Rp
=
R㜠d㡰

æ
h ö gB = g × ç҈ - ÷ Rø è 꾀矄
ᨨᆠ
�矄
犨�� 
⼄盐泐

 
⼄盐ᨨ
 
ᨨ矄

� 鼀 矄
g �矄
矄鼀

 矄
ᆠ矄
g A = gB

R
w
æ
h ö
gh Þh= Þ g - Rw
= g ç҈ - ÷ Þ Rw
=
g Rø è R





R
p

㡰 ෌�

꾀ᆠᆠ�ᨨ
盐
ᨨ
��泐


�� 

ᨨ鼀
 矄 矄盐 ᆠ
�矄

æ ö = ç҈ ÷
è ø

҈

= 





黨


R


æ ö � = � ç҈ + ÷ è  ø

䰘


=

(

-


æ ö Þ

黨 = 黨

ç҈ + ÷ è  ø

-


᲼矄





桌

)


-҈


㦨





맔



҈

├鼀
㦨




맔
҈


鼀 㡰 ␔�
ᨨ�ᨨ
ᨨ

ᨨ
�盐

ᆠᆠ�ᨨ
盐
ᨨ ��泐
泐
矄
�Ἐ
㦨
�

w

ᆠᨨ矄

f ␔ 
f
�矄
�盐

 
犨�

ᨨ
ᆠ  �
�
��泐 
ᨨ矄
矄
f 㦨
黨
꾀

ᨨ 
ᨨ� 矄
矄
w
� ᆠ矄矄
�Ἐ
犨�
ᆠ矄矄  ᆠ

犨�� 

␔
㦨
鼀�
ᆠ矄矄
b㡰 ��ᨨ
ᨨ
ᆠᆠ�ᨨ
盐
ᨨ
��泐

g
犨� �矄 ᆠ
Ἐd Ἐ
ᨨ鼀
ᆠ 
ᨨ
 


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EBD_8341 PS-84

Physics

ᔼ

d < R
g = ᔼ
d
㦨
R

gs
㦨


Gm R
Gm

҈


R
Gm

҈



Þ

Q

g
㦨


҈
a  x㡰 ҈
a

GMm G ҈
M 㡰m GMm G ҈
M 㡰m + KE =
a
a
a
a é ҈ ҈
҈ ҈
ù KE = GMm ê + ë
a
a
a
a úû

-

é҈ +

-
- ҈
ù Þ KE = GMm ê úû
a ë ҈ 黨
GM é
� ù Þ mv
= GMm ê ú Þ v = a


a ë û

Rp � Rp

 �GM
a 㡰 ෌�


෌��ᨨ 
�
Ax
V¥ =
EG =
 x + a
㡰⏌
Vx x r r dV = ò ò EG × d x Þv=





¥

V¥

x

Ax

Þ Vx - V¥ = - ò

 x + a
㡰⏌



¥

dx

A A -
=
 x
+ a
㡰҈⏌
 x + a
㡰҈⏌
㡰 㝐矄矄
ᨨ
矄鼀
鼀 
ᨨ
 
ᨨ
�盐矄
x
 
 �ᆠ├ 矄矄 dx. æ x
ö dm = r ´
px
dx = r
çç ҈ -
÷÷ ´
px
dx è R ø 㝐矄矄
ᨨ
 
  \ Vx =

R








 
g¢
㦨


GM



黨黨 R㡰
R �盐矄
ᨨ
 

矄 � ├矄
泐
҈奠


gἘ æ R ö =ç ÷

Þ

gἘ
>
g g è

黨黨 R ø ҈
҈



×
҈

⏌矄㡰 矄� � g¢

㦨 g

Rw

ᆠᨨ矄

f \

꾀
⼄盐ᨨ


f

㦨



\

g¢
㦨 g

Rw



g¢
㦨

\ w

㦨


a


A

 
ᨨ� 
犨 
���ᨨ
�
ᨨ
ᨨ
 矄 ᆠ ᆠ
ᆠ
ᨨ 
�


ᨨ
GM G ҈
M 㡰 =
x ҈
a - x 㡰
҈
Þ = Þ
x = ҈
a - x Þ x =
a



�㡰 x ҈
a - x㡰 矄� �
ᆠᨨ 矄�ᨨ
ᨨ
 �泐

犨


GMm

GM

A

a x

㡰 ᆠ㡰 ᨨ鼀

mg
㦨


҈
M

M


�㝐

 
ᆠᆠ�ᨨ
盐
ᨨ
��泐

ᨨ� 
㝐
犨 �ᆠ Rp �矄
 
鼀�
ᨨ� 
ᨨ
 
犨�
 
�矄



ᨨ

҈
é R p ⏌҈
ù -
g pM = g p ê҈ ú = ҈[҈ -

҈] =

黨 鼀矄 R p úû êë \ ᨨᆠ
㦨
鼀矄矄
ᨨ
犨�
맔
gpM
㦨
҈


맔


黨
㦨
҈


 ҈҈


㡰

҈

�.e.

g
µ
d
d
黨
ᆠ㡰 ␔
├ ᨨ犨

�Ἐ
㦨
�

w
ᆠᨨ矄

q � = � - w

 ෌�

� Ἐ = �
�
w =
� ҈
w= =

´



´ ҈
 ҈ =


´ ҈
- ⏌矄




=
´
´ ҈

R
´҈

=
´҈
� m ҈

b㡰 ෌�


R p = e = ҈
҈



鼀矄矄
ᨨ
 
犨� Rp 㦨
҈

맔
҈


맔
҈
�
㦨
҈


├� 꾀ᆠᆠ�ᨨ
盐
ᨨ
��泐

  ҈
M 矄盐 ᆠ
ᨨ
 
 
 

g p = prGRP 錐 
 
ge
㦨
prGRE  g p Rp ҈
҈ = ҈ 鼀矄 
\ Þ
g p = = = ҈
ge RE ҈
ᔼ
d
>
R

�
㦨


Topic-4 : Gravitational Field, Potential and Potential Energy

�.e.

g
µ d
矄�� 
� 㡰


d

g
㦨
҈



맔
҈

⏌矄 R

r

æ x
ö M =
pr
çç x
-
÷÷ dx R ø
è

ò



r dx

x

�

r r -
 �R ෌��ᨨ 
�
Þ M =
pr


E=

GM

r




=

æ r r� ´
pr
ç ç  - �R
r
è G

ö ÷÷ ø

WWW.JEEBOOKS.IN

PS-85

Gravitation





æ r r ö Þ E =
pGr
ç -
÷ ç  �R ÷ è ø dE E
�矄
鼀 �鼀盐鼀
犨 
=
dr æ ҈ r
ö dE Þ =
pGr
ç -
÷ =
Þ r = � R ç  �R ÷ dr  è ø b㡰 ෌��ᨨ
�

 
矄盐 ᆠ Gm E=
r Gm Gm \ E҈ =
҈   E
=

r҈ r
ᨨ鼀
 
��鼀
��
�
⼄盐矄�ᨨ
E҈
= E

r҈
㦨
҈鼀

R

㦨

m
�� 㡰


\

�






Ef
㦨


黨


矄 

=

Y

GM

a


D

A

r




Mv
㦨
矄盐 
ᨨᆠ r ᨨ犨矄
ᆠ  \







҈ö Mv GM æ =
+ ÷
ç è
ø æ a ö a çè ÷ø


O

-෌㝐 æ ҈҈
ç
 çè



ù

ú úû

和ᨨ� 矄 

ö ෌㝐 ÷ = -҈҈ ÷
ø

 ᔴ�泐

෌㝐 
෌㝐 =ᆠ�泐 =


和ᨨ
ᨨ �

 
ᆠ 
ᨨ
ᆠ�泐 = 矄  - ᆠ�泐

B

҈ ö æ çè҈ + ÷

ø

Þ v=

GM a

GM ҈ ö æ çè҈ + ÷ = ҈
҈
a

ø

C

rdV 㡰m GMm = òa 㡰 F = r r
R

)

  泐 é ù  = - ê ò ᔼ  + ò ᔼ 泐 泐 ú 㦨
 é ᔼ
+ ᔼ
泐 ù 泐 û ë êë
úû
㦨
 éë � ( 䰘 -
) + ҈
( - -
) ùû ᔼ=-

a

GM a

҈҈෌㝐 æ  ෌㝐 ö -෌㝐 -ç÷=
è
ø GM R
 r
㡰 ҈

ᆠ㡰 꾀矄

V =  
R ෌
ᆠ㡰
鼀ᨨ矄
ᆠᨨ矄泐
�ᆠ矄
 
ᆠᨨᆠ
��ᨨ
ᨨ
vr㡰
҈҈
㡰 ෌��ᨨ 
�

ᔼ = ��ˆ + ҈
ˆ
⏌├� =-

(

a


�

Þ v
=

R

-෌㝐 é
æ ö = ê - ç ÷ è
ø
 êë

盐
ᨨ
ᆠ�泐

ᨨ �

ᨨ� 
P

AC a
a = =


矄盐 
ᨨᆠ
ᨨ
 
ᨨ泐 GM
ˆ GM
ˆ GM
ᆠᨨ矄
�°�ˆ + 矄�
�° ˆ 㡰 B
㦨

� +
 + a a a
㡰


a


GMm æ
҈ ö GMm ç  + ÷
㦨
 R è 
ø
R 㡰 盐
ᨨ
ᆠᨨ鼀
矄ᨨ�
矄 

ᨨ �

ᨨ� 
P Ef

E�
㦨






㡰 +

GMm ⏌
GMm ⏌
GMm GMm  
㦨
 æRö æ R ö
R
R
ç ÷
ç ÷ è
ø è
ø


GMm
GMm =
R R \
�  ᆠ
犨
� ��
 
� 
 �泐


Q r=

GM


GMm
R




㦨


æm ö ҈ Þ
ç ҈ ÷ =
è m
ø GM GM G 
M 㡰 ᆠ㡰
E g =
= +

a 㡰 a 㡰 a
b㡰 AC = a


ᆠ㡰 ᔼ ��
���ᨨ 
ᨨ �
 �泐

E�
㦨
 � 
���ᨨ 
ᨨ �
 �泐


E҈ æ r
ö æ m҈ ö
æ
ö æ m҈ ö =ç ÷ ç ÷ Þ =ç ÷ ç ÷  è ҈ ø è m
ø E
è r҈ ø è m
ø

Þ F =

䰘






k
pr
dr æ ҈ö
pkGm = -
pkGm ç ÷ = = mG ò

è ø r R r

r 矄� �
犨ᨨ 矄
矄ᆠᨨ 
犨

犨
 mv


pkGm = R R ᨨ
v

㦨
ᔴ
ᆠᨨ 矄
㡰 T
pR
pR

�鼀
�ᨨ

T = =
ᨨ

㦨

㦨
ᆠᨨ 矄 
R v
C

X

҈



㦨
- éë � + ( -
) ùû = ҈  ⏌ ├� �



ᆠ  �
�
���ᨨ 
ᨨ �
҈
⏌├�
 ᆠ
ᆠ  �
�
���ᨨ 
ᨨ �
 �泐
҈
 b㡰 ␔
ᆠ ᨨᨨ矄
  ᆠ
ᨨ� 

� � �泐

犨 
ᨨ  �泐
ᨨ
 
矄泐矄鼀
�矄
ᨨ
和ᨨ

� ��
 �泐
ᨨ
 
矄泐矄鼀
㦨



҈ ҈ ෌鼀 ҈鼀
鼀 ҈ ҈
+ 鼀




 ᨨ鼀
ᆠᨨ 矄�ᨨ
ᨨ
 �泐
ᔼ ��
 �泐
㦨
� 
 �泐 ҈ ҈ ෌鼀 ҈ 鼀
\
= 鼀 ҈ ҈
+ 鼀






� 
 �泐
㦨



ᨨ

෌鼀 ҈ 鼀
҈ ҈ 鼀 ҈ ҈
+ 鼀 ҈ 

=




҈㡰



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EBD_8341 PS-86

Physics

泐
ᆠᨨ 矄�ᨨ
ᨨ
� 
鼀ᨨ鼀 盐鼀  鼀 鼀 鼀҈҈ + 鼀

=

ᨨ
҈ = -
Þ 
=  ҈ ҈ 
鼀҈ 鼀
盐� �
盐
ᨨ


�
⼄盐�ᨨ
҈㡰

犨
�

GMm ù é êëQ ෌��ᨨ  ᨨ �  �泐 =  r úû

Þ




æ 鼀  ö
෌鼀҈鼀
+ 鼀
çç - ҈ ҈ ÷÷ =  è 鼀
ø


鼀҈鼀
҈ + 鼀҈ ҈
෌鼀҈鼀
= 鼀


鼀҈҈


҈ =


෌鼀

 鼀҈ + 鼀
㡰

= 鼀


҈䰘



෌  鼀҈ + 鼀
㡰

㝐矄矄
ᨨ
 
 
ᨨ

ᨨ
 
矄  㝐 䰘 㝐Ἐ = 㝐= 㝐



 ᨨ
���ᨨ 
�
盐
ᨨ
 
 
ᨨ

ᨨ
  矄 

F҈ =

GM r҈




 
F

㦨


҈�


F
ᨨ



҈ =
F
r


GM

r

r҈
GM R

´r

F҈
㦨




´ r҈

 
F

㦨


GM

Y

=

҈ r

A 㜠




㡰

æ æ
ö ö çQ M = ç pR ÷ r
÷ è ø ø è

æ
ö mv
é GM ù = mê
 
M
㦨
ç pR ÷ r
ú
r è ø ëR û

X

Z


���ᨨ 
�
盐
ᨨ
鼀矄矄矄

꾀
 


�졠
�矄 ⼄盐
 
ᨨᨨ矄�

꾀矄
 
 
ᨨ
 矄
ᆠ�ᆠ矄
�矄
Ӹ㜠
^ ᨨ
㜠 �矄
矄ᨨ
ᨨ �

 泐
ᨨ� 
ᨨ
 矄
犨ᨨ
ᆠ�ᆠ矄
犨� 
⼄盐
盐
ᨨ
鼀矄矄
M
 
鼀矄矄矄
M
 
m


Þ

v µ r

�


v  r
�
�矄

矄�� 
� 
矄矄� �
 ᨨ盐�
ᨨ���
㡰 ᔴ  �
�
ᨨ �
 �泐 B \ DU
㦨
Uf

U�

Topic-5 : Motion of Satellites, Escape Speed and Orbital Velocity

R

é ҈ ҈ ù GMm A 㦨


GMm
ê 㦨
ë
R R úû
R GM R
Þ

GM
㦨
gR
Q g
㦨

R O gR
m mgR = \ DU
㦨

R
҈

҈



㡰 矄� �
犨
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
 �泐

ᨨ
 �泐

�� 
҈


㦨
ᨨ
 �泐

 GM E m ҈ GM E m ҈  + mV

= + mV
҈
R

R

B 




㡰

O




ᨨ



\



GM

r


´ r
R R F҈ r҈ = \ F
r
҈


ᆠ
㡰

���ᨨ 
�
쟼㡰
�  矄�泐

 
ᨨ� 
O �



ᆠ 
ᨨ

矄ᨨ�
矄 
�矄
ᨨ

ᨨᆠ
ᆠ� �
ᨨ

矄 鼀矄矄
m

ᆠ

O F
㦨
m

E
㦨
m

맔


㦨

ᨨ犨

y

졔
z

㦨


矄 矄
 
⼄盐�ᨨ
ᨨ

ᆠ�ᆠ
犨� ᆠ 







㡰
 
�盐矄


盐 �矄
 
 
ᨨ
 
ᆠ�ᆠ
�矄  �ᆠ盐
ᨨ
x㜠 �矄
\

mv
é GM ù
犨 
M
�矄
 
ᨨ
鼀矄矄
ᨨ
  = m´ê ú r ë r
û 矄 �ᆠ
矄泐矄鼀
\

v µ

GM

ᨨ
r


R

 
���ᨨ 
�

F
㦨


䰘 ෌㝐


҈

ᆠ㡰 ᨨ
r ³ R =

᲼Q 

㦨
҈

├鼀⏌矄
�� 桌


b㡰
ᨨ
r
>
R

 
���ᨨ 
�

F
㦨


\


෌   鼀҈ + 鼀
㡰 ᆠ㡰 
鼀矄矄
ᨨ
矄鼀
矄 
犨 �ᆠ
矄
ᨨ

鼀ᨨ㡰
�矄
鼀 �盐矄 =  ᨨ鼀
��盐㡰
㝐 㝐 鼀 Þ鼀= =

æ ö
p p ç ÷   è
ø

෌㝐 Ἐ

҈ ö V

V
黨 æ ҈ = Þ V
= V

+ gR çè ÷+ ҈
ø

�

黨 Þ V = V

+ gR » ҈
km ⏌ s �

和�鼀�泐

= - 鼀҈

҈


GM E R

҈


㡰 泐
 �盐
鼀ᨨ鼀 盐鼀
ᆠᨨ 矄�ᨨ

mr鼀� v鼀 = mr鼀 v鼀�

v鼀�

r鼀�

r鼀  

v鼀 ෌�

v鼀� =

v鼀


\

r鼀� v鼀� ҈ = = r鼀 v鼀


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PS-87

Gravitation



㡰 졠�
矄
ᨨ
 
ᨨ泐
犨 
�
ᨨ矄
泐
ᆠᨨ矄 ᨨ
 
矄盐 ᆠ
ᨨ
 

GM


�㡰 R 

G
㦨
���ᨨ 
ᆠᨨ 矄  쟼矄ᆠ
矄
ᨨ鼀
 
矄盐 ᆠ
ᨨ
 

�


V
=


GM R ��� �
�㡰
泐
��㡰

犨
 Ve =





GM ҈ R =
GM
R

V
= Ve 





��㡰

ᆠ㡰

hv =

Q
p

K
 
G

ᆠᨨ 矄 矄 \ T
µ R
r r 㡰 ᨨ鼀
犨
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
鼀ᨨ鼀 盐鼀

p� = p f m҈u҈
졔
m
u

㦨
MVf mv ö æ çè mv + ÷
ø �v Þ vf = = m

ᔴ泐

vf


v�
\

犨�

��ᆠ b㡰 ⧤
쟼

ᨨ
矄�
�矄
ᨨ


矄盐 ᆠ
 

�� 
ᨨ鼀
 �泐
ᆠᨨ 矄�ᨨ

矄盐 ᆠ
졔
쟼҈
㦨
 

 V


෌㝐 鼀 ෌㝐 鼀 + 쟼҈ =   (  + )

æ ҈ ෌㝐鼀 ҈ ö ´  Þ 쟼҈ = ෌㝐  鼀 ç ÷ Þ 쟼҈ = (  + )  è   + ø ෌��ᨨ 
ᆠ�ᨨ

Re

෌
㦨
鼀ᆠ
㦨


鼀
=

R鼀

쟼
=

V鼀�


GM Re

졠�
ᨨᆠ�泐

V
=

ᨨ鼀
 �泐
ᆠᨨ 矄�ᨨ



GMm ҈ æ  ö GMm ҈
+ mç V÷ = + mV鼀�

Re R鼀

è
ø ᨨ鼀
 �盐
鼀ᨨ鼀 盐鼀
ᆠᨨ 矄�ᨨ





䰘





҈㡰

 VRe = V鼀� R鼀



㡰
和ᨨ� �
⼄盐�ᨨ
҈㡰
 

㡰
犨
�
R鼀
㦨
Re 㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

鼀矄矄
 矄�泐
ᨨ

矄 �ᆠ k � 泐
�矄
矄

r 㝐矄矄

M = ò rdV

ÞM =

r = R


ò


R


k
pr
dr
Þ M =
pk ò r dr r



pkR

=
pkR

GMm = mw

R = FC 㡰 FG = R







ᨨ

M =

M R


pkR G
= w
R Þ w =
pKG


æQ w =
p ö Þ ç ÷


T ø è R R
p
p R
pR
pR \T = = = Þ T
= w
KG KG
pKG


m

෌㝐 鼀 鼀
= (  + ) (  + )


෌㝐  鼀

(  + )

෌㝐 鼀 鼀
=

(  + ) 쟼҈
㦨
쟼
ᔴ泐


= ҈ Þ =  = 


├鼀 




GM A RA 犨 
MA
 
RA

 
鼀矄矄
 
�盐矄
ᨨ
 
 
A
꾀ᆠᆠᨨ� �
ᨨ
��
ᨨ鼀 M R M B = A
RB = A


GM A M
G A RA n
\ VA = = =҈ \ VB = GM


VB
RA A RA

Þ n
㦨

M M R u b㡰 R Þ R v m ҈  GMm ҈
 GMm mu
+ = mv +
R

R ҈  GMm Þ m v
 u
㡰 =

R 㡰 쟼矄ᆠ
ᨨᆠ�泐
ᨨ
 
 
A
�矄
VA =

Þ  =  = 盐
 v
=

GM
R

෌㝐

\ vrad =




�㡰 m´v = ҈
v æ mö çè ÷ø ҈


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EBD_8341 PS-88

Physics

쟼ᆠ� �

ᨨᆠ├
ᨨ
鼀矄矄


⧤� �ᆠ
 �泐
ᨨ
矄�



m ҈


GM m GM 黨m ´ = ´ vt Þ Vt
=
҈ ҈

R ҈

R ⧤� �ᆠ
 �泐
ᨨ
ᨨᆠ├
҈M
VT + Vr
⧤쟼 rᨨcket =
҈
҈ m æ GM GM ö 㡰҈

+
҈ = ´ ´ ç u
 ÷
҈
è R R ø vT m æ
GM
҈ GM ö M M ´ ҈

ç u  + 㦨
÷ è




R ø R ҈

R ҈҈黨 GM ö æ 黨M = �m ç u
 ÷ ҈
è


R ø 㡰 �鼀
�ᨨ
ᨨ
ᨨ盐�ᨨ
ᨨ
矄�

pr T= v GM r r v= \ T =
p r =
p r GM GM 和盐矄�盐� �
 
盐矄

犨
�

҈



㦨 鼀 


\

(

黨


)




㡰 ´ ҈
҈


T =
p




䰘 ´ ҈
-҈҈ ´
´ ҈





㦨
䰘
҈



矄


;


҈䰘

Þ ҈҈
ᨨ盐�ᨨ 矄
҈

㡰 쟼矄ᆠ
ᨨᆠ�泐
vc =


GrV = R


GM = R


矄ᆠ


GS ´
pR = R


prGR




prGRm  R

 ෌�


pR =

´ pRm
ᨨ
Rm =  
Ἐ ᨨ
鼀ᨨᨨ

vc =

Ἐ \ ve =

҈҈



v
æ Rö prG ç ÷ = c è ø 



҈
mve E v
v E =
= Ἐe
= e = ҈
ᨨ



E 㦨
v ҈
EἘ ҈ æ ö
e mv Ἐe vc çè
÷ø
r b㡰 鼀�ˆ + 鼀ˆ =
鼀  r  Þ  = ˆ� + ˆ













҈ ෌㝐 = ´
b㡰 졠�

ᨨᆠ�泐


㦨
҈


� + 㡰 » �

쟼矄ᆠ
ᨨᆠ�泐
㡰
�矄

 =


� + 㡰 »
�

෌㝐 

᲼Q



桌

D =  - 
= 
- ҈㡰 �

vR

҈

b㡰 꾀
�� 

ᨨ鼀
ᆠ 
ᨨ


ᨨ�
ᨨᆠ�泐

෌㝐 r 泐
� ᆠ�
ᨨ
 �泐
ᆠᨨ 矄�ᨨ

æ ෌㝐鼀 ö ⧤쟼
ᨨ
�鼀
졔
çè  ÷ =
+
 ø Q 꾀
� � �泐

쟼
㦨
⧤쟼
㦨

㡰 v=




෌㝐鼀 æ ෌㝐 ö 㦨ç 鼀
㦨
鼀
ᨨ ⧤쟼
ᨨ
�鼀
㦨
  ÷ø è ҈�
㡰 
㝐
�矄
鼀矄矄
ᨨ
矄
鼀
�矄
鼀矄矄
ᨨ
鼀ᨨ� 泐
 �泐
ᆠᨨ �ᨨ
犨


 
¥. ෌㝐鼀 ෌㝐鼀 ҈  + + 鼀 矄
=
+
 

෌㝐
´


䰘 ´ ҈
҈҈ ´  ´ ҈
҈ = \ 
㦨
 ҈
҈҈ ;


´҈
� 鼀 ⏌ 矄 ҈

b㡰 꾀泐� �
 �泐
ᆠᨨ 矄�ᨨ

GM e m GM e m ´  ´҈
� ҈ mVs
=
Re


� ´҈

Re

Vs
GM e é  ´҈
� ù = ê҈ + ú
Re ë

� ´҈

û ö
Gme æ
GM e ö æ Vs = ҈ ç = ҈҈

├鼀 ⏌ 矄 ÷ ççQ Ve = ÷ ÷ Re è Re ø è ø \ Vs
㦨 ҈ ´ ҈҈

»

鼀 矄 ҈䰘
b㡰 ᨨ






 
ᨨ�
ᨨᆠ�泐
�矄
� 쟼矄ᆠ
ᨨᆠ�泐
㦨

� \

鼀� �鼀盐鼀
� ᆠ矄
�
�矄
ᨨ�
ᨨᆠ�泐 ҈



㦨

�

�
㦨
�



҈㡰 ᆠ㡰 꾀ᆠᆠᨨ� �
ᨨ
盐 �矄
犨
ᨨ ෌��ᨨ
෌��ᨨ 
ᨨᆠ

㦨


⧤� �ᆠ
 �泐
ᨨ
矄�
꾀



꾀
㦨 鼀 꾀 꾀

㡰 ᨨ

矄�
ᨨ�� �
ᆠᨨ矄
ᨨ
 


ᨨ� ᨨᆠ�泐
�矄
��
泐


=




 r æ ö æ ö Þ = ç ÷ +ç ÷ = è
ø è
ø


҈



҈


෌㝐 鼀´

꾀 Þ
㦨
෌㝐
㦨
҈ 
鼀 ´


꾀矄ᨨ 盐



෌㝐鼀  졔 㡰


mv

GM
GM m = ҈黨
㡰 F = m
ᨨ

r Lr Lr

GMm é
p ù æ
p ö mr ç ÷ = Qv = r w   w = T úû Lr êë èT ø Þ T µr

쟼

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PS-89

Gravitation




㡰 꾀矄
犨
├ ᨨ犨







-GMm ෌��ᨨ 
ᨨ �
 �泐
㦨
r  
ᨨ�
ᨨᆠ�泐


v

㦨
෌㝐 ⏌ + ҈ ෌㝐鼀 ҈ ෌㝐 ෌㝐鼀 쟼 = 鼀

= 鼀

  ෌㝐鼀 æ ҈ ö - ෌㝐鼀





= ç - ҈÷ =  è
ø
-෌㝐鼀 +⧤
쟼� =
E�
㦨
Ef

҈
맔
⧤쟼
ᨨ
矄ᆠ
҈ GMm ҈ ᨨ mv

㦨
´
R
GMm GMm GMm =\
R R R+h ҈ ҈ Þ R 졔 h = 2R ᨨ


h = R =Þ R+h
R æ
෌ ö
�
ç

ᨨ
鼀ᆠ  �ᆠ
 �泐
鼀矄矄
m

a ç  㝐҈ 졔 㝐
㡰 ÷÷ è ø 鼀�犨泐
ᨨ� 
犨
 
ᆠ 矄
ᨨ

 
鼀ᨨᨨ

⧤쟼
ᨨ�
㦨


�෌㝐鼀
b㡰 
�ᨨ
犨
矄ᆠ
矄
V e
 

 ᨨ
鼀� �鼀盐鼀
⼄盐�
 �泐


⧤ =


҈


ᨨ�
ᨨᆠ�泐
Ve
㦨

V


├� �ᆠ
 �泐

 
�鼀
ᨨ ᆠ�ᨨ

ì é GM ҈M
GM ҈m GM
m ù é GM ҈M
ù ü - 㦨
 í ê úý d d ⏌
d ⏌
úû êë d ûþ îë

҈ ҈ mVe
= m
V 㡰
= mV






㡰 
h

 
�� 
ᨨ
犨 �ᆠ
 
盐
�矄矄 犨�
 
�� 
ᆠᆠ�ᨨ
盐
ᨨ
��泐
�矄
矄

KE =

æ è

g҈
㦨
g ç҈ +

hö ÷ Rø




Þ

g hö æ
㦨
� ç҈ + ÷
è Rø

Gm  M҈ + M
㡰 㦨
� � �
 �泐㡰 d ⏌
⧤� �ᆠ
 �泐
⼄盐�
ᨨ
矄ᆠ
 
鼀矄矄
�m
ᨨ
� � �泐 ҈ Gm \  M҈ + M
㡰 mV e
=
d
犨 
Ve
�矄
 
ᨨᆠ�泐
犨�
犨 �ᆠ
鼀矄矄
m
�矄
ᨨᆠ

G M҈ + M
㡰 Þ Ve
㦨
d 㦨





Þ

h
㦨
R


GM
㦨
v N
�� 㡰



�㡰 R ᨨ犨
泐� �
ᆠᨨ 矄�ᨨ
ᨨ
 �泐
� ᆠ� ᨨ矄矄
ᨨ
├� �ᆠ
 �泐
㦨
��
�
���ᨨ 
ᨨ �
 �泐 GMm æ GMm ö ҈
ç mv
㦨
 \
R R ÷ø
è

␔
├ ᨨ犨
矄ᆠ
矄

ve
㦨


GM R ᔴᨨ鼀� �
⼄

�㡰


��㡰
犨
� N
㦨

\





\

vp ve

gp

=

g

´

Rp R

gp




�㡰

ve

=

\ vp =



´
 ´ = = ҈҈

҈҈ ´

҈҈

  ´ ve = ´҈҈├鼀⏌矄 =  ├鼀⏌矄 ҈҈ ҈҈






h㡰
꾀ᆠᆠᨨ� �
ᨨ
⧤Ἐ矄
犨
T
µ R \

T҈


R҈
= T

R





R҈ = R +
R = 䰘 R
 
R

㦨


�R
졔
R
㦨

�
R T҈
㦨


ᨨ盐矄 \


gR


෌�

㦨



��㡰 ge ҈҈ g p r p Rp
꾀矄ᨨ



g = p G rR

\ = ´  r ge R r é ù
R 

p = �� 㡰ú êQ \ = ´ ҈҈  R ë r  û Rp 
\ =




















���㡰 R


ᨨ鼀
�㡰

��㡰


���㡰 vp









��㡰

v
㦨


㡰 ␔
├ ᨨ犨
 
矄ᆠ
矄
v =

 㡰
ᨨ鼀
ᆠᨨ 矄�ᨨ
ᨨ
鼀ᆠ  �ᆠ
 �泐 GMm ҈ æ GMm ö mv
+ ç ÷= - R+h R
( ) è ø


䰘







´



T



=

䰘 ´ 䰘 ´ 䰘 ´ R 
� ´ 
� ´ 
� ´ R 


Þ
T

㦨







矄
犨
 �
�矄
ᨨ
ᨨ
 
⼄盐ᨨ�
 
෌ᨨ矄�ᨨ 泐
矄�
�矄
盐 ᆠ 
ᨨ
 
⼄盐ᨨ�
 
b

㡰 
 矄
P
 
Q

 
矄鼀
盐 � ᨨ鼀  矄�泐
�r
 
矄盐 ᆠ
矄
A
 

A
矄ᆠ�泐

  
鼀矄矄
ᨨ
P, MP

m




éAù


m = r´ pr  = r´ p ê ú   ë
p û


é
Aù


鼀矄矄
ᨨ
MQ
㦨
r´ p ê ú  ë
p û

⏌


⏌


=
鼀

\ 
鼀矄矄
ᨨ
 
R
㦨


鼀
졔
鼀
㦨
黨
鼀 ᔼ
 
�盐矄
ᨨ
P
㦨
r


 
�盐矄
ᨨ
Q
㦨

r ⏌
⏌
ù é æ
Aö æ Aö =
ç êQ rQ = ç ú ÷ ÷ è
p ø è
p ø êë úû ҈⏌  
�盐矄
ᨨ
R
㦨
黨 r

éQ M R = M P + M Q ù ê ú     êë rR = r + 
r 㡰 = 黨 r úû

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EBD_8341 PS-90

Physics

꾀矄
犨
├ ᨨ犨

矄ᆠ
ᨨᆠ�泐
ᨨ鼀
 
  Ve
㦨


vQ
㦨


vR 㦨

黨






MB㦨҈
M


GM P
GM =
\ vP
㦨
Rp R
GM Q


G 黨 鼀㡰 ҈⏌

黨

r

⏌


T҈ æ R҈ ö ҈ =ç ÷ 㦨
T
è R
ø
㡰 꾀
矄ᨨ  
�

ᨨ�� �
矄ᆠ
矄�ᨨ
ᨨ
  
 � ᆠ矄
犨�� 
矄矄
矄

�矄
�

矄
ᨨ





ᨨᆠ
ᆠ� �
ᨨ
矄ᨨ  
�矄
盐��矄
�
ᨨ�� �

ᆠ矄矄泐
ᆠ �
ᨨᆠ
꾀泐� �
鼀ᆠ  �ᆠ
 �泐
ᆠᨨ 矄�ᨨ


ᨨ
 �泐

A
㦨
ᨨ
 �泐

B ⧤
쟼
㡰A 졔

쟼
㡰A 㦨

쟼
㡰B


µ
 \



҈
GM é -GMm ïì
æ 黨黨 R ö ïü ù GMm m´ +ê R - ç í ýú = ÷  è ø
҈

ï ú R R+h êë
R ïî þû

B

㝐 �鼀盐鼀
�� 
� 
泐  
ᨨ泐
ᨨ鼀
 
矄盐 ᆠ
ᨨ  
 㡰




ve A R

R ҈

黨黨R ҈




 
ᨨᆠ
ᨨ
ᆠ�ᨨ
�矄
ᨨ


�矄 ᆠ
x
ᨨ鼀
  ���
矄


GM S m

㦨
-

=






Þ

҈
M

M

=

Þ x
㦨

a x ҈
a - x㡰
x ҈
a - x㡰 
v
 ᨨ
ᨨᆠ�泐
犨�
犨 �ᆠ
 
ᨨ泐
ᨨ
鼀矄矄
m
�矄 �
矄ᨨ
 
�
ᆠᨨ矄矄矄
P
ᨨ �
OP


 �泐
�矄 ᆠᨨ 矄



 �泐
ᨨ
 
矄泐矄鼀
ᨨ
 矄
鼀矄矄矄㡰
� ��泐 㦨
� 
 �泐
犨 
m
�矄

P GM B M S GM B m GM S m ҈
+ mv ҈
a
a
a



和ᨨ� �
ᨨ
⼄盐�ᨨ

犨 �
h
㦨
黨黨
�
R


和鼀
矄

x

GM B m

= 黨҈⏌ v P



ᆠ㡰
ᨨᆠ
ᨨ
矄�
�矄
犨泐矄
ᨨ犨矄

犨 �ᆠ ᆠ矄
 
矄�
犨�
 
���ᨨ 
ᨨᆠ
F

  ᨨ
ᆠᆠ�ᨨ
ᨨ
矄�
S
�矄
犨泐矄
�ᆠ
ᨨ犨矄
ᆠ  ᨨ
 





ᨨ⼄盐
ᨨ
 �矄
���ᨨ 
ᨨᆠ
F
ᨨ盐
ᆠ 
ᨨ 
�矄
ᨨ

 ᨨ

 �盐
鼀ᨨ鼀 盐鼀
ᨨ
�

鼀� �盐
 
�ᆠ�ᨨ 㡰
ᨨ
S
ᨨ盐
ᆠ 
ᨨ

�矄 ᆠᨨ 矄 
 ᨨ盐� ᨨ盐
꾀矄
 
ᨨᆠ
F
�矄
ᆠᨨ 矄�
�
盐

  ᨨ 鼀ᆠ  �ᆠ
 �泐
ᨨ
矄�
鼀� 矄
ᆠᨨ 矄 

和
ᨨ S
�矄
鼀 �鼀盐鼀
犨 
�
�矄
矄
ᨨ

 
鼀� �鼀盐鼀 犨 
�
�矄
 矄
b㡰 Q 졠�
ᨨᆠ�泐


Þ

a O'

҈
a

V҈ R

GM ҈
ᨨ


V
µ
= \


㦨
R ҈ V
R҈ R L҈ m҈v҈ R҈
´
´҈ ҈
㦨

㦨

㦨
L
m҈v
R
҈´҈´
҈ GMm ⧤� �ᆠ
 �泐

K
㦨

R k҈ m҈ R҈
´

´ \

㦨

㦨

㦨
҈´ ҈ ҈ k
m
R
ᨨ鼀
⧤Ἐ矄
犨
ᨨ
 泐
鼀ᨨ�ᨨ






P


ᨨᆠ
ᨨ
鼀矄矄
m
盐
ᨨ
���
矄
㦨
ᨨᆠ
ᨨ
鼀矄矄
m
盐
ᨨ
矄鼀
矄


V
㦨


҈


O
a ���
矄


G 
鼀㡰 =
v P
r

=

RQ


Gm r

MS=M




GM B M S GM B m GM S m ҈
a
a
a

᲼Q M B = ҈
M 錐 M S = M 桌  �GM
a 

�㡰

ᆠ �
ᨨᆠ
�矄
ᨨ�
泐
 
���ᨨ  盐 v mv
GMm A m = \  R + h㡰  R + h㡰
\ v
㦨


h

B

Þ

GM v

㦨
R+h

R

M

E

҈ ҈
GM ve =

R ҈ æ
GM ö GM Þ
R
졔

h
㦨

R
\ h
㦨
R
㦨





├鼀
\ ç ÷=
è R ø R+h ��㡰 


 
矄
犨�
犨 �ᆠ
 
矄�
�矄
  矄盐 ᆠ
ᨨ
 


ᔼ
 
矄�
�矄
矄ᨨ

�矄 ├� �ᆠ
 �泐
�矄
ᨨ

␔ 
�
矄
泐
ᨨ
 
쟼
�矄
ᨨ �
 �泐
ᆠ矄矄
 
ᆠᨨ 矄
� ᨨ ├� �ᆠ
 �泐
\ 
쟼
㡰A


쟼
㡰B
㦨
⧤
쟼
-GMm æ -GMm ö ҈
-ç = mv Þ è R ÷ø

R ෌�

v
㦨


GM = gR = 黨

´


´ ҈

R 㦨
䰘黨


鼀⏌矄
㦨
䰘
黨

├鼀⏌矄 �

 
� ��
ᨨᆠ�泐
 
矄 ᨨ盐

��
ᨨ
ᆠ
�ᆠ 
�


ᆠ �
ᨨᆠ
�矄
ᨨ�

泐
 
矄盐  ���ᨨ 
ᆠ�ᨨ
ᨨ
 
犨ᨨ
鼀矄矄矄
Þ v
㦨


m

FR
㦨
F
+ F
+
F
ᆠᨨ矄

°





㦨
 F =  G \

mv
G
= r a

m´m a











O

m


a⏌ Ö

F

FR

m

F

a

m

v

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PS-91

Gravitation

æ mv
ö = ᆠ �
ᨨᆠ÷ ç è r ø

㝐矄矄
ᨨ
 
� �

dM
㦨
s
pxdx
㦨


�盐矄
ᨨ
 
ᆠ�ᆠ盐


r
㦨


Gmr

a

a
a 㦨
 



R


K dr



Þ Gm

K
pr
mG

rr 㡰 K r K = = ᨨ


m m
pr m G
pr m
G 㡰 㝐矄矄

盐 �

ᨨ
 
矄 

M M 鼀矄矄 = = s
㦨
 p
R㡰
- R㡰
㡰 䰘 pR

盐矄
ᆠᨨ 矄�

� �
ᨨ
�盐矄
x
 
 �ᆠ├ 矄矄
dx
矄 矄 ᨨ犨
�
 
��盐
\





r=

䰘pR





R

ò

R

xdx ҈
R
+ x



R GM
p é
GM ҈
R
+ x
ù = 

- �㡰
ëê ú û 䰘R 䰘 pR R ␔ᨨ├ᨨ 
�
鼀ᨨ� �

盐 �
鼀矄矄
ᨨ鼀

ᨨ
¥
㦨
V¥

VP

æ -
GM ö
GM 

- �㡰 ÷ = 

- �㡰 䰘R è 䰘R ø 䰘㡰 ᨨ
ᨨ� 
鼀矄矄

�矄 ᆠ
r
㦨

l

ᨨ WP¥
㦨



ç 


GMm 
l 㡰
㝐

+

Gm
l



㦨
ma

鼀

鼀 l

r
㦨
l ᨨ
ᨨ� 
鼀矄矄

�矄 ᆠ
r
㦨
l

GMm l 㡰
\


K dM = dr Gm


pr
dr r =


R㡰 +  x 㡰




GM
p

VP
㦨



mv
GMm GMm GMm ҈ = Þ mv
= ÞK=
r

r
r r
Kr K \ M = Gm �  �� �
 
ᨨ m ⼄盐�ᨨ
犨


ἘrἘ
犨
� dr r dM dM
K = dr Gm M

\




=-

和ᨨ� �

犨
�

b㡰 ෌��ᨨ 
盐
ᨨ
 
鼀矄矄
ἘM Ἐ
矄 
�
 
矄  ᨨ
�盐矄
Ἐr Ἐ

ᨨ�
 
⼄盐�
ᆠ �
ᨨᆠ
ᨨ
�ᆠ ᨨ
鼀矄矄
�m
ᨨ
盐
�

ᆠ�ᆠ盐



ᨨ

GdM

-

R

a
pr
p a ⏌  = = p



㦨
Gm v Gm a

҈


ò

VP
㦨


 G ma

Topic-6 : Miscellaneous (Mixed Concepts) Problems

䰘pR


ᨨ �

ᨨ� 
P
盐
ᨨ
矄 


Gm v =
=

Þ
v =
a a ´  a

�鼀
�ᨨ
ᨨ
ᆠ�ᆠ盐
鼀ᨨ�ᨨ





pMxdx





-

Gm
l


GMm 
l㡰




+


㦨
ma

Gmm l




=

GMm l㡰






Gmm l


䰘M é҈ ҈ ù Þ m
㦨
\ K
㦨
䰘 ú


ë 黨 ҈
û b㡰 ᨨ鼀
ᆠᨨ 矄�ᨨ
ᨨ
鼀ᆠ  �ᆠ
 �泐
-GMm GMm ҈
+ mv 㦨


졔


L L \


m
㦨 M
ê 

ᨨ


҈

GMm mv 㦨

L

GM
GM =
L L

V=

\ 

㝐

ᔴ 

鼀



㝐

P
R dx
R

R x

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EBD_8341 PS-92

Physics

Mechanical Properties of Solids

8

Topic-1 : Hook's Law & Young's Modulus ҈


-A

ᆠ㡰 ෌�


U =

r




+

B

�


r҈


ᨨ
⼄盐���盐鼀


p ´ 
´ ҈
- 㡰
ᆠ㡰 Dl҈
㦨
Dl
ᨨ


dU = - A -
r -䰘 㡰㡰 + B  -҈
r -҈ 㡰 =
dr
A ҈
B
A ҈ Þ
= 䰘 - ҈ Þ =
҈
B r r r F=

æ
B ö \
和�ᨨ
犨
鼀ᨨᆠ盐矄

r = ç è A ÷ø





Fl҈




Fl
pr

y


҈⏌


和矄矄҈
=

- A
A
- A
+ =
B
B
B FL F FL =
ᆠ㡰  La
DT㡰
㦨
\ a
㦨
AYT pr YT AY ᔴᨨ �ᆠ� 
ᨨ
ᨨ盐鼀
 矄矄�ᨨ

F g
㦨
a
㦨


pr YT ᨨ 盐矄㡰

和矄矄 æ Dl ö çè ÷ø L


s

 
矄矄矄

ᨨ
ᨨ ��ᨨ
Dl  =

é҈ ҈ù Dl  = s ê + ú ë Y҈ Y
û

sL҈ sL
+ Y҈ Y


᲼Q
L҈
㦨
L

㦨
҈鼀桌

æ YY ö s = Dl ç ҈
÷ è Y҈ + Y
ø





N æ ҈

´

ö =


´ ҈
- ´ ç ´ ҈
黨 =
´ ҈


è ҈

÷ø m ᆠ㡰 ෌�
�盐矄
ᨨ
犨�

r
㦨


鼀鼀 㝐矄矄
ᨨ
 
ᨨ
鼀
㦨


├� F mg 和矄矄
= = A pr 㡰


R ´䰘

=

҈
�



´


҈
� ᨨᆠ҈ = ҈ 

҈
㡰

和矄矄

=

҈
� ᨨᆠ
= 




㡰

x ᆠ  �
�
 �
= ᨨ��� 
 �



x
�矄
 
�矄ᆠ鼀 
ᨨ
矄ᆠᨨ 
ᨨᆠ├
和�

=

ᨨ
矄鼀
鼀�





ᨨ


䰘




҈㡰


矄矄矄҈ 矄矄矄
= 矄� ҈ 矄�


҈
�

҈

�

Ӹᨨ盐 �Ἐ矄
鼀ᨨ盐盐矄

Y =





ᆠ  �
�
 � ҈

� ´ ҈
-
= ᨨ��� 
 � ҈

҈ ᨨ
矄ᆠᨨ 
ᨨᆠ├


A B +

÷=
B ⏌ A
B ⏌ A ø

和

ᨨ


和� ҈
=

҈⏌
ö

矄矄

= 
҈ ´ ҈

⏌鼀


\
R
㦨
҈
䰘�
鼀鼀 b㡰 ᨨ
矄鼀
鼀�
 
�ᨨ
ᨨ
矄矄矄
ᨨ
矄�
�矄
矄鼀 ᨨ
�矄
ᆠ盐























=





=

pr҈
y҈

ᨨ �
 �泐


æ æ
B ö U çr = ç ÷ è Aø è


´ 
҈p

=

=




㡰
x






� ´ ҈
-


҈ 和ᨨ� �
犨
�

x
㦨



�
ᆠ鼀 ᆠ㡰 ᔴᨨ 矄�

矄鼀
鼀 

ᨨ
�盐矄




졔
 꾀
⼄盐���盐鼀
ᆠ  �
�
 � ᨨ
 
犨� 



㦨

҈

ò  = ò




㝐� 


é
ù + ú 泐 pê  ë û



├� �
�鼀�
ᨨ鼀


ᨨ


é ù 㝐� ê ú 㝐� ê ҈  ú 㝐� D = ´ = 
ú
p泐 ê é
ù ê ú p 泐 + úû êë êë  úû




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PS-93

Mechanical Properties of Solids


⼄盐���盐鼀
  
 �
ᨨ
犨�
㦨

졔
D

= +



æ ҈ 㝐� ö =  ç҈ + ÷ p Ӹ è  pӸ
ø 㝐�

҈




㡰 ෌�


lc
㦨

鼀錐

ls
㦨
҈
鼀錐 Yc
㦨
҈
맔
҈
҈҈
⏌鼀
 
Ys
㦨


맔
҈
҈҈
⏌鼀






DL = α DT L 꾀矄
 �
�矄
ᆠᨨ 矄 㡰 㦨


맔
҈
҈҈
맔
҈
҈
맔
҈
�
맔
҈


㦨




맔
҈


 æ
鼀㝐 ö b㡰  矄�ᨨ
�
 
犨�

= ç ÷� è鼀+㝐ø ᨨᆠ ⏌  矄�ᨨ


鼀㝐 � = 和矄矄
㦨
꾀 꾀鼀 + 㝐㡰
鼀 ´
鼀㡰�











=

㝐
㦨


鼀
�� 㡰 꾀鼀 +
鼀㡰

҈鼀

鼀

Ts



Ts
ᆠᨨ矄




和�
㦨


黨


҈




鼀

鼀� �=



= 鼀꾀 꾀 ᆠ㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ
l矄  泐 Dl矄 = 
矄 = 
矄 = ᆠ
=䁸 l  泐 Dl  꾀矄

泐 =

Dl 矄 =

Dl  =

鼀�l 矄 p矄

泐矄

p

泐 

҈


Þ

Dlc æ Tc ö æ lc = ç ÷´ ç Dls è Ts ø è ls

Tc ҈ = Ts 

Þ

Yc ҈´ ҈
҈҈ ҈ = = Ys
´҈
҈҈


 

Fl ADl 

As
㦨
Ac

Dl
㦨


Fl AY

ö æ Ys ö æ ҈ ö æ  ö æ
ö ÷ ´ ç ÷ 㦨
ç ÷÷ ´ ç ÷ =
÷ ´ çç ø è Yc ø è  ø è ҈ ø è ҈ ø

├� �
矄矄矄

s
㦨




맔
҈
䰘
鼀
꾀
ᨨ
ᆠᨨ矄矄㜠矄ᆠ�ᨨ


㦨
҈0

ᆠ鼀


s=

T ml w
= A A

(

)



´ ҈
䰘 A ml w
£

´ ҈
䰘 Þ w
£ A ml

F A Dl l


F A

lc  = ҈ ls

Ts = Tc



㦨
㝐lw


⏌ p 
㡰
⏌ p
Ӹ= = Dl҈ ⏌
 Dl
⏌  㡰 矄� �

Y㦨


\

Þ

㝐 �鼀盐鼀
 �盐
矄
w
㦨
䁸

Dl 矄 p

泐  = 矄 矄 =
\
Dl 
㝐�
l 
 ᔴ p

泐 

҈



Ts Tc  =



㝐矄矄
ᨨ
 
ᨨ泐

鼀
㦨
҈

├�


᲼Q  =
㝐�桌

ᆠ㡰 矄� �

Y
㦨


Þ

҈�

㡰 ෌�


␔�
 �

l
㦨



鼀

㝐�l 矄

҈҈


Tc
ᆠᨨ矄



꾀
⼄盐���盐鼀

Ts

ᆠᨨ矄



㦨
Tc
ᆠᨨ矄



\






᲼Q 矄 = 㝐 +
㝐㡰�桌


㝐�l 

Tc 


҈




ᨨ鼀

Y
㦨


l l Þ Dl = 꾀Dl 꾀泐








矄矄矄 㡰 Ӹᨨ盐 �Ἐ矄
鼀ᨨ盐盐矄
Ӹ
㦨
矄�

和矄矄
㦨
Ӹ
´
矄�

和矄矄
�
矄
犨�
㦨
꾀�
矄矄盐 矄矄盐
㦨
矄矄矄
㦨
Ӹ
맔
矄�

(

´ ҈
)(҈
) = ҈

Þ
w £ 䰘

Dl \
=
Dl҈



´ ҈ Dl F l = =

-
-
l A Dl ҈
´ ҈


=
´ ҈
҈҈  ⏌ 鼀

T⏌A T ´l T l 㡰 矄� �


Y
㦨

Þ

Dl = = ´ Dl ⏌ l A´Y Y A æ T ö l \ Dl µ
çQ �矄 ᆠᨨ 矄  ÷ è Y ø A

l
�矄
�矄
�
㡰
 ᆠ
�矄
  矄�ᨨ
꾀

Þw




-


鼀
㦨


⏌矄 ҈
´  ҈



b㡰
ᨨ鼀
�

 
鼀 �鼀盐鼀
矄矄矄
 

ᆠ
犨� 矄   ᨨ
├� �
�矄
�
 
Q


和�

㝐 矄�
㡰 㝐 矄�
턼㡰

㝐 �鼀盐鼀 矄矄矄
ᨨ
턼

和矄矄

㝐 矄矄矄 ᨨ



矄�
ᨨ
P
�矄
鼀ᨨ
 
Q
  ᨨ
P
�矄
鼀ᨨ
盐ᆠ�


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EBD_8341 PS-94

Physics

矄矄矄 和ᨨ

��
矄�

矄矄矄
�矄
鼀ᨨ
ᨨ
Q
矄�

\
Ӹ턼
>
Ӹ


Q
Ӹ =

矄� �
⼄

�㡰


��㡰
\


Topic-2 : Bulk & Rigidity Modulus and Work Done in Stretching a Wire ҈






㡰 ᔼ
ᨨᆠ
F
ᆠ矄
ᨨ �
 
 �

ᨨ
 
犨�
ᨨ
ᆠᨨ矄矄㜠 矄ᆠ�ᨨ
A

 
 �泐
矄ᨨ
�
盐 �
ᨨ盐鼀
ᨨ
犨� �矄
��
泐

�









  ᨨ鼀
ᨨᆠ 㦨 㦨 꾀 
p 㡰 꾀 和矄矄
㦨
맔矄�

㡰

和矄矄 㦨

㦨 

F X ö ҈ F F æ = ´ ´
çèQ 矄矄矄 =  
矄�
㦨 ÷ø A AY
A AY ҈ F
҈ F
´ ҈
҈ F
´ ҈
= =
A
Y
pd
㡰
Y
pd
Y ᔼ
u҈
 
u


 
 矄��矄
ᨨ
犨ᨨ
犨�矄

 

dr mg =

 ᆠ�ᨨ 
ᆠ鼀 
�
�盐矄㡰 r Ka


p     㦨  
p 㡰 p 


҈ 쟼 �泐
 矄�泐
㦨

矄矄矄
맔
矄�




=

dr mg = r Ka

d҈ d u҈ æ d
ö ҈
= (
) Þ ҈ =

҈ =ç ÷ Þ d d u
è d҈ ø

㡰 ␔ 

ᆠ盐
�矄
矄ᆠ 
盐
ᨨ
 �
l

 
  矄ᨨ
 �泐
�
�
㦨
D├

쟼
Þ


p 㡰
 


p 
 ᨨᆠ

ᨨ
盐矄
 
ᆠᨨ├
㦨 m 㦨 m
p  
㦨

pm㡰D ᆠ㡰 和 � �
矄�
�矄
ᆠ
ᨨ �
 
矄�
矄盐 ᆠ
ᨨ  
盐 ᆠ 
�矄├

ᨨ
 
 
ᨨᆠ矄
 
ᨨ
 
�矄├ 
 
ᨨ �
 
ᆠ�ᆠ盐鼀  ᆠ
ᨨ
 
�矄├

 
  ᨨ
ᨨᆠ
 
ᨨ
�矄
ᆠ 
ᨨ 泐




mv
L ҈ æ YA ö ҈
ç ÷ DI 㡰
= mv




Þ y =
è Lø
D Dl 㡰




鼀
㦨





├� 
㦨



鼀矄҈ 
㦨





鼀 꾀
㦨
p

㡰⏌
㡰 
㦨


맔
҈

鼀 Dl 㦨




鼀 y=


盐矄
矄矄盐鼀
 
 
矄 � �
矄矄矄
ᨨ �
 
矄� 矄盐 ᆠ
ᨨ
 
�矄├
�矄
盐 � ᨨ鼀

 

=





´


´



´



ᆠ㡰

Þ

ᨨ盐鼀
ᨨ
矄 

V =

æ҈ è


矄盐 ᆠ

ö ø

㦨


黨䰘 ´ ҈

; 
 ´ ҈







�㡰


 pR 

҈ æ YA ö
x
çè L ÷ø ␔ᨨ├
ᨨ 
W
㦨
犨 

K
㦨


dV dr = V r

꾀

-
-

㦨

� ´҈
´ ç ´ ҈
÷ ´

 ´҈
´
p





ᆠ�ᨨ 
ᆠ  �
�
ᨨ盐鼀


矄盐 ᆠ

ò

æö ÷ è
ø

mg æ dV ö aç ÷ èV ø

dV mg = V Ka

s 鼀 꾀 = s 鼀

㦨
ò s 鼀
꾀 = s鼀

p ç

㦨



맔
҈


⏌鼀


ᨨ盐鼀�ᆠ 矄矄矄 盐├ 鼀ᨨ盐盐矄
⧤ = ᨨ盐鼀�ᆠ 矄�

K=

ò

 鼀 =

矄盐 ᆠ

p ´ 
´ ҈
´



和ᨨ

ᨨ
�矄
҈







ò










��㡰

\

YA L

҈
Kx
 

x
㦨
  矄�ᨨ
�
犨�

҈ æ YA ö
÷x
è L ø

W
㦨
ç

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PS-95

Mechanical Properties of Fluids

Mechanical Properties of Fluids

9





Topic-1 : Pressure, Density, Pascal's Law and Archimedes' Principle ҈


= A҈



㡰r
°c g

㡰 ᔼ
⼄盐���盐鼀

mg = Fe FB = V r
g
 
鼀矄矄
㦨
ᨨ盐鼀
맔
 矄�泐




ᆠ鼀



p R  - r  㡰r
g = pRr w g   r
䰘 ෌�

�
 矄�泐


= rw



ᔴ



\



r 黨 ҈ r
r = Þ ҈- =  Þ =   R  R R 
䰘 ҈⏌
r æ
ö r Þ = ç ÷ Þ҈-  = è ø R 
䰘 R

Þ ҈-

r


҈黨 Þ  = ҈=
䰘
䰘 R



�


R


pr
R

 ´ = pR rl  � 



x҈ x + rSx
㡰 g ×




U f = rSx f 㡰 g ×

xf








´


泐
ᨨ盐鼀
ᆠᨨ 矄�ᨨ


x҈

xf xf

Sx҈ + Sx
= S 
x f 㡰 x +x xf = ҈



x









=

ù rSg é x҈
x

- x҈ x
ú
= rSg  x҈ - x
㡰
ê +
êë

úû


\ rl =


r
�

㡰 
P҈

P

 
P

 
矄矄盐

ᨨ� 矄
M

N
 
O 矄ᆠ�泐
M P҈ 㦨

r F҈ N P
㦨�rg F

 㦨
҈� g P
O 矄矄盐
�矄
��
泐
P
㦨
rgh ᨨ犨

P҈
㦨


Q

h
㦨

㡰 P

㦨
rg�㡰 P 
㦨
rg҈�㡰



㦨
҈�
rg

 P҈ + P
㡰 A
 P + P 㡰 ᨨᆠ
ᨨ
ᨨ犨


F
=
A
F �rg � ҈ \ ҈ = = = F


rg




ᨨᆠ
ᨨ
盐


F҈ =

␔ 

�矄
ᨨ 
ᨨ矄矄
�
ᨨ �
 �泐
ᨨᆠᆠ盐
�
犨 
ᆠᨨ鼀
矄鼀


DU = U � - U f éæ x
x
ö ù DU = rSg êç ҈ +
÷ - x
f ú
ø êëè
úû


éx x
æ x҈ + x
ö ù ҈ Sg = r + ê ú ç ÷







è
ø úû êë


ᆠ㡰 ᨨ
鼀� �鼀盐鼀
 矄�泐
ᨨ
�⼄盐�

矄ᨨ�
矄 
矄
ᨨ
ᨨ ᆠᨨ鼀泐
�鼀鼀矄㡰
�
 
�⼄盐�
mg
㦨
FB
矄ᨨ
V�鼀鼀矄
㦨
Vᨨ㡰

é r r� ù
Þ
pr
ê 
ú = pRrl êë  �R úû


㡰 ᔼ ��
ᨨ �
 �泐


� 
ᨨ �
 �泐



ᔴ


 pR r l 
é ù æ r ö êr r 㡰 = r
ç ҈ 
÷
< r £ R �� ú êë úû è R ø
R æ r ö
Þ ò r

p ç ҈ 
÷
r
dr = pR r l  R ø
è

ᆠ㡰 ෌�

�盐矄
ᨨ
�
盐
㦨
҈
ᆠ鼀
犨
ᆠᆠ�ᨨ
ᨨ
盐

a
㦨
黨


ᆠ鼀⏌矄



U҈ = rSx҈ 㡰 g ×

r
°c

= = ҈

҈ r
°c 䰘黨


҈
ᆠ鼀

ᨨ ò rdV =


\ r =


黨 R = R
黨

盐ᨨ泐  r犨
㦨
҈
�
ᆠ鼀
p 
p ᨨ盐鼀
V = r = ´ ҈㡰 =

҈黨 ᆠ鼀 a   F盐ᨨ泐  F盐ᨨ泐  - mg = ma Þ m = mg g 졔a V rw g 㡰 V rw 

҈黨㡰 ´ ҈

҈黨 \m = = = = =

҈�� a 黨

g+a ҈

҈ ҈+ ҈+ g 黨




҈


ᆠ鼀

␔ 
ᆠ泐� 
�矄
ᨨ� �
�
犨



ᔴ 
 盐矄
= Ah
 h҈ 㡰r
°c g = A҈


҈㡰r
°c g








é æ r ö  ù
䰘 Þ ê҈ - ç ÷ ú rw = rw è Rø ú
ëê û 

ᆠ㡰 ␔ 
ᆠ泐� 
�矄
ᨨ� �
�
犨


ᔴ 
 盐矄
= Ah
 h҈ 㡰r
°c g

䰘


b㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

҈
졔


㦨



黨
맔


졔


҈ Þ ҈
졔


㦨




矄矄盐
�
犨ᨨ
鼀矄
ᨨ
盐㜠盐
�矄
⼄盐 \ 

졔
r├�

҈㡰
졔
r犨� ҈


҈㡰
㦨


졔
r犨�
᲼r├
㦨
 矄�泐
ᨨ
├ᨨ矄 


r犨
㦨
 矄�泐
ᨨ
犨桌




҈㡰

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EBD_8341 PS-96

Physics

Þ r├�

҈㡰
졔
r犨� ҈

r犨�
맔


҈㡰
㦨
r犨� Þ



맔
҈

맔


҈

졔
҈



맔
҈

맔
҈

҈



맔
҈

맔


҈
㦨
҈



맔 ҈

맔

Þ ҈




 ҈


㡰
㦨



\ ҈




㦨








㡰 ᨨ鼀
⼄ 矄
҈㡰



㡰 ҈

㦨


�
 


㦨




\

 ᆠ

  矄�ᨨ
ᨨ
 
矄� �
犨 
�
�矄
�
⼄盐���盐鼀
�矄



= ҈



҈

� � = =


 

㡰 
V҈
㦨
ᨨ
鼀�
ᨨ盐鼀
ᨨ
ᆠ泐� �ᆠ
矄  V

㦨
ᨨ
� 矄�
ᨨ盐鼀
ᨨ
ᆠ泐� �ᆠ
矄 
  y
㦨
ᆠ�ᨨ
ᨨ
V

ᨨ盐鼀
�
犨�
犨


ᨨ
犨�� 
㦨
 盐矄
ᔴᨨ ��ᨨ
ᨨ
ᨨ�ᨨ 㡰

҈


æ V҈ + V
ö ÷
҈㡰g è
ø

\

V҈rc g
졔
yV
㡰
҈㡰g
㦨
ç

ᨨ

V y
㦨


�
졔


�

rc㡰 ҈



҈
鼀

h҈

㝐� æ 꾀s ö ç҈ ÷ ├ è
㝐 ø

V


和ᨨ

�
rc




�

 
y
>


� 㡰 盐├
鼀ᨨ盐盐矄


h



㦨






´ ҈㡰 ´ g = M ᆠ盐 g ҈



㡰 ᨨ鼀
꾀ᆠ �鼀矄
� ᆠ�  盐矄
㦨
犨

ᨨ
盐�
�矄ᆠ Fᨨᨨ鼀

Fᨨ
㦨
Vrg Þ Fᨨᨨ鼀
㦨
Fᨨ
졔
Vrg














æ
V ö Mg = ç ÷ rwg è � ø æ M ö
rw
rw ᨨ
ç ÷ =
ᨨ
r = � èV ø � ␔ 
ᨨᆠ├
ᨨ矄
盐泐
�
犨
 
ᨨ�

 

a ᆠᨨ矄a

矄� a 
矄�
a


҈䰘


矄� a

꾀

\
 R ᆠᨨ矄 a + R 矄� a 㡰d
g

=  R ᆠᨨ矄 a - R 矄� a 㡰d҈ g d҈ ᆠᨨ矄 a + 矄� a ҈ +  a = = Þ
d
ᆠᨨ矄 a - 矄� a ҈ -  a

҈



 꾀� = 㝐�


᲼Q
鼀矄矄
㦨
 矄�泐
맔
ᨨ盐鼀桌

 Þ ├
= 㝐� - s 꾀�


Þ
=

㝐� -

s꾀�
= 㝐� æ҈ - 꾀s ö ç ÷ ├ è
㝐 ø ├

㡰 ␔�� 
ᨨ
ᆠ泐� 
㦨 盐 盐矄
盐
ᨨ
盐
�⼄盐� 졔
盐 盐矄
盐
ᨨ
ᨨ犨
�⼄盐�


b㡰 矄矄盐
�
�鼀
ᔼ
 B
㦨
矄矄盐
�
�鼀
ᔼᔼ

A hr҈g
㦨
hr
g Þ

r҈
㦨
r
 ᆠ
矄ᆠ� �ᆠ
��泐
ᨨ
�⼄盐� ᔼᔼ
㦨
sp
��泐
ᨨ
�⼄盐�
ᔼ
㦨
҈
҈ 



㡰

h P҈ ᔼᔼ A

h P
ᔼ B

 æ  ö ç- ÷   è  ø æ Dl ö Þ = - Þ ç ÷ =    è l ø

꾀矄
犨
├ ᨨ犨

盐├
鼀ᨨ盐矄
 =

ᆠ㡰 ᨨ鼀
��盐


├
+  = 㝐�

├
+ s

V

æA ö æ Aö æ  ö æ Aö æ L ö D ç ´ L ´ g ÷ = d ç ÷ ç L ÷ g +
d ç ÷ ç ÷ ´ g è � ø è
ø è� ø è �ø è
ø �d \ D
㦨





d҈

h

㦨
rg
᲼pR

h
졔
V桌

V  æV ö rV 㡰 g = ç ÷ rᨨ� g + rwg ᨨ
rᨨ�
㦨
rw =


rw
è
ø � ᆠ㡰 矄矄盐

�  ᆠ
꾀
鼀盐矄

矄鼀
ᨨ鼀
ᨨ
 
矄�矄
ᨨ 
�
⼄盐���盐鼀
d


P҈

㦨
P҈
맔
A
졔
Vrg 㦨
hrg㡰
맔
pR
㡰

졔
Vrg

҈



a

҈҈


҈�


ᆠ㡰

Mg = Fb҈ + Fb


҈



㡰 ␔ 
 
ᆠᨨ�
矄
ᨨ鼀
 
ᨨ
ᨨ
ᨨᆠ├
� ᨨ
犨
  ᨨᆠ├
鼀ᨨ矄
盐犨矄
ᆠ盐矄
 
犨�� 
ᨨ
ᨨ� �
ᨨ泐 ᆠᨨ鼀矄
矄矄
 
 ᆠ
l
ᆠ矄矄

꾀矄ᨨ

�
犨�
�矄ᆠ

ᨨ盐鼀 ᨨ
犨

⼄盐
ᨨ
�矄
ᨨ犨
ᨨ盐鼀

犨 矄
犨 
�
�矄
ᨨ
  ᨨᆠ├
�
�矄ᆠ矄
鼀ᨨ
ᨨ盐鼀
 
ᨨ
ᨨ犨
ᨨ盐鼀
ᆠ盐矄  矄�泐
ᨨ
ᆠᨨ�
�矄
�
 
 矄�泐
ᨨ
犨

 ᆠ
h
犨�
矄ᨨ ᆠ矄





�㡰


鼀
鼀矄矄
矄 ᨨ盐

ᆠ

 




��㡰 �
㡰
맔
҈㡰
맔
g
㦨
Mᆠ盐
졔
m㡰g 和盐ᆠ� �
⼄盐�ᨨ
�㡰
ᨨ鼀
⼄盐�ᨨ
��㡰

犨
� Þ 鼀�
㦨
�
㡰
맔
�
҈



㡰
㦨
҈
�
맔


䰘
맔
҈

� Þ 鼀
㦨

䰘
�
├� 黨


҈

b㡰 ␔ 

ᨨ泐
ᨨ矄
 
 
犨�� 
ᨨ
 
ᨨ泐
㦨
盐 盐矄

\ �
㡰 ´

҈
҈
� ´ ҈
� - ҈

҈ ´ ҈
� 㡰 D =
㦨
҈
��
맔
҈
�
 D ⏌ 

҈

⧤


㝐�

æ - ö l æ r� ö l Dl = ç ÷ =ç ÷ è  ø è  ø =

҈
 ´ ҈
´ � ´ ҈
 黨

䰘
´ ҈


҈ ´ =



´ ҈
- 

\ Dl 㦨





鼀鼀
;





鼀鼀

WWW.JEEBOOKS.IN

PS-97

Mechanical Properties of Fluids ҈黨


K
㦨


-DP DV ⏌ V

Mg DV Mg \ = A V AK V f - V� 㡰 Mg V� - V f Mg Þ





Þ
= = V� AK V� AK

 pR - p  R - d R 㡰 Mg  Þ  =
 AK pR  dR Mg R - ᲼ R - R
dR桌 Mg = Þ =


Þ
 R  AK AK R 矄
␔ 
 
ᨨᆠ├
ᨨ
�ᆠ
鼀矄

 

矄 ᨨ
犨�
盐�鼀泐
矄� ├
�

 
犨

␔ 

矄 ᨨ
矄� ├矄

�
犨�
�矄ᆠ
犨
⼄盐
ᨨ �矄
ᨨ犨
ᨨ盐鼀

盐
犨 

矄 ᨨ
犨矄
鼀
�
�ᆠ

� �矄ᆠ
鼀ᨨ
ᨨ盐鼀
ᨨ
犨
 
�矄
ᨨ犨
ᨨ盐鼀
ᆠ盐矄 d
>
d犨
DP 㦨


犨 






Q ᨨ盐鼀
㦨

҈











㦨


GMr R P

\

k WFS R r


pRrg
pRrg ᨨ x = Þ
kx =  k
�


ᆠ㡰 
d

 
 矄�泐
ᨨ
 
鼀�
ᨨ
 
矄  ␔�� 
ᨨ
矄  㦨
 盐矄
盐
ᨨ
�
졔
 盐矄
盐
ᨨ
ᨨ� Vdg
㦨


V V d � g + dᨨ� ´ g



Þ d
㦨





d� + dᨨ�


=

҈

+


= 䰘

� ⏌ ᆠ鼀


b
ᆠ㡰 ␔ 
 
ᨨᆠ├
ᨨ
鼀矄矄
m
�矄





A � 矄�
 
�⼄盐�

盐 盐矄
FT
犨�
ᆠ ᨨ
 
鼀矄矄
犨 �ᆠ
犨�
ᆠ矄
  m � �
ᨨ
A
�



犨�

矄矄
 


├�
꾀ᆠᆠᨨ� �
ᨨ
犨ᨨ Ἐ矄
 �
犨

ᨨ
ᆠ F F  
泐
ᆠ�ᨨ

 
�矄
⼄盐
  B ᨨᨨ矄�
ᆠ�ᨨ
和ᨨ
泐
ᆠ�ᨨ

⼄盐
ᨨᆠ
犨�

 
ᨨ
 
�⼄盐�
ᨨ
  ├
犨 �ᆠ
犨�
� ᆠ矄
 
� �
�
B
�



犨�

鼀ᨨ  
�
├�
㡰 
犨 ᨨ
矄泐矄鼀
矄
泐
盐 
��泐

矄ᨨ
g
㦨

꾀ᆠᆠᨨ� �
ᨨ
꾀ᆠ �鼀矄
� ᆠ�  盐矄
㦨
犨�� 
ᨨ
盐�
�矄ᆠ
㦨
鼀矄矄
ᨨ
盐�
�矄ᆠ㡰
맔
g \  盐矄
㦨


ᆠ㡰 ᔴᨨ 矄�
 
⼄盐���盐鼀
ᨨ
犨ᨨᨨ
ᆠ泐� �ᆠ
ᨨᆠ├
ᨨᆠ矄
ᆠ� �
�
 
ᨨ犨 犨
�ᆠ�ᨨ

T


䰘








r

p
㡰
dr

GM r



r

ò rdr

R R


졔



W 㦨
p 
r 㡰 ´ h ´








= p ´
r

GM r é

ù R -r \ P
㦨
û
R ë é
黨R
ù
êR  ú 䰘R ҈
úû P  =  ⏌
㡰 êë
 = = ҈
= \

P  =
⏌ 㡰 é

R ù �R

êR  ú 黨 黨 ûú ëê
é
黨R ù êR  ú
� ûú ҈
P  =  ⏌ �㡰 ëê = =  
P  =
⏌ �㡰 é

R
ù
҈ êR  ú
� ûú ëê






pR
rg 
pRrg  



㡰

ᆠᨨ鼀
矄泐矄鼀
�矄
矄
矄 ᨨ犨
�
��盐

x

 

ᨨ ��ᨨ
ᨨ
 
矄� �
꾀
⼄盐���盐鼀

ᨨ
盐
矄  W
졔 FS
㦨
FB


r

R r




T

P҈ ´ p 
r 㡰

�㡰 ␔�� 
ᨨ
犨ᨨᨨ
ᆠ泐� 

 ò dp =


Þ
kx
㦨


鼀矄矄

 ᆠ


ᨨ
犨
犨�

 矄�泐

矄
␔ 
鼀盐
�矄
� ᆠ矄

 
 矄�泐
ᨨ
鼀ᆠ盐泐
r㡰 ᆠ矄矄
矄
rt
㦨
r

҈

gt㡰
 
 ᆠ

 

ᨨ
鼀ᆠ盐泐
�

ᨨ鼀
盐
 㡰

� ᆠ矄矄 Q hr
㦨
ᆠᨨ 矄  矄




ᨨ
犨
犨�
鼀�
 
矄鼀
�
 
ᨨ 
␔ 
 
鼀
� ├矄
矄ᨨ鼀
犨
ᨨ鼀
 
ᨨ 

�矄
犨��  � ᆠ矄矄
 
  ᨨ
 
ᨨ
犨�
矄� ├
盐 

꾀 
 �矄
矄� ├� � ᨨ
 
ᨨ
犨�
�矄ᆠ
 
矄鼀
ᨨ盐鼀
ᨨ
犨
�
ᨨ 
矄 盐 ├
泐
鼀
b

ᆠ㡰

盐矄
ᆠᨨ 矄�

鼀 
鼀矄矄
鼀
矄 ᨨ犨
�
 
矄  ᨨ�ᨨ
ᨨ
 
��盐




P

p


P
졔
dP㡰

pr


FB



pR rg
졔
kx
㦨
pR
r㡰g  

r ´g 

hr g 

P҈

P


W

P


��㡰 ᨨᆠ
盐
ᨨ
矄矄盐
P҈㡰
ᆠ
泐
�⼄盐�
ᨨ
�� 
h ҈ ᨨ
 
犨ᨨᨨ
ᨨᆠ├ F҈
㦨
P҈
맔
p

r㡰

㦨
᲼P

졔
h҈rg桌
맔
p 
r㡰
���㡰 ᨨᆠ
ᆠ� �
ᨨ
 
盐犨
�ᆠ�ᨨ
盐
ᨨ
矄矄盐
P
 
ᨨ鼀
ᨨ犨
 
犨ᨨᨨ
ᨨᆠ├
 
鼀ᨨ矄 �ᆠ
矄矄盐 F
= P
´ p é
r 㡰
- r
ù + P
´ p r 㡰


ë û



= [ P
+  h҈ + h 㡰rg ] ´ p ´ r
+ P
pr


ᔼ
矄�盐�ᨨ
҈


 
�
ᨨ
�矄� �
 
ᨨᆠ├

F
= F҈
졔
W

[ P
+ h҈ + h㡰rg ] ´ ( p ´ r
) + pr
P
= [ P
+ h҈rg ] ´
pr
+


黨


�h p ´
r
hrg

ᨨ


h҈ =  

b㡰 꾀��
ᆠᨨ 矄�� �
⼄盐���盐鼀
ᨨ
犨ᨨᨨ
ᨨᆠ├


ᨨ
ᨨ犨 犨
ᨨᆠ
㦨
ᨨ
ᨨᆠ
盐犨矄 ␔

ᨨ
ᨨᆠ├
졔
ᨨᆠ
盐
ᨨ
鼀ᨨ矄 �ᆠ
矄矄盐
㦨
ᨨᆠ
盐
ᨨ 矄矄盐
ᨨ
�⼄盐�
졔
ᨨᆠ
盐
ᨨ
鼀ᨨ矄 �ᆠ
矄矄盐

r p ҈
r
㡰 ´ g + P
p ´ ҈
r
㦨
᲼h
rg
졔
P
桌
p
᲼҈



㡰r
桌  졔
P

맔

r


WWW.JEEBOOKS.IN

EBD_8341 PS-98

Physics

犨 
h

㦨
�� 
ᨨ
 
犨

�
矄�盐�ᨨ



ᨨ
犨 �ᆠ
  ᨨᆠ├
鼀� 矄
�
�矄
ᨨ��� 
ᨨ矄��ᨨ


 ᆠ
犨 
矄ᨨ 矄

盐
�
犨

 

ᨨ
犨
矄 




\ h

㦨
h 黨





҈





g + l d 犨ᨨᨨ g = l d 犨 g

㡰 ᔼ
矄�盐�ᨨ



犨 
 
�� 
h

ᨨ
犨

�矄
盐  ᆠ矄

 
 
盐犨
ᨨᆠ
ᆠ� �
ᨨ
 
犨ᨨᨨ
ᨨᆠ├ ᆠ矄矄


ᨨ
ᨨᆠ
ᨨ犨 犨
鼀� 矄
 
矄鼀

�矄 �  ᆠ
犨�

ᆠᨨ鼀 矄
泐
 
ᨨ鼀
ᆠ�ᨨ
泐
   ├
犨
ᨨ
 
犨ᨨᨨ
ᨨᆠ├

 ᆠ
 
ᨨᆠ├
ᨨ矄
ᨨ
鼀ᨨ矄 盐
 
鼀� 矄

�矄
ᨨ��� 
ᨨ矄��ᨨ
㡰 꾀矄
 
泐ᨨ矄�ᆠ
ᨨᆠ
 
泐
�⼄盐�
꾀
ᨨ
 
ᆠ泐� 
ᨨ鼀 
矄�矄
 泐
ᆠ ᆠ矄
ᨨ盐
 
 

盐
�矄
ᨨ
b㡰 ᔼ
⼄盐���盐鼀

盐ᨨ泐 
ᨨᆠ
㦨
犨�� 
ᨨ
 
ᨨ泐 Þ hArAAg
졔
hBrB Ag
㦨
hA 졔 h 졔 hB㡰
A
rC g 犨 
rC
㦨
 矄�泐
ᨨ
ᆠ泐� 㡰

æ hAr A + hB rB ö ÷ - hA + hB 㡰 rC è ø


 
矄�
ᨨ
�
ᨨ
ᆠ盐

l

␔ 
鼀矄矄
m
㦨




�
�矄
ᨨ
  ᆠ盐
ᨨ
犨ᨨᨨ

Þ l d 犨ᨨᨨ = l d 犨 -






�㡰 ␔ 
 
鼀矄矄
m
㦨




�
�矄
鼀ᨨ

l d 犨ᨨᨨ = l -
㡰 l
d 犨



��㡰 ᨨ鼀
⼄

�㡰
 
��㡰

l d 犨 -


= l -
㡰 l
d 犨 Q
d 犨 = ҈ 㡰

l -


= l
l -
㡰
Þ l = ҈
ᆠ鼀

\

h
㦨
ç

�



泐
�矄 ᆠ
x
 
ᆠᨨ盐鼀
ᨨ
鼀ᆠ盐泐

�矄ᆠ


和盐矄�盐� �
盐
ᨨ
hA

hB
rA

rB
 
re
 
矄ᨨ� �
犨
�

h 㦨



�
ᆠ鼀 ᆠ㡰 
ᆠᆠ�ᨨ

a
㦨


A

F盐ᨨ泐  - Mg

p1

M

mg

҈鼀

q



�鼀

├� �
鼀ᨨ鼀 
ᨨ盐
ᨨ� 
O

Þ p

㦨






�㡰

꾀矄ᨨ
FT
㦨
犨

ᨨ
盐�
�矄ᆠ
㦨
᲼l - x 㡰 A桌 ´rw g



��㡰 꾀 
m 㦨
l A㡰

�rw




���㡰 ␔ 
A
�矄
 

ᨨ
ᆠᨨ矄矄
矄ᆠ�ᨨ
ᨨ
 
ᨨ
ᨨ鼀
⼄

�㡰

��㡰
 
���㡰




 Vl
Vs
㦨
ᨨ盐鼀
ᨨ
�⼄盐�
 
矄ᨨ 
矄ᆠ�泐
 
dl
ds 㦨
 矄�泐
ᨨ
�⼄盐�
 
矄ᨨ 
矄ᆠ�泐 ␔ 
 
矄ᨨ 矄
犨
�
 
ᨨ

 
犨�� 
ᨨ
矄ᨨ 矄
犨  ᆠ
泐
 
盐ᨨ泐 
ᨨᆠ



� ´ 䰘
d � ´ g
� - x





��㡰

Þ p҈
졔
҈
d�
맔
g
㦨
p






���㡰

ᨨ鼀
⼄

�㡰

��㡰
 
���㡰


� ´ 䰘
´ d � g


� + x



�

� =
\ q
㦨

� ҈ - x

䰘
䰘

和� ᆠ

ds
>
dl




\


Vs


Vl





�㡰

p҈
맔
A
졔
҈

맔
A
맔
d�g
㦨
p

맔
A

l æ l - xö 矄� q 㦨
᲼l - x㡰 A桌rw g ´ ç 矄� q è
÷ø


Vsds
㦨
Vl dl


� ´ 䰘
d � ´ g
� + x

M
�矄
 
鼀�㜠ᨨ� 
ᨨ
盐
AB. 꾀
⼄盐���盐鼀 p҈맔
A
졔
mg
㦨
p

맔
A



l = ҈ 鼀 \ ҈

x㡰

㦨


�


Þ


x
㦨



黨
鼀 ᨨ鼀
 
��鼀

ᆠᨨ矄 q =

B

ᨨ
�
矄 
�
ᆠᨨ盐鼀
턼 p 맔

�
맔
A
㦨
p

맔

�

x㡰
맔
A

O


l A㡰

�rw g ´

45 cm

B

Þ p҈㦨


鼀�

l æ l - xö 矄� q mg ´ 矄� q = FT ç è
÷ø


Q


꾀

 

ᨨ
ᆠᨨ矄矄㜠矄ᆠ�ᨨ
ᨨ
 
盐

x
�矄 ᆠ
 
ᆠᨨ盐鼀
ᨨ
鼀ᆠ盐泐

�矄ᆠ
�
 
盐
�矄 
�ᆠ泐
꾀泐� �
P҈V҈
㦨
P
V
ᨨ
�
矄 
�
ᆠᨨ盐鼀
꾀 p
맔

�
맔
A
㦨
p҈
맔

�
졔
x㡰
맔
A

x

q

M

Q'

p2

45 cm

P

10 cm 10 cm

ᔴᨨ 矄�
 
⼄盐���盐鼀
ᨨ
 
 ├
� �

ᨨ� 
�졠 ᨨ
⼄盐���盐鼀

F 
㦨



 

t 
㦨



F҈

P'

M

é hAr A + rB h + hB 㡰 - h + hA + hB 㡰 rC ù


㦨
ê ú g rC h + hA + hC 㡰 ë û g


㦨
盐犨矄




x

A

Þ

+ ҈
d � ´ g =


� ´ 䰘

� ´ 䰘
+ ҈
=
� + x
� - x


� ´ 䰘
´ d � ´ g
� - \
㦨


黨�
ᆠ鼀


Topic-2 : Fluid Flow, Reynold's Number and Bernoulli's Principle ҈


㡰 矄� �
 ᨨ盐�Ἐ矄
⼄盐�ᨨ

҈ ҈ P҈ + ru҈
+ rgh҈ = P
+ ru

+ rgh




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PS-99

Mechanical Properties of Fluids





ᨨ
ᨨ�ᨨ 
�

h҈
㦨


 
h

㦨


 
├� �

҈҈
㦨




P P҈
㦨
P

P

㦨


犨
�


p 

� 㦨 p



҈ P ҈ P ҈ ҈ Þ P + ru
= + rV
Þ + ru
= rV






P Þ V = u
+ r b㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ


ᨨ
ᆠᨨ矄矄㜠矄ᆠ�ᨨ

A

aA
㦨


ᆠ鼀

 

B

aB
㦨



ᆠ鼀

ᨨᆠ�泐
ᨨ
�⼄盐�
ᨨ犨

A

㦨
VA
 

B

㦨
VB 矄� �
⼄盐�ᨨ
ᨨ
ᆠᨨ � 盐�泐
aAVA
㦨
aBVB

VA
㦨


VB Þ
VA
㦨
VB ᨨ犨

盐矄� �
 ᨨ盐�矄
⼄盐�ᨨ

҈ ҈ ҈ PA + rVA
= PB + rVB
Þ PA - PB = r VB
- VA





(






� 㦨 


和盐矄�盐� �
 
盐
ᨨ


�
⼄盐�ᨨ
�㡰


� 졔
� 㦨
�
ᨨ

=  é  ù
ê + ú
ë û 䰘


㡰 �

ᆠ鼀




x

�

( DP ) ´
= ( 䰘

) ´
鼀⏌矄 = ҈
䰘 ´ ҈

ᆠ鼀⏌矄

҈�

҈�

ᨨ盐鼀
ᨨ犨

턼
㦨
aB
맔
vB 㦨



맔
҈


맔
VB
㦨

䰘

ᆠ鼀⏌矄
»

䰘


ᆠ鼀⏌矄 㡰 ᨨ鼀
 
⼄盐�ᨨ
ᨨ
ᆠᨨ � 盐�泐 A҈v҈㦨
A
v


v҈
 
v


 
ᨨᆠ��矄

犨ᨨ
 矄
ᨨ
�
꾀҈
 
꾀


 

ᨨ
�

犨ᨨ
 矄





v҈ A
p 


㡰 黨 = = =
҈
v
A҈ p 


㡰 b㡰 矄� �
 ᨨ盐�矄
⼄盐�ᨨ

Þ

\


- ҈
=
�


҈㡰 꾀ᆠᆠᨨ� �
ᨨ
 
⼄盐�ᨨ
ᨨ
ᆠᨨ � 盐泐

҈ P + v҈
- v

㡰 + rgh = P


Þ v

= v҈
+
gh Þ v
= v҈
+
gh

꾀҈҈ = 꾀

꾀҈
鼀鼀
= 꾀
҈
鼀鼀


쟼⼄盐�ᨨ
ᨨ
ᆠᨨ � 盐�泐 꾀҈҈
㦨
꾀



ᨨ鼀
⼄盐�ᨨ

㡰



æ ҈� ö
҈
ᆠ鼀
㡰
҈
鼀⏌矄㡰
㦨
꾀
㡰
ç ҈㡰 +
´҈
´ ҈

÷ è ø ҈


맔
҈
㦨
꾀

맔



ᨨ


=


҈ ҈





éQ � = ҈
鼀 ⏌ 矄
= ҈
 ᆠ鼀 ⏌ 矄
ù ê ú êë  = � ᆠ鼀 úû ҈
和ᨨ� �
犨
�
҈ =




䰘
= p 
 = ( p´
´҈

) ´
�

䰘
´҈

䰘

Þ
� = Þ
� =


p

p

 ᨨ
 


犨 �ᆠ
犨
ᨨ犨矄
 ᨨ盐�
 

䰘
´ 䰘
䰘

´ 䰘

Qp
=҈
) Þ = »


鼀 (

´

´҈

´

´



�





ᨨ
 
ᆠ 
ᨨ
 
ᨨ � �
ᨨ鼀
 

ᨨ 犨
�
 
 ├
�矄
ᆠᨨ矄
ᨨ




鼀 㡰 꾀ᆠᆠᨨ� �
ᨨ
 ᨨ盐�Ἐ矄
� ᆠ�


҈
҈ r ҈ 졔 r � 㦨 r 





҈
졔
� 㦨 


� 졔
� 㦨 



꾀҈ 

= = 꾀
҈ ҈


æ
ö
 ç ҈ ÷ - ( ҈ ) =
´҈
´ � è ҈
ø

b㡰 

ᨨ盐鼀
�ᆠ
ᨨ犨


Þ
� =





㡰

盐� �
 �矄
盐
ᨨ


�
⼄盐�ᨨ
҈㡰

㦨






 �
ᨨ
 
�
ᆠᨨ盐鼀
ᨨ
鼀ᆠ盐泐
�
 
盐
�矄
P
졔
x
㦨
P
Þ P
㦨
䰘


x㡰 Þ

맔
A
맔
䰘

㦨
䰘


x㡰
맔
꾀
맔
�


x㡰 \ x
㦨



盐矄

 �
ᨨ
�
ᆠᨨ盐鼀

㦨
�




㦨
҈

ᆠ鼀
㡰 꾀ᆠᆠᨨ� �
ᨨ
 ᨨ盐�矄
 ᨨ鼀


҈ ҈ ҈ + r҈
= 
+ r





҈
-
\ 꾀

㦨

㦨
�
맔
҈
�
鼀



�


�
x㡰

)








P



ᆠ鼀

æ ҈ V
ö V
Þ DP = ҈


çVB
- B ÷ Þ DP = �

´ B ç

÷ø
è











Þ VB =

҈




盐
㦨
꾀҈҈ = 꾀

= 黨



´ ҈
= 䰘
�ᆠᆠ ⏌ 矄


㡰 ᨨ鼀
 ᨨ盐�Ἐ矄
 ᨨ鼀

ᨨ� 
҈
 

h

҈







�㡰

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EBD_8341 PS-100

Physics

҈ rV҈



㦨


졔


졔


҈ rV




P

졔
rgh
㦨
P҈




�㡰

꾀 
⼄盐�ᨨ
ᨨ
ᆠᨨ � 盐�泐
A҈V҈
㦨
A
V
ᨨ

V

㦨


A҈V҈ A


҈



҈҈


҈



L
p

҈�


҈



\ P

㦨
�


Pa ᆠ㡰 

�矄ᨨ
�矄
盐矄 


矄

v҈
㦨
�
鼀⏌矄 
�
ᆠᨨ鼀矄
ᨨ盐
ᨨ
ᨨ
犨�

矄
v
ᨨ鼀
� ᆠ�
ᨨ
ᆠᨨ � 盐�泐
a҈v҈
㦨
a
v
Þ



pr҈
v҈ = pr

v


Þ

r҈ v҈ = r
v


Þ

(

)
´ � = (҈)
´ v


\

v
㦨





鼀鼀矄҈
㦨


鼀矄҈

㡰 PX - PY =

҈ ra va



X

Y Z



\

҈ ҈ rl vl
= ra va
Þ vl =



\

ᨨ盐鼀
ᨨ犨

µ

ᨨ鼀
⼄盐�ᨨ
ᨨ
ᆠᨨ � 盐�泐


和   =  ´ 
和 - 和 㡰 =  ´ 和 ҈䰘





��㡰

ᨨ鼀
⼄矄

�㡰
 
��㡰


-  =

҈



ra ´ va rl

ra rl

㡰 
ᨨ盐鼀
ᨨ犨

Q㡰
ᨨ

� ᆠᨨ鼀矄矄�
盐�
�

矄泐
ᨨ犨
鼀� 矄
ᆠᨨ 矄  ᨨ鼀
⼄盐�ᨨ
ᨨ
ᆠᨨ � 盐�泐
av
㦨
ᆠᨨ 矄  \

�

鼀鼀

҈


v
A
㦨�ᆠ鼀


A҈㦨҈
ᆠ鼀


盐 PZ
㦨
PX


҈ æ ҈

䰘 ö ҈ 䰘 r ç  - 
 ÷ = r  = r

\ 
㦨
黨 ø
黨
è黨
 
㡰 ᔼ ��泐

矄矄盐
ᨨ
�
ᆠᨨ盐鼀

P҈ = ҈
� N ⏌ m
ᨨ
犨
P҈
㦨
҈
�
鼀 

  V҈ = �

- H 㡰 A ᨨ盐鼀
V҈ = �

- H 㡰 A

犨  A
�矄
 

ᨨ
ᆠᨨ矄矄㜠矄ᆠ�ᨨ
ᨨ
  H 矄矄
� 泐

 
ᨨ盐鼀
ᨨ
�
ᆠᨨ盐鼀
ᨨ 犨
V

㦨
�






㡰
꾀
㦨



꾀

ᔼ
P
�矄
 
矄矄盐
ᨨ
�
 

ᔼ ��泐

҈

v҈㦨҈鼀矄


和 和


 

P


҈ PZ - PY = rl vl






�㡰



Þ=

P҈㦨



Pa

҈ = P
+ ´҈


´




҈ ҈ 
+ r
=  + r


҈ r
- 
 㡰


\ 
�
�� 
ᨨ
犨

盐
ᨨ
 
ᨨ � �
ᨨ
ᨨ� �ᆠ
㦨








㦨


鼀鼀 꾀ᆠᆠᨨ� �
ᨨ
⼄盐�ᨨ
ᨨ
ᆠᨨ � 盐�泐 A҈v҈
㦨
A
v



Þ

҈

맔
҈
㦨
�
맔
v

Þ
v

㦨

鼀⏌矄 ᨨ犨

盐矄� �
 ᨨ盐�Ἐ矄
 ᨨ鼀

҈ Þ



+ ´҈


´҈









ᆠ㡰 ෌�


v҈
㦨
҈


鼀矄҈ A҈
㦨
҈


鼀
v

㦨
ᨨᆠ�泐
ᨨ
犨
矄鼀



҈�
鼀
ᨨ犨
 
 A

㦨
䁸 ᨨ
ᆠᆠ盐� �
v


盐矄� �
v


u

㦨

as u
㦨
҈
鼀⏌矄錐

s
㦨
҈
�
鼀


a
㦨
g
㦨
҈

鼀⏌矄


 
v
㦨
v

㦨
䁸 v


҈
㦨


맔
҈

맔


҈�

Þ

v
㦨
v

㦨


鼀⏌矄 泐
⼄盐�ᨨ
ᨨ
ᆠᨨ � 盐�泐

v҈A҈
㦨
v
A
v A ҈ ´ ҈
-
= � ´ ҈
-� m
\ A

㦨
҈ ҈ = v

黨㡰 꾀泐� �
 ᨨ盐�Ἐ矄
⼄盐�ᨨ


-  =

� 泐

҈ ҈ P҈ + rv҈
= P
+ rv






gy 㡰 ´ L
= 
g ´
y 㡰 pR
R
㦨


P
= 黨

´ ҈

N ⏌ m


\
҈
� ´ �

- H 㡰 A = 黨

´ ҈

㡰 ´ 

A Þ 
㦨




鼀鼀

㡰 꾀矄
犨
├ ᨨ犨

ᨨᆠ�泐
ᨨ
 盐
V
㦨

gh ᨨ鼀
⼄盐�ᨨ
ᨨ
ᆠᨨ � 盐�泐
A҈V҈
㦨
A
V


Þ

\

V
= 

A

PV ҈ ҈ = P
V





��㡰

Q h =  -

�
�
=


䰘� 鼀 㡰

L

㦨

pR



\

P
+ ҈
 ´ ҈
´

꾀矄矄盐鼀� �
 
鼀盐
鼀� 矄 ᆠᨨ 矄 

ᆠᆠᨨ� �

ᨨ
ᨨ泐矄
犨

ᨨ鼀
⼄

�㡰


��㡰
gh
´ ҈
´


䰘� v

㦨

ᨨ

v

㦨
㦨
�

鼀
⏌矄

҈ - 

҈㡰
æ aö ҈- ç ÷ è Aø

҈



P





= ҈
� ҈




\




鼀鼀



졔
rgh
졔


Q
㦨
a
맔
v 㦨
ᆠᨨ 矄 
ᨨ

a
µ


҈ v

犨 
a
㦨

ᨨ
ᆠᨨ矄矄㜠矄ᆠ�ᨨ
 

v
㦨

ᨨᆠ�泐 Þ ᔼ
v
ᆠ矄矄
a
� ᆠ矄矄
 
�ᆠ
㜠
矄
␔ 
矄鼀
ᨨ
犨
鼀ᨨ矄
盐

�矄
矄
v㡰
ᆠ矄矄
    ᨨ
�a
� ᆠ矄矄
�


 
犨
矄矄
ᨨ盐
矄

ᨨ盐 �
␔ 
矄鼀
ᨨ
犨
ᨨ鼀
ᨨ矄
�
鼀ᨨ矄
ᨨ犨

�矄
矄 � ᆠ矄矄
 
  ᨨ

ᨨ
ᆠᨨ矄矄㜠矄ᆠ�ᨨ
ᆠ矄矄
෌�
A҈
㦨
p
맔


맔
҈

鼀㡰


A

㦨
p
맔
҈맔
҈

鼀㡰





v҈
㦨



�
鼀⏌矄

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PS-101

Mechanical Properties of Fluids v҈

A


A҈

A ´L´D´g H H � P
㦨
P

졔
dg + ´
d ´ g +

A �d ù é æ H L ö + ÷ dg

êQ D = Þ P
㦨
P

졔
ç
úû è
ø ë
b㡰 �㡰 꾀泐� �
 ᨨ盐�Ἐ矄
 ᨨ鼀 éH ù æH ö P

졔
ê ´ d ´ g + ç - h÷
d ´ g ú è ø
ë
û

v


h

x ᨨ鼀
⼄盐�ᨨ
ᨨ
ᆠᨨ � 盐�泐


A҈v҈
㦨
A
v
\

v

㦨


H -
h㡰 ҈
g 㦨
P

졔

d 㡰v
Þ v
㦨


��㡰 ᨨ
�ᆠ
鼀ᨨ�ᨨ
ᨨ
�⼄盐�
� �
ᨨ鼀
ᨨ
�鼀
├
ᨨ ᆠ
 
�⼄盐�
ᨨ
 
ᨨᨨ鼀 uy 㦨



Sy
㦨
h

ay 㦨
g

ty
㦨
t

A҈ p ´ 
´ ҈
- 㡰
´


� v҈
㦨

㦨


鼀⏌矄
A
p ´ ҈ ´ ҈
- 㡰

ᨨ鼀 h
㦨



h ҈
gt

Þ t

㦨
g


S
㦨
ut
졔


 ᆠ
ᨨ�ᨨ 
 �


h
´ ҈

� =
´ =
鼀 g ҈


x 㦨
v

맔
t
㦨
v

҈黨


෌�


r
㦨
҈



├�⏌鼀

h҈
㦨

鼀

h

㦨
�
鼀 A҈
㦨


맔
҈
鼀


A

㦨


맔
҈

鼀


v҈
㦨
҈
鼀⏌矄 ᨨ鼀
⼄盐�ᨨ
ᨨ
ᆠᨨ � 盐�泐

H
H 㜠h


A҈v҈ =

� 鼀⏌矄 A҈v҈
㦨
A
v





\
v

㦨
A


v

  ᆠ 
ᨨ

쟼

꾀泐� �
 ᨨ盐�Ἐ矄
 ᨨ鼀


P
 
Q p҈

p
㡰
㦨
rg
h


h҈㡰



x = v × t 㦨
H -
h㡰

dx 㦨

dh dx ҈ = ᲼H


h㡰h桌҈⏌

{H


h晤 dh
҈ H ᲼H


h㡰桌
҈⏌

᲼H


h桌
㦨


Þ h
㦨


  ᆠ

x
犨�

鼀 �鼀盐鼀

h
㦨



҈ ҈  ù

 é҈   꾀  r v
- v҈ 㡰 = ´ ҈
ê - ҈ú = - ´ ҈
⏌鼀


ë
û















㦨



䰘�
맔
҈

⏌鼀 \ ␔ᨨ├
ᨨ 
⏌
ᨨ

泐
矄矄盐 ҈



㦨

黨


맔
҈




䰘�
맔
҈

⏌鼀
㦨

黨


�
맔
҈

⏌鼀
㡰 �㡰 和� ᆠ
 
ᆠ泐� 
�矄
�
⼄盐���盐鼀
�
 
�⼄盐� \ ␔�� 
ᨨ
ᆠ泐� 
㦨
盐 盐矄
盐
ᨨ
盐
�⼄盐�
졔
盐 盐矄 盐
ᨨ
ᨨ犨
�⼄盐�

d L


A⏌� F D


d


鼀 �鼀盐鼀
盐
ᨨ
x

é æ  H ö ù H xm
㦨
êH -
ç è
÷ø úû
ë

H⏌



㦨


H⏌
A

҈
H H
H  H ´ = =





Topic-3 : Viscosity and Terminal Velocity

mg

L


Pᨨ h x

g
h ´
㦨
H -
h 㡰 h g
���㡰 ᨨ
� � �
 
盐
ᨨ
h
ᨨ
犨 �ᆠ
x
�矄
鼀 �鼀盐鼀


҈ r v

- v҈
㡰


ᨨ犨

犨ᨨ├
ᨨ ⏌ᨨ

泐
��泐
ᨨᆠ矄 㦨
rg
h


h҈㡰
㦨
҈

맔
黨


맔

㦨

黨


맔
҈

⏌鼀








h ҈
҈
at Þ h 㦨
gt Þ t 㦨
g

Pᨨ

҈


㡰 矄� �

v


u

㦨

gh

Þ v
-

=
gh Þ v =
gh A A L A L ´ L ´ D ´ g = ´ ´
d ´ g + ´ ´d ´g � �
�



d d �d + = Þ D
㦨



��㡰 ᔴᨨ 矄�� �
�ᆠ
⼄盐���盐鼀
ᨨ
犨ᨨ
�⼄盐�矄
 
  ᆠ泐� 


ᨨ
矄矄盐

 
ᨨᨨ鼀
ᨨ
 
ᆠ泐� 
㦨
꾀鼀ᨨ矄 �ᆠ 矄矄盐
졔
矄矄盐
盐
ᨨ
�⼄盐�
ᨨ
 矄�泐
d
졔
矄矄盐
盐
ᨨ �⼄盐�
ᨨ
 矄�泐

d
졔
矄矄盐
盐
ᨨ
ᆠ泐� 
᲼␔�� ⏌꾀桌

鼀� 
ᨨᆠ�泐


VT =


r
r - s 㡰 g h 黨

꾀 
� �
 ᨨ盐�
h
 
ᨨᆠ�泐
矄 ᨨ盐

⼄盐
ᨨ
鼀�  ᨨᆠ�泐

\
gh =


r
r - s 㡰 g 黨 h

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EBD_8341 PS-102

Physics

Þh=




r
g r - s㡰

҈h


㡰
䰘
맔


��� �

⼄盐�ᨨ
��㡰
泐
�㡰



r
g
r - s㡰

҈ h


Þ
gh =

҈ R
= Þ R҈ =
R

R҈

Þ h a r




 pr
㦨
pr  

ᨨ

r
㦨


鼀� 
ᨨᆠ�泐

v
µ
r


ᨨ盐鼀
V =

R






v҈ r҈
\
v =
r





æ r
ö æ R⏌ö ҈ ᨨ v

㦨
ç ÷ v҈ = ç ÷ v҈
㦨
黨 è R ø è r҈ ø








ᨨ

v҈ v

㦨
黨



= =



䰘 鼀

䰘 ´ ҈
´

҈� ´ ҈
-


r
r

s㡰g 黨h
r҈
s - r҈ 㡰 g VP 黨h҈ r
s - r҈ 㡰 h
=
= ҈
´ VQ
r
s - r
㡰 g r
s - r
㡰 h҈ 黨h


�





᲼
-


桌
´ = 

�㡰 ᲼
- ҈

桌  ҈









r
r
 s҈ 㡰 � 黨h҈

Q

r҈

s

㦨
s҈

r
㡰

r1

҈ s1



r2

�

s2

҈


ᆠ㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

矄矄盐
� 矄�

҈矄
矄ᨨ
盐


DP҈ = P҈ - P
=


҈ =


T R҈

꾀 
DP
= P
- P
=



=




�㡰


T R





��㡰

R=

T 
d  r㡰 g

b㡰 꾀矄
犨
├ ᨨ犨
 

㡰 ␔ 
 
盐
�矄
ᆠ 
盐ᨨ泐 
ᨨᆠ
㦨
ᨨᆠ
盐
ᨨ
矄盐 ᆠ
 矄�ᨨ

q

꾀��
VP


VQ





�.e.

��
矄
VP
 
VQ

ᨨᨨ矄�
ᨨ
ᆠ ᨨ 


Topic-4 : Surface Tension, Surface Energy and Capillarity

R

d  r㡰 g
Þ 


ᆠᨨ矄 q = r�

� = 䰘
�

␔ r� ᆠᨨ矄 q� ᆠᨨ矄҈�° ҈ = ҈


= = ᆠᨨ矄 q␔ ᆠᨨ矄
° r␔
æ ö q �

ᆠᨨ矄 æ � ö r ␔ æ � ö =ç ÷ç ÷ç ÷ ␔ çè ␔ ÷ø çè r� ÷ø çè ᆠᨨ矄 q ␔ ÷ø ҈ ҈
= 䰘
� ´ ´ =


= ҈

�



r
s
 r҈ 㡰 � 黨h


\

㡰 ᨨ
 
ᨨ矄
ᨨ

�
⼄盐���盐鼀
盐犨
ᨨᆠ
ᨨ
ᨨ
㦨 ᨨ犨 犨
ᨨᆠ
ᨨ
ᨨ

Þ T=




h VP = ҈ h
VQ







pR dg  pR rg  
 Þ T
pR㡰
= pR 
d  r㡰 g 

␔ 

�矄
�盐矄
ᨨ
矄 
ᨨ犨 犨
鼀� 
ᨨᆠ�泐

VQ =




T

pR =



㡰
和� ᆠ
矄� �
�矄
盐
r҈


s҈
 

r



s
ᨨ
ᨨ� �


犨�� 
ᨨ
矄泐矄鼀
㦨

盐 盐矄 r҈
졔
r
㡰
Vg
㦨
s҈
졔
s
㡰Vg 犨
鼀� 
ᨨᆠ�泐

VP =


T ᆠᨨ矄 q rgr


´


� ´

-҈

VT
㦨









=

V҈ R҈
R

= =  = ҈ V
R
 R


\

b㡰 ෌�
꾀 �
ᨨ
ᆠᨨ ᆠ
q
㦨

和盐 ᆠ
 矄�ᨨ

T
㦨



�
鼀҈ �盐矄
ᨨ
ᆠ�泐
盐

r
㦨


҈�
鼀鼀
㦨


҈�
맔
҈
鼀  矄�泐
ᨨ
鼀 泐 
�ᨨ�

r
㦨


䰘
├�
鼀 ᔴ�泐
�矄

h =

b㡰 꾀ᆠᆠᨨ� �
ᨨ
ᨨ�ᆠ�矄
 ᨨ鼀
ᨨᆠ�泐
ᨨ
 
V

㦨

gh =
´ 黨

´ � @ 黨

鼀矄 㡰 꾀矄
犨
├ ᨨ犨

鼀� 
ᨨᆠ�泐


 pR 

rq

T맔dl

ᨨᆠ
盐
ᨨ
 ᆠ矄矄
矄矄盐
㦨

盐 盐矄 꾀ᆠᆠ矄矄
矄矄盐
�
�
盐
㦨








T
p R  rw g p r
㡰 = R T


Þ
r =


R
rw g

rw g
Þ
r = R T T

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PS-103

Mechanical Properties of Fluids



㡰 꾀ᆠᆠ�ᨨ
盐
ᨨ
��泐
ᆠ  �矄
犨�
 



꾀ᆠᆠᨨ� �
ᨨ
�
�矄
⼄盐�ᨨ

PV
㦨
nRT

ö æ �¢ = � ç҈ - ÷ è ø

PAVA
㦨
nARTB Þ


p



㡰
㦨
nARTA


�㡰 
 PBVB = nB RTB Þ

҈
´ p 



㡰 = n B RTB …
��㡰 

矄矄盐


㦨
r�  ᆠ
�ᨨ

䰘


æ ö �矄 ç҈ - ÷ 泐 è ø

㡰

C

��� �
⼄

��㡰
泐
�㡰

R

a


q
졔
a b


M


҈
´ p



㡰 n 
= B
nA ҈
´ p



㡰  nB \ =
nA

q

P

҈




S ö æ 꾀矄ᨨ
ç P
÷ + hrg = P
R ø è





Þ

hrg
㦨



S R

T= ҈҈



S
S ᆠᨨ矄 ( q + a
) = Rrg brg

҈ pR = k ´ pr  \


R = K r 
㡰  

é R

ù
҈⏌ \
D
㦨

ps ê k ´
⏌ - R ú =
pSR éëk - ҈ùû ë k û



҈ ´ ҈
-
㡰
éë҈
a ⏌ - ҈ùû
p

\
҈

= ҈
a ⏌ - ҈
�ᆠ� �

a =



\
a =

T 
㡰 矄矄盐

PA
㦨
P�
졔
RA Q 쟼 ᆠ矄矄
矄矄盐
盐
ᨨ
矄盐 ᆠ
 矄�ᨨ
T㡰
�
矄ᨨ
盐

҈


㦨
҈
a⏌




Þ



҈





ᆠ鼀



ᆠ鼀 



T
´




= +

Þ PA = ҈
N ⏌ m RA




T
´



=
+ 和�鼀�泐

PB
㦨


졔

㦨
҈

⏌鼀




RB

\
㦨
҈

鼀鼀
㦨

䰘
҈ 

b

ᆠ㡰 ᨨ
ᆠ矄
ᔼ


=

�
㦨


鼀

PA =
+

⏌



ᆠᨨ矄 q҈
´

䰘� ´ ᆠᨨ矄
° = = 䰘
�ᆠ鼀 r �
´ ҈
-
´ ҈


´ ҈


ᨨ
ᆠ矄
ᔼᔼ

B ҈ v

犨ᨨ
鼀҈ 
䰘
㡰꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

 
犨
ᨨ
 
�鼀
矄

�矄ᆠ
ᨨ  �ᆠ├ 矄矄
Ἐ Ἐ
� �
矄鼀�ᆠ�ᆠ盐
�矄

㦨
⏌
矄矄盐

 
ᨨᨨ鼀
ᨨ
�矄ᆠ
㦨
矄矄盐
盐 ᨨ
矄盐 ᆠ
 矄�ᨨ

҈ =

㦨

T⏌r㡰

A

rhrg ҈� ´ ҈
-� ´ ҈� ´ ҈
-
´ 黨

´ ҈
= = ҈
҈ 鼀��



é ҈ ҈ ù ҈ ù


é \r� = ê + = ú = ê
+ ⏌
úû ë ë ҈
û







´


䰘 ҈
´҈

Þ
= Þ = = =  r� r� ҈
´ ҈
҈

´҈




a ⏌ - ҈ùû ᲼Q K
㦨
҈
a桌 \
D
㦨

pSR éë҈


黨


᲼ᆠᨨ矄 q = ᆠᨨ矄
° = ҈桌

æ ҈ ҈ ö r� = ç + ÷ è ҈
ø ҈ ҈ >


\ ҈

 


㦨
⏌





DU = S᲼k ´
pr -
pR 桌

\
҈

㦨

p ´


T ᆠᨨ矄 q rgr

和盐 ᆠ
 矄�ᨨ


h=




[Q TA = TB ]

҈
҈㡰 ෌�


�盐矄
ᨨ
ᆠ�泐
盐
r
㦨



҈�
ᆠ鼀
㦨
҈�
맔
҈
�
鼀鼀 h
㦨
҈�
ᆠ鼀
㦨
҈�
맔
҈


鼀鼀 矄� �

h =


R

 
�盐矄
ᨨ
 
鼀 �矄ᆠ盐矄
ᨨ鼀
犨�

ᆠᨨ 矄   �
q
 
P


 
鼀ᨨ矄 �ᆠ
矄矄盐
aö b æ b ᔼ
Dᔴ㝐
ᆠᨨ矄 ç q + ÷ = Þ R
㦨
aö
ø R æ è çq+ ÷
ø è

\

҈

맔


҈



ᆠᨨ矄 q

´

䰘� ´ ᆠᨨ矄

° = = 
䰘�ᆠ鼀  r�
´ ҈
-
´ ҈


´ ҈



ᆠᨨᆠ�ᨨ
�
 
�� 
ᨨ
犨
ᆠᨨ盐鼀
�矄
�
 
盐
盐
ᨨ
犨�� 
ᨨ
犨
ᆠᨨ � 
�
 
鼀 �矄ᆠ盐矄
犨�

�   ᨨ
ᨨ
ᆠ矄矄
ᔼ
ᆠ矄
ᔼᔼ

�
 
ᆠ�泐
ᨨ� 
�矄
�
ᆠ鼀
ᨨ
 
犨
矄盐 ᆠ  
犨
�
ᆠ�泐
犨�
ᨨ
ᆠ
 
�  ᆠ

␔
犨� ᆠ
ᨨ 泐
�

䰘�
ᆠ鼀

s ᆠᨨ矄 q 


ᆠ㡰 꾀矄
犨
├ ᨨ犨
h = rrgeff ҈ 꾀矄
�r
� ᆠ矄矄

h
ᆠ矄矄
h
µ
r ᲼
ᨨ 
鼀矄
鼀� � �
ᆠᨨ 矄 桌

WWW.JEEBOOKS.IN

EBD_8341 PS-104

҈



Physics

꾀矄ᨨ
h
µ
s 盐 
�
� 
�矄
�ᨨ� �
盐
犨�

ᆠᆠ�ᨨ
�a  
g
㦨
g
졔
a

꾀矄
g
� ᆠ矄矄

�h
ᆠ矄矄
꾀矄ᨨ
h
µ ᆠᨨ矄
q
ᨨ
h
µ
q ᆠ㡰 �ᆠ
ᨨᆠ
盐
ᨨ
矄盐 ᆠ  矄�ᨨ

q FV 㦨
T ᆠᨨ矄
q㡰
맔

pr r

҈黨


\

q


pr
T ærö




= T ç ÷ ´
pr = è Rø R

Þ ҈



Þ Þ



ᨨ
⼄盐���盐鼀
ᨨ
犨�
�
�ᆠ
�ᆠ�ᨨ

ᨨ鼀

ᨨ泐 ��鼀
ᨨ
 
犨�
矄 ᨨ犨
�
 
��盐


T ᆠᨨ矄
q \
Tl
ᆠᨨ矄
q
㦨
llg ᨨ
T
ᆠᨨ矄
q
㦨
lg

y a

\

T

q q

T
矄�
q

lag
T ´ y
㦨
lg ᨨ T
㦨
a
y

㡰 \

ᨨ
R
㦨
҈



맔
҈

鼀 b㡰 和盐 ᆠ
 �泐
㦨

맔

p


犨 �
y



a

҈



҈




pr
T pR rg =  R
æ  r T ö  � ´҈
-
㡰
´

҈҈ = ´ R
= ç 㦨


҈

맔
҈
҈
ç
rg ÷÷
҈


´ ҈
è ø 

ᆠᨨ矄
q
㦨



T rv


Topic-5 : Miscellaneous Problems



´ ҈


´ ҈
- 㡰
=

䰘 ´ ҈
-
 㦨


҈҈ ´
´ 䰘 ҈䰘



T
T ´ A



Þ
R
㦨

R rv

R

㡰 ␔ 
 
ᨨ
�矄
ᨨ盐
ᨨ
ᆠ
ᨨ鼀
 
ᨨ ␔�� 
㦨
�ᆠ
ᨨᆠ
盐
ᨨ
矄盐 ᆠ
 矄�ᨨ
mg
㦨
FV㡰 \

rAv

㦨


�盐矄

犨 �ᆠ
盐矄
矄
ᨨ鼀
 
� �

㦨


rù é êQ ᆠᨨ矄 q = R ú ë û ҈�



盐
犨�
矄
ᨨ鼀
 
� �

犨 
 
ᨨᆠ
盐
ᨨ
� 㦨
rAV

�
 
盐
⼄盐矄
 
矄盐 ᆠ
 矄�ᨨ
ᨨᆠ

T⏌R � 矄�
 
盐


҈ dr = ᆠᨨ 矄  r dt
pR  d é m ù
㦨
ᆠᨨ 矄  m dt ê
 ú ê pR ú ë û

෌�




d -  R 㡰
㦨
ᆠᨨ 矄  dt dR dR µR R   -  R -
㡰
㦨
ᆠᨨ 矄  \


dt dt R

b㡰 ␔
├ ᨨ犨
 
 ᆠ矄矄
矄矄盐
�

矄ᨨ
盐
�矄
� 矄泐 ᨨᨨ�ᨨ 
ᨨ
�矄
�盐矄

�.e.

P
㦨


T

T
矄�
q w
㦨
lg




␔ 
 
盐
�矄
ᨨ
 

盐矄� �
 ᨨ盐�Ἐ矄
 ᨨ鼀

҈ ҈ P + P
+ rv҈
+ rgH = rv

+ P


҈

Þ P
졔
rgH
㦨
rv
- v҈ 㡰



T r


矄ᨨ
盐

 
҈
矄 矄鼀
�盐矄
矄
ᆠᨨ鼀
ᨨ
  矄ᨨ
盐

 


�� 㡰


 ᨨ
 ᆠ矄矄
矄矄盐

҈
҈ �矄
鼀ᨨ
 ᆠ

�
ᨨ犨矄
ᨨ鼀
 
҈
ᨨ
 


 
 
ᨨ盐鼀
ᨨ
矄ᨨ 盐

 
҈
ᆠ矄矄
ᆠ㡰 ᔼ
ᨨᆠ
ᆠ矄
ᨨ
犨
ᨨ犨矄
�� 

犨 

矄矄
ᆠᨨ � � � 犨
�矄
��

ᆠᨨ 矄 
ᆠᆠ�ᨨ
a
ᨨ犨矄
 
�� 

꾀矄


ᆠ�ᨨ
 
犨
 矄

ᨨᆠ
ᨨ
 
犨
ᨨ
矄矄
ᨨ犨矄 �� 

꾀矄

ᆠ�ᨨ
 
犨
盐矄 矄
 
犨
ᨨ犨矄
 
꾀ᆠᆠᨨ� �
ᨨ
犨ᨨ Ἐ矄

犨
ᨨ
鼀ᨨ�ᨨ


矄ᨨ
ᨨ
犨
�矄 矄
矄 ᨨ犨
�
��盐


ᨨ鼀
⼄盐�ᨨ
ᨨ
ᆠᨨ � 盐�泐
A҈v҈
㦨
A
v
v҈
㦨


A
v
A҈

P

᲼ 
v

㦨
v
桌


҈ éê
æ A
ö ùú v v r \ P
졔
rgH
㦨

ç
÷
ê è A҈ ø úû ë
҈
é æ A
ö ù ê v ҈ r ç ÷ ú
P
졔
rgH
㦨


ê è A҈ ø ú ë û




P
졔
rgH
㦨
DP 꾀ᆠᆠᨨ� �
ᨨ
ᨨ�矄矄盐�Ἐ矄
⼄盐�ᨨ

p DP 㡰a p DP㡰a
Q
=

Þ

h
㦨

Ql
hl




é æ A ö
ù ҈ p  P + rgH 㡰 a
p = ´ rv

ê҈ - ç
÷ ú ´ a

Ql
Ql
ê è A҈ ø ú ë û p ҈
é d
ù
é A
d
ù h= ´ rv
ê҈ -
ú ´ a êQ = ú

Ql
ëê D ûú ëê A҈ D ûú

\


h
㦨


黨


a

ma

矄盐 





mg

P
�矄

�ᆠ
ᨨ
犨



矄盐 ᆠ
ᨨ
犨
矄 ᨨ盐
矄
�
矄

黨

ᨨ
F矄盐 

㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

��ᨨ
ᨨ
 
 矄�泐
ᨨ
�
犨� �� 

ᨨ鼀
 
�ᨨ盐 


r 㡰 = r
-




꾀

㦨
҈


鼀

鼀矄矄
鼀
㦨




├� \ 鼀�
㦨
�r҈

㡰 Þ




맔
�
㦨
r҈� 꾀

㦨
҈�

鼀

鼀矄矄
鼀
㦨





㡰├� \ 鼀�
㦨
�r҈�
㡰 Þ





㡰�
㦨
r
�

r�

鼀�

WWW.JEEBOOKS.IN

PS-105

Mechanical Properties of Fluids

 ö -
⏌



-  r
æ Þ ç҈ = = ÷= r҈


è


ø - ҈ ⏌
 �
�
´


Þ ҈=҈ \ = =











�


҈ -



-

=

�






\

v'

㦨


V'd'mg

Vdmg

ᔼ
矄ᆠᨨ 
ᆠ矄

犨 
鼀盐
�矄
� ᆠ矄
泐
DT
v'd'Hgg = V'd'm g

黨


dἘ vἘ = m 㦨
ᆠ�ᨨ
ᨨ
ᨨ盐鼀
ᨨ
鼀
矄盐鼀�
�

Þ V Ἐ d ἘHg 鼀ᆠ盐泐
㦨 x'
矄泐㡰 \

x Ἐ d Ἐm ´ d Hg d Ἐm ´ d Ἐ Hg ҈ + g
DT 㡰 ҈ + g
DT 㡰 = = = x d Ἐ Hg ´ dm d Ἐ Hg ´ d Ἐm ҈ + g҈DT 㡰 ҈ + g҈DT 㡰
DT㡰҈

㦨
҈
졔
g

DT㡰
҈
졔
g҈ 㦨
҈
졔
g

DT㡰
҈

g҈
DT㡰
㦨
҈
졔

g

 g҈㡰
DT 
g҈g

DT㡰


xἘ - ҈ = ( g
- g҈ ) DT


Þ x

㡰 和� ᆠ
 矄�泐
ᨨ

d㡰


 矄�泐
ᨨ
�⼄盐�
dL㡰 盐 盐矄
>
犨�� 
\ 

矄
犨�
�ᨨ
� 矄�
�⼄盐�
\

�ᨨ
㦨


ᨨ

a
㦨


Vd L g - Vdg Vd

ᨨ

t
㦨


v gt҈ = a
a

t
㦨


gt҈ dt҈ =
d L - d 㡰 æd -d ö
ç L ÷g d è ø

v =
gh


ᨨ

t

㦨
t҈
졔


ᨨ



㦨


d d 㦨

h h 1 2




��㡰


´ dt҈
d L - d 㡰

é

ᨨ t

㦨
t҈ ê҈ +

ë

ù d ú d L - d 㡰 û

t҈dL dL - d

æd -d ö 㡰 和� ᆠ
 
�ᨨ
㦨
ç L ÷ g
�矄
ᨨ
ᨨᨨ㜠�ᨨ  è

d

ø

ᨨ
�矄ᆠ鼀 

 
鼀ᨨ�ᨨ
ᨨ
 

�矄
ᨨ
矄�鼀
鼀ᨨ �ᆠ
ᆠ㡰 ᔼ
d
㦨
dL
 
 
�ᨨ
= �ᨨ
㦨



和� ᆠ
  
矄�├矄
 
犨
矄盐 ᆠ
犨�
矄ᨨ鼀
ᨨᆠ�泐

�
犨�
ᆠᨨ � 盐 犨�
 
矄鼀
ᨨᆠ�泐
v
㦨
gt҈⏌

�
ᨨ犨 犨
�ᆠ�ᨨ


dx

x A




�㡰

\ 
�鼀
ᨨ
�矄
ᨨ
�⼄盐�
矄盐 ᆠ
㦨
t
㡰 Q t

㦨
�鼀
├
泐
 

ᨨ
ᆠᨨ鼀
ᆠ├
ᨨ
 
ᨨ矄��ᨨ

ᨨ鼀
犨 �ᆠ
�
犨矄
矄
ᨨ t

㦨
t҈
졔

t
矄盐矄�盐
t
ᨨ鼀
��㡰

ᔼ

 
 
ᆠ矄矄
d =
h҈h
= ҈

鼀 ᔼ
geff
㦨



 
ᨨ
犨
├矄
ᨨ盐
矄
 
犨�

ᨨ
矄矄盐 �  ᆠ
d w 
盐矄
ᆠᨨ 矄�

矄鼀
鼀  鼀矄矄
dm


�矄 ᆠ
x
ᨨ鼀
A
矄 矄 ᨨ犨
�
 
��盐

ᔴ �
ᨨᆠ ⼄盐�
ᨨ
 
鼀矄矄
dm
ᨨ
ᨨ
㦨 H dm㡰
xw
\ ᨨ
ᆠ �
ᨨᆠ ⼄盐�
ᨨ
 
鼀矄矄
ᨨ
 �
L
ᨨ ᨨ

æ dL - d ö ÷g è d ø

ᨨ a
㦨
ç

᲼Q ᨨᆠ�泐
ᨨ
 

盐矄
 ᨨ
�
ᆠᨨ�矄 犨�
�⼄盐�

V
㦨
gt҈⏌
桌


h҈ g

㦨

h҈h
geff

> g geff
㦨
g geff


g

盐 盐矄 - 犨��  鼀矄矄


 

ᆠᨨ鼀
ᨨ
矄
� 矄�
�⼄盐�
�
�鼀
t
\

ᆠ㡰 ᨨ�ᨨ 
�矄 ᆠ
d
㦨
v
맔
t

ᔼ





xἘ- x = ( g
- g҈ ) DT x

u
æ dv ö ÷
ᨨ

 
㦨
h꾀
h dx è ø u
æ dv ö 犨 

㦨
ᨨᆠ�泐
�� 
ç ÷ h è dx ø ҈
µ
h錐
µ
꾀錐

µ
m
 

µ
h


gh
´

( )

pd
´ H ´ r´ g




ᆠ

㡰

㦨

h꾀 ç

㦨


ö pd
´ dx÷ ´ xw

ø

w
L
pd
pd
w
´ L
\ ´ H ´r´ g = r´ ´
ᨨ

H
㦨

g




T +D T

T

ò
çè r ´

�矄
ᆠ �
ᨨᆠ
�矄
ᨨ�
泐
 
犨�� 
ᨨ
�⼄盐�
ᨨ �� 
H ␔�� 
ᨨ
�⼄盐�
ᨨ
�� 
H

V' v

Læ

ᨨ
ᆠ �
ᨨᆠ
㦨


pd
´ dx


L pd
´ w
ò x dx


L
pd ´ w
´ 㦨
r ´



v'd'Hgg

V

ò


 dm㡰 xw

犨 

dm
㦨
r ´

㦨
r ´

d v = m = x 矄泐㡰 V d Hg vdHgg

䰘


㦨



ᆠᨨ ��ᨨ
ᨨ
ᨨ�ᨨ

vdHgg
㦨
Vdmg ᆠ�ᨨ
ᨨ
ᨨ盐鼀
ᨨ
鼀
矄盐鼀�
�
鼀ᆠ盐泐 㦨






L

L

dm

B

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EBD_8341 PS-106

Physics

Thermal Properties of Matter

10

Topic-1 : Thermometer & Thermal Expansion ҈


㡰
Ӹᨨ盐 �矄
鼀ᨨ盐盐矄
Ӹ
㦨






Dl Dl Þ = aD \Ӹ = 꾀 ( aD ) lD

l ᆠ㡰 꾀矄
犨
├ ᨨ犨

盐├
鼀ᨨ盐盐矄

㡰 
L'҈
 
L'


 
 � 矄
ᨨ
 
犨�
犨 

鼀盐
�矄
ᆠ  �
泐
DT °ᔴ
꾀
T ᔴ


Leq = L҈ + L


⏌ 꾀 矄矄矄
㦨
꾀 Dl ⏌ l) ( 矄�

矄� �

ᆠᨨ �ᆠ� 
ᨨ
� 
  矄�ᨨ


�


a=

꾀
T + D°ᔴ L' = L' + L' eq

҈




\
Leq ҈ + a ⼄ DT 㡰 = L҈ ҈ + a҈DT 㡰 + L
҈ + a
DT 㡰 Þ  L҈ + L
㡰҈ + a ⼄ DT 㡰 = L҈ + L
+ L҈a҈DT + L
a
DT

a҈ L҈ + a
L
L҈ + L
㡰 ᔴ  �
�
 �
ᨨ
 
鼀
犨�
Dl㡰
犨 
�矄 鼀盐
�矄
ᆠ  �
泐
DT
�矄
��
泐 Dl = l aDT 

a
㦨
ᔴᨨ �ᆠ� 
ᨨ
� 
  矄�ᨨ



Dl
㦨




奠

DT
㦨
҈
ºᔴ



Dl -� \a = =
Þ a =
´ ҈
l DT ҈

´ ҈
ᨨ盐鼀
ᆠᨨ �ᆠ� 
ᨨ
  矄�ᨨ

g = a =
´ ҈
-�

䰘


M Qr = V DV -� -
´ ҈


㦨
gDT
= 
´ ҈
´ ҈
´ ҈

㡰 =
´ ҈
V ᨨ盐鼀
� ᆠ矄
泐




奠
  ᨨ
 矄�泐
ᆠ矄
泐



奠
ᨨ 盐矄㡰
D鼀
㦨
Dᨨ
 
A
㦨
pr

㦨
p҈
㡰

㦨
p
맔
҈

FL L a
DT
㦨
AY F ´


ᨨ



맔
҈
�
맔



㦨
 p ´ ҈
-
㡰 ´ ҈
҈҈ f \ F
㦨


p
N
\ m = 㦨

p
㦨





├� g 㡰 ᔴ  �
�
 �
�
ᨨ
ᨨ矄

矄鼀
�

Dl҈ = D l






Þ a ⼄ =








la҈D q҈ = la
Dq
a҈ Dq
= a
Dq҈
q  
=  ҈

 
q
㦨


ᔴ

é a҈
ù êQ = ú ë a
 û

Þ

DV P = V K g= 3a

V
㦨
V

҈
졔
gDt㡰

[Q L Ἐ = L ҈ + a DT 㡰]





DP æ -DV ö çè ÷ V ø

K=

DV P P P = gDt \ = gDt Þ Dt = = V
K gK aK ᆠ㡰
盐
ᨨ
 鼀
 


ᆠ  �
�
 �
Dl㡰
㦨
l
a
D




�㡰 ᨨ鼀
矄矄矄 Ӹᨨ盐 �矄
鼀ᨨ盐盐矄
Ӹ㡰
= ᨨ ��盐� 
矄�

꾀 Dl Ӹ= Þ = Dl l l 꾀Ӹ

D =

l 꾀Ӹ

ᨨ鼀
⼄
�㡰



҈
ᔴ

l = l a D

꾀Ӹ


㦨
꾀Ӹ
a D

㡰 꾀
矄泐
矄


ᨨ犨

�矄
矄鼀
ᨨ鼀
P
ᨨ
Q  
ᨨ鼀
Q
ᨨ
S

㝐
⧤





턼

⧤ 

和





ᔴ


⧤꾀 (


- )
⧤꾀 ( - ҈
) = Þ

㦨
҈

ᔴ l l


鼀盐
��  \



 ҈

- ҈
= \

㦨
҈

  ҈

 ᨨ
ᆠ  �
鼀盐


ᆠᨨ矄矄㜠矄ᆠ�ᨨ
㝐
�矄  � 
ᨨ鼀

�矄 D
㦨
҈

쟼  矄�ᨨ
�

矄鼀
鼀 
 �
� 
�矄 l
㦨
 a D
㦨
 a (҈
) ҈

\
ò dl = ҈
a ò 


҈ \
Dl
㦨
҈

맔
҈


맔
҈
�
맔

㦨
䰘

맔
҈


�
鼀


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PS-107

Thermal Properties of Matter 黨


� �
ᨨ
 泐
矄ᆠ

    -   㦨
ᆠᨨ 矄 
ᨨ

矄ᆠ矄

ᆠ㡰

҈



Þ

mg ⏌ A mg l = Dl ⏌ l ADl 꾀矄ᨨ
Dl = l a DT ᨨ鼀
�㡰
 
��㡰

҈҈


l
+ l҈ a a + a s l҈ as = =
Þ as l҈ l҈ + l
a a + a s 㡰 ᨨ
⼄盐��b�盐鼀
犨h
鼀盐

°ᔴ
��
㡰  盐矄
㦨
␔

ᨨ
ᨨ泐 \ K҈Vd
g
㦨
Vd҈g Þ K҈
㦨


d҈ d






�㡰 K҈Vd
g

d҈

K
Vd ' '
g d'҈

V

Vd g

d


V'










҈

㡰

㡰

ᨨ
⼄盐��b�盐鼀
犨h
鼀盐
� ᆠ矄矄
ᨨ


°
��

b㡰 ␔ 
 
鼀盐
�矄
� ᆠ矄
 
 矄�泐
犨� ᆠ矄
\ d҈Ἐ
㦨
d҈
҈
졔
gFe
맔


㡰   d
Ἐ
㦨
d

҈
졔
gHg
맔


㡰 꾀��
盐 盐矄
㦨
␔

ᨨ
ᨨ泐 \ K
VἘd
Ἐg
㦨
VἘd҈Ἐg \

҈



é ù d
d҈ ú= K

ê êë҈ + g Hg ´

úû ҈ + g Fe ´



é ҈ + g Fe ´

ù ú= \ K

ê êë҈ + g Hg ´

úû ᆠ㡰 W҈
㦨
mg

Vda g

W

㦨
mg 
V'd'a g
㦨
mg

V
҈
졔
�

gb㡰



é ҈ + �
g b ù 㦨
mg


V dag
ê ú ë҈ + �
g a û ෌�
gb


ga \ ҈


҈
졔
�

gb


҈
졔
�

ga





ᨨ


\
m =

da g ҈ + �
g a 㡰

҈ + �
g b
W҈
ᨨ
W҈


W
㡰 꾀矄
ᆠᨨ 矄 
ᨨ盐鼀
�矄
 鼀ᨨ鼀

犨ᨨ├矄
ᨨ

ᔴ Ἐ矄
犨





��㡰

YA a DT ҈
҈҈ ´ p҈
- 㡰
´ ҈
-� ´҈
= =p» g ҈







㡰 ᨨ盐鼀
ᆠᆠ�泐
ᨨ
├

V

㦨
�


ᆠᆠ

Vb = V
+ V
g ├ DT ␔ 
├
�矄
�泐
�
犨�
Vm
ᨨ盐鼀
ᨨ
鼀ᆠ盐泐


Vb҈ = Vm + Vm g m DT  �
ᨨ盐鼀
V
- Vm 㡰 = Vb - Vm҈ 㡰

Þ V
g ├ = Vm g M ᨨ

Vm =

\Vm =

V
g ├ gM

�

´
´ ҈
-


=


ᆠᆠ
҈
� ´ ҈
-
ᨨ盐鼀

V
㦨
Ibh

DV Dl Db Dh = + + V l b h g
㦨
ᆠᨨ �ᆠ� 
ᨨ
ᨨ盐鼀
  矄�ᨨ 㡰 Þ
g
㦨
�
맔
҈
�
졔
�
맔
҈


졔
�
맔
҈

㦨



맔
҈

⏌ C \ 盐
ᨨ

C
㦨





҈䰘
b

㡰

L


 
ᨨ��� 
 �
ᨨ
 
矄�
ᔴᨨ㜠 �ᆠ� 
ᨨ
� 
  矄�ᨨ
ᨨ
矄矄
�矄
�
 

 
ᨨ
ᆠᨨ
�



a B > a C






矄矄 \ g=

LB
㦨
L
҈
졔
aB
DT㡰
R
졔
d 㡰q  㦨
L
҈
졔
aB
DT 㡰 꾀��

LC
㦨
L
҈
졔
aC
DT 㡰
㦨
Rq

\


҈ + g Fe ´

K d҈ Þ ҈ = K
҈ + g Hg ´

d






�㡰

mg l mg = Al a DT A a DT

҈






㡰

V'd' g

d'


҈

Y=

҈�


Þ

F Dl ⏌ A l

\
Y =



-
䰘 °Ӹ -  -҈

㡰
䰘 泐 + ҈

= Þ = 䰘 -
䰘 -�
-  -҈

㡰 ҈

҈҈
\ Ӹ
㦨





Ӹ ҈

ᆠ㡰 ␔ 
 
 � 矄
ᨨ
ᆠ
ᨨ
� ᆠ矄矄
泐
 
矄鼀 鼀ᨨ盐  \ Dl a = Dl s Þ l҈a a t = l
a s t a l
aa l Þ
+҈ = a +҈ = l҈ as l҈ a s

㡰 ␔
├ ᨨ犨
 

Y =

ᨨ



(R + d )q Rq

ᔴᨨ



҈ + a B DT = ҈ + aC DT

q

R+d = (҈ + a B DT ) (҈ - a C DT )
R ᲼泐
� ᨨ鼀�
  矄�ᨨ 桌

d = ҈ + ( a B - aC ) DT - a B aC DT 㡰
R d d = ( a B - a C ) DT ᨨ R = ᨨ
R ( a B - a C ) DT ҈ ҈
 
R µ
\ Rµ a B - aC DT ҈

ᔼ ��泐


鼀盐
T  ᨨ泐 
ᨨᆠ FB
㦨
鼀�
ᨨ
Axrl g = ALrb g ᨨ


҈+

\
xr l = Lrb





�㡰

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EBD_8341 PS-108

Physics 꾀



F'B

r

o





rl

Þ


ᨨ

A Ἐ xr Ἐl g = ALrb g

r Ἐl = rl ҈ - gDT 㡰

Þ

\

A҈
졔

a DT㡰
x
rl ҈ - gDT 㡰 g
㦨
ALrbg

Þ

xrl ҈ +
a DT 㡰҈ - gDT 㡰 = Lrb

Þ

xrl ҈ +
a DT 㡰 ҈ - gDT 㡰 = xrl




᲼ ᨨ鼀
⼄

�㡰桌

҈


ᨨ

�ᔴ

Þ K҈ 
㡰 = K
�
㡰 = K 

㡰






ᨨ

黨�ᔴ 矄





h҈
㦨
�


ᆠ鼀

h

㦨
�҈
ᆠ鼀

h
㦨

黨
ᆠ鼀 矄矄盐

B
㦨
矄矄盐

C P
㦨
d黨� ᔴ
gh҈

d� ᔴ
gh
㦨
P

졔
d� ᔴ
gh


d黨� ᔴ
gh

ᨨ

K҈ K
K Þ K҈
K

K = ҈



҈� = = ҈

҈� Þ K҈
K =


㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

ᨨ 

ᨨ
�矄
├� �ᆠ
 �泐 �矄
ᆠᨨ 
� ᨨ

�
 
犨ᨨᨨ
Þ

ᨨ

ᔴ

҈

- 䰘
㡰 K҈ A 䰘
-

㡰 K
A 

-
㡰 K A = = l l l

Þ K҈ ҈

- 䰘
㡰 = K
䰘
-

㡰 = K 

-
㡰



ᔴ

㡰 꾀矄
 
ᨨ矄

� �ᆠ

矄ᨨ
 泐

矄鼀
 �
l㡰  

ᨨ
ᆠᨨ矄矄㜠矄ᆠ�ᨨ
A㡰

泐

ᆠᨨ

ᆠ
�
矄�矄
和ᨨ


ᆠ盐 
犨�

矄鼀
ᨨ

ᨨ矄


Þ

꾀



b W
- W҈ 㡰 W
- W҈ + W
- W
㡰 t
- t҈ 㡰 W
- W
㡰

æ DQ ö æ DQ ö æ DQ ö 
ᆠ盐 
= ç ÷ =ç ÷ =ç ÷ t t D D è ø AB è ø BC è Dt øCD




҈

gl =

Topic-2 : Calorimetry and Heat Transfer

Þ ҈
졔

a DT

g DT
㦨
҈ Þ g
㦨

a ᨨ

g�
㦨


µ
矄  矄�泐
ᨨ

�⼄盐�
ᆠ矄矄
矄
鼀盐
�矄矄
ᆠᆠᨨ� � ᨨ
ᨨ�
d

㦨
dt
҈
졔
gt㡰
犨 
g
㦨
ᆠᨨ㜠 �ᆠ� 
ᨨ
ᆠ盐�ᆠ   矄�ᨨ
ᨨ
�⼄盐�


ᨨ

W
- W҈ ҈ + g l DT W - W҈ ҈ + g l t
- t҈ 㡰
Þ

= = W
- W
҈ + bDT W
- W
҈ + b t
- t҈ 㡰

Þ W


W҈㡰
᲼҈
졔
b
t



t҈㡰
㦨
W


W
㡰
᲼҈ + g l t
- t҈ 㡰桌

ᨨ犨

AἘ
㦨
A
҈
졔

aDT㡰

矄

V ҈ + b DT 㡰 ´ d l ´ g



㡰 ҈ + g l DT

W
- W҈ Vd l g ҈ + g l D T 㡰 = W
- W
V ҈ + bDT 㡰 d l g

\ FB Ἐ = mg

�ᔴ ᨨ 黨�ᔴ





�㡰

��� �
�㡰
泐
㡰

犨
�

꾀
鼀盐
T
졔
DT  
ᨨ盐鼀
ᨨ
 
ᆠ盐
� ᆠ矄矄 盐
 
 矄�泐
ᨨ
�⼄盐�
ᆠ矄矄
矄ᨨ

盐ᨨ
犨 �ᆠ  
ᆠ盐
�矄
�鼀鼀矄
�
 
�⼄盐�
鼀� 矄
矄鼀


҈黨


  dl = dl Ἐ ҈ + g l DT 㡰

W


W

㦨


mg

鼀�





���㡰

ᨨ鼀
⼄

��㡰

���㡰
 
�㡰

x

x

꾀矄ᨨ

V Ἐ
㦨
V
҈
졔
b DT㡰

҈
҈ mv ´ = ms DT



d黨� ᔴ
h҈
졔
h㡰
㦨
d� ᔴ
h

졔
h㡰

d黨�°C h
+ h d
⏌ (҈ + 黨�g ) h
+ h = = ᨨ d �°C h҈ + h d
⏌ (҈ + �) h҈ + h

҈ + �g �҈ +
黨 ҈

҈ = = = ҈ + 黨�g �


+
黨 ҈
҈

҈

҈
ᨨ ҈
졔
黨�g
㦨
҈

҈

졔
�
맔
҈

҈
g


҈
´
´ ҈

ᨨ 黨�g  �

黨
g =


҈

ᨨ
g =
黨
黨҈ g
´ ҈

\ ᔴᨨ �ᆠ� 
ᨨ
� 
  矄�ᨨ
a
㦨
=   \ a
㦨



䰘
맔
҈
�

ᔴ




gl

 
ᆠᨨ �ᆠ� 
ᨨ
ᨨ盐鼀
  矄�ᨨ
ᨨ
ᆠ盐 犨�� 

 
犨�� 
㦨
盐
 盐矄




ᨨ

W


W҈
㦨
 ´  l ´ �





�㡰

W


W

㦨
 Ἐ´  Ἐl ´ �





��㡰

v

҈
´
҈
= =
䰘
�°ᔴ
´ s
´


´

 ´ ҈


㡰 
�ᆠ
鼀矄
盐
ᨨ
犨

 
鼀ᨨ盐 
ᨨ
�ᆠ
鼀矄
㦨
m�ᆠ

Þ DT =

mw sw Dq = m�ᆠ L�ᆠ







=



\ m�ᆠ =




´



´
�



´ ҈
�

mw sw Dq L�ᆠ

=



҈䰘 ├�
㦨

҈
䰘
�

㡰 
��
泐
犨
= mw Cw T鼀� - Tw 㡰




































=


´҈´ ҈ -
�㡰 
├
泐
矄鼀
㦨
m
L矄鼀
졔
m Cw Ts  T鼀� 㡰 㦨
鼀
맔
�


졔
鼀
҈㡰
맔
҈


҈㡰
㦨
鼀
맔
�


졔
鼀
҈㡰
맔

黨㡰 ᨨ鼀
 
� ᆠ�
ᨨ
ᆠᨨ�鼀

ᨨ矄
㦨

�� 

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PS-109

Thermal Properties of Matter

\ 


㡰҈ -
�㡰 = m ´ �

+ m҈㡰
黨㡰 쟼 �泐
盐



Þ ҈


= m

黨㡰 Þ m »

�




㡰 矄� �
 
� ᆠ�
ᨨ
ᆠᨨ�鼀泐

Þ M
�

㡰
졔
M
맔
҈
맔
҈





㡰
㦨




맔



졔




맔
҈
맔


Þ




M
㦨







Þ
M
㦨


g



㦨

��
鼀
맔
矄Dq 㦨
鼀҈矄矄矄
Dq1 졔
鼀

矄犨
졔
Dq2㡰 Þ
҈黨

맔
和
맔
҈




҈
�㡰
㦨
҈


맔
黨

맔

҈
�




㡰

㡰 M҈C�ᆠ
맔
҈
㡰
졔
M҈L = M
Cw
�
㡰

和ᨨ� �
犨
�
졔




맔





맔

҈
�




㡰

ᨨ
M҈
맔
ᔴ�ce
㦨

�㡰
맔
҈

졔
M҈L = M

맔
҈
맔
�


和
㦨
黨҈

├�҈├҈

�
M
Þ = -� M҈

҈
㡰 q d

d

䰘


㡰 H҈
㦨
H

q





æ q
- q ö æ q - q҈ ö æ q + 黨q
ö = kA ç ᨨ q = ç ҈ ᨨ
k 㡰 A ç ÷ ÷ è d ø è d ø è ҈
÷ø 㡰 쟼 ᆠ�
 鼀
ᆠᨨ 盐ᆠ��泐
ᨨ
矄泐矄鼀 ⧤ ⼄ = =

= 黨





k

k q҈

⧤҈꾀҈ + ⧤
꾀
꾀҈ + 꾀


⧤҈p
㡰 + ⧤
p
㡰


⧤


=

⧤҈ + ⧤



㡰 
鼀
�鼀
ᨨ
�ᆠ
�矄

ᨨ鼀
� ᆠ�
ᨨ
ᆠᨨ�鼀 
�� 
泐
�ᆠ㡰
㦨

ᨨ矄
泐
犨㡰 \




맔


҈
맔
鼀
졔
鼀



㡰
맔


㦨
�

맔




맔


䰘

鼀
㦨





졔




鼀
㦨



҈

҈



ᆠ㡰 
ᨨ矄矄
㦨

��
㦨
鼀和Dq 和ᨨ

鼀꾀和꾀Dq꾀㦨
鼀和Dq Þ ҈


맔
和꾀
맔
҈



黨
㡰
㦨
�

맔
和
맔
黨


䰘�㡰

 和
 ᨨ犨

҈


맔
和꾀
맔
҈



q㡰
㦨
�
맔
和
맔
q

�
㡰


和꾀
㦨
҈
�和
Þ
和꾀
㦨


҈

꾀

鼀

ᨨ

鼀

ᨨ 
矄ᨨ盐ᆠ 
矄ᨨ� ҈
鼀 ҈҈
㡰 
҈
⧤ ҈
⧤ æ 턼 ö ├꾀D

çè ÷ø 㦨  l

⏌


⏌




⏌


⏌
턼



⏌


졠 

҈

´ �  


㦨

� ᔴ 맔
㦨

�
҈

㡰 꾀ᆠᆠᨨ� �
ᨨ
� ᆠ�
ᨨ
ᆠᨨ�鼀泐
턼��
㦨
턼盐矄



맔
和
맔
҈�




㡰
㦨
҈�

맔
҈
맔





䰘㡰
졔

�
맔





䰘㡰



맔
和
맔
҈҈

㦨
҈�

맔
҈
졔

�
맔
҈ 和ᆠ� �ᆠ

ᨨ
盐鼀� �盐鼀 D 턼 =

҈ ´
� ´ 䰘 =


⏌├�㜠 ᔴ


´ ҈҈
҈�
㡰 꾀ᆠᆠᨨ� �
ᨨ
� ᆠ�
ᨨ
ᆠᨨ�鼀泐

ᨨ矄
㦨

��

҈


맔


҈

䰘�㡰
㦨
҈


맔


҈
맔

�
졔
҈䰘

맔
҈
맔

� ҈

T
䰘�

㦨

�

졔
䰘
�

㦨

҈



Þ
T

䰘�
㦨

҈
T
㦨


� C

和=

҈



b㡰  

ᆠᨨᨨ
㦨
\





ᆠᨨᨨ㡰
㦨


æ ö

맔
çè ÷ø
맔
҈



q㡰
㦨
q

�
㡰




q
㦨

q

҈





㦨
�q
Þ
q
㦨


ᔴ

⏌


ᔼ
矄泐
矄
鼀盐
�  ᆠ
犨

 
턼


⧤҈꾀҈

p
㡰


D 꾀 ҈

҈

´ � = =
꾀
�



꾀


⧤҈p
+ ⧤
᲼p 
㡰
- p
桌


p

(

҈)( 黨

)

= 黨
␔⏌鼀
҈ ҈

ᆠ㡰 
矄ᆠ� �ᆠ

ᨨ
盐 ├ ᨨ犨
鼀

�矄
꾀ᆠᆠᨨ� � ᨨ
� ᆠ�
ᨨ
ᆠᨨ�鼀泐


ᨨ矄

㦨

M�ᆠ
Lf
졔
m�ᆠ





㡰
Cw
㦨
m矄鼀
Lv
졔 m矄鼀
҈





㡰
Cw

҈ æ 턼 ö ├D

çè ÷ø
㦨
꾀  l

҈䰘


mc D T V rc DT = t t



҈
´ ҈


´


´҈
 ´

 ´

´



\
ᆠᨨᨨ
㦨



䰘␔ b㡰 꾀矄 Pt
㦨
mCDT 和ᨨ
P
맔
҈

맔



㦨
mC
҈




�㡰   P
맔
��
맔



㦨
鼀


��㡰 ��� �
⼄盐�ᨨ
�㡰
泐
��㡰
犨
�

҈
C ´҈

= �� L \ L
㦨
��

ᆠ
⏌�


WWW.JEEBOOKS.IN

EBD_8341 PS-110

҈



Physics

ᆠ㡰 
ᨨ

ᨨ犨
�矄
��
泐



҈



p
D r =
p ᲼
-  - D 㡰
桌

⧤꾀q҈ - q
㡰 l ␔ 

⧤
㦨
ᆠᨨ �ᆠ� 
ᨨ
 鼀
ᆠᨨ 盐ᆠ��泐 l
㦨
 �
ᨨ
ᨨ
 
꾀
㦨

ᨨ
ᆠᨨ矄矄㜠矄ᆠ�ᨨ
ᨨ
ᨨ

턼
㦨


҈

ᔴ

和

Þ r
D  = ᲼
-
+
D - D
桌

Þ r
D  =
D


T rL


㡰 꾀矄
 
矄盐ᨨ盐 � �
�矄
� �ᆠ

矄矄
�矄
� �ᆠ �鼀
├
ᨨ
ᆠᨨᨨ
ᨨ
犨
 
�⼄盐�
 ᨨ鼀

ᔴ
ᨨ

� ᔴ㡰 �矄
矄鼀


鼀� 盐矄

  ᨨ

矄矄
ᔴ


ᔴ

æ 턼 ö æ 턼 ö =ç ÷ çè ÷ø  犨 è  ø �⼄盐�

ᔼ
 
盐 ᆠ�ᨨ
鼀盐
�矄


 

턼ᔴᨨ
㦨
턼矄矄
졔
턼和

鼀犨 ᔴ 犨 + ␔㡰D 鼀l ᔴl + ␔㡰 D

=   ␔
㦨
犨

⼄盐� 
ᨨ
 
矄矄㡰 ᨨ


鼀 犨 ᔴ 犨 = 鼀 l ᔴ l



黨
´
҈

- T 㡰



ᨨ







´
´ T -
㡰

҈
´
´ T -
㡰
졔
҈ ҈
Þ






㦨


졔



Þ
㦨


ᔴ

㦨


҈黨


\
和ᆠ� �ᆠ

ᨨ
�⼄盐�


ᔴl
㦨




黨
´
´

=


ᆠ⏌矄

ᆠ㡰 ᔼ
 
��
ᨨ鼀


�
鼀盐
ᨨ
矄 
\

Qᔴᨨ =

 = ( 
-

) =


鼀盐
ᨨ
矄盐ᨨ盐 � �

矄盐
㦨

ᔼ ��
鼀盐
ᨨ
矄 

� ��
㦨

和ᆠ� �ᆠ

ᨨ
 
鼀�
ᨨ
 
矄 
�矄
矄


ᆠ = a 

盐 �
鼀矄矄
a
㦨

ᆠᨨ 矄 㡰 꾀泐� �
ᨨ鼀盐


(

 s꾀

=

- 矄盐  㝐ᆠ

)

s
p
é

Þ

= 
) - (
) ùú (  ê ë û  㝐a (  ) 
Þ
 =

㝐a
䰘

 s
p
´





和ᨨ� �
犨
�


�鼀
├
ᨨ
 
矄 
ᨨ
ᆠᨨᨨ
ᨨ犨
鼀盐





㝐a

æ ҈
ö  ç ÷ ҈
p s è  ø


㡰 
T

 
� 
矄泐
矄
鼀盐
ᨨ
  ᆠ├
ᨨ泐
ᔼ
矄泐
矄
쟼 �泐
ᨨ矄
㦨
쟼 �泐
��  sT


T

㡰
맔

pR

㦨
IpR
㡰 =

᲼
D
�矄
泐
矄鼀桌

ÞR=

ᔴᨨ 

㡰 ␔ 
�盐矄
�矄
ᆠ矄
泐
D






\ �
䰘 ´ ҈
-
éêT
- ( 

) ùú ´
= 黨҈
ë û \ T
㦨


⧤

㦨




鼀␔ᔴ␔ 鼀l

�
´ ҈ =

� ├ᆠ ⏌ ├� ҈



㡰 쟼⼄盐� 
 鼀
矄�矄 ᆠ
�
ᆠᨨ ��盐�ᨨ 㜠ᔼ

L L  L + = KA
KA
KA 쟼⼄盐� 
 鼀
矄�矄 ᆠ
�
ᆠᨨ ��盐�ᨨ 㜠ᔼᔼ R = R҈ + R
=

R l KA
KA ᨨ

҈ ҈ ҈ RP = = s = + = + 

� KA RP R҈ R
l l �



鼀
矄�矄 ᆠ
�
ᆠᨨ ��盐�ᨨ ᔼᔼ

RP
�矄


�
�鼀矄 矄矄
 
 鼀
矄�矄 ᆠ
�
ᆠᨨ ��盐�ᨨ

ᔼ

Rs ts 黨 = s =
s

�

�


ᆠ㡰 꾀矄
矄 ᨨ犨
�
 
��盐

 
 ҈


⏌矄 
�� 
泐
 
犨
ᨨ
�矄
�矄 鼀盐 㦨
҈




҈

㡰
㦨




⏌矄
l ᨨ犨

 

⼄盐�
ᨨ
�矄  
鼀盐
ᨨ
犨
ᨨ鼀

䰘 ᔴ
ᨨ
䰘䰘 ᔴ Q
㦨
mc
Dt
㦨


맔





맔
�

























 ᆠ
�鼀
⼄盐�
ᨨ
��
턼 ҈



⏌矄 鼀ᨨ盐 
ᨨ
 \



�t P = t s Þ t P =


´



´ �
Q = 㦨
�


矄
㦨


鼀�



矄






�
㡰 ҈
ᔴᨨ�
�矄
 
鼀ᨨ盐 
ᨨ

⼄盐�
ᨨ
�矄 鼀盐
ᨨ
҈
�鼀
ᨨ
犨
ᨨ鼀
҈

� ᔴ
ᨨ
҈�
� ᔴ

䰘

鼀鼀
ᨨ
�



㡰 ␔鼀� �
ᨨ
�矄矄
ᨨ
盐
盐
ᨨ
�鼀 
�矄
�鼀�泐 盐
ᨨ
��ᨨ

꾀
鼀�盐鼀
�矄
⼄盐�
ᨨ
ᆠᨨ ᆠ�ᨨ

ᨨᆠ矄矄

꾀矄

盐
�矄
鼀ᨨ矄
ᆠ盐


ᨨ鼀
  �鼀 
�矄
 矄鼀�
 ᨨ盐�
��ᨨ
t
㦨


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PS-111

Thermal Properties of Matter
䰘


㡰 꾀ᆠᆠᨨ� �
ᨨ
␔� Ἐ矄 �矄ᆠ鼀 
犨

҈


lm
맔
T
㦨
ᆠᨨ 矄  ᨨ鼀
 
�

l m < l m
< l m҈







҈

 T1 T2

T

\ T
>
T

>
T҈


鼀盐
ᨨ
和盐

T
�矄
�� 
 
 
ᨨ
犨� � ᆠ
T

犨 �ᆠ
�
盐
�矄
�
 
盐 �矄
�鼀 
㜠T҈
㡰 ᔼ

ᆠᨨ鼀� �ᨨ
ᨨ
ᨨ矄 KP
㦨
K҈
졔
K

㦨
K
졔
K
㦨

K

ᨨ

꾀

҈

ᔴ K

ᨨ


ᔴ

l ⧤





꾀

ᔼ
矄�矄
ᆠᨨ鼀� �ᨨ

ᨨ

æ -dT ö æ dT ö ᆠ㡰 ᨨ鼀
 
�

ç > è dt ÷ø x çè dt ÷ø y

æ -dT ö ÷ µ 鼀�矄矄��泐
㡰 dt ø


ᨨ
ᆠᨨᨨ� �

ç è

KK K K - K
= KS = ҈
㦨

K
K҈ + K


҈

ᔴ ⧤

b㡰 
矄
犨 
�├�
ᨨ
犨



ᔴ
矄
ᨨ

ᔴ 鼀盐 㦨
mCw DT
㦨
�
맔
҈
맔



㦨
҈


├ᆠ 
⼄盐�
泐


├�
�ᆠ




ᔴ
ᨨ
ᆠᨨ 
� ᨨ


├�
ᨨ �ᆠ


ᔴ 㦨
mC�ᆠ DT
㦨


맔


�
맔



㦨



├
ᆠ
 ᆠ
鼀� � �
҈






㦨


├ᆠ
ᨨ

犨�
鼀
鼀
├�
�ᆠ 

ᔴ
ᨨ
犨


ᔴ Q
㦨
mL
Þ


㦨
m
맔



Þ m
㦨
҈
├�

 ᨨ
 
鼀ᨨ盐 
ᨨ
犨


ᔴ 㦨
�
├�
졔
҈├�
㦨


├� 꾀 
鼀ᨨ盐 
ᨨ
�ᆠ


ᔴ
㦨



҈
㦨
҈
├�


\ Ex
>
Ey 꾀矄ᨨ
ax
>
ay
矄
�ᨨᨨ
矄ᨨ矄

�ᨨᨨ
鼀�矄
ᨨ

꾀




ᔴ

l

l





K A ҈

㡰 KA ҈

㡰
 
q

= l
l ҈ q
KA ҈

㡰 = ´ = \ q҈
l
K A҈

㡰
ᆠ㡰 ᨨ鼀
�

K
ᨨ
ᨨ�� �
鼀盐

T
� ᆠ矄矄
� 泐 矄
Q
㦨
mc
DT

 ᆠ
T

t
�
犨�


矄�� 
�  � ᆠ� 
ᨨ
�鼀
 �矄
盐� �
ᨨ�� �
Q
㦨
mL

鼀盐
鼀� 矄
ᆠᨨ 矄 
�
ᨨ�� �
�矄
ᆠᨨ鼀
  �
犨�


矄�� 
� 
矄
ᨨ
�鼀
 �矄
꾀 
 

鼀盐
� ᆠ矄矄
� 泐
T㜠t
�
犨�
 
矄�� 
� 
� ᆠ� 
ᨨ
�鼀
 �矄
b㡰 ᨨ鼀
犨� Ἐ矄
�矄ᆠ鼀 
犨 l鼀T
㦨
ᔴᨨ 矄  ᆠᨨ 矄  \ T= lm q҈
㦨



黨






C C C ü ì
T =
T = í\ TA = ý -䰘 B -䰘 C  ´ ҈

´ ҈
� ´ ҈
-䰘 þ î

y x

҈






㡰 꾀ᆠᆠᨨ� �
ᨨ
⧤�ᆠ ᨨ Ἐ矄
犨

�ᨨᨨ
矄ᨨ矄

�ᨨᨨ 鼀�矄
 

 ᆠᨨ矄
꾀
��
鼀盐
�
 
盐 ᆠ㡰

矄� ᆠ
�
矄ᨨ矄
鼀ᨨ  �泐

�
鼀�矄
鼀ᨨ
��ᨨ 矄
矄
犨
 
 ᆠ
�矄
  �� 矄
 
�
�矄
 
├矄
ᨨ泐
� ��泐






b㡰 
q°ᔴ

 
鼀盐
ᨨ
盐 ᆠ�ᨨ

B


Q
�矄  

ᨨ犨� �

矄ᆠᨨ 
ᨨ鼀
A

黨
C
ᨨ
B

q C ᨨ

ᆠᆠᨨ盐 
ᨨ
鼀盐
�  ᆠ
\

Q
㦨


KA 黨
- q 㡰 l

꾀 
矄鼀
ᨨ
C
ᨨ
B.





�㡰

Q=

KA黨
- q㡰 l A

꾀��
ᨨ鼀
矄  Ἐ矄
犨 Q
㦨
sAT


\ QA
㦨
s.p


맔
҈

㡰

맔
QB


㦨
s.p


맔
҈

㡰

맔



҈ ´ ҈
-



Q D

C



ᨨᔴ


�
´ ҈
-



  QC
㦨
s.p


맔
҈

㡰

맔
 ᆠ
QB
>
QC
>
QA

C

黨
ᨨᔴ




C


턼


ᨨᔴ

qᔴ ᨨ

턼






� ´ ҈


B

C 黨
ᨨᔴ

-



\



ᨨ犨� �

矄ᆠᨨ 
ᨨ鼀
B
ᨨ
D

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EBD_8341 PS-112

Physics

KA q -
㡰



��㡰 l ��� �
⼄

��㡰
泐
�㡰 q

㦨


Þ
q =

° 黨
- q  ᆠ
鼀盐
ᨨ
 
盐 ᆠ�ᨨ
q
㦨


ᔴ b㡰 꾀ᆠᆠᨨ� �
ᨨ
␔� Ἐ矄
�矄ᆠ鼀 
犨

lT
㦨
ᆠᨨ 矄  ᨨ鼀
�
l҈


l


l

\ T҈
>
T
>
T

㡰 졠
®
꾀

 
鼀盐
ᨨ
�ᆠ
ᆠ  �矄
ᨨ鼀
҈
ᔴ
ᨨ

ᔴ
꾀
®


�ᆠ


ᔴ
鼀矄
� ᨨ
犨


ᔴ

®
ᔴ

犨


ᔴ
ᆠ  �矄
� ᨨ
犨

҈

ᔴ
ᔴ
®


犨

҈

ᔴ
ᆠ  �矄
� ᨨ
矄鼀

҈

ᔴ


黨



Q
㦨


�




҈⏌ 
⏌ù æ ҈ö DT esAT
esT
é æ m ö k êp ç ú = = 㦨
çè ÷ø m Dt mc mc ê è
prø÷ úû ë

 ᨨ
ᨨ
 
��
犨ᨨ
ᨨ�矄


ᨨ


DT҈ ⏌ Dt҈ æ m
ö = DT
⏌ Dt
çè m҈ ÷ø




鼀


꾀
ᨨ ᔼᆠ

ᔴ ҈

ᨨ ␔

䰘


lmT
㦨





맔
҈


nmK

U
E









´ ҈

nmK = ҈
 nm



K

U҈

U l
鼀


黨黨 �

㡰

黨黨黨 ҈
黨黨 ҈


㡰 ҈�

㡰

l

T - Tc

E ⏌ t ҈


㡰




㡰


= = =

�
�

㡰


҈
㡰


ᨨ


E 㦨

�

맔


㦨
҈



␔ t

A 2a

a B

a

C

TC

2T

=

K2 R

K1

2R l

E

= sT
´ A = sT ´
pr t ᨨ




�㡰

T

Tc -
T Þ T - Tc =
Tc -
T a
a Tc  Þ T = Tc 
+ ҈㡰


Þ = T
+҈
҈
ᆠ㡰 
K
㦨
 鼀
ᆠᨨ 盐ᆠ��泐
ᨨ
 
矄泐矄鼀

ᨨ
 矄 
ᨨ


矄ᆠᨨ 
 ᨨ盐�
 
ᆠᨨ鼀�  矄泐矄鼀
㦨

 矄 

矄ᆠᨨ 
ᨨ鼀
鼀�
犨�  鼀
ᆠᨨ 盐ᆠ��泐
K҈
졔

 矄 

矄ᆠᨨ 
ᨨ鼀 鼀�
犨�
 鼀
ᆠᨨ 盐ᆠ��泐
K



ᔼ
ᨨᨨ犨矄
 
 
 �泐
�
犨

黨黨
nm
ᨨ
�


nm 犨�

矄矄
 
 
鼀�
犨
黨黨黨
nm
ᨨ
҈



nm
�



U҈


U

ᨨ
U

>
U҈
㡰 ᨨ鼀
和  Ἐ矄
犨
 �泐
�

矄ᆠᨨ 
泐
 ᆠ├
ᨨ泐

E =
�
犨  T
㦨
�


K

r
㦨


҈

鼀 t \
�

㦨

ps
�

㡰



҈
㡰
E 犨 
T
㦨
҈



K

r
㦨





鼀

= 䁸 t E \ =
p s ҈


㡰




㡰
t ��� �
⼄

��㡰
泐
�㡰

犨
�

DQ k A DT = Dt l

æ DT ö ÷



























= ç è a øCB

ᨨ
 
矄ᆠ盐鼀 lm
㦨


҈⏌ 

æ DT ö ᨨ
矄�矄
AC
 
CB ç ÷


è
a ø AC

Þ


犨 �

 
├

æ ҈ö = ç ÷ è ø

b㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

鼀盐

B =
T > 鼀盐

A
㦨
T
矄ᨨ

ᨨ犨
ᨨ鼀
B
ᨨ
A

A
ᨨ
C
 
C ᨨ
B ᨨ
矄泐
矄
ᆠᨨ ��ᨨ

D턼 D
�矄
矄鼀



矄盐�

㡰 꾀ᆠᆠᨨ� �
ᨨ
␔� Ἐ矄 �矄ᆠ鼀 
犨


҈⏌ 

꾀泐� �

ᆠᨨ 盐ᆠ�ᨨ
ᨨ鼀盐


 ҈

ᨨ 和鼀


ᨨ ␔

졠 ᨨ ҈
ᔼᆠ

㡰 꾀ᆠᆠᨨ� �
ᨨ
和  Ἐ矄
犨
DQ
㦨
esAT

Dt 꾀矄ᨨ

DQ
㦨
mc
DT ᨨ
mc
DT
㦨
esAT

Dt

KADT K҈ A҈DT K
A
DT = + l l l



Kp

R㡰
㦨
K҈p
R 졔 K
p
᲼
R㡰


R
桌

K ҈ + K




㡰 ᨨ鼀
� ᆠ�
ᨨ
ᆠᨨ�鼀泐
ᨨ


K 㦨



ᨨ矄
泐
矄鼀
㦨

�� 
泐

犨
졔
ᆠᨨ�鼀㡰 mL
졔
m
맔
c
맔
҈





㡰
㦨
҈
҈

맔
c
맔




҈�㡰 m
᲼�


졔
҈
맔


桌
㦨
҈
҈

맔
҈
맔

� m
㦨


҈
├�





��㡰





b㡰

Q Q
nCv DT ҈ = = Þ Q
= ҈ g Q҈ nC p DT g

Þ

Q
=

䰘
= �
ᆠ ҈



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PS-113

Thermal Properties of Matter






㡰 ᨨ鼀

鼀矄
㦨
㝐
㦨




=


鼀矄


黨


RT m

 ´

҈
´
黨
´ ҈


=
� ҈黨
´ ҈黨


b㡰  
 鼀
⼄盐���盐鼀

q A - q B = 
°C ෌� 㡰 KA
㦨

KB ෌� 㡰

KA q + l A q ᔴ 㦨
KA + l

\






q C
㦨


qᔴ

q꾀 KB qB l


KB l

q

KA

KB

l

l


q A + q B
q A + q A - 
q A - ҈
㡰 = =   

\ q A - qC = ҈
°C  ᆠ

 
鼀盐
�  ᆠ
ᆠᨨ矄矄
 
泐 A
㦨
҈
C 黨㡰
꾀ᆠᆠᨨ� �
ᨨ
矄  Ἐ矄
犨

P µ T



ᨨ


P = P
T
\
ᨨ�
P = ᨨ�
P
+ ᨨ�
T


\
ᨨ�


P =
ᨨ�
T P


P =
ᨨ�
䰘

= ҈ …
�㡰 꾀

㦨


䰘 ᔴ
㦨
䰘


⧤

ᨨ�
P
꾀

㦨

䰘
䰘 ᔴ
㦨



⧤
r ᨨ�e =
ᨨ�



=
ᨨ�
䰘

´
㡰 r















=
éëᨨ�
䰘

+ ᨨ�


ùû =
ᨨ�
䰘

+
= ҈ +
= 黨
䰘

㡰 ᨨ鼀
�㡰
和  㜠ᨨ鼀


犨

P
㦨
sAT


 
��㡰 ␔� Ἐ矄
�矄ᆠ鼀 
犨
㦨
lm
맔
T
㦨
ᆠᨨ 矄 

PA AA = PB AB

L ҈�
= =
䰘
°C C � �҈




㡰 
ᨨ

ᨨ犨
犨�

矄鼀


\


ᨨ

ᨨ犨
犨 
R =

A l
= A´ B TB
AB l
A ҈ é


´


ù
ê ú


é R
P l A é AA =ê ´ ú =ê A´ Bú =
l B ë AB PA û ë ҈
êë RB PA úû l \ A
㦨

lB 黨㡰 꾀ᆠᆠᨨ� �
ᨨ
矄  㜠ᨨ鼀


犨


p (
r ) L҈ L K A =
\ ҈ =
= =
[Q L҈ = L
= L ] K҈ A҈ K
A
K
A҈ pr


û

�



ᨨ ᨨ  �泐 � 泐 꾀 ᨨ ᨨ  �泐 � 泐 


æT ö æ r ö = ç ҈ ÷ ´ç ҈ ÷ 㦨


sT

pr
㡰 è T
ø è r
ø æ l m
㦨
çç l è m҈







2

æ ҈�

ö æ
ö
㦨
黨 㦨
ç è �

÷ø çè ҈
÷ø

和ᨨ
ᨨ犨
ᆠ�
泐




鼀

 㦨
҈



␔⏌鼀
㡰




鼀
㡰
㦨




␔ 㝐矄矄
ᨨ
�ᆠ
㦨




�
㦨






├� 
⼄盐�
ᨨ
鼀
�ᆠ
㦨
鼀 㦨





㡰


맔
҈
�㡰
㦨
黨



맔
҈


 ᔼ
t
�矄
 
�鼀
├
ᨨ
 
�ᆠ
ᨨ
鼀
Eù é \ 


㡰t
㦨
黨



맔
҈


 êQ P = t ú ë û




ù ö æ r҈ ö é T҈ l m
泐 ␔�
Ἐ矄
犨 ú ÷ ç ÷

êQ = ÷ è r
ø ê T
l m úû ҈ ë ø


L KA




҈ ù


҈ PB ù


sT҈

pr҈
㡰

dQ 鼀

�  ᆠ ҈ = = ( T
 T҈ )  鼀
矄�矄 ᆠ R dt









҈

=
ᨨ

R҈
㦨
R
R҈ R


TA


\







҈

҈ ´ 

´ ҈
� ´�+ ҈


҈




\鼀= =
� Þ


= 鼀 ´ ҈

� + 
҈

� �


䰘
°ᔴ㡰

C

 
矄ᆠ� �ᆠ

ᆠᆠ�泐
ᨨ
�⼄盐�
 
 
 
 

ᨨ
ᨨ盐�矄�ᨨ
ᨨ鼀
� ᆠ�
ᨨ
ᆠᨨ�鼀泐

ᨨ矄
㦨

��

mCSC
D
㦨
mC
DT
졔
mL ᨨ
mCSC
҈҈




㡰
㦨
�ᔴ





㡰
졔
�


�㡰 ␔ 

mC
㦨
鼀矄矄
ᨨ
ᆠᨨ�鼀 SC
㦨
矄


ᨨ
ᆠᨨ�鼀 꾀��

犨 


�
�⼄盐�
�矄
ᨨ盐
 
⼄盐���盐鼀 鼀盐
�矄
�
ᔴ mCSC




�
㡰
㦨


ᔴ
�



㡰


��㡰 ᨨ鼀
⼄

�㡰


��㡰 ҈



ᔴ
㦨

�
ᔴ
졔
� Þ


= 鼀 ´

 ᆠ

 
�矄
�矄


�



㡰 꾀矄
 
�矄
ᨨ
ᨨ 

 ᆠ  �
�
 �矄
ᨨᆠ矄矄

和ᨨ 
矄盐�
㦨

盐矄
�
ᆠᨨ � �
鼀
�鼀矄
ᨨ
�ᆠ ᨨ鼀
� ᔴ
ᨨ

ᔴ
졔

盐矄
�
ᆠᨨ � �
҈
�鼀
ᨨ
�ᆠ

ᔴ
ᨨ
犨


ᔴ ᨨ





㦨
mcD q
졔
ml

ᨨ
�


t
㦨


黨


´ ҈

矄 = 
矄 º �
�鼀�







t

 
鼀盐
ᨨ
 
�  ᆠ
 
 
矄泐 矄
�矄
ᆠ 




 矄 

矄ᆠᨨ 
 ᨨ盐�
A Q҈㦨
K҈
A
҈



t㡰



 矄 

矄ᆠᨨ 
 ᨨ盐�
B Q
㦨
K

A
t


㡰 꾀
矄泐
矄
K҈
A
҈



t㡰
㦨
K
A
t


㡰 Þ 


҈



t㡰
㦨




t


㡰 Þ 



t
㦨

t
Þ t
㦨



ᔴ

WWW.JEEBOOKS.IN

EBD_8341 PS-114

Physics

�



ᔼ
├矄
t
矄ᆠᨨ 矄
ᨨ
 
矄盐矄 ᆠ
ᨨ
矄ᨨ�� 泐
�� 㡰


 ᨨ
ᨨ

矄
�
t
矄ᆠᨨ 矄
㦨
P
맔
t 㦨
mL 盐矄�ᨨ

P ´t \ L 盐矄�ᨨ
㦨
m

��



ᨨ

 矄 

ᨨ

P=

P=

鼀盐 �   鼀 矄�矄 ᆠ




t ⏌
pR K

t=






pKTR
P 쟼 �泐
�

矄ᆠᨨ  dT sT
A
㦨
mc dt Þ


r ´ p r cdT  dt 㦨

= sT A s T
´
pr
mcdT


҈




ᨨ
 ᆠ

�








l -t 㡰 ù é êQ 鼀 矄�矄 ᆠ㦨 KA ú ë û

T


pKTR
P �.e.

 �ᆠ├ 矄矄
t
矄 ᨨ盐 Þ

�黨


⧤

졠
® 꾀
鼀�
�矄
�
矄ᨨ�
矄
ᨨ 泐
鼀盐
�矄 � ᆠ矄� �

ᨨ鼀
꾀
® 
鼀�
�矄
泐
矄ᨨ�
 
泐 �⼄盐�
和� ᆠ
P
�矄

ᨨ� 
犨
A
 
B

  ᨨ
 
鼀� �矄
泐
矄ᨨ�
 
泐
�⼄盐�
 矄㡰
쟼 �泐
�

矄ᆠᨨ 
泐
 
�矄
矄  E҈
㦨
esT
A
㦨
es




㡰

맔

p
맔
҈҈
맔
҈҈





㦨
҈



맔
p
맔
҈
҈


맔
es 쟼 �泐
�

矄ᆠᨨ 
泐
 
矄ᆠᨨ 
矄  E

㦨
es
맔




㡰

맔

p
맔


맔







㦨
҈



p
맔
҈
҈

맔
es \
E҈ = E


ᆠ

㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ


턼
ᨨ犨矄
ᨨ 泐
ᨨ鼀
 
ᨨ
��   ᨨ盐�
 
ᨨᆠ├矄

 ᆠ

ᨨ犨
 ᨨ盐�
矄
꾀
 
쟼 
 
矄鼀
꾀

턼꾀

rrc dT s T
ᔼ �� �
ᨨ
矄�矄

ò
dt =

�䰘


\

�



꾀

rrc rrc é ҈ ù = s ò


T
s êë T  úû




( ) (

)

\

sT

맔

pR

㦨
҈



맔

pr
ᨨ

ᔴ

ᨨ

T
=

ᨨ

T
=


⧤

 �⧤

턼ᔴ

턼 

턼쟼 쟼

RA =

L ҈
L
=
RB = =
K 
L㡰W
KW K  LW 㡰 KW

RC =


L ҈
L
=
RD = =
K 
LW 㡰
KW �K  LW 㡰 �KW

҈ L =
K 
LW 㡰

KW ᨨ犨

DT
㦨
QR 和� ᆠ
 
矄�矄 ᆠ
ᨨ

ᨨ犨
�矄
矄
ᨨ
矄
E

  鼀盐
�  ᆠ
ᆠᨨ矄矄
E
�矄
矄鼀矄
꾀矄ᨨ DTC DTC QC = = =
KW  DTC 㡰 RC ҈⏌
KW DTB DTC  KW  DTC 㡰 QB = = = ᲼Q
DTB
㦨
DTC桌 RB
⏌  KW
RE =

҈


´ r
sR


DTC �KW DTc 㡰 DTD = = ᲼Q DTD = DTC 桌 RD
⏌ �KW
 KW  DTC 㡰 � KW  DTC 㡰 QB + QD = +


KW  DTC 㡰 = =
KW DTC 㡰 = QC


҈


´ ҈
� ´ ҈
҈҈㡰


QD =





�

䰘 ´ ҈
㡰䰘 ´ ҈
㡰

҈
´


� ´ ҈




ᨨ T
㦨
�


⧤
�

䰘 ´
黨 ´ ҈





⼄盐�
ᨨ
҈


�
ᨨ
�ᆠ



ᔴ
ᨨ
ᆠ  �
� ᨨ 犨


ᔴ
㦨
mL
㦨
҈


맔



맔



㦨









矄
泐


�
ᨨ
犨


� ᔴ
ᨨ
ᆠ  �
�矄 鼀盐
ᨨ

ᔴ
㦨
mcDT
㦨



맔




맔

�
㦨
҈
�


  ᆠ
ᆠᨨ鼀
�ᆠ
犨�
ᨨ
鼀

矄ᨨ
 
� 
鼀盐 ᨨ
 
鼀� 盐
犨�


ᔴ


l KA


 
犨�
ᨨ
矄矄

␔



和

T
=

쟼
⧤

␔
├ ᨨ犨
 
 鼀
矄�矄 ᆠ

㦨


E

쟼 �泐 = sT
´
pR
�鼀 쟼 �泐
ᆠ�
ᨨ

 盐 �
�鼀
㦨
҈



맔

pr





턼 

턼
꾀 㦨
턼
 졔
턼
ᔴ 졔
턼
 㦨
턼쟼

҈



҈

dT

rrc é ҈ ҈ ù t
㦨
ê ú  黨s êë ҈

㡰 


㡰 úû 䰘rrc 䰘rrc » = ҈
䰘҈rrc t
㦨


䰘
´ ҈
㡰 s 䰘
´ ҈
�

䰘 ´ ҈
-
㡰  ᆠ
�鼀
⼄盐�
ᨨ
 
鼀盐
ᨨ
 
矄 
ᨨ ᨨ
ᨨ
҈


├
㦨
҈
䰘҈
rrc 꾀ᆠᆠᨨ� �
ᨨ
和  Ἐ矄
犨
쟼 �泐
�

盐 �
�鼀 R 
盐 �

㦨
sT
r \

⧤

ᔴ


 é
ù êëQ m = r ´  pr   A =
pr úû

Þ dt 㦨


t




⧤






�.e.
QC
㦨
QB
졔
QD 

ᆠ

㡰 和� ᆠ
 
鼀盐
ᨨ
ᆠ├
ᨨ泐
 
ᆠ 鼀
鼀� 矄 ᆠᨨ 矄 
矄ᨨ
 �泐
鼀�
㦨
 �泐
矄ᨨ
泐
ᆠ├ ᨨ泐


WWW.JEEBOOKS.IN

PS-115

Thermal Properties of Matter





b㡰
쟼 �泐
鼀�

矄ᆠᨨ 
泐
ᨨ泐
A
 
B

矄鼀



\
㦨
eAs TA A
㦨
eBs TB
A ҈⏌


æ eA ö ´ T
㦨
҈黨

⧤ \
TB
㦨
ç A è e ÷ø

҈
䰘

=


黨� ├�
\
㝐矄矄
ᨨ
ᆠᨨ � 
㦨



黨�
├�

҈






ᨨ矄
泐
矄鼀
㦨

�� 
泐
犨 msL 盐矄
㦨
mwc DT ᨨ
m =

Þ ms 㦨


B

꾀ᆠᆠᨨ� �
ᨨ
␔� Ἐ矄
�矄ᆠ鼀 
犨

l m µ

҈ T


䰘


lm㡰A
TA 㦨
lm㡰B
TB

Þ

l m 㡰 A TB �


= = l m 㡰 B TA ҈黨






�㡰

和� ᆠ
鼀盐
ᨨ
A
�矄
鼀ᨨ
  ᨨ
lm㡰A
�矄
矄矄 \ lm㡰B

lm㡰A
㦨
҈
맔
҈


鼀 �� 㡰



��㡰 和ᨨ� �
⼄

�㡰
 
��㡰

犨
� lB
㦨
҈
�
맔
҈


鼀






�



ᨨ矄
泐
矄鼀

҈

ᔴ
ᨨ
ᆠ  �
ᨨ
҈

ᔴ
犨 Q҈
㦨 mL
㦨



�
맔





맔
҈



㦨
҈
҈




 
ᨨ矄
泐
҈

ᔴ
犨
ᨨ
ᆠ  �
ᨨ

ᔴ
犨 Q

㦨
MCDT
㦨



�
맔





맔
҈


㦨

҈




 
⼄盐�
泐



�
├�
ᨨ
�ᆠ
ᨨ
ᆠ  �
�矄
鼀盐 ᨨ鼀

�
K ᨨ

䰘
K Q
㦨
m
맔
C�ᆠ
맔
DT
㦨



�
맔

҈


맔




㦨
҈

黨


 
⼄盐�
泐



�
├�
�ᆠ


䰘
⧤
ᨨ
ᆠᨨ 
� ᨨ



� ├�
犨


䰘
K Q

㦨
mL 盐矄�ᨨ
㦨



�
맔


맔
҈



㦨
҈�҈




 ᨨ鼀
 
ᨨ

�
�矄
ᆠ
 
Q҈
졔
Q

>
Q
盐 Q҈
졔
Q



Q
졔
Q

矄ᨨ
犨 ᨨ
�ᆠ
犨�
ᨨ
鼀


 ᨨ
 
� 
鼀盐
犨�


䰘
⧤
ᨨ

ᔴ

m

 
鼀矄矄
ᨨ
 
ᆠᨨ �  꾀ᆠᆠᨨ� �
ᨨ
� ᆠ�
ᨨ
ᆠᨨ�鼀泐


ᨨ矄
㦨

��


ᨨ矄
泐
ᆠᨨ � 
㦨
msdT
ᨨ
dQ
㦨
mA
졔
BT㡰dT Q





ᨨ

ò
dQ = mò�

( A + BT ) dT

ᨨ

é BT
ù Q = m ê AT + ú
úû ëê �



ᨨ

é
´ ҈
Q = m ê҈

T +
ëê








mwcDT

҈ ´



´

= 㦨



҈


├� L fus �

´ ҈
 ´





 �泐
鼀�

矄ᆠᨨ 
泐
ᨨ
矄 矄
犨�
 矄鼀
矄
 
鼀盐
 
矄盐 ᆠ


矄鼀
␔
├ ᨨ犨
 
Q = mcDT

dQ mcdT = dt dt 和� ᆠ
Q
�矄
矄鼀
 
c
�矄
矄ᨨ
矄鼀
ᨨ
ᨨ
ᆠᨨ㡰


ᨨ



dQ ҈ µ dt m 㝐矄矄
ᨨ
ᨨᨨ犨
矄 
�矄
矄矄
矄ᨨ
ᨨᨨ犨
矄 
犨�
ᆠᨨᨨ 矄



෌�

�奠
ᨨ
 

�矄
矄ᨨ
泐
 
ᨨ矄ᆠ


 ᨨ
䰘�奠

�矄
盐矄
�
鼀� �
ᨨ


ᔼ ��
鼀
㦨

䰘 ᔴ 㝐� �
ᨨ� 
ᨨ

㦨

䰘 ᔴ
\ D
㦨

䰘


䰘
㦨


ᔴ \




䰘�㡰
맔


҈ Mv
㦨
Mc
DT
졔
ML


҈


䰘�㡰
맔
v 㦨




맔



졔

㡰
맔




v
㦨
҈

黨

鼀⏌矄
黨
�㡰 ᔼ
��ᨨ
㜠
꾀


�矄
矄ᨨ
泐
 
鼀�

 ᆠᨨ 矄 
鼀盐
 
�矄
盐矄
�
 矄
ᆠ  �� �
ᨨ鼀 矄ᨨ�
ᨨ
�⼄盐�
ᔼ
��ᨨ
ᔴ

�矄
矄ᨨ
泐
 
鼀�


ᆠᨨ 矄  鼀盐
 
�矄
盐矄
�
 矄
ᆠ  �� �
ᨨ鼀
�⼄盐�
ᨨ
�矄
��㡰  

ᨨ
ᨨ�矄�ᨨ
㦨


 

ᨨ
盐矄�ᨨ 㡰 ���㡰 
矄ᨨ
DE
� �ᆠ矄
 
 
鼀盐
ᨨ
  矄ᨨ�
�� 矄
ᨨ
�矄
�㡰 
矄ᨨ
ᨨ
졠꾀
>
矄ᨨ
ᨨ
ᔴ

�矄
� �ᆠ矄
   
�矄

�矄
�
矄ᆠ� �ᆠ

䰘

෌�

鼀
ᆠᨨ 盐ᆠ��泐 KX 㦨


黨

ᆠ⏌矄ᆠ㜠ᆠ鼀㜠 ᔴ KY
㦨





ᆠ⏌矄ᆠ㜠ᆠ鼀㜠 ᔴ ᔴ ᲼ 桌 ᔴ

 



ù T
ú ûú �





é 

) ´ ( �

) ù ( ê ú ᨨ Q = m ҈

( 

 �

) + ҈

êë úû ᨨ Q
㦨
m
᲼






҈


桌
ᆠᨨ� ᨨ
Q
㦨


҈



m
ᆠᨨ�


�㡰 
�� 
泐
�ᆠ
�
鼀� �
㦨
mL \
Q҈
㦨


҈
맔






㦨





ᆠ 
�� 
泐
犨
ᨨ
ᨨ
�ᆠ
㦨
msDT \
Q

㦨


҈
맔
҈



맔

䰘
㦨

䰘


ᆠ \
ᨨ

�� 
㦨





졔

䰘


Q҈
졔
Q

㦨
҈
䰘


ᆠ


��㡰 \

ᨨ矄
㦨

�� 
ᨨ

҈



m
㦨
҈
䰘



꾀 ᲼

ᨨᔴ桌

Ӹ

 

 ᲼ 桌 Ӹ 

쟼 ᲼҈
ᨨᔴ桌

Ӹ ᲼ 桌

ᨨ鼀
��盐


KY A 

- TB 㡰 K X A TB - ҈
㡰 KY A TB - ҈
㡰 = + l
l
l 和ᨨ� �
 
ᨨ
⼄盐�ᨨ

犨
�
TB
㦨

ᔴ ᨨ

犨�
ᨨ犨
 ᨨ盐�
ᔴ 꾀矄
C
�矄

ᨨ� 

 
鼀�
ᨨ
쟼
  ᨨ
鼀盐
 TB + TE 
°C + ҈
°C = =

°C

和�鼀�泐
鼀盐


�矄
矄ᨨ


ᔴ


C
=

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EBD_8341 PS-116

Physics



Topic-3 : Newton’s Law of Cooling ҈


T҈ - T
éT + T ù = K ê ҈
- T
ú t ë
û

犨 
q

�矄
 
鼀盐
ᨨ
矄盐ᨨ盐 � �




҈
㦨
�
ᔴ



㦨


ᔴ 

ᨨ
㦨


ᔴ


㦨



和
㦨
�
鼀� 盐矄

ᨨ犨

ᨨ
犨
ᆠᨨᨨ矄
ᨨ鼀


ᔴ
ᨨ
�
ᔴ
�
҈

鼀� 盐矄


�
-

æ �
+

ö = Kç -

÷ �㝐�

è
ø



- �
é

+ �
ù = -⧤ ê - q
ú ҈

ë û




�㡰



- T æ

+ T ö = Kç -

÷ � è
ø ��� �
⼄

��㡰
泐
�㡰

犨
�

�
-

é �
+

ù = -⧤ ê - q
ú ҈
ë
û




��㡰





�� - q
҈ =




- q
҈
�� - q
=


- q





= 
°ᔴ





- ҈
q
=


-
q

q
=



æ dT ö
÷ = esAT dt ø

æ dT ö çè - ÷ø dt B

q
= ҈
°ᔴ �


dT es ´ A ´ T
dT ҈ = Þµ dt r ´ Vᨨl
´ S dt rS

æ dT ö çè - ÷ø dt A




T Þ


- �T = T �

b㡰 
ᨨ

ᨨ矄矄
㦨
mS çè

-




��㡰

��� �
⼄盐�ᨨ 矄
�㡰
泐
��㡰
犨
�



- T

+ T -

T = = ҈
�
+

-

�


\T =




�㡰

꾀��

�
ᆠᨨᨨ矄
ᨨ鼀
�
ᔴ
ᨨ


ᔴ
�
 
҈

鼀� 盐矄



T

 
鼀盐
ᨨ
矄 
 
 
�
鼀� 盐矄




Þ

- T =

泐
犨ᨨ 矄
犨
ᨨ
ᆠᨨᨨ� �

q҈ - q
é q + q
ù = -⧤ ê ҈ - q
ú 
ë û

b㡰 ᨨ鼀
犨ᨨ Ἐ矄
犨
ᨨ
ᆠᨨᨨ� �


Þ

b㡰

ᆠ㡰 꾀ᆠᆠᨨ� �
ᨨ
犨ᨨ Ἐ矄
犨
ᨨ
ᆠᨨ� �
 
鼀盐 �ᨨ矄
ᨨ
ᆠ矄� �
犨�
�鼀
ᨨ 㜠� 泐
Topic-4 : Miscellaneous (Mixed Concepts) Problems

҈


r S ҈




= B´ B = ´ r A S A
´ ҈







㡰 ᨨ犨


P=

dQ d dT = mc 㡰T =  H 㡰 dt dt dt d éT
҈ + bt҈
㡰ù û dt ë

æ dT ö æ dT ö Þ ç- ÷ > ç- ÷ è dt ø A è dt ø B




=  H 㡰

和ᨨ

꾀
ᆠᨨᨨ矄
ᨨ犨

矄

㡰 꾀ᆠᆠᨨ� �
ᨨ
犨ᨨ 矄
犨
ᨨ
ᆠᨨᨨ� �


P =  H 㡰T
=

æ q҈ - q
ö æ q҈ + q
ö - q
÷ çè ÷ø = K çè ø t
æ

- �
ö æ

+ �
ö -
�÷ çè ÷ = K çè ø ҈
ø
æ �
- q ö æ �
+ q ö -
�÷ ÷ = K çè ø ҈
ø


 
çè

��� �
⼄

�㡰
泐
��㡰


҈


= Þ q =



� ᔴ @
 ᔴ �
- q㡰 q

\

H 㡰 =







�㡰

盐 t҈
=







��㡰

\

t
=

bt -

犨 
H
㦨

ᆠᆠ�泐



Pt 
T
b







�㡰

T t 㡰 - T
bT


[T t 㡰 - T
] bT








��㡰

ᨨ鼀
⼄

�㡰


��㡰



P [T t 㡰 - T
]
P [T t 㡰 - T
] =   T
b b T
b
T



H 㡰 =



WWW.JEEBOOKS.IN

PS-117

Thermal Properties of Matter



b㡰 ᔼ
 
�矄

 
ᨨᆠ
�矄
ᆠ
盐
ᨨ
矄矄盐
  �
 
矄ᆠᨨ 

 
ᨨᆠ
�矄
盐
ᨨ
矄盐 ᆠ
 矄�ᨨ
T


턼 = s
 ×
D

꾀 ᨨ犨
ᨨ鼀
⼄

�㡰

P0 2R

FT h/2 P0 졔

h rg 2

鼀矄



= s꾀
-

㡰 

Þ

 s꾀 ᲼
+ D 㡰
-

桌 =  鼀矄

=








hrg ö éæ ù


ᨨᆠ
�矄
ê ç P
+ ÷ø ´ 
R ´ h㡰ú -
RT è
ë û \ ᨨᆠ
㦨

P
Rh
졔
Rrgh



RT 㡰 ␔ 

矄 �ᆠ
ᨨ泐
�矄
├
� 矄�

 ᆠ泐
ᆠ├ ᨨ泐
 
 
ᨨ

�
泐
 
ᨨ泐
�矄
⼄盐
ᨨ  
ᨨ
 

ᆠ├
ᨨ泐
ᔼ
 �矄
⼄盐矄�ᨨ
 
��
ᨨ�ᨨ 矄

犨ᨨ �
矄

 
ᨨ盐 ᨨ�ᨨ 矄
ᆠᨨ �
e
�
ᆠ
ᨨ
s
ᆠ㡰 ᨨ

��
鼀矄矄
ᨨ
�矄

ᆠᨨ 矄 
矄矄盐


ᨨ

턼 = s
 ×
D

꾀 和� ᆠ


ᨨ
ᨨ矄矄
ᨨ

㦨

ᆠ�

矄ᆠᨨ  䰘


㦨
⧤⏌꾀㡰
鼀矄㡰
D

᲼⧤
맔鼀矄
㦨





맔҈
桌

DV ҈ ҈ = d ᨨ dT = ҈ = ᨨ T T V DT


⼄盐�ᨨ
矄 矄

ᆠ �盐
泐ᨨ
ᨨ
  ᨨ鼀
xy
㦨
c

�ᆠ
泐
�
c㡰


�


黨㡰
ᔴ

l

҈

ᔴ

䰘




ᨨ
h
ᨨ犨
ᨨ鼀

ᨨ



dQ҈ dm҈ KA


= Lf = lx dt dt ᨨ
h
ᨨ犨
ᨨ鼀

ᨨ









�㡰

é æ D ö ù 턼 = s
+ D 㡰
-

㡰 = s

êç҈ + ÷ - ҈ú 꾀


ø êè ú ë û

éæ D ö ù = s

êç҈ +
÷ - ҈ú


ø úû êëè


é ù ҈
ö
æ ê AT ҈ s + - ҈ú t
ç 㦨 ÷ êëè T
ø úû

é

ù






㦨
sAT
ê T ú ´ t =


´҈´ 

´҈ =
҈
J ë
û

P=

쟼 �泐
� = sAT
- sAT

�鼀

dp




Þ

ᆠ㡰 쟼 �泐
�
泐
 
ᨨ泐
㦨
sAT
- T

㡰  ᲼ ᨨ

ᆠ├
ᨨ泐

㦨
҈桌 㦨 sA᲼
+ ҈
㡰
- T

桌t







�㡰

dm҈ dm
ù dQ
dm
KA 

é = = = Lvap





��㡰 ê ෌�
ë dt dt úû dt ҈
x - lx dt ��� �
⼄

�㡰
泐
��㡰
 
矄ᨨ� �
犨
�

l
㦨
黨 

㡰 
ᨨ
ᨨ矄矄
ᨨ



턼 = s 꾀
-

㡰 

䰘

´ 꾀 䰘

´ � ´҈
-
�

� = = = ⧤ ´ 鼀矄
 ҈
- ´



\ D
㦨


 Þ D =

 ҈


l


ù s꾀
éêæ D ö ú


´ ç ҈ + ҈ ÷ 鼀矄


ø êè ú ë û



= ⧤D 錐  æ ö
s꾀
 ᔴᨨ 矄  ᨨ 犨ᨨ Ἐ矄 犨 ᨨ ᆠᨨᨨ� � ÷ ç⧤ = ç ÷ 鼀矄 è ø  ⧤ Þ
s꾀
= 鼀矄㡰 꾀 ᨨ鼀
⼄

�㡰

ᨨ




ᔴ

᲼Q 턼
㦨
鼀矄D 桌

 s꾀
=


×
D

 鼀矄

V V + DV V \ = 㦨
ᔴᨨ 矄  T T + DT T VT
졔
VDT
㦨
VT
졔
TDV
ᨨ VDT
㦨
TDV

꾀




��㡰

 \
dT = sA
T
㡰

\
dp = sA
T
 㡰 dT



\
DP =
sAT
 
矄
盐鼀
ᨨ泐
�矄
ᨨ

ᆠ├
ᨨ泐

和ᨨ
ᨨ�ᨨ
㡰
  㡰

� ᆠᨨᆠ
쟼 �泐
�
µ
꾀

犨 
꾀
�矄
 
矄盐 ᆠ

ᨨ
  ᨨ泐

 ᆠ
ᨨ�ᨨ
ᆠ㡰
�矄
ᆠᨨᆠ


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EBD_8341 PS-118

Physics

Thermodynamics

11

ᨨᆠ
ᨨ
矄� �
泐
�矄
㦨
kx kx \ P
= P҈ +
꾀
㦨

ᨨ
ᆠᨨ矄矄

矄ᆠ�ᨨ
ᨨ
�矄ᨨ 㡰



��㡰 A ␔ 
V

㦨

V҈

T

㦨
T҈

Topic-1 : First Law of Thermodynamics ҈






㡰 D
鼀� 矄
矄鼀
ᨨ
ᨨ
 矄
꾀ᔴ
 
꾀 D턼꾀ᔴ
㦨
D␔꾀ᔴ
졔
D꾀ᔴ Þ



㦨



졔
D꾀ᔴ Þ 꾀ᔴ
㦨


 \ D꾀
㦨
D꾀ᔴ
㦨


 D턼꾀
㦨
D꾀
졔
D␔꾀 㦨
҈


졔



㦨



 㡰 ᨨ盐鼀
ᨨ
犨
ᨨ矄
ᨨ
ᆠ  �

ᨨ
犨ᨨ├
�矄
ᨨ 
ᨨ

ᨨ
泐
 
矄泐矄鼀
␔
㦨

㡰 꾀ᆠᆠᨨ� �
ᨨ
�矄
犨
ᨨ
 鼀ᨨ泐 鼀�ᆠ矄

\


盐� �
 �矄
盐
ᨨ
P

�
⼄

��㡰
犨
� PA kx  P҈ = P҈ + Þ kx = ҈ A

V
 V҈
V҈  ҈ V҈ x= = = A A A 쟼 �泐
矄ᨨ
�
 
矄� � PV ҈ ҈ = kx
= ( kx )( x ) = ҈ ҈


㡰 ᔴ  �
�
�  
 �泐


턼 㦨  졔 ␔ 






�






䰘


ᨨ
ᔼ矄ᨨᆠ ᨨ�ᆠ
ᨨᆠ矄矄
턼 㦨   D
㦨
mᆠ
㦨


맔

҈


맔



㦨
҈

䰘
├
㡰 꾀矄
犨
├ ᨨ犨
ᔼ矄
犨
ᨨ
 鼀ᨨ泐 鼀�ᆠ矄㡰 D턼 = D盐 + D犨 Þ
D턼 = D盐 + D ᨨ
҈�
= D盐 + ҈

(҈ -
) 㦨
D盐 - ҈

\
D盐 = ҈�
+ ҈

=
�


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 
�  
 �泐
ᨨ
 
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泐

�

 㡰 ᨨ鼀
 
�矄
犨
ᨨ
 鼀ᨨ泐 鼀�ᆠ矄 dQ
㦨
dU
졔
dW 
dW
㦨



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㦨
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矄� ᆠ
dQ





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
, dQ
�矄
��
矄ᨨ
dU
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犨 
鼀盐
T
ᆠ矄矄

㡰 꾀泐� �
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犨
ᨨ
 鼀ᨨ泐 鼀�ᆠ矄
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
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㦨
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졔
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㦨
DU�af
졔



\
DU�af
㦨



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Q�bf 㦨 DU�bf 졔 W�b 졔 Wbf 㦨



졔
�

졔
҈

Q�b 졔 Qbf
㦨

�




�㡰 꾀矄ᨨ
Q�b = DU �b + W�b \
Q�b
㦨
҈


졔
�

㦨
҈�




��㡰 Qbf Q�b g  Q�b 

= = =
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⼄

�㡰


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Q�b Q�b ҈�
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 �矄
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ᆠᨨ 矄 
�
ᆠ盐盐鼀

\
DW
㦨

꾀矄ᨨ

Q
㦨


矄
 
ᨨᆠ矄矄
�矄
��ᆠ

 ᆠ
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�矄
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DU
㦨


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鼀� 矄
ᆠᨨ 矄 


b

ᆠ㡰 
矄� �
�矄
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泐
x
ᨨ
� �
 
�矄
P҈ PV Ἐ ҈ἘT҈

x


P҈

P
ἘV
ἘT
kx

kx

x
+ x

㡰 꾀矄
�矄
�矄
�
鼀ᨨ ᨨᨨ鼀�ᆠ
PV PV ҈ ҈ =

\
T҈ T


PV ҈ ҈ = P
´ (
V҈ ) Þ P =  P
҈ T҈
T҈

f æ ö P
V
 PV ( ҈ ҈ ) = ç P҈ ´
V҈  PV ҈ ҈ ÷
㦨
P҈V҈ ø

è
ᆠ㡰 꾀��

犨 
V

㦨
V҈
 
T

㦨

T҈ 

ᨨ鼀
⼄

�㡰
P ( V ) PV
҈ ҈ =
҈ Þ P
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V x=
҈ = ҈ A A ᨨ鼀
⼄

��㡰
PA
kx P҈ = P҈ + Þ kx = ҈   A ␔ᨨ├
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泐
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ᨨ
鼀ᨨ矄 
졔 쟼 �泐
矄ᨨ
�
矄� �
҈ ҈ æ P A ö æ
V ö Wg = P҈ Ax + kx
= P҈ (
V҈ ) + ç ҈ ÷ ç ҈ ÷

è  øè A ø ҈ 䰘 PV =
PV ҈ ҈ + PV ҈ ҈= ҈ ҈   㡰 DQ
㦨
Wg
졔
DU 䰘  = PV ҈ ҈ + ( P
V
 PV ҈ ҈) 
䰘 æ
ö = PV ҈ ҈ + ç P҈ ´ V҈  PV ҈ ҈÷ ø 
è b

ᆠ㡰


�矄

ᆠ矄
�
ᨨ盐鼀
盐� �
鼀� �
ᨨ

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矄 

䰘
⧤

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泐
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 ᆠ

ᨨ�ᨨ
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盐� �
鼀� �
�


dQ �矄 ᨨ矄��㡰
 
矄
犨

矄

犨ᨨ├
ᨨ 
泐
�ᆠ㜠犨 矄泐矄鼀
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㡰
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 鼀ᨨ泐 鼀�ᆠ矄
dU
㦨
dQ

dW ᆠ  �
�
�  
 �泐
ᨨ
�ᆠ㜠犨
矄泐矄鼀

dU
犨�
 ᨨ矄��
ᨨ
�  
 �泐
犨�
� ᆠ矄
DU =








�㡰

WWW.JEEBOOKS.IN

PS-119

Thermodynamics 黨


҈



꾀㜠

錐
㜠
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矄錐
㜠
 꾀㡰 ᨨᆠ矄矄
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ᆠ矄矄 P
µ
T
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W
㦨

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P
㦨
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V
� ᆠ矄矄
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µ
T
矄ᨨ
T
� ᆠ矄矄

 ᆠ
D␔

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 
D턼
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㡰
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®
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㦨


黨 PV ᨨ


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㦨
҈

\

㦨
R 黨 PV ᨨ
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
㡰
㦨
҈ 꾀
\
꾀
㦨
R �



 
ᨨᆠ矄矄
�矄
�矄ᨨ 鼀






\ D
㦨

ᨨ犨

D턼
㦨
D
졔
␔











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␔
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�
 �矄
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矄ᨨ
W
��
�


犨ᨨ鼀
ᨨ 
ᨨ
 
�矄
 
DQ
��
 ᆠ

�矄
ᨨ矄
꾀㡰㜠⼄㡰


꾀矄
 
�
�矄
  矄
�
ᆠ盐盐鼀

 
  ᆠᨨ � 
�矄
� 矄盐
  ᨨ
W
㦨




Q
㦨


 
ᆠᆠᨨ� � ᨨ
�矄
犨
ᨨ
 鼀ᨨ泐 鼀�ᆠ矄 D
㦨
턼
졔
␔
Þ

D
㦨

 ᆠ
 
�矄
ᨨ
ᆠ  �
�
 
鼀盐
ᨨ
 
�矄
ᨨ T
�矄
ᆠᨨ 矄 
㡰㜠

㡰


෌�
P µ

PV
PV
� ᆠ矄
和ᨨ
T nR � ᆠ矄矄
ᨨ盐鼀
� ᆠ矄矄
矄ᨨ
W
� ᆠ矄矄
ᨨ鼀
Q
㦨
DU
졔
W

Q
� ᆠ矄矄
 ᆠ
 
�矄
�� 矄

҈҈
ᨨ鼀
 
��
PV
�

ᔼ
ᨨᆠ矄矄
J
®
K
ᨨ盐鼀
V 㦨
ᆠᨨ 矄 
p
�矄
ᆠ矄� �
 
P
µ
T
 ᨨ

T
矄 ᨨ盐 矄ᨨ
ᆠ矄
\
W
㦨
Pdv
㦨



DV
㦨
DQ




 ��㡰 ᨨ鼀
DQ
㦨
DU
졔
DW ᔼ
ᨨᆠ矄矄
K
®
L p
㦨
ᆠᨨ 矄 
㦨
nCvDT
V
�矄
� ᆠ矄� �
和ᨨ 鼀盐
矄 ᨨ盐
矄ᨨ
� ᆠ矄
\
DW
㦨
Pdv
>



DU
㦨
nCvDT >


 
Q
㦨
mCDT
>

ᔼ
ᨨᆠ矄矄
L
®
M V
㦨
ᆠᨨ 矄 
P
�
ᆠ矄矄
ᨨ
T
� ᆠ矄矄 \
W
㦨



DU
>


 
Q
>

ᔼ
ᨨᆠ矄矄
M
®
J V
�矄
ᆠ矄� �
\
DW



T
�矄
矄ᨨ
ᆠ矄� �
\ DU




 
DQ



҈


ᆠᨨ矄ᨨ � �
㜠
�
ᨨ
   꾀  ��
㜠
�
�矄
矄 ᨨ犨
�
��盐
ᨨᆠ矄矄
꾀
®


꾀
矄�� 
� 
�
V㜠T �
� �ᆠ矄
V
µ
T
\ 矄矄盐
�矄 ᆠᨨ 矄 

ᨨ
V
  ᔴ T

� ᆠ矄� �
 ᨨᆠ矄矄

®
ᔴ
ᨨ盐鼀
�矄
ᆠᨨ 矄 

和� ᆠ
 
鼀盐 �矄
ᆠ矄� �

和ᨨ
矄矄盐
犨�
矄ᨨ
ᆠ矄
ᨨᆠ矄矄
ᔴ
®
꾀

鼀盐
�矄
ᆠᨨ 矄 
盐
ᨨ盐鼀 I ᆠ矄矄

 ᆠ
矄矄盐
犨�
� ᆠ矄
矄
P µ
V 㡰㜠⼄

矄㡰


ᨨ鼀
PV
㦨
nRT Þ
T
㦨


Topic-2 : Specific Heat Capacity and Thermodynamical Processes ҈


҈

ᨨ


㦨
ᆠᨨ 矄  V
ᨨ

nRTV
㦨
ᆠᨨ 矄  \

맔

㦨
ᆠᨨ 矄  꾀矄
 
�矄
  矄
�矄
ᨨ盐鼀
� ᆠ矄矄
矄ᨨ
鼀盐
ᆠ矄矄 ␔
├ ᨨ犨
 
턼
㦨


ᔴ
D













�㡰 ᨨ

ᨨ泐ᨨ�ᆠ
ᨨᆠ矄矄 ᔴ
㦨ᔴ
졔




  
㦨
ᆠᨨ 矄  ҈㜠





㦨
ᔴᨨ 矄 
\

㦨

 \
ᔴ
㦨
ᔴ
졔 㦨
ᔴ


㦨


㦨
҈㜠


턼
㦨
ᔴ
D
㦨

맔
맔
D


�


D
�矄
��

Q
�矄
��
矄ᨨ

�矄
ᨨ矄
꾀矄
ᨨ盐鼀
� ᆠ矄矄

矄ᨨ
鼀盐
ᆠ矄矄
��

҈ P µ
⏌
Þ PV
⏌
㦨
ᆠᨨ 矄 
V
\
n
㦨

    - \ᔴ
㦨
ᔴ
졔
㦨
졔
㦨 
㦨


-҈

҈ æ - ö \
턼
㦨
nCV
DT
㦨
ç ÷ Δ è
ø 꾀矄
D
�矄
��

턼
�矄
ᨨ矄��

和ᨨ
�矄
�� 矄



b㡰 鼀盐
ᆠ  �
DT
�矄
矄鼀
ᨨ

 
ᨨᆠ矄矄矄 A ® B錐 A ® C
 
A ® D DU = nCv DT
㦨
矄鼀

E AB = E AC = E AD





␔ᨨ├
ᨨ 

W = P ´ DV AB ®
ᨨ盐鼀
�矄
� ᆠ矄� �
Þ WAB >
AD ®
ᨨ盐鼀
�矄
ᆠ矄� �
Þ WAD <
AC ®
ᨨ盐鼀
�矄
ᆠᨨ 矄 
Þ WAC =
ᆠ㡰 ᔼ
��ᆠ
ᨨᆠ矄矄 PV g = ᆠᨨ 矄  g æ mö æmö \ P ç ÷ = ᆠᨨ 矄  çèQ V = r ø÷ èrø 꾀矄
鼀矄矄
�矄
ᆠᨨ 矄 

\ P µ rg

ᔼ
P�
 
Pf

 
� ��
 
� 
矄矄盐
ᨨ
 
�矄
  r�
 
r f

 
� ��
 
� 
 矄�泐
ᨨ
 
�矄



g Pf æ r f ö 䰘⏌� =ç ÷ = 
㡰 P� è r� ø nP Þ � = 
� 㡰䰘 ⏌� =
䰘 P�

Þ n =
䰘 = ҈









b㡰 盐矄� �
ᨨ
�盐鼀
ᨨᨨ
�矄
�矄�
 
�
 �矄 ᨨᆠ矄矄
DQ =


矄ᨨ
��ᆠ
㡰 ᨨ鼀
 
ᆠᨨ矄ᨨ � �
㜠
�
��
�
⼄盐矄�ᨨ
ᨨᆠ矄矄
xy
® ᔼ矄ᨨ�ᆠ
  矄�ᨨ
ᨨᆠ矄矄
yz
® ᔼ矄ᨨᆠ ᨨ�ᆠ
矄矄盐
ᆠ矄矄㡰

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EBD_8341 PS-120

Physics

ᨨᆠ矄矄
zx
® ᔼ矄ᨨ 鼀
ᆠᨨ鼀矄矄�ᨨ

 ᨨ

ᆠᨨ矄ᨨ � �
PV
�
�矄
矄
矄 ᨨ犨
�
��盐

҈



҈҈


�


b㡰 ෌�

V҈
㦨
҈
�

P҈
㦨
҈
鼀 V

㦨

�

g
㦨
҈



g g 矄� �

PVr
㦨
ᆠᨨ 矄 
Þ
PV ҈ ҈ = P
V
҈



æ҈ö Þ P
= P҈ ´ ç ÷ è ø

҈ atm


��� PV  P
V
\
␔ᨨ├
ᨨ 

W = ҈ ҈ g ҈ ҈ æ ö ´  ÷҈

҈
� ´ ҈
� ´ ҈
 ç҈´҈ 


��� ø è =
㦨
黨

҈
J






=

\ = ҈�


҈



g ҈

Þ T µ V ҈+

盐矄� �
�㡰

Þ T

҈+g µV


҈+g

҈+g æ
V ö
Þ =ç ÷ = 
㡰
T� è V ø b㡰 DUᆠ
㦨

DUᆠ㡰
㦨


҈

㡰
㦨
҈


 Q
㦨

�

졔



㦨
҈

 ᨨ犨
Q
㦨
DU
졔
W ᨨ ҈

㦨
҈


졔
W ᨨ W
㦨
҈

 ᆠ㡰 꾀矄
 
ᨨᆠ矄矄
�矄
�矄ᨨᆠ ᨨ�ᆠ
矄ᨨ

䰘

 턼 = ᆠ  DT = ´ ´

= 黨
= 黨
´

҈ ; 䰘







b㡰 ␔

��


Tf

黨


҈



V




�㡰 T 쟼⼄盐�ᨨ
ᨨ
��ᆠ
ᨨᆠ矄矄
�矄 TVg҈
㦨
ᆠᨨ 矄  ҈ Þ T µ g-҈ V





҈


ᔴᨨ矄矄
盐
ᨨ
W
㦨
黨

�
J ᨨ 盐矄㡰
␔
├ ᨨ犨
 
 �ᨨ
�鼀


Tµ

䰘


҈



ᆠ㡰 ᔼ  
 �泐
 矄
ᨨ 泐
ᨨ
� ��
 
� 
矄 和ᨨ

D꾀
㦨
D 꾀矄ᨨ
D턼
㦨
D
졔
␔ 꾀矄
␔꾀
>
␔
Þ D턼꾀 >
D턼 b㡰 和盐ᨨ矄
鼀ᨨ盐 
ᨨ
犨
ᨨ

㝐
�鼀



҈�


㝐㡰
�鼀
犨
ᆠᨨ 
� ᨨ
�ᆠ
矄ᨨ


ᆠᨨ 矄盐鼀
�
ᨨᨨ�ᨨ
㦨

矄
�
盐矄�ᨨ

㝐
맔

㦨
҈�


㝐㡰
맔
矄 㝐
맔


҈
맔
҈


㦨
҈�


㝐㡰
맔



맔
҈
� Þ
㝐
㦨



� 㡰 
®
ᔼ矄ᨨ矄�ᆠ


®
ᔼ矄ᨨ 鼀

ᆠ
®
꾀��ᆠ

®
ᔼ矄ᨨᆠ ᨨ�ᆠ b㡰 ᨨ
犨ᨨ├
ᨨ 
泐
 
�矄
盐� �
 
ᆠ泐ᆠ
�矄
⼄盐 ᨨ

ᨨ
� �
꾀ᔴ
҈ \ D␔ = ´
´ � = ҈

b㡰 쟼⼄盐�ᨨ
ᨨ
��ᆠ
ᆠ  �
�矄

g-1
㦨
ᆠᨨ 矄  䰘 䰘 盐
g =

犨
�

g - ҈ = - ҈ � �


�

b㡰 ␔ᨨ├
ᨨ 


҈ ␔
㦨
D
㦨
D
㦨
´

҈´ 䰘
;
黨҈
㡰 쟼⼄盐�ᨨ
ᨨ
 
ᔴ
  = 
-
 -

㡰 
盐矄� �

㦨


鼀盐

=

=


 -




+

 
҈´ Q


㦨
҈
鼀ᨨ
�� 㡰


é

ù ê� ú êë 
úû




 � =
Þ�=
Þ  = 
 



 é �

�
ù
� 



=
ê� ´
´ 
ú = ë

҈
û
҈䰘
ᆠ㡰 
⼄盐�ᨨ
ᨨ
 
� 
�矄

é ҈ æ V ö
ù P = P
ê҈ - ç
÷ ú êë
è V ø úû

 ᨨ

é

҈ù

P


␔ 
҈
㦨

Þ ҈
㦨


ê҈ - ú = ë
û
␔ 


㦨


é ҈ æ ҈ ö ù æ 䰘 P
ö Þ 

㦨


ê҈ - ç ÷ ú = ç ÷ ë
è
øû è
ø PV P V é PV ù DT = T
- T҈ = ҈ ҈ -

êQ T = nR nR ë nR úû æ ҈ ö æ ҈ ö æ P
V
䰘 P
V
ö DT = ç ҈ ҈ - P
V
㡰 = ç ÷  PV ÷ç nR
÷ø è ø è nR ø è
�P V �P V =

=

Q
n
㦨
҈㡰
nR
R

ᆠ
ᨨ

q ᨨ

ᨨ

q ᨨ ᨨ

- 
 +  


졔


㦨


盐 
㦨





\

㦨
 
㦨



ᨨ



᲼矄ᨨ
㦨


-


ᆠ
㦨

桌 



�㡰


��㡰

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PS-121

Thermodynamics





졔


㦨



\



졔


㦨














���㡰 

ᨨ
鼀盐
ᨨ

鼀 �鼀盐鼀

㦨

 �  �� �

⼄

���㡰
泐
�
犨
� 


졔


㡰
㦨


 

\


㦨








  



-


㦨


㦨




P 
㦨
᲼ ᨨ鼀
�㡰桌 \ P=


黨

\
鼀
㦨
᲼ ᨨ鼀
���㡰桌
㡰 쟼 �ᆠ� ᆠ泐
ᨨ

 �� 
�矄
��
泐 ᔴ 犨
㦨 ҈㜠  㦨 㦨 㦨 h㦨 � 턼 ᔴ ᔴ �
Q
ᔴ

ᔴ
㦨
㡰 � ᨨ
鼀ᨨ ᨨᨨ鼀�ᆠ
�矄
ᔴ  㦨

ᆠ㡰 Q PV r = ᆠᨨ 矄  � �
㦨
ᆠᨨ 矄 
Þ
�⏌
㦨
ᆠᨨ 矄 
\
g =  ᨨ鼀
�㡰


��㡰


҈



҈黨








�



ᨨ
 
犨ᨨ
矄
ᨨᆠ矄矄

(

)



)

W = p� V f  V� = ҈
� 䰘 ´ ҈
 = 䰘 ´ ҈



(

)

҈ æ҈ f

ö Pf V f  PV çè ´ ҈
҈
÷ø � � =
g ҈
  黨
D U =
´ ҈
=  ´ ҈




ᨨ鼀
�矄
犨
ᨨ
 鼀ᨨ泐
鼀�ᆠ矄 Q

W
㦨
DU
䰘
黨
´ ҈

 = �

 \ Q = 䰘 ´ ҈
 ´ ҈
=

҈


㡰 t =  æ ö 

2p
ç ÷ èø 㝐  tµ

꾀矄

g҈
㦨
⧤ 和ᨨ

t
µ
g
졔
҈⏌
g +҈

 ᨨ

⼄
㦨
DU =










҈æU ö 㡰 꾀矄

P = ç ÷ èV ø 
盐
= ⧤

 ҈
和ᨨ

 = ⧤

 盐 ҈ = ⧤

᲼꾀矄

㦨
盐
桌 ᨨ
 
  p = ᆠᨨ 矄   ҈

 ᨨ

µ
b㡰 ᔼ
VT
� 㜠ᨨᆠ矄矄


ᔼ矄ᨨ�ᆠ

鼀盐
� ᆠ矄矄
ᆠ
ᨨᆠ矄矄


꾀��ᆠ

矄矄盐
ᆠ矄矄
ᆠ
ᨨᆠ矄矄


ᔼ矄ᨨ�ᆠ

ᨨ盐鼀
ᆠ矄矄

ᨨᆠ矄矄


꾀��ᆠ

矄矄盐
� ᆠ矄矄


 ᨨ
ᨨᆠ矄矄矄
ᆠᨨᆠ泐
矄 
�
P㜠V
��鼀
㡰
㡰 ᔼ
ᆠ泐ᆠ�ᆠ
ᨨᆠ矄矄

ᆠ  �
�
ᨨ
�  
 �泐
�矄
ᨨ
Dᆠ泐ᆠ�ᆠ
㦨

�R DT Dᔴ
㦨
nCv DT = ҈ ´
␔ 

ᔴ
㦨
鼀ᨨ
矄ᆠ� �ᆠ


ᆠᨨ 矄 
ᨨ盐鼀
ᨨ
ᔴ

D
㦨




⧤ \ Dᔴ
㦨
�

ᆠ㡰 ෌�


犨ᨨ├
ᨨ 

W
㦨



 ᨨ

ᨨ
鼀ᨨ矄
ᨨ
�矄

m
㦨

ᨨ
�ᨨ鼀�ᆠ
�矄
g
㦨
҈

␔ᨨ├
ᨨ 
盐� �

��ᆠ
ᆠ  �

mR T҈ - T
㡰 g -҈
´

 DT 㡰
´

 DT 㡰 Þ


㦨

㦨
҈

- ҈




´



㦨



⧤ Þ DT
㦨

´


�
b㡰 ᨨ盐鼀
ᨨ
 
�矄 鼀 =
   矄� �
 g = ᆠᨨ 矄  W
㦨


��ᆠ

(






҈

��

�


҈


Ἐ  æ Ἐö = =ç ÷  Ἐ è  ø






g

ᨨ

҈


㦨

㡰g 䰘 \ g = = ҈

� 㡰 ᔼ ��泐

ᨨ盐鼀
 ҈
㦨
�

l

鼀盐
㦨
T҈
  � g =  ᨨ
鼀ᨨ ᨨᨨ鼀�ᆠ
�矄㡰  �

l ҈


盐鼀
ᨨ
鼀ᨨ矄
ᨨ
�矄

n = =



l
� 泐
 
��ᆠ
ᆠᨨ鼀矄矄�ᨨ 㡰


㦨


䰘l ᨨ
��ᆠ
ᨨᆠ矄矄
T҈V҈g-҈ = T
V
g-҈

æ V҈ ö \
T
= T҈ ç ÷ è V
ø

g -҈

�

-҈

æ �

ö 
⏌  = T҈ ç ÷
㦨
T҈ 
㡰 =
T҈ è

䰘 ø

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EBD_8341 PS-122


䰘


Physics

␔ᨨ├
ᨨ 
�
��ᆠ
ᨨᆠ矄矄 ҈ R T 
T҈ ] nR DT
[ ҈ 黨 = = RT҈ W= � g -҈
é ù êë  ҈úû b㡰 ᔼ
 
�矄
ᨨᆠ矄矄
WAB
�矄
졔

矄
DV
�矄
ᨨ矄��

�
  矄ᆠᨨ 
ᨨᆠ矄矄
WBC
�矄


矄
DV
�矄


 

盐   
ᆠ盐
ᨨ
矄ᆠᨨ 
ᨨᆠ矄矄
�矄
鼀ᨨ \ 
␔ᨨ├





 
矄ᨨ
P
>
P҈
C

P

꾀��ᆠ

P1

A

ᔼ矄ᨨ 鼀 B

P2


黨






�


b㡰 ᨨ鼀
 
P

T
�
犨
� 
AB
ᨨ


�矄ᨨ 鼀 ᨨᆠ矄矄

AC
�矄
��ᆠ
ᨨᆠ矄矄
��

꾀矄ᨨ
ᨨ

  矄�ᨨ

ᨨᆠ矄矄

 
矄ᨨ
ᨨ
��ᆠ
ᆠ盐
�矄
鼀ᨨ
ᨨ
犨
ᆠ
矄泐  
 

盐 
 
P

V �
ᨨ
�矄ᨨ 鼀
ᨨᆠ矄矄
�矄 鼀ᨨ
 
��ᆠ
ᨨᆠ矄矄
ᨨ
矄鼀
� ᆠ矄
�
ᨨ盐鼀㡰
 ᆠ
�
㡰
ᆠᨨᆠ泐
�ᆠ矄
ᆠᨨ矄ᨨ � �


 �
㡰 ᨨ
ᆠ泐ᆠ�ᆠ
ᨨᆠ矄矄錐 Qᆠ泐ᆠ�ᆠ
㦨
WAB
졔
WBC
졔
WCA
㦨
҈


졔


졔
WCA
㦨
�
 \ WCA
㦨

�
 b㡰 ᨨ
��ᆠ
ᨨᆠ矄矄
PV g
㦨
ᆠᨨ 矄  꾀矄ᨨ
ᨨ
鼀ᨨ ᨨᨨ鼀�ᆠ
�矄
�├
�盐鼀

g
㦨
҈

䰘 ᨨ
�ᨨ鼀�ᆠ
�矄
�├
ᨨ 泐�

g
㦨
҈



g-҈

æV ö = T
V
Þ T҈ = T
ç
÷ è V҈ ø V҈
T҈ = T�
T
= aT�
g
㦨
҈


 ᨨ
�ᨨ鼀�ᆠ
�矄㡰
V
= 
҈

-҈


aT é҈ù é҈ù \ T� = aT� ê ú = aT = �




\

㦨


�ê �ú
ë 
û
ë û 



㡰 ᨨ
��ᆠ
ᨨᆠ矄矄

TV g -҈ =
ᆠᨨ 矄  ᨨ

T҈V҈g -҈ = T
V
g -҈ 䰘 V T҈ =

°ᔴ +
䰘 =
黨 ⧤

V
= ҈
 
g = � ҈



⏌�

䰘


\ T҈V҈g ҈ = T
V
g ҈ ҈

-҈

æ ö çV ÷ Þ T
= 

㡰 ´ ç ҈ ÷ Þ T
㦨

맔҈
㡰


çç V҈ ÷÷ è ҈
ø ᔼ
�矄
⼄盐�ᨨ

PV
㦨
nRT

nRT P Þ V 㦨
kT
矄� ᆠ
矄矄盐
�矄
ᆠᨨ 矄 
ᨨ
�矄ᨨ�ᆠ ᨨᆠ矄矄㡰 和ᨨ

盐� �
�矄ᨨ�ᆠ
ᨨᆠ矄矄 V

㦨
kT



�㡰


㦨
kTf


��㡰 ��� �
�㡰
泐
��㡰 ҈ T
=
Tf

V





⏌

 dPV dV ᨨ

�矄ᨨ 鼀
ᨨᆠ矄矄錐

PV
㦨
ᆠᨨ 矄  \ PdV
졔
VdP
㦨

dP =K \ P
㦨
盐├
鼀ᨨ盐盐矄㡰 dV ⏌ V

b㡰 盐├
鼀ᨨ盐盐矄

K
㦨


盐� �
 
ᨨᆠ矄矄

ᆠ  �
�
�  
 �泐 � DU = NCV DT = � ´ ´

 ´

 ;

´ ҈
 
㦨 X ├
\ X =


҈
҈
㡰
ᨨ

��ᆠ
ᨨᆠ矄矄


g҈
㦨
ᆠᨨ 矄 

\ V=

和� ᆠ

g�ᨨ鼀�ᆠ


g鼀ᨨ ᨨᨨ鼀�ᆠ \P�ᨨ鼀�ᆠ
>
P鼀ᨨ ᨨᨨ鼀�ᆠ  ᆠ
�
҈
�矄
ᨨ
�ᨨ鼀�ᆠ
�.e.

ᨨ 泐�
 
�


�矄 ᨨ
鼀ᨨ ᨨᨨ鼀�ᆠ
�.e.

ᨨ
�盐鼀
㡰 ␔ᨨ├
ᨨ 
�矄
⼄盐
ᨨ

盐 
 
ᆠ盐
ᨨ
PV ��鼀

 ᆠ
ᨨ鼀
 
��鼀
w

>
w҈
>
w
㡰 ᨨ鼀
TVg
҈
㦨
ᆠᨨ 矄  � ᨨ
鼀ᨨ ᨨ鼀�ᆠ
�矄
g
㦨

 ᆠ
TV
⏌
㦨
ᆠᨨ 矄  
⏌
⏌ Q
V
µ
L㡰 \
T҈ L҈ = T
L


æL ö T  ᆠ

҈ = ç
÷ T
è L҈ ø

g -҈

æ ҈ö Þ
黨 = T
ç ÷ Þ T
=
黨҈
㡰
⏌ � ; 䰘
⧤ è ҈
ø DT = 䰘
-
黨 =

 ⧤

҈






g-҈

æV ö T҈ V҈ 㡰 g -҈ = T
ç ҈ ÷ è ҈
ø

P

҈


㡰 
犨ᨨ├
ᨨ 
盐� �
 
ᆠ泐ᆠ
㦨

 ᆠᨨ矄
�
  P

V
ᆠ盐 
㡰 ᨨ

��ᆠ
ᨨᆠ矄矄

泐� �
TVg҈
㦨
ᆠᨨ 矄 

T҈V҈g-҈




V1














Tf
㦨

T

㦨



맔


맔
҈
㡰



㦨҈
҈

K ҈
䰘䰘
ᨨ
҈
䰘
㡰  �  ҈ ᨨ
鼀ᨨ ᨨ鼀�ᆠ
�矄

g =   ҈⏌
ᔼ
��ᆠ
ᨨᆠ矄矄 
P҈҈g
㦨


g

Þ

҈


҈


҈



҈ 
҈㡰�⏌ = 

҈ 㡰�⏌


WWW.JEEBOOKS.IN

PS-123

Thermodynamics

Þ 
=

�⏌

҈ æ ҈ ö
çè
÷ø

=

҈ ҈









ᆠ㡰 
PV
�
ᨨ
 
��
VT
�
�矄
��
ᨨ犨


 ҈҈ - ҈ 
҈ 㡰 ҈҈ - 

҈

= 꾀 
␔�
= � g -҈ -҈
   ⏌
㡰 黨 = ҈ ҈ = ҈҈
⏌
ᔼ
�矄ᨨ 鼀
ᨨᆠ矄矄



㡰 ҈






㡰





黨






҈










⧤
쟼 = ᔴ鼀� ´


ᨨ鼀

T҈V҈g

҈
㦨
T
V
g

҈ ᨨ

T

㦨
T҈
㡰

⏌�
㦨


T


\ ⧤
쟼 = ᔴ鼀� ´
=







㡰 ␔ᨨ├
ᨨ 
盐� �
ᆠ泐ᆠ�ᆠ
ᨨᆠ矄矄
҈®
®®
҈㡰
W



㦨

 ᆠᨨ矄
�
 
ᨨᨨ
=





= Q 


㦨








\





= \ ␔ᨨ├
ᨨ 
W
= [ 矄 = ҈] ҈


㡰 ᨨᆠ矄矄
҈
®


�矄
�矄ᨨ�ᆠ ᨨᆠ矄矄


®

�矄
�矄ᨨᆠ ᨨ�ᆠ ᨨᆠ矄矄

®


�矄
�矄ᨨ�ᆠ ᨨᆠ矄矄


®
҈
�矄
�矄ᨨᆠ ᨨ�ᆠ  ᆠ
ᨨ
��ᆠ
ᨨᆠ矄矄
�矄
�鼀ᨨ
ᆠ㡰




 D턼҈®
 =  ᔴ D  =  ᔴ (

-
)  =  ᔴ


 D턼
®  =  D   =  ᔴ  D  =  ᔴ 
 ᔴ � D턼҈®
\ = = D턼
® ᔴ  


㡰 D턼®
= ᔴ


ᔴ


D턼҈®

\ = =


҈ D턼®


ᔴ









b㡰 ᨨᆠ矄矄
꾀
ᨨ
 

TA
㦨
TB
\
UA
㦨
UB 꾀矄ᨨ
P

V

㦨

맔


V









WAB = nRT
ᨨ� e


�


V 


V


P

␔ᨨ├
ᨨ  Þ PB =

҈



æ ö ҈

\
 = 
ç
÷ = 
(
)  è ø 㦨



㡰҈


㦨






㦨
黨



␔ᨨ├
ᨨ 
盐� �
 
ᨨᆠ矄矄
  - ҈ ҈ 黨


´ 
⏌
㡰 - 

҈

␔=

= = ҈- g ҈ - ҈


盐



㦨



矄

㦨
�
졔
҈
㦨

㡰 ҈ (

) \ ␔= = -҈
\

␔
㦨
҈
RT




�
⧤
쟼

ᨨ
 
�矄
鼀� 盐


 
㡰



㡰


æ ö ␔�矄ᨨ
㦨




m
ᨨ�҈
ç
÷ è ҈ ø æ
 ö Þ ␔�矄ᨨ
㦨
҈҈
 ç ҈ ÷ =
҈҈ ᔼ
è ҈ ø ␔
  ᔼ
҈
\ �矄ᨨ = ҈ ҈ = 
= 
黨 ␔� 黨 ҈҈
҈
和ᨨ

= =҈
䰘䰘䰘
»҈
䰘
黨 


�㡰 ᨨ
��ᆠ
  矄�ᨨ

P P PVg
㦨
Pa
맔

V㡰g
Þ Pa
㦨
g = ҈

䰘

ᨨ
�矄ᨨ 鼀
  矄�ᨨ

P P
맔
V
㦨
P�
맔

V
Þ P�
㦨

Pa

P = ´ = \
㦨





P�
҈

䰘 P
҈

䰘 T盐㡰 
矄ᨨ
ᨨ
P㜠V
ᆠ盐
�矄
鼀ᨨ
ᨨ
��ᆠ ᨨᆠ矄矄
 
ᨨ
�矄ᨨ 鼀
ᨨᆠ矄矄

ᨨ鼀
 
�
�
�矄 ᆠ
 
矄ᨨ
ᨨ
B
�矄
�
 
 
矄ᨨ
ᨨ
A. 和ᨨ
  �矄ᨨ 鼀
ᨨᆠ矄矄
�矄
矄 
泐
 
ᆠ盐
A


b

㡰 꾀��ᆠ
ᆠᨨ 矄 
ᨨ
 
�矄
鼀� 盐
� 䰘

҈ᔴ҈ +
ᔴ
� ´
+ ҈ ´
g鼀 = = = ҈



҈ᔴ ҈ +
ᔴ
� ´  + ҈ ´ �

ᨨ

��ᆠ
ᨨᆠ矄矄

Pv g
㦨
ᔴᨨ 矄 




V



V
V


ᔴ

꾀 T


T


㦨
P
V
ᨨ� 
᲼Q P0V0 = nRT0 桌


ᨨᆠ矄矄
ᔴ
�矄
ᨨ
ᆠ

 ᨨ
ᨨ
盐�鼀 
ᆠ


鼀

C
�� �
P
 
T
b

㡰 㡰 ᨨᆠ矄矄
A

B
�矄
ᨨ
�矄ᨨ 鼀

ᔼ
ᆠ矄
ᨨ

�矄ᨨ 鼀 ᨨᆠ矄矄
犨
�

ᆠ �盐
泐ᨨ
�

㜠
��鼀
㡰 ᔼ
ᨨᆠ矄矄

®
ᔴ
®


D
�矄
��


ᆠ矄矄  
ᨨ盐鼀
矄ᨨ
ᆠ矄矄

  ᨨ
␔
�矄
��

ᨨ鼀 �矄
犨
ᨨ
 鼀ᨨ泐 鼀�ᆠ

턼
�矄
��
�



 
�矄
 
ᨨ矄矄


WWW.JEEBOOKS.IN

EBD_8341 PS-124






Physics ᔼ
턼҈®

㦨
ᔴD ҈®

㦨
ᔴ

�㡰

ᆠ㡰 ␔ 꾀
>
␔ ᔴ

 ᨨ
犨ᨨ├
ᨨ 
盐� �
 꾀
®

®
ᔴ
�矄
ᨨ矄��
㡰 ␔ᨨ├
ᨨ 
�矄
ᆠᨨᆠ├犨�矄
ᆠ泐ᆠ
�


��鼀
�矄
ᨨ矄��


b

ᆠ

㡰 㡰 ᨨ

 鼀
ᨨᆠ矄矄矄

ᆠ  �
�矄
�  
 �泐
�


ᆠᨨ 矄 
矄矄盐
ᨨᆠ矄矄

� é  

㡰 

ù = ҈´ ê
᲼Q

㦨
桌
ë úû � � æ ö �
= 

= ç
÷ =

è  ø
\
ᔼ



DU = nCv DT 犨  DT = (T
- T҈ )

b㡰 꾀ᆠᆠᨨ� �
ᨨ
�矄
犨
ᨨ
 鼀ᨨ泐 鼀�ᆠ矄


DQ = DU + DW


䰘


ᔼ

��ᆠ
ᨨᆠ矄矄
DQ =

ᨨ

= DU + DW ᨨ
DU = DW ᆠ㡰 ᔼ
 
�矄ᨨ 鼀
ᨨᆠ矄矄

DT =
\
DU
㦨

Q DU
㦨
nCv
DT 㡰 ᔼ
 
��ᆠ
ᨨᆠ矄矄

DQ =

㡰 ᔼ㡰 꾀�b�ᆠ
ᨨᆠ矄矄


ᨨ
 ᆠ  �
ᨨ

├矄
ᆠ 犨�
矄盐ᨨ盐 � �矄



黨


ò P × dV =


 ᆠ
犨ᨨ├
ᨨ 
�矄
ᨨ
ᔼ㡰
ᔼ
�矄ᨨ�ᆠ
ᨨᆠ矄矄
矄矄盐
鼀� 矄
ᆠᨨ 矄 
W = P × DV ¹
f f DU = nR DT = ᲼ P DV 桌 ¹




\ DQ = nC p DT ¹



҈ ᲼Q 

=
�� 桌   \ ᔼ
®
턼 ᔼᔼ
D҈®
®
㦨
D҈®

졔
D
® 㦨
ᔴD ҈®

졔
ᔴD
®   = ҈´  - � 㡰҈®
+ ҈´ ᲼ - � 桌
®

  é 
ù ´

- 
´

ú 㦨
᲼


- 

桌 + ê

ë
û ҈ 㦨



㦨

맔
RT
㦨
RT
 ᔼᔼ
®
ᔼᔼᔼ
턼҈®
®
㦨
턼҈®

졔
턼
® 㦨
ᔴD ҈®

졔
ᔴD
® �  é  ù = 

+ ҈ ´ ê
´

- 


㡰 ú

ë
û







= ´ =


  ᔼᔼᔼ

和 =



l
+
= l
[Q D҈®
=
]   ᔼ

 ᆠ㡰 ᨨᆠ矄矄
ᔼ
�矄
��ᆠ
  ᨨ
D턼
㦨

ᨨᆠ矄矄
ᔼᔼ
�矄
�矄ᨨ�ᆠ
P
㦨
ᆠᨨ 矄 
  ᨨ
␔
㦨



҈㡰 㦨




㡰
㦨



ᨨᆠ矄矄
ᔼᔼᔼ
�矄
�矄ᨨᆠ ᨨ�ᆠ
V
㦨
ᆠᨨ 矄 
  ᨨ ␔
㦨



҈㡰
㦨

ᨨᆠ矄矄
ᔼ
�矄
�矄ᨨ 鼀

鼀盐
T
㦨
ᆠᨨ 矄 

\D盐
㦨

㡰 ᆠἘ矄
ᆠᨨᆠ�ᨨ
ᨨ
 
矄
ᨨ
矄ᨨ盐 
�
� �矄
�矄

ᨨ
��ᆠ
ᨨᆠ矄矄
P

V
ᆠ盐
�
��ᆠ
ᨨᆠ矄矄
�矄
矄
 
�矄ᨨ 鼀 b㡰 ␔ᨨ├
ᨨ 
�
�矄ᨨᆠ ᨨ�ᆠ
ᨨᆠ矄矄
�矄
ᨨ
ᨨ
犨 �ᆠ
犨 �

�ᆠ
� 
�
㜠
�

�
s㡰
㡰 D
㦨
D턼

D W
㦨

PDV
㦨
P
V


V҈㡰
㦨
PV

졔
PV҈ 犨 �ᆠ
�矄
 
ᨨ鼀盐
ᨨ
�矄ᨨ�ᆠ
ᨨᆠ矄矄

矄ᨨ
�
P㡰 㡰 ␔෌쟼
㦨








㡰
㦨

҈


␔෌
㦨








㡰
㦨






Pf Vf  PV � � P


㡰 - 


㡰 ␔ 㡰��ᆠ =
㦨



= � ҈ g ҈ æVf ö æV f ö ␔ ෌㡰
�矄ᨨ 鼀
= nRT  ç ç V ÷÷ = P
V
 çç V ÷÷ è � ø è � ø

=

㡰 ᔼ
␔҈®
®
㦨
␔҈®

졔
␔
® ᲼␔
®
㦨



DV
㦨

桌 㦨


맔


졔
졠

= 

=

éQ = ҈
  ␔
® �矄ᨨᆠ ᨨ�ᆠ ù ê ú ë ᨨ ᆠ  � � ᨨ盐鼀
D =
û

\ ᔼ

 ᔼᔼ
D҈®
®
㦨
D҈®

졔
D
® 㦨 졠 + ᔴ D = D [ ᨨᆠ矄矄
҈ ®

�矄ᨨ 鼀�ᆠ
D
㦨

]



ö  æ = ҈´ ç
- ÷ =
᲼ ᨨ
鼀ᨨ ᨨᨨ鼀�ᆠ
�矄

㦨
桌
è  ø \
ᔼᔼ

ᔼᔼᔼ
턼҈®
®
㦨
␔҈®
®
졔
D҈®
® ᨨ鼀
�矄
犨
ᨨ
 鼀ᨨ泐 鼀�ᆠ矄
= l
+

= [ l
+ ]  \
ᔼᔼᔼ



ᔼ
턼҈®

㦨
␔҈®

졔
D҈®


ᔼᔼ㡰 ᔼ矄ᨨh鼀
ᨨᆠ矄矄


鼀盐
鼀� 矄
ᆠᨨ 矄  f \ DT =
Þ DU = nR DT Þ DU =

ᨨ
ᆠ  �
�
�  
 �泐
᲼DU
㦨

桌
ᔼᔼᔼ㡰
ᔼ矄ᨨᆠ ᨨ�ᆠ
ᨨᆠ矄矄
ᨨ盐鼀
鼀� 矄
ᆠᨨ 矄 






Vf

+ Pdv V�


 =
҈
l
+ ᨨ  
RT =

 㦨
nRT



Þ DQ =


DV =
Þ W =

ᆠ㡰 ᔼ

␔҈®
® = ␔҈®
+ ␔
-

�



�҈


�

�


�



WWW.JEEBOOKS.IN

PS-125

Thermodynamics




㦨





ᨨ�

�
㦨
҈





ᨨ�
㡰
�矄
 
ᆠᨨᆠ
ᨨ�ᨨ

㡰 ᨨ鼀
DQ
㦨
msD

DQ




DT = = = �
°C ms ҈ ´


æ ҈ ö � 㡰 DT = V gDT = çè ÷ 黨 ´ ҈
( �
) 黨


ø 㦨
�
맔
҈
䰘
鼀 \
W
㦨
p
DV
㦨
҈
㡰�
�
맔
҈
䰘㡰
㦨



� 㦨
҈



㡰
ᨨ�


��


(

)

ᆠ㡰 DU = DQ  W = (






�) J = ҈黨黨黨黨
黨� J �



VB

ò

V 㦨
ᆠᨨ 矄 
ᔼ矄ᨨ�ᆠ
ᨨᆠ矄矄㡰 T VA VB \ = TA TB é VB ù V =
ú Þ TB
㦨
B ´ TA 㦨


맔



㦨




K êQ VA ë VA û 㡰 
矄ᨨ
ᨨ
矄 ᔼ
ᨨᆠ矄矄

A
ᨨ
B
�矄

�矄ᨨ�ᆠ
ᨨᆠ矄矄㡰 � \ Q
㦨
nCpDT

㦨


맔
R ´ 


㦨
҈�


R
� é ù êQ C p =
R ᨨ 鼀ᨨ ᨨᨨ鼀�ᆠ �矄 ú ë û 
�矄
矄ᨨ
ᔼ
ᨨᆠ矄矄

B
ᨨ
C
�矄

�矄ᨨ 鼀
ᨨᆠ矄矄㡰
\ ᔼ  
 �泐
ᆠ  �
dU
㦨


Q
DT
㦨

\ ᨨ鼀
�矄
犨
ᨨ
 鼀ᨨ泐 鼀�ᆠ矄
dQ
㦨
dW Vf \ Q
㦨




맔
nRT
ᨨ�҈

V� 㦨





맔


맔
R
맔




맔
ᨨ�҈


㦨

䰘



맔
ᨨ�҈



맔
R
㦨

҈


R 矄ᨨ㡰 ᔼ
ᨨᆠ矄矄

C
ᨨ
D
�矄

�矄ᨨᆠ ᨨ�ᆠ
ᨨᆠ矄矄㡰 dW
㦨


Q dV
㦨

æ ö \ Q
㦨
nCv
DT
= n ç R ÷ ( TA - TB ) è
ø  R 㦨


맔
맔



㡰
㦨

黨


R
矄㡰

ᔼ
ᨨᆠ矄矄

D
ᨨ
A
�矄ᨨ 鼀
ᨨᆠ矄矄㡰 Vf \ Q
㦨




맔
nRT
ᨨ�҈
V� æVf ö 㦨




맔


R
맔



맔
ᨨ� ç ç V ÷÷ 㦨


҈


R è � ø 
�矄
矄 ᆠ㡰 ᨨ
犨ᨨ├
ᨨ  W
㦨
QA
®
B
졔
QB
®
C
졔
QC
®
D
졔
QD
®
A 㦨
҈�


R
졔

҈


R㡰

黨


R
졔

҈


R㡰
㦨




R 㡰 盐鼀
ᨨ
鼀ᨨ矄

n
㦨



T҈
㦨


⧤ 盐� �
 
ᨨᆠ矄矄
꾀
®
 pT
㦨
ᆠᨨ 矄 
ᨨ
p
V
㦨
ᆠᨨ 矄 
㦨
K
矄泐㡰 \

p=

K V

VB

K V

dV

㦨

K VB  VA 桌 㦨

᲼ KVB  KV A 桌 㦨

᲼

( P V )V
B B

B

( P V )V
A A



A桌

㦨



᲼PBVB

PAVA桌 㦨



᲼nRTB

nRTA桌
㦨


nR ᲼T҈


T҈桌 㦨


㡰

㡰
R㡰
᲼






桌
㦨

҈


R \ ␔ᨨ├
ᨨ 
ᨨ
 
�矄
�
 
ᨨᆠ矄矄
AB
�矄
҈


R
꾀 
鼀hᨨ 

㡰 ᔼ
 
��
V㜠T
�

AB
�矄

矄�� 
�  \

�䰘


ò

\ WA® B = p
 = VA VA

ᔴ




P҈




҈

҈

T҈

꾀


T҈



pV
㦨
nRT \ pdV 졔 Vdp = nRdT

( nRT )


dp = nRdT


�㡰 P ᨨ鼀
 
��
ᆠᨨ ��ᨨ

pT
㦨
ᆠᨨ 矄  pdT
졔
Tdp
㦨




��㡰 ᨨ鼀
쟼⼄矄

�㡰
 
��㡰

犨
� pdV
㦨

nRdT ᨨ pdV 졔

\ WA® B =

ò

pdV =
nR

TB

òT

dT =
nRTB  TA 㡰

A

㦨

nR
T҈


T҈㡰
㦨

㡰

㡰
R㡰







㡰 ᨨ
WA® B = ҈


R 㡰 ᨨ

㦨
ᔴᨨ 矄


㝐ᨨ

ᆠᆠ�泐 R R R R C= + = + � ҈ g ҈ ҈ x ҈ ҈ 



㦨
ᆠᨨ 矄 
ᨨ
҈⏌


㦨
ᆠᨨ 矄  ҈ \x =
Þ C = 
�R 턼꾀
®

㦨
ᔴ
D
㦨


�
㡰







㡰
㦨


҈


ᨨᆠ矄矄


ᔴ

ᨨᆠ矄矄
�矄
�矄ᨨ�ᆠ æ� ö 턼
®
ᔴ
㦨
ᔴ
D
㦨

㡰
çè R ÷ø
 ᔴ

㡰
æ� ö 㦨


çè R ÷ø 
҈

҈㡰
㦨
� 㡰







㡰
㦨
҈�



�矄
矄ᨨ ᨨᆠ矄矄
ᔴ

꾀

ᨨᆠ矄矄
�矄
�矄ᨨ 鼀

æ PC ö è PA ÷ø

D
㦨


\
D盐
㦨


和ᨨ

턼ᔴ
®
꾀
㦨
␔ᔴ
®
꾀
㦨
ᔴ
 ç

æ
P҈ ö
㦨

㡰
 㡰


㡰
 
㡰
㦨
҈


맔


黨
㦨

҈㡰
 ç è P҈ ÷ø 턼ᔴ
®
꾀
㦨

҈



矄ᨨ㡰

WWW.JEEBOOKS.IN

EBD_8341 PS-126

�



Physics

n
㦨
ᨨ

ᨨ
鼀ᨨ矄
㦨


꾀㡰 
ᆠᨨ鼀
ᨨᆠ矄矄
ᨨ
㜠
��鼀
�矄
矄 ᨨ犨
ᨨ犨






ᔴ � �

ᆠᨨ 矄  ᨨ盐鼀


꾀��ᆠ ᆠᨨ鼀矄矄�ᨨ

҈

꾀

҈




㡰
�㡰
ᨨ
犨ᨨ├
ᨨ  W
㦨
WAB
졔
WBC
㦨


҈

ᔼ矄ᨨᆠ ᨨ�ᆠ
ᨨᆠ矄矄㡰 

 PV ҈ ҈ - P
V
㡰 +
 g - ҈㡰 ᲼
Q WBC
㦨
PDV
㦨
P
맔


㦨

桌 g

æV ö 꾀ᆠᆠᨨ� �
ᨨ
ᨨ�矄矄ᨨ Ἐ矄
犨


P

㦨
P҈ ç ҈ ÷ è V
ø g ù æV ö ҈ éê P҈V҈ - P҈ ç ҈ ÷ V
ú \ W
㦨
g -҈ ê ú è V
ø ë û g -҈ é V҈ æ V҈ ö ù æ ҈ ê �ö P҈V҈ - P҈
V


ç ÷ ú çQ g = ÷ 㦨
ø g -҈ ê V
è V
ø ú è ë û
⏌ é æ V ö
⏌ ù æ V҈ ö ù   éê ҈ ú P V P V P V W
㦨

㦨
҈ ҈ ê҈ - ç ÷ ú ÷ ҈ ҈ ҈ ҈ç V

ê V ê ú è
ø úû
è ø ë ë û ��㡰 ᨨ
ᆠ  �
�
�  
 �泐


DU
㦨
DUAB
졔
DUBC 㦨
Q

W é æ V ö
⏌ ù  P V 㦨
Q

҈ ҈ ê҈ - ç ҈ ÷ ú
ê è V
ø ú ë û ᲼ᆠᆠᨨ� �
ᨨ
�矄
犨
ᨨ
 鼀ᨨ泐 鼀�ᆠ矄桌 é B ® C Q = DU BC +
ù ê A ® B Q = DU + W ú AB ë û ���㡰 ᨨ
ᨨᆠ矄矄
BC


DUBC
㦨
nCv
DT
㦨
Q ᲼Q WBC
㦨


桌  R = R
ᨨ
鼀ᨨ ᨨᨨ鼀�ᆠ
�矄
Cv
㦨
g -҈
Q R
DT Þ
DT
㦨
\ DUBC
㦨
Q
㦨


맔

R
꾀ᆠᆠᨨ� �
ᨨ
ᨨ�矄矄�ᨨ Ἐ矄
犨

g -҈

ᨨ
 
ᨨᆠ矄矄
AB

TAVB

æV ö ᨨ TB
㦨
TA ç ҈ ÷ è V
ø

�黨


g-҈

g -҈

= TBVB

PV æ V ö = ҈ ҈ç ҈ ÷ nR è V
ø

g-҈

P g ҈-g PV � ⏌ V
-
⏌  \ TB
㦨
҈
V҈
V
= ҈ ҈
R
R � ⏌  -
⏌  PV V
Q  ᆠ

TC
㦨
TB
졔
DT
㦨
҈ ҈ +
R R 盐鼀
ᨨ
鼀ᨨ
n
㦨
҈

ᨨ
鼀ᨨ ᨨᨨ鼀�ᆠ
�矄

Cp
㦨


�R R
Cv =



㡰 ␔ᨨ├
ᨨ 
泐
 
�矄

w
㦨
꾀
ᨨ
DABC ҈ ҈ W
㦨

맔
矄
맔
�� 
㦨

V

맔

P

㦨
P
V


㡰 
ᆠ
泐
 
�矄
�
 

CA
�



�矄ᨨ�ᆠ ᨨᆠ矄矄 DQCA
㦨
nCp
DT
㦨
҈
맔
�R⏌
㡰
TA
 TC㡰

PV
P
V

TA =

I ´R I´R � R é P
V

P
V
ù � = - P
V
DQCA 㦨
ê ú
ë R
R û 
矄ᨨ
泐
 
�矄
�
 

AB
�.e


�矄ᨨᆠ ᨨ�ᆠ ᨨᆠ矄矄 DQAB
㦨
nCvDT
㦨
҈
맔
R⏌
㡰
TB

TA㡰 R é P
V
P
V
ù = P
V



㦨

êë ҈ ´ R ҈ ´ R úû ᆠ㡰 꾀矄
DU
㦨


�
ᆠ泐ᆠ�ᆠ
ᨨᆠ矄矄

 ᆠ

ᨨ鼀
�矄
犨
ᨨ  鼀ᨨ泐 鼀�ᆠ矄
DQ
㦨
DU
졔
DW \ DQ
㦨
D␔ DQAB 졔
DQCA
졔
DQBC
㦨
DW PV PV DQBC
㦨
P
V




=



é
P
ù 㡰 쟼⼄盐�ᨨ
ᨨ
� 
BC
P
㦨

ê ú
V
졔
�P

ë V
û (Q y = mx + c ᨨ
P = mV + C )

TC
㦨


RT
᲼ ᨨ
ᨨ 
鼀ᨨ桌 V
P \ RT
㦨


V
+ �P
V V
P
㦨


ᨨ
鼀 �鼀盐鼀錐






�㡰


P dT =


-
맔

V
졔
�P

㦨

錐 V
dV

�V




��㡰
 ᆠ
ᨨ鼀
⼄盐�ᨨ
�㡰
 
��㡰

\ V
㦨





-
P
æ �V
ö æ �V ö ´ç + � P
ç
÷ RT鼀
㦨
è
÷ø è
ø V

�
�P
V

� P
V
+ = 㦨


P
V

맔
҈



� P
V
\ T鼀
㦨

R V VA  = ҈

C =



෌�

 VB VB TA
㦨



K

TB
㦨
䁸

TD
㦨
䁸

h
㦨
䁸

n
㦨
҈

ᨨ
�ᨨ鼀�ᆠ
�矄

g
㦨
҈

ᨨ
��ᆠ
ᨨᆠ矄矄
A
®
B g -҈ = TBVB g-҈ æV ö ᨨ
TB = ç A ÷ TA è VB ø

ᔴ

g -҈ TAVA

 꾀 

= ҈
㡰


⏌�

´ 

= 黨
黨 K

\ ᨨ
�矄ᨨ�ᆠ
ᨨᆠ矄矄
B
®
C
矄矄盐
㦨
ᆠᨨ 矄  VB VC æV ö =

ᨨ



TC
㦨
TB
ç C ÷
㦨
黨
黨
᲼
桌
㦨
҈
҈

K TB TC è VB ø

WWW.JEEBOOKS.IN

PS-127

Thermodynamics ᨨ
��ᆠ
ᨨᆠ矄矄
C
®
D

VA V V =
Q
A = ҈
 
C =
錐 \
VC VB VB g -҈ TCVC


҈


= TDVDg -҈

ᨨ

ᨨ

鼀ᨨ矄

n
㦨
҈

R
㦨


҈ 鼀ᨨ҈
k҈

TA
㦨
҈



⧤

g -҈ ҈
҈
é҈ù é VC ù = 㦨
䰘黨҈K \ TD
㦨
TC
ê ú 㦨
҈
҈

êë
úû 

㡰҈⏌ � ë VD û ᨨ
B
®
C
ᨨᆠ矄矄



矄ᨨ
Q҈
㦨
nCp
TC 
TB㡰 gR 䰘 ⏌ �㡰 R TC - TB 㡰 = ҈ ҈
҈
- 黨
黨㡰 㦨
n g -҈ 
⏌ �㡰 䰘R ´ 黨
黨 @ ҈

R 㦨

ᨨ
D
®
A
ᨨᆠ矄矄



矄 R T - TA 㡰 Q

㦨
nCv
TD

TA㡰
㦨

n g -҈ D �R R ´
黨҈ 䰘黨҈ - 

㡰 = 㦨
҈

⏌ �㡰
Q ᨨ

�矄
 ᆠ  �
�
��ᆠ
ᨨᆠ矄矄矄㡰
nR T - TA 㡰 ᨨ犨

WAB
㦨
g -҈ B �R R ´

黨 黨

- 

㡰 = 㦨

⏌ �㡰
WBC
㦨

nR
TC

TB㡰
㦨
҈맔
R
҈
҈


黨
黨㡰
㦨
黨
黨
R nR R TC - TD 㡰 = + ҈
҈
- 䰘黨҈㡰 WCD
㦨
g -҈ 
⏌ �㡰 �R ´ ҈

䰘
㦨

�R �R W 
㦨
黨
黨
R
졔

҈

䰘



黨㡰
㦨
黨
黨
R
졔

맔

҈


㦨
黨
黨
R
졔
҈

�
R
㦨
҈黨�

R ҈黨�
R \ 쟼 �ᆠ� ᆠ泐
㦨
҈


맔
W ⏌Q҈㡰
㦨
҈


맔
㦨

҈

奠 ҈

R � �㡰 T�
㦨

䰘
졔

䰘
㦨



K錐

g
㦨




 ᨨ
鼀ᨨ ᨨᨨ鼀�ᆠ
�矄㡰  和� ᆠ
 
�矄
  矄
��ᆠ泐

矄ᨨ
盐矄� �
⏌�

T�V�g -҈ = T f V fg -҈






�㡰 WAB
��ᆠ
ᨨᆠ矄矄㡰

nR᲼TA - TB 桌 g -҈ g ҈

g ᨨ鼀
T
P
㦨
ᆠᨨ 矄 

WAB
㦨


g ҈-g

TA PA

g ҈-g

= TB PB

æP ö Þ ç A÷ è PB ø

҈-g

g-҈

g

æT ö = ç A ÷




�㡰 è TB ø ҈-� ⏌ 

éP ù g é ù �⏌ Þ TB
㦨
TA
ê A ú
㦨

�

⧤
= ҈


ê ú ë
û ë PB û ҈ ´

҈᲼҈


-
�
桌 \ WAB
㦨

㦨
҈
䰘

 �⏌ -҈ ��㡰 H
ᨨ矄
�
h
ᨨᆠ矄矄

®
ᔴ
�矄ᨨᆠ ᨨ�ᆠ
ᨨᆠ矄矄
QBC
㦨
nCv
DT
㦨
nCv
TB

TC㡰 P P

ᨨ
� 
TC

犨
盐矄
B = C
ᨨ盐鼀
V
㦨
ᆠᨨ 矄 㡰 TB TC ҈ é ù ê PC  PA ҈ ú PC ҈ ´ TB = ´
�
=

� K êQ = = ú Þ TC
㦨 PB
ê PA
PA
ú  ë û  \ Q
㦨
҈맔

맔


҈
᲼

�


�
桌
㦨

�
黨


���㡰 ᔴ
®

 
꾀
®


��ᆠ
ᨨᆠ矄矄矄
g

æ T ö҈ g P \ PC҈  gTCg = p҈D  gTDg Þ C = ç D ÷ PD è TC ø




�㡰

PA҈ g TAg = p҈B  g TBg g

æ T ö҈ g P Þ A =ç B÷


��㡰 PB è TA ø 㝐盐�泐� �
쟼⼄矄

�㡰
 
��㡰

犨
� g

g -҈

�⏌ -҈ æV ö éV ù = ҈
黨 K Þ Tf
㦨
T�
ç � ÷
㦨


ê ú çV f ÷ ë
V û è ø ��㡰 ␔ᨨ├
ᨨ 
�矄
��ᆠ
ᨨᆠ矄矄 -nR T
- T҈ 㡰 -
´

҈҈
黨 - 

㡰 = W
㦨
g -҈ �⏌  -҈ +

҈ ´ ҈҈҈ ´  = +
䰘
䰘 㦨

꾀ᆠᆠᨨ� �
ᨨ
�矄
犨
ᨨ
 鼀ᨨ泐 鼀�ᆠ矄 DQ
㦨
DU
졔
DW 盐
D
Q
㦨

矄
 
ᨨᆠ矄矄
�矄
��ᆠ㡰 \ DU
㦨

DW
㦨


䰘
䰘
 ���㡰 ␔ᨨ├
ᨨ 
泐
 
�矄
W
㦨

䰘
䰘
  꾀 ෌�
TA
㦨
҈



⧤
҈ PB
㦨
PA 錐
PC
㦨
PA















  

CP � = g
㦨

 ᨨ
鼀ᨨ ᨨ
ᨨ鼀�ᆠ
�矄㡰 CV 

PC PA æ TDTB ö҈ g =ç ÷ PD PB è TC TA ø ᨨᆠ矄矄矄

®
ᔴ
 

®
꾀

�矄ᨨᆠ ᨨ�ᆠ
V
㦨
ᆠᨨ 矄 㡰 PC TC P T =   A = A

 ᨨ

PB TB PD TD 㝐盐�泐� �
 矄
犨ᨨ
⼄盐�ᨨ 矄

犨
� PC PA TC TA =


�㡰 PB PB TB PB ᨨ鼀
쟼⼄矄

���㡰
 
�㡰

犨
 g

g

æ TC TA ö҈ g æ TC TA ö æ TD TB ö҈ g æ TC TA ö =ç =ç ç ÷ ÷ ᨨ ç ÷ ÷ è TB TD ø è TD TB ø è TB TD ø è TC TA ø TC TA =҈ Þ TD TB



ᔴ 

WWW.JEEBOOKS.IN

EBD_8341 PS-128





Physics

TC TA (

�)(҈


) = ᨨ TD = TB
�
ᨨ TD
㦨
�


⧤ ᨨ
 
ᆠ泐ᆠ�ᆠ
ᨨᆠ矄矄
��

ᨨᆠ矄矄
㜠ᔴ QBC = nCP
TC

TB㡰


҈ ö æ䰘 ö æT = ( ) ç R÷ ç A  TA ÷ =  RTA è
øè
ø
㦨
�

�RTA ᨨᆠ矄矄
ᔴ㜠꾀

㦨
ᆠᨨ 矄  \
WCA
㦨

ᨨ
QCA
㦨
DUCA
㦨
nCV
TA

TC㡰


鼀

҈鼀

꾀



ᔴ





⧤




⧤



Wᨨ├
ᨨ 
ᨨ
ᨨᆠ矄矄
A
ᨨ
B
ᔼ矄ᨨb�ᆠ
ᨨᆠ矄矄㡰 WAB
㦨
nR
TB

TA㡰

㦨
nR
맔
҈


㦨


맔




맔




㦨
҈



 Wᨨ├
ᨨ 
ᨨ
ᨨᆠ矄矄
C
ᨨ
D
ᔼ矄ᨨb�ᆠ
ᨨᆠ矄矄㡰 WCD
㦨
 WAB 㦨

҈


J

㦨

҈

nR
㦨





맔




㦨

҈



 Wᨨ├
ᨨ 
ᨨ
ᨨᆠ矄矄
B
ᨨ
C
ᔼ矄ᨨh鼀
ᨨᆠ矄矄㡰

PB WBC
㦨



nRT
ᨨ�҈

PC
㦨



nR
맔




ᨨ�҈
㦨

䰘䰘


nR ҈ 㦨
��



맔




㦨


҈


Wᨨ├
ᨨ 
ᨨ
ᨨᆠ矄矄
D
ᨨ
A
ᔼ矄ᨨh鼀
ᨨᆠ矄矄㡰 WDA
㦨



nRT
ᨨ�҈








PD ҈ 㦨



nR
맔



ᨨ�҈
PA


㦨



䰘
黨nR 㦨


҈�


맔




㦨


�黨
�



ᨨ
犨ᨨ├
ᨨ 

W
㦨
WAB
졔
WBC
졔
WCD
졔
WDA 㦨
҈҈�
 㡰 ᨨ鼀
�矄
犨
ᨨ
 鼀ᨨ泐 鼀�ᆠ矄
DU
㦨
Q

W ᨨ
ᆠᨨ鼀
ᆠ泐ᆠ

DU
㦨

\ Q
㦨
␔
㦨
҈҈�
 㡰 
犨ᨨ├
ᨨ 
W
㦨
҈҈�
 ᆠ㡰 DU
㦨



和� ᆠ

 
ᨨᆠ矄矄
�矄
ᆠ泐ᆠ�ᆠ
�



DT
㦨

㡰 
p㜠V
 
p㜠T
��鼀矄

矄 ᨨ犨
ᨨ犨 P PA PA


P PA

A

C VA

( pV ) A

B

PA



VA

A C

TA


B


VA

TA

㡰 ᨨᆠ矄矄
A㜠B
T
㦨
ᆠᨨ 矄  ҈ \ P µ
V �矄
ᨨ盐
矄ᨨ
p 犨�
ᆠᨨ鼀

V nRTA RTA 꾀矄ᨨ

VA = = PA PA DU AB =


( pV )C =

Þ\ TC =

æ
V ö \ Q AB = WAB = nRTA ᔼ ç A ÷ è VA ø 㦨
RTA
�

㡰
㦨




RTA


�


T ö æ� öæ 㦨

㡰
ç R÷ ç TA  A ÷ = 
䰘�RTA è
øè
ø ᔼ

ᆠ泐ᆠ�ᆠ
ᨨᆠ矄矄
DU
㦨

\
Qnet
㦨
QAB
졔
QBC
졔
QCA
㦨


�

RTA �㡰 ᨨ
��ᆠ
ᨨᆠ矄矄



g


҈
㦨
ᆠᨨ 矄  T TV g -҈ = �


V 㡰 g -҈

Þ

㦨

�


㡰g ҈


├� �
ᨨ�
ᨨ
ᨨ
矄�矄

ᨨ�


㦨
g

҈㡰
ᨨ�
�


\ g
㦨
҈

盐
g
㦨
҈
졔





Þ ҈


㦨
҈
졔
f f


= �
�
ᨨ
ᨨ鼀㡰


��㡰 ᨨ
��ᆠ
ᨨᆠ矄矄
ᨨ犨
盐矄� � PVr
㦨
ᆠᨨ 矄  Þ f
㦨
g

g

PV ҈ ҈ = P
V


P ҈҈

�


㡰 ␔ᨨ├
ᨨ 
ᨨ
��ᆠ
ᨨᆠ矄矄 æ P ö PV - ç �


V 㡰 è ҈҈

÷ø PV P V ҈ ҈

= 㦨
҈

�
PV W
㦨
g -҈ ҈

- ҈


꾀泐� �
�
�矄
⼄盐�ᨨ
PV
㦨
nRT Þ P

㦨


P

҈



=

PV ҈

´ ҈
6 ´




 ҈
㦨
�

鼀ᨨ矄 = = RT

 ´ 

 �R R ෌�
Cp 㦨

\

Cv =




黨
맔
҈



ᨨ

 �泐
�矄
矄盐�

ᆠᨨ 矄 
ᨨ盐鼀  ᆠ
ᨨ鼀 Q
㦨
nCv
DT Q


黨 ´ ҈

= \ DT
㦨
㦨
䰘�
⧤  nCv �
 ´ ´


\ � 
鼀盐
㦨



졔
䰘�
㦨

䰘�
⧤ ᨨ犨
ᨨ鼀
෌泐
盐矄矄ᆠἘ矄
犨
P҈ P
P҈T
=

Þ P

㦨
T҈ T
T҈ \ n
㦨


҈

´ ҈

´
䰘�
㦨



맔
҈


鼀



䰘
ᔼ
��ᆠ
ᨨᆠ矄矄


犨ᨨ├
ᨨ  ҈ ᲼ P V - PV 桌 ␔
㦨
҈- g

҈ ҈ 

P҈
㦨
҈
�
⏌鼀


V҈
㦨

l
㦨


맔
҈

鼀 =

WWW.JEEBOOKS.IN

PS-129

Thermodynamics g

æV ö P

㦨
P҈
ç ҈ ÷
V

㦨

l 㦨


맔
҈
 è V
ø  � ෌�
Cv
㦨
R
\
Cp
㦨
R ᲼Q C p - Cv = R桌

g=

\

Cp Cv




= ҈

䰘 ҈

䰘

é
ù 㦨
҈
�
맔
㡰҈

䰘
㦨





맔
҈
�
⏌鼀
ë û ҈ \ W
㦨

᲼




맔
҈
�
맔


맔
҈


҈
�
맔


맔
҈
桌 ҈ - ҈

䰘 \ P

㦨
҈
�
ê ú


҈
�
㦨

黨䰘
҈
 ᲼҈
�





桌
㦨



䰘 -


䰘 
��
矄��
� �ᆠ矄
 
 
�矄
�矄
ᆠᨨ鼀矄矄
W
㦨


Topic-3 : Carnot Engine, Refrigerators and Second Law of Thermodynamics ҈






P

҈㡰




B

C

P


P


D

A

V



V


V

ᨨ鼀
 
��盐
␔ᨨ├

W =
P
V

��

Q� = WAB + WBC = n × CV DTAB + nCP DTBC R n� R TB - TA 㡰 + TC - TB 㡰 =n



R �R ö æ   CP = çèQ Cv = ÷

ø

 � =  PBVB - PAV A 㡰 +  PCVC - PBVB 㡰

 � = ´ ᲼P
V
- P
V
桌 + ᲼
P
V
- P
V
桌

҈�
҈ P
V
= P
V
+ P
V
=


P
V

W = = 쟼 �ᆠ� ᆠ泐

h =
҈ Q�
҈ PV


h奠 =






» ҈黨

҈

W ␔ᨨ├
ᨨ  = 
矄ᨨ SQ Q҈ + Q
+ Q + Q
=

�























= Q҈ + Q 

Q҈
㦨
҈黨҈�


Q

㦨





 
Q
㦨
҈
�
 ҈黨҈� -

+ ҈
� + Q
\ =

� ҈黨҈� + ҈
�

ᆠ㡰 쟼 �ᆠ� ᆠ泐

h =

�






Þ ҈黨҈� -

+ ҈
� + Q
= ҈


Þ Q
= ҈


-



Þ Q
= -Q = -黨

 Þ Q = 黨

 㡰 ᨨ
ᆠ ᨨ
 ��ᨨ Q҈  Q
쟼 �ᆠ� ᆠ泐
= Q҈ ␔ 
Q҈
㦨

ᨨ矄
ᨨ鼀
矄ᨨᨨ盐 � � Q

㦨

矄ᨨ
ᨨ鼀
矄ᨨ�

ᨨ犨
鼀盐
Q  Q
w 꾀矄ᨨ
҈ = Q҈ Q҈ ҈ 犨 Þ = ҈
턼҈ Þ Q҈
㦨
w
맔
҈

㦨
҈


J 和ᨨ

턼҈

Q

㦨
w Þ Q

㦨
Q҈
w Þ
҈



҈

㦨
Q

㦨
黨

J b㡰 
QH
㦨

├
泐
�矄
 ��  QL
㦨

ᆠ
泐
�矄
 ��  Q2
㦨

ᆠ
泐
矄ᆠᨨ 
 ��  ␔ᨨ├
ᨨ 
泐
҈矄
 �� 
㦨
犨ᨨ├
ᨨ 
泐


 ��  W
㦨
QH

QL
㦨
QL

Q
Þ

QL
㦨
QH
졔
Q
q q
= H +
qL qL 


 
鼀盐
ᨨ
ᆠᨨ
矄ᨨ�
ᨨ
�矄
 �� 



�
ᆠ ᨨ
 �� 
QH T҈ Q T =   L = QL T Q
T
T T Þ
= ҈ +
盐矄� �
�㡰 T T T +T Þ
T = T҈ + T
Þ T = ҈

T
b㡰 矄� �

n
㦨
҈ T҈ T
҈ n
㦨

㦨
҈ T҈
T
-

T  
= ҈  T҈ 졠
矄ᨨ� �

犨
� T҈
㦨
黨黨 ᔴ
 
T

㦨
䰘 ᔴ b㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

h҈
㦨
h

㦨
h



\҈
= ҈  = ҈


҈




 ᲼Q

 �� 矄

⼄盐泐
 �ᆠ� 桌



Þ
= =


҈
 Þ
= ҈ 




�㡰

 =






��㡰

ᨨ鼀
�㡰
 
��㡰


=  ҈

㡰

҈




҈


㦨
 ҈
㡰

WWW.JEEBOOKS.IN

EBD_8341 PS-130

䰘


Physics

҈ 
犨 꾀 =

 턼҈


  ␔ =  

h =


턼
꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ
␔꾀
㦨
␔

㡰 h꾀 =

҈



턼 -  ҈ \ ҈= ҈´
= 턼

҈ -



\
=

 + 


ᨨ
h =




+


㦨
�

⧤
㡰 ෌�

鼀盐
ᨨ
ᆠᨨ
ᨨ泐



㦨

�

⧤ 鼀盐
ᨨ
ᨨ
ᨨ泐錐
҈
㦨



⧤ 
ᆠ�

Q

㦨
�


ᆠ
犨ᨨ├
ᨨ 

␔
㦨
䁸 =





쟼 �ᆠ� ᆠ泐
㦨
҈ 

黨


҈



㡰 D
㦨
鼀
㦨
�
맔


맔
҈

㦨
턼矄� ├

턼矄� ├

㦨 矄� ├ 턼矄ᨨ盐ᆠ 矄ᨨ盐ᆠ

҈


-


´ ҈

奠 ҈




h =

奠 ᨨ犨

ᨨ
ᆠ ᨨ
ᆠ泐ᆠ





´


҈


㦨




⧤ 㦨






䰘


=

҈


\

= -°ᔴ b㡰 
�矄
 ᆠ
ᨨ鼀
 
矄ᨨ盐ᆠ
�

꾀
 
꾀
�矄

DQ =

 æ P
V
ö � æ
P
V
ö R + R
çè R ÷ø
çè R ÷ø

æ ҈ö  � P
V
+
P
V

= ç ÷ P
V
è
ø

҈

ᆠ㡰 ᔴᨨ � 
꾀 ᔴᨨ � 
 㝐矄矄
ᨨ
�矄
㦨
mA 㝐矄矄
ᨨ
�矄
㦨
mB ᔴ  �
�
矄矄盐
㦨
DP ᔴ  �
�
矄矄盐
㦨
҈
�DP Þ

PAV

㦨


mA RT M

PἘA

㡰
㦨


矄ᨨ盐ᆠ ´ 턼矄� ├

矄� ├

쟼 �泐
ᆠᨨ 矄盐鼀
泐


Þ PA

PἘA
㦨
Þ D
㦨


෌�

矄ᨨ盐ᆠ 㦨
䰘 ᔴ 졔
䰘 㦨 

⧤


  PB

P'B 㦨


 æ 



ö - ҈÷ = ҈

䰘 ´ ҈
�  ␔ᨨ盐盐
㦨
� ´ 
´҈
ç è
䰘 ø ҈҈
㡰 
 ᨨ泐
ᆠ  �
ᨨ
 
ᨨ泐
�
 
犨ᨨ
ᆠ矄矄
�矄 矄鼀
矄
 ᨨ泐
�矄

矄
盐 ᆠ�ᨨ


PBV

㦨


mA RT M

æ 矄ᨨ盐ᆠ ö - ҈÷ \ 犨 ᨨ盐盐 㦨 턼矄ᨨ盐ᆠ 㜠 턼矄� ├
= 턼矄� ├ ç è 矄� ├ ø

矄� ├ 㦨
ᔴ 졔
䰘 㦨
䰘 ├

턼

= 턼҈ ҈




=
҈







T
W W
�
= =

Þ
҈  T҈ Q
+ W 

Q
+ W

Q
�

´


J
㦨




 = ␔
㦨
� � b㡰 ␔ᨨ├㜠ᨨ 
␔㡰
㦨


꾀ᆠᆠᨨ� �
ᨨ
� ᆠ�
ᨨ
ᆠᨨ�鼀泐 
��
㦨
턼꾀
㦨
턼ᔴ 㦨
ᔴ 꾀
졔
ᔴ ᔴ  � =   - 꾀 㡰 +  ᔴ -  㡰

 � = 


- 

㡰 + 


-

㡰

҈ = 


␔


鼀
 �ᆠ� ᆠ泐
ᨨ
 �� 
h㡰
= = =

҈� 턼 �� ҈

\ 턼矄ᨨ盐ᆠ =

b㡰 ෌�

턼҈
㦨
҈



 턼

㦨






҈
㦨

҈
䰘 ᔴ
㦨




⧤



㦨
䁸 h
㦨
䁸 쟼 �ᆠ� ᆠ泐
ᨨ
ᆠ ᨨ
 �� 
␔ h= ´҈

奠 턼҈ 턼
- 턼҈ ´҈

奠 ᨨ
h = 턼҈

P'B

V㡰
㦨


mB RT M

m A RT m A RT MV M 
V 㡰

m A RT
MV





�㡰

mB RT mB RT MV M 
V 㡰

mB RT
MV ��� �
⼄

��㡰
泐
�㡰 ҈
�
D
㦨


mB RT M





��㡰

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PS-131

Thermodynamics

҈�


 m ҈
�DP mB Þ
= B

Þ

mA
㦨

mB =
mA DP MA 
䰘黨҈㡰 ෌�

矄ᨨ

Q

㦨
mL
㦨



맔
҈


㦨





ᔴ

鼀盐
ᨨ
�ᆠ

T

㦨

䰘
⧤

鼀盐
ᨨ
矄盐ᨨ盐 � �
T҈
㦨

䰘
졔

䰘
㦨



⧤ 쟼 �ᆠ� ᆠ泐
㦨


҈



Þ p
㦨


w Q҈ - Q
T҈ - T


-
䰘 = =
㦨
Q
Q
T

䰘

T
W = T҈ Q҈



҈


Þ ҈ = 黨

Q҈ Þ

҈䰘



҈


= Þ Q҈ = ҈


 Q҈

 ᨨ

 �泐
�
泐
 
 �� 
ᨨ
 
ᨨ犨 鼀盐
矄ᨨ�

턼

㦨
턼҈

␔
㦨
҈




҈



㦨







J 㡰
꾀矄
 
ᆠ泐� 
�矄
ᨨ

�矄
ᨨᨨ鼀
 
矄

矄鼀
ᨨ 
�矄
ᨨ

 ᆠ
 
矄矄盐
� 矄�
 
ᆠ泐� 
�矄 鼀ᨨ矄  �ᆠ
矄矄盐



 ᨨ盐� ᨨ盐
 
矄ᨨ犨
盐� � ᨨᆠ矄矄






+ P


P
´
L p é P ´ pR
ù P
= ´
L = ê

ú ´
L é Mg ù êë P
pR - Mg úû P


ê
ú pR
û ë

P

맔


맔
p
㡰
㦨
px
맔
pR


Þ

x =

Q҈ -




䰘 = Þ Q҈ =
䰘黨҈ ᔴ




䰘 





㡰 ෌� 錐
T҈
㦨
黨


K

T

㦨


K

W
㦨
҈



J 矄� �

҈ 

- Mg


pR
0 和� ᆠ
 
ᆠ泐� 
�矄ᨨ 鼀泐
ᆠᨨ 盐ᆠ� � \
鼀盐

T
㦨
ᆠᨨ 矄  꾀泐� �
P҈V҈
㦨
P
V


Þ

҈



㡰 
x

 
�矄 ᆠ
ᨨ
 
�矄ᨨ
ᨨ鼀
 
ᨨ
和 꾀
⼄盐���盐鼀 Mg
㦨
P


p㡰
pR



҈黨


ᆠ㡰 꾀
⼄盐���盐鼀

p
㦨 P Þ
p
㦨
P

졔
L


H㡰
r�





�㡰

꾀矄ᨨ

P

맔
pR

L
㡰
㦨
p
᲼p

L


H㡰桌


(Q PV ҈ ҈ = P
V
)

L
P































L
- H 和盐矄�盐� �
 �矄
盐
ᨨ
E 
ᨨ鼀
�㡰
 
��㡰 P L
P
= P
+  L
- H 㡰 r g P L
- H Þ L
P

㦨
P

L


H㡰
졔
L


H㡰

r g Þ r�
L


H㡰

졔
P

L


H㡰

L
P

㦨

Þ p
㦨


P


p H

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EBD_8341 PS-132

Physics

Kinetic Theory

12

Topic-1 : Kinetic Theory of an Ideal Gas and Gas Laws ҈


䰘


㡰 ᔴ泐
ᨨ鼀
��盐

꾀
㦨 ҈꾀
졔
鼀�

× 꾀

ᨨ


꾀l
㦨



l҈




l





Þ

PAV

㦨


× 꾀 + 鼀� 꾀l҈







҈꾀

꾀


鼀�


æ ҈ ҈ö

ç  ÷
㦨
鼀� è l
l҈ ø




\

鼀
㦨
� ç l × l ÷ è ҈
ø b㡰 ෌�

鼀盐
T�
㦨
҈䰘
졔

䰘
㦨

黨

K

鼀盐
Tf 㦨

䰘
졔

䰘
㦨



K 꾀鼀ᨨ矄 �ᆠ
矄矄盐

P

㦨
҈
맔
҈
�
Pa ᨨ盐鼀
ᨨ
ᨨᨨ鼀

V

㦨


m �  ᆠ
�
盐鼀
ᨨ
鼀ᨨᆠ盐矄

nf

n�
㦨
䁸 矄� �
�
�矄
⼄盐�ᨨ

PV
㦨
nRTN
㡰


㦨
꾀ᨨ�ᨨἘ矄
盐鼀

n=

\

nf  n�
㦨


�




P
V
æ ҈ ҈ ö - ÷  ç R è T f T� ø


҈´ ҈
� ´ 
҈ ö æ ҈ ´



 ´ ҈

 ç
㦨
÷

҈
è 


黨
ø
� 㦨



�
맔
҈
ᆠ㡰 ᨨ鼀
㜠
�


µ



PV 
㡰 RT

Þ

҈



㦨
ᆠᨨ 矄 
 
矄矄盐
�矄
� ᆠ矄� �
ᨨ鼀


ᨨ
҈ 

矄ᨨ
ᨨ�ᨨ
ᆠ㡰
矄 矄
ᆠᨨᆠ
㜠
�
㡰 ҈㝐҈
㦨
҈
 



㝐

㦨



 㝐  \ ҈´ ҈ = ҈ 
㝐



҈


҈ = ´ =






\ 
黨   
㡰 꾀

�矄
 矄
矄

�
�矄

ᨨ犨
矄矄盐
 
�� 鼀盐
㡰 꾀ᆠᆠᨨ� �
ᨨ
ᨨ泐Ἐ矄
犨

PV
㦨
ᆠᨨ 矄 
\ PdV 졔 VdP 㦨

Þ

mA RT




















M

ᔴᨨ � 
 㝐矄矄
ᨨ
�矄
㦨
mB ᔴ  �
�
矄矄盐
㦨
҈
�DP PBV

㦨


mB RT M

mA mB RT RT P'B

V㡰
㦨
M M mA RT mA RT Þ PA

PἘA
㦨
MV M 
V 㡰 m A RT Þ D
㦨




�㡰
MV mB RT mB RT   PB

P'B 㦨
MV M 
V 㡰 ҈
�
D
㦨




��㡰 mB RT
MV ��� �
⼄

��㡰
泐
�㡰  m ҈
�DP mB Þ
= B

Þ

mA
㦨

mB =
mA MA DP ᨨ 盐矄㡰⏌
ᔼ ��泐
��ᨨ
�矄


 �矄 ᆠ
ᨨ

鼀
ᨨ鼀
 
ᨨ


鼀盐


㦨



⧤ ҈
鼀 
㦨
꾀
㦨




꾀
㦨


҈
맔




 ᨨ
ᨨ 㡰 
�矄ᨨ
矄 � 矄
泐
맔
鼀 
 
� 
鼀盐
�矄


 


꾀

 㡰
㦨


҈



҈





\ 
꾀 = =
 
҈꾀 =
- ҈

- 㡰 ҈

+ 㡰 PἘA

㡰
㦨


æ l҈  l
ö





ᆠ㡰 ᔴᨨ � 
꾀 㝐矄矄
ᨨ
�矄
㦨
mA ᔴ  �
�
矄矄盐
㦨
DP

PdV æ ҈ ö æ dV ö æ ҈ ö Þ b
맔
P
㦨
҈ = -V 錐 b = - ç ÷ ç = è V ø è dP ÷ø çè P ÷ø dP

 ᆠ
�
犨
b
 
P
犨�


ᆠ �盐
泐ᨨ






� 泐

꾀 - ҈꾀 = 鼀� =
 =

Þ



æ ҈ ҈ ö ç ÷ è ҈

-
+ ø

æ
ö =
 ҈
çè ҈
-
÷ø




ᔴ
맔



졔
鼀�



㦨



ᔴ



 Þ ´ ´ 

+
 = ´


҈
҈

æ 黨

ö æ ö Þç +
 ÷ =  ç ÷ ҈
è ø è ҈
ø


 ö æ
ö æ æ 


 ö

Þ ç 
+ =
 Þç + ÷ = ÷  ø çè ҈
-
÷ø  ø ҈
è è ҈
ö æ
ö æ Þ
 ç  + ÷ç =
  ø è ҈
-
÷ø è 黨 + 㡰 
㡰 Þ =҈  ҈
-
㡰 Þ ҈

졔



㦨







Þ �

졔
҈





㦨

\
㦨
҈
䰘

»
 ᆠ
�矄 ᆠ
ᨨ鼀
ᨨ
犨 
ᆠ 
⼄盐���盐鼀
㦨


졔


㦨


鼀

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PS-133

Kinetic Theory 黨


�㡰

矄� �
�
�矄
⼄盐�ᨨ

PV = nRT

Þ PV ҈ ҈ = nR ´
�


P

V҈ =

�n R ´









᲼Q T҈ =
�
⧤桌




�㡰

᲼Q T
=



⧤桌




��㡰

��� �
⼄

�㡰
泐
��㡰



´
�
P҈ P ҈ = Þ ҈ = P
�
P
� ´



҈



\

P
= �
P҈





RT kT = ҈҈

´ ҈
 ᨨ


= ҈҈

´ ҈
 M m 꾀矄
鼀盐
�矄
ᆠᨨ 矄 㡰

DP V


�㡰 P ᔼ
矄ᆠᨨ 
ᆠ矄

盐矄� �
�
�矄
⼄盐�ᨨ
��

�


P DV = -nR DT nR DT P 쟼⼄盐� �
�㡰
 
��㡰

犨
� DV = -




��㡰



ᔴᨨ鼀� �
 
ᨨ
⼄盐�ᨨ
犨�
 DT  = C  DP 

犨


 � 鼀ᨨ ᨨᨨ鼀�ᆠ
 㡰
=
+  - 㡰



҈ � = + 



ᨨ犨

ᆠ  �
�
ᨨ
├� �ᆠ
 �泐
ᨨ
 
�矄


T 㡰 PV
㦨
nRT
ᔼ
�矄
⼄盐�ᨨ 㡰

nRT V

෌�
VP

㦨
ᆠᨨ 矄 ᨨ鼀
�㡰
 
��㡰

T
= ᆠᨨ 矄  ᨨ
T µ v V Q V҈
㦨

V
 
T҈
㦨
T V
V T҈
T

=
T Þ T

㦨
T҈

= T \ = V҈ V V҈ V
҈  矄㡰 ᨨ

�ᆠ盐
鼀盐
T


V
µ P ᨨ盐鼀
�矄
�
ᨨ
矄矄盐
P҈ \ p҈


p






�㡰



��㡰

D = 턼 =

\

҈



Topic-2 : Speed of Gas, Pressure and Kinetic Energy ҈






 kT =
´ ҈
-҈

P
㦨


ᆠ鼀
ᨨ
�

V
㦨


ᆠ鼀 PV PrgV
´ ҈

´ 黨

´
N= = ;
´ ҈
҈

KT KT ´ ҈
-҈
 RT ᆠ㡰 Vrms = M v҈ T҈ 
䰘 + ҈
䰘㡰




= = = = = �

� v
T

䰘 +
䰘㡰 �

䰘


� � v҈ = ´


= ҈

� 鼀⏌矄




҈






ᆠ㡰 矄矄盐

P
㦨


҈ mN
Vrms  V

 mN 㡰T V ᔼ
 
�矄
鼀矄矄
 
鼀盐

ᆠᨨ 矄 
 

P
µ
V鼀矄㡰

µ
T 和ᨨ

ᨨᆠ
µ
V鼀矄㡰

µ
T �



盐
ᨨ
q
㦨
҈ ᨨ






P=

㡰 Q






ᔴ =



㝐

 ´

҈
´ 

㝐  ´

҈
´ 

»
´ ҈
- ├�
㝐= ҈黨
´ ҈黨



�矄
�矄




(҈黨
)
=

㡰 ෌�


⧤
쟼
鼀
㦨


\ v
=

鼀矄 (  ) 㝐 꾀

鼀矄 ( 꾀 )
㦨
㝐 
㦨


㦨

҈
㡰 쟼 �泐
矄矄ᨨᆠ�
犨�

鼀ᨨ矄
ᨨ
�ᨨ鼀�ᆠ
�矄


 =




ᨨ
 �泐
犨 

鼀ᨨ矄
ᨨ
�ᨨ鼀�ᆠ
�矄
ᆠᨨ 
� ᨨ

DT 

⧤ V = = = ҈�
⧤⏌鼀 nR DP
鼀

Þ P
㦨



\ T
㦨
҈


⧤

� � = 


쟼 �泐
矄矄ᨨᆠ�
犨�

鼀ᨨ矄
ᨨ
鼀ᨨ ᨨᨨ鼀�ᆠ
�矄

DP nR DT V =V Þ DT = DP nR P P

҈҈


 ´ ҈

´ ҈
-
 T

= ҈҈

´ ҈

´ ҈
- ҈鼀矄 㝐
㡰 矄� �

鼀矄
㦨
㝐҈ ᨨ


DV = -

C=

҈


´
´ ҈


´ ҈

=
N ⏌ m
҈ ´҈ ᆠ㡰 v鼀矄
㦨
ve =

҈�
㡰


ᔼ
�矄
ᆠ矄
ᨨ鼀
�
�矄
⼄盐�ᨨ

PV = nRT

P D V + V DP =


ᨨ盐 矄㡰




ᨨ
ᆠ  �
ᨨ
鼀ᨨ鼀 盐鼀
盐� �
ᆠᨨ�矄�ᨨ

mv   mv 㡰
mv N 㦨

= Dt Dt N ´ 
mv 㡰 矄ᨨ
矄矄盐 P = Dt ´ A

r 鼀矄矄
ᨨ
 
�矄

黨


ᆠ㡰

鼀矄 =

҈



㡰

矄� �
V鼀矄
㦨


v鼀矄  �盐鼀㡰 v鼀矄 �ᨨ 㡰

=

RT ҈ Þ Vrms µ M M

M �ᨨ

M �盐鼀


㦨




= ҈
» 
҈



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EBD_8341 PS-134

҈҈


ᆠ㡰

Physics ᨨᆠ�泐
ᨨ
矄ᨨ盐 
�

�矄
v =




 ҈



҈


҈



=

g 
㝐  g  㝐 


gp = p

 VP
꾀 

�
├� �ᆠ
 �泐
ᨨ
�矄ᨨ盐矄
鼀ᨨᆠ盐矄 ҈
҈ æ ö  E
㦨
mvrms = m ç v
p ÷ = mv
p

è
ø


 䰘 ⏌�´
= � ⏌  ´

�

=

b㡰 ␔ 

 ᆠᨨ矄
�矄
�矄
ᆠᆠ
�
 
ᨨ矄�� 㜠�ᆠ�ᨨ
 
 
矄矄盐
ᨨ
 
�矄
ᆠ矄矄
ᨨ �
  ᨨ矄��
㜠 �矄
 
ᨨᨨ犨矄
 
⼄盐�ᨨ

DP
㦨

r
a dx 和ᨨ
鼀ᨨ
矄矄盐
ᨨ
 

矄�
 
矄矄
矄矄盐
ᨨ
 
ᨨ  矄�




ᆠ㡰 ᨨ鼀


㦨

Þ

㦨

ᨨ


µ


 Q

 


矄鼀
㡰

 ᨨ

�

�矄
ᨨ盐

矄矄盐
矄ᨨ
ᆠᨨ鼀矄
犨ᨨ
�鼀矄

�




҈

Vrms 㡰҈ V T = = ҈ Þ b㡰 Vrms 㡰



T
Vrms 㡰


Þ

ᨨ鼀
 矄
 矄矄�ᨨ 矄

犨
ᆠ
ᆠᨨ ᆠ盐
  vp

v


vrms

gRT M

҈ =

Þ V鼀矄㡰

㦨

V Vrms 㡰



꾀矄ᨨ

鼀矄
㦨


Topic-3 : Degree of Freedom, Specific Heat Capacity, and Mean Free Path ҈


ᆠ㡰 ᨨ
�
ᨨ
ᨨ鼀
f
㦨

졔


㦨
� nfRT � RT

ᨨ
 �泐

U = =

Cp

䰘 꾀 
g = = ҈+ = ҈+ = � � Cv f





b㡰 㝐



l =

V

҈�


b㡰 꾀ᆠᆠᨨ� �
ᨨ
㝐 犨Ἐ矄
�矄�盐�ᨨ
ᨨ
矄

�
矄

҈



ᨨ
鼀ᨨᆠ盐矄
ᨨ

�
�矄

µ
\ 
ᨨᆠ�泐
ᨨ
ᨨ 泐�
鼀ᨨᆠ盐矄
犨�

矄鼀
�
꾀
矄
犨 矄
ᔴ
矄
 
鼀盐
ᨨ
A
 
C

矄鼀 
҈㡰

ᨨᨨ鼀
鼀
矄⼄盐
矄
�矄
��
泐

RT M 

M
㦨
㝐ᨨ
鼀矄矄
ᨨ
�矄
鼀ᨨᆠ盐 T
㦨
鼀盐
ᨨ
 
�矄
鼀ᨨᆠ盐 vrms =

盐
v鼀矄 = 


␔

��
v = v



\ ҈䰘


҈



RT
㝐 


=

RT
M 


Þ







҈





=

�䰘 Þ T
=
҈ ⧤



 矄㡰
␔
├ ᨨ犨
 

鼀矄
矄 R
T 㡰 RT v
㦨


\ vἘ
㦨

v
 
vἘ
㦨
M M ⏌
g ҈RT CH
㡰҈ M҈ g҈ M
= = ´  矄㡰
C M҈ g  H e 㡰
g
RT
M


䰘⏌�
䰘 

´ = ´ ´
= �⏌
� �
�  矄㡰 ᨨ

�ᆠ盐
鼀盐
ᨨᨨ
鼀
矄⼄盐
矄
҈ Vrms
µ
M �



Vrms
犨�

�  
盐矄
ᨨ
�  
�矄矄




㝐 �



鼀矄
 矄
ᨨ
鼀盐
 
鼀ᨨ
鼀矄矄
 
 ᆠ
鼀矄 矄
犨�

�  
ᨨ
�  
�
�矄矄
ᆠ

㡰 ␔
├ ᨨ犨
 

 矄㡰 㝐和
矄

鼀矄
=

v=


RT 錐 vrms = pM

kT R

\t =

l ҈ Þtµ T kT m

ᆠ㡰 꾀矄
犨
├ ᨨ犨
Cp
g= = ҈ +

犨 
f
㦨
�
ᨨ
ᨨ鼀 Cv f 꾀㡰 㝐ᨨ ᨨ鼀�ᆠ

f
㦨



� =   㡰 �ᨨ鼀�ᆠ
���
鼀ᨨᆠ盐矄

f
㦨
�
䰘 \ g = ҈+ = � � ᔴ㡰 �ᨨ鼀�ᆠ
ᨨ 㜠���
鼀ᨨᆠ盐矄

f
㦨
䰘
黨 \ g = ҈+ = 䰘 䰘 㡰 �ᨨ鼀�ᆠ
���
鼀ᨨᆠ盐矄

f
㦨



\ g = ҈+ =
 㡰 
�
ᨨ
ᨨ鼀

f
㦨

졔

㦨


ᨨ
�ᨨ鼀�ᆠ
ᨨ 㜠�  鼀ᨨᆠ盐
ᔼ  
 �泐
ᨨ

鼀ᨨ
ᨨ
 
�矄

鼀盐
T
\ g = ҈+

T






㦨


҈黨


҈


pnd
犨 

d
㦨
�鼀
ᨨ
 
鼀ᨨᆠ盐 n
㦨
盐鼀
ᨨ
鼀ᨨᆠ盐矄

盐 �
ᨨ盐鼀 l 盐

鼀
�鼀
ᨨ
ᆠᨨ�矄�ᨨ

t = v鼀矄


RT RT
  vp 㦨
M M

f
nRT = RT = RT

b㡰 
Cp
 
Cv

 
矄ᆠ� �ᆠ

ᆠᆠ�泐
ᨨ
 
�矄
 ᆠᨨ 矄 
矄矄盐
 
ᨨ盐鼀
꾀
ᆠᨨ 矄 
矄矄盐


⼄盐�

U =

�


DQ҈ = nC p DT Þ ҈

= nC p × �





�㡰

꾀
ᆠᨨ 矄 
ᨨ盐鼀


⼄盐�

DQ
= nCv DT Þ


= nCv ×҈






��㡰

��� �
�㡰
泐
��㡰

犨
� Cp
҈

C p �
= × Þ =


Cv ҈

Cv 

WWW.JEEBOOKS.IN

PS-135

Kinetic Theory g=

Cp

Cv

=



=҈+  f

 ᆠ
�
犨
t ⏌矄



f
㦨
�
ᨨ
ᨨ鼀㡰

Þ f =





䰘


҈҈


㡰 ᨨ
 �泐
ᨨ
 
�矄
鼀� 盐
f҈n҈ RT҈ f
n
RT
+ E鼀�
=

� � =  ´ RT + ´ RT = ҈�RT

㡰 꾀矄
犨
├ ᨨ犨
鼀



 � R
 
CP҈ = R

� 䰘 CP = R ᨨ
ᨨ 泐�


CV
= R
 


� 䰘 n
R +
n
R N҈CP҈ + N
C P


= ҈黨 nR ´
g=
Þ
g =  �
҈nR㡰 N҈CV҈ + N
CV
n
R +
n
R

æC ö ҈黨 \ç P ÷ = C è V øm�xture ҈

҈ N æ ö
ç ÷ pd
èV ø 
N
㦨
ᨨ

ᨨ
鼀ᨨᆠ盐 V
㦨
ᨨ盐鼀
ᨨ
ᆠᨨ �  d
㦨
�鼀
ᨨ
鼀ᨨᆠ盐 盐
PV = nRT = nNKT

҈



N P ҈ KT = =n Þ l= V KT
pd
P ᨨ
ᆠᨨ 矄 
ᨨ盐鼀
 
 ᆠ
ᆠᨨ 矄 
盐鼀
 矄�泐
n ᨨ
�矄 P 鼀ᨨᆠ盐矄
�矄
ᆠᨨ 矄 
T 和ᨨ
鼀


鼀� 矄
矄鼀
꾀矄
鼀盐
� ᆠ矄矄
ᨨ

ᨨ
ᆠᨨ�矄�ᨨ
� ᆠ矄矄
矄ᨨ
 �ᨨ

�鼀
ᆠ矄矄
㡰 和ᆠ� �ᆠ

ᨨ
�矄

ᆠᨨ 矄 
ᨨ盐鼀 ҈ Cv = fR錐 f =
�
ᨨ
ᨨ鼀
ᨨ
�矄
꾀
�ᨨ鼀�ᆠ㡰
㦨
�

 矄�ᨨ 
졔


ᨨ�ᨨ 㡰

黨


ᨨ
�矄

�ᨨ鼀�ᆠ㡰
�
��ᨨ
ᨨ

 矄�ᨨ 
졔


ᨨ�ᨨ 㡰

��ᨨ 
�
ᨨ
ᨨ鼀
� CvA
R � 䰘 B \ Cv = R
 ᆠ
= =
CvB 䰘 R 䰘
ᨨ 盐矄㡰

㝐


ᨨ

�矄
鼀ᨨᆠ盐
�矄
��
泐

҈


҈



҈�



p d
n 

n
㦨
盐鼀
ᨨ
ᆠᨨ�矄�ᨨ 矄

盐 �
ᨨ盐鼀
d
㦨
�鼀
ᨨ
 
鼀ᨨᆠ盐 ᔼ
�
矄
ᨨ
鼀ᨨᆠ盐
�矄

 

\ tµ

҈
pnd
v

=

҈
pnd
M҈

M RT

d
M \ t҈ = ´

t
M
d҈ d


æ ççQ v = è

҈





æ

҈ ö ´ç ÷ = ҈

黨 ҈

è


䰘 ø

ᆠ㡰  �ᨨ

�鼀
 t㡰 µ  

v µ T

\tµ

҈ T

RT M

ö ÷÷ ø







n҈᲼Cv҈ 桌 + n
᲼Cv
桌 n҈ + n


R �R ù é ê
´
+ ´
ú 㦨
ê ú
+ ê ú ë û 㦨


҈

㦨


҈
맔



㦨
҈䰘


⏌鼀ᨨ├ � Cv ҈ ҈ = = b㡰 F
㦨

㦨
C p r 䰘 ⏌ �㡰 䰘 䰘 䰘 ´ ҈
�
W = ҈- = ᨨ Q
㦨

W =
㦨
�


䰘 䰘 Q ᆠ㡰 
㦨

�
맔
҈

鼀 


㦨
҈
鼀ᨨ
ᨨ
졠



㦨



⧤ 鼀矄
㦨




鼀⏌矄 ҈ \ l=
p


҈



҈ = =



p


t l


´


´



 ´ ҈




p ´ ҈
-҈
´


黨
� ´ ҈
-


ᆠᨨ矄矄
盐
�
 
��
ᨨ�ᨨ
�矄
㦨
҈
҈
ᆠ㡰 꾀鼀ᨨ盐 
ᨨ

⼄盐�
턼㡰
ᨨ
�矄
 
鼀盐
 ᆠᨨ 矄 
ᨨ盐鼀 턼
㦨
ᔴD




�㡰 꾀鼀ᨨ盐 
ᨨ

⼄盐�
턼҈㡰

ᆠᨨ 矄 
矄矄盐 턼҈
㦨
ᔴD




��㡰 =




=

ᆠ㡰 ᲼Cv桌鼀�

㦨


꾀�
�鼀


l 㝐

�鼀

t = v Þ t=

҈

n҈Cv + n
Cv

ᨨ

҈

l=

n҈C p + n
C p


� n҈ n n +n + = Þ +
= ҈
Þ
� g m ҈ g ҈  ҈ g
 ҈ g m ҈
҈ ҈   � ҈䰘 黨
´ � Þ gm = = ҈



Þ g 鼀 ҈= Þ + = ҈
҈
҈
gm ҈

� R


v

㡰 矄� �

g鼀� 盐㦨


҈

Þ

\C A =


�矄

矄�� 
� 
犨 �ᆠ
�矄 T ᆠᨨᆠ泐
�ᆠ
泐
�
矄 ᨨ犨
�
ᨨ�ᨨ
ᆠ㡰
㡰 �盐鼀
�矄

鼀ᨨ ᨨᨨ鼀�ᆠ
�矄
 
졠 泐�
�矄

�ᨨ鼀�ᆠ
�矄
ᨨ
�盐鼀

CV҈ =

l=





҈

��� �
⼄盐�ᨨ
��㡰
泐
�㡰

犨
�

鼀
  ҈

Þ t µ 矄 

æ 䰘ö Þ Q҈ = Q㡰 ç ÷ è �ø

\

Q҈ C p = Q Cv

C p 䰘ö æ çQ g = C = � ÷ è ø v

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EBD_8341 PS-136

҈䰘


Physics

b㡰
g A =  
g B =

҈



CP
黨 = = ҈

< ҈


�ᨨ鼀�ᆠ㡰 Cv








g 鼀� 盐


҈
= = ҈

 > ҈


҈ 䰘

b㡰 矄� �

t=

\ µ 

҈
p
�

ᨨ
ᨨ 鼀ᨨᆠ盐矄 ù é êë\ = úû ᨨ盐鼀

\
盐
=

R
R ᨨ
�ᨨ�
M
㦨


㡰
Cp  Cv
㦨
b =

a = ҈
ᨨ
a
㦨
҈
b \ b









�   㡰 + ´ �





ᔴ ᔴ

æ
ö 䰘 = ç҈ + ÷ = ᔴ è � ø �
䰘


㡰 ᨨ

ᨨ泐ᨨ�ᆠ
ᨨᆠ矄矄

\ ᔴ - ᔴ = ҈-

҈-

\

҈ - = \ ҈=

ᔴ - ᔴ ᔴ - ᔴ ᔴ = ᔴ +

  ᨨ

ᨨ
㦨
҈


 ᆠ㡰 
 矄 
Q = nC
D
矄
�矄
�
ᆠᨨ矄
矄矄

ᨨ
ᨨ盐
 
鼀矄
矄

鼀盐
矄 ᨨ盐



�鼀矄
�



Ἐ

\ =

㦨


矄
鼀矄
㦨
RT ⏌ M

҈� � ´ R ´ ´ (
T - T )




=

䰘 é CP ù êë\ CV = γ d�atᨨm�c = �
C p - Cv = R úû







=


㡰
⏌



㦨
҈
黨
⧤
 � 
鼀盐㡰 ᔴ  �
�
�  
 �泐
D =

æ  öæ
� ö =
ç ÷ç ÷ -҈҈҈㡰 = -

䰘 ├ è
øè  ø

D




ᔴ - ᔴ ᔴ - ᔴ

Q ᔴ  - ᔴ  = 㡰

㡰 矄� �
⼄盐���ᨨ
ᨨ
 �泐

犨


C=

� ᨨ
鼀ᨨ ᨨᨨ鼀�ᆠ
�矄
g = 





ᔴ - ᔴ  - ᔴ - ᔴ  - ᔴ + ᔴ  = ᔴ - ᔴ ᔴ - ᔴ


KT = mCT


ᨨ
턼
㦨
҈





㦨
҈

├ ᆠ㡰 ᔼ

��ᆠ
ᨨᆠ矄矄

g҈
㦨
ᔴᨨ 矄 







ᨨ

҈҈g҈
㦨


g҈



㡰
⏌
㦨


㡰
⏌




Þ

æ
ö = g = ç҈ + ÷ è ø

犨 

�矄
�
ᨨ
ᨨ鼀

æ  ö =   =  =   ( ) ├ ÷

è
ø 꾀
 æ鼀ö  æ
ö 


㦨
 ç ÷ =  ç ÷ 㦨 ´
´҈
´ = ҈
� ´҈

è r ø
è
ø







㡰 
�ᨨ
ᨨ
矄ᆠ� �ᆠ
矄

ᆠᨨ 矄 
矄矄盐
ᔴ 㡰
  ᆠᨨ 矄 
ᨨ盐鼀
ᔴ㡰

ᔴ


㦨
҈�

ᆠ㡰 鼀
 �泐
ᨨ

鼀ᨨᆠ盐

\
턼
㦨


R M

ᨨ
泐ᨨ�
M
㦨

㡰
Cp  Cv 㦨
a =

㦨
 ç






㡰 꾀矄
犨
├ ᨨ犨

Cp

Cv
㦨
R
犨 
Cp
 
Cv

鼀ᨨ
矄ᆠ� �ᆠ 
ᆠᆠ��矄 ᨨ
Cp  Cv
㦨


㡰  = ҈ ҈ +






ö  = ÷÷ ø鼀�


� 䰘 g ҈ =
g
=
 
矄ᨨ� �
犨
�


㦨

 �
�


ᔴᨨ 矄�� �
 矄�ᨨ 
 
ᨨ�ᨨ 
鼀ᨨ矄

�矄
ᨨ ᨨ鼀
҈
㦨
�
 


㦨



g
g
-҈


g
-҈

盐� �
 
盐
ᨨ
҈
㦨





㦨


ç

҈ �

 = ´ »
´҈

ᨨ


´҈


 




҈


ö ÷÷ ø 鼀�

҈g҈ + g -҈ = ҈

҈ + g҈ - ҈

æ ᔴ çᔴ è 



=  =  ´ ҈

㡰
㡰 = ´  ´ ҈






ᔴᨨ 矄�� �
�矄
�矄
鼀ᨨ ᨨᨨ鼀�ᆠ
�




㦨
 쟼 �泐

쟼
㦨
黨
맔
҈









æ ᔴ = çç è ᔴ

෌矄
꾀
矄
鼀ᨨ
 
�㜠�矄
ᨨ
ᨨ鼀
㡰 쟼 �泐
ᨨ
 
�矄

쟼

=

҈黨


b㡰 矄� �
ᨨ鼀盐



黨




 ´҈

´҈

 ´



´҈


䰘 ´҈


\
ᔴ
㦨
黨
�
⏌├�⧤ b㡰 b㡰 졠
��� �
矄鼀
鼀ᨨ盐 
ᨨ



ᆠᨨ 矄 
矄矄盐

  �矄
ᨨ
ᆠ  �
�
鼀盐
ᨨ
鼀ᨨ ᨨ

�
 
ᨨ泐ᨨ鼀�ᆠ


æ ᨨ

ᨨ
鼀ᨨᆠ盐矄 ö  D턼㡰  = mᔴ  D ç m = è 꾀ᨨ�ᨨ Ἐ矄
ᨨ
÷ø ᨨ

D µ

҈

ᨨ

ᨨ
鼀ᨨᆠ盐矄

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PS-137

Kinetic Theory ҈


҈ U
㦨
nfRT
犨 
f

㦨
盐鼀
ᨨ
�矄
ᨨ
ᨨ鼀


䰘


㡰










䰘㡰


犨ᨨ├
ᨨ 
ᨨ
�矄
 

矄盐�
ᨨ
 
�矄
 ᨨ

T 
 
� 
⼄盐���盐鼀
鼀盐
ᨨ
 
�矄
�
 
矄矄


ᨨ
�  
 �泐
ᨨ
�矄矄
鼀�
矄鼀


fO
= �
 
fAr
㦨
 æ� ö æ ö U = U O
+ UAr =
ç RT ÷ +
ç RT ÷ è
ø è
ø

\ 






\ ᆠ㡰

ᨨ
�  
 �泐
U
㦨
҈҈


꾀�
 矄�ᨨ 
├� �ᆠ
 �泐
ᨨ

�
�矄
鼀ᨨᆠ盐

�矄


 ├ 犨 �ᆠ
 矄
ᨨ
鼀盐
ᨨ 泐

 ᨨ

�


鼀盐
�矄
矄鼀

 矄�ᨨ 
├� �ᆠ
 �泐
ᨨ
졠

 

ᨨ
犨�

⼄盐
㡰 ᨨ

�
�矄

㦨


ᔴᨨ �ᆠ� 
ᨨ
ᨨ盐鼀
  矄�ᨨ

æ D ö = ᔴᨨ 矄  ç ÷ = è D ø 
꾀�
 矄�ᨨ
⧤
쟼

ᨨ
졠

㦨


�



u҈ + u
= u Ἐ҈ + u Ἐ
ᨨ

n҈Cv DT҈ + n
Cv DT
= n҈ + n
㡰CvT

Þ 

҈㡰Cv 


㡰 + 


�㡰Cv 


㡰 = 

҈�㡰CvT \T = 黨


 ├








҈


=

n҈Cv + n
Cv
҈

n҈ + n


�


Cp Cv

=

R = ҈
�
R

T盐㡰 ᔴ泐

Cp
>
Cv
Cp

Cv
㦨
R Þ CP = CV 졔 R 

b

ᆠ㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ
ᨨ

ᨨ
鼀ᨨ
ᨨ
泐ᨨ�
㦨
ᨨ

ᨨ
鼀ᨨ
ᨨ �盐鼀
㦨
҈

ᨨ
�  
 �泐

盐

Vs 㡰m�x = Vs 㡰 He

㡰 꾀ᆠᆠᨨ� �
ᨨ
 
犨
ᨨ
⼄盐���ᨨ
ᨨ
 �泐
 
� ├� �ᆠ
 �泐
矄矄ᨨᆠ�
犨�
ᆠ
�
ᨨ
ᨨ鼀

鼀ᨨᆠ盐 �矄



=


҈ ├


ᔼ
 �矄
ᆠ矄

졠

 


ᨨ

犨ᨨ
�矄
ᨨ


g m�x M He ´ M m�x g He

ᨨ�ᨨ 
├� �ᆠ
 �泐
 
矄� ᆠ
 
鼀盐
�矄
矄ᨨ
矄鼀



´ KT


�.e.

҈


҈
 
�ᨨ
ᨨ
 
�
ᨨ�ᨨ 
├� �ᆠ
 �泐

´ KT




㡰 �矄ᨨ
A
�矄

ᨨ
鼀ᨨ

  ᨨ

犨�

矄盐�
 ᆠᨨ 矄 
矄矄盐 \ DQA
㦨
nCpDTA



�㡰 �矄ᨨ B
�矄

� 

  ᨨ

犨�

矄盐�

ᆠᨨ 矄  ᨨ盐鼀
\ DQB
㦨
nCvDTB



��㡰 盐 DQA
㦨
DQB �� 㡰 \

nCpDTA
㦨
nCvDTB

DTB 㦨

g DTA㡰 㦨
҈

㡰


⧤㡰 \ DTB
㦨



⧤

æ Cp ö ÷ DTA è Cv ø

\

DTB
㦨
ç

᲼g
㦨
҈


�ᨨ鼀�ᆠ㡰桌

 � R + ҈´ R

=
R ҈+҈


҈ ҈ n҈ + n
n n = + = ҈ +
Þ
䰘 g 鼀� - ҈ � g 鼀� x ҈ g҈ - ҈ g
 ҈ -҈ -҈  �  ҈
 � = +
㦨


Þ

g 鼀� - ҈ = \
g 鼀� =

g 鼀� - ҈



æ ö æ �ö ҈㡰 ç R÷ + ҈㡰 ç ÷ R è
ø è
ø = =
R ҈+ ҈

Cp
ᨨ
 
鼀� 盐㡰
㦨
Cv 졔 R 㦨 R

\ g 鼀� 盐 =

n҈ + n


҈´

f҈ f  � nRT +
nRT Þ u = RT + RT =
RT


d u
RT \
꾀�
�  
 �泐

鼀ᨨ
㦨
㦨

㦨

RT
n
gRT ␔
├ ᨨ犨
 
V矄ᨨ盐 
㦨
M

g҈ =

Cv ᨨ
 
鼀� 盐㡰

n҈Cv҈ + n
CV


=

� 
鼀 矄
�矄
�矄
鼀ᨨ ᨨᨨ鼀�ᆠ
ᨨ
Cr = R ҈ 
䰘 � g
=
鼀 矄
�矄
�矄
�ᨨ鼀�ᆠ
ᨨ
Cv = R
�


b㡰




=




䰘 ⧤ 

㝐ᨨ
矄ᆠ� �ᆠ

ᨨ
 
鼀� 盐

ᆠᨨ 矄 
ᨨ盐鼀

Cv =

 
�矄
ᨨ
ᨨ鼀
ᨨ
 矄�ᨨ 
鼀ᨨ�ᨨ 㡰
ᨨ犨
ᆠ矄
�
矄矄盐
� ᆠ矄矄
 
ᨨ盐鼀
ᔼ
� ᆠ矄矄
鼀


ᨨ
 
鼀ᨨᆠ盐矄

꾀矄ᨨ
�
⧤
쟼
ᨨ矄
ᨨ
 
ᨨ
 
�矄

矄ᨨ
鼀ᨨᆠ盐矄
ᨨ
ᆠ
ᆠᨨ鼀ᨨ  
ᨨ 鼀� 盐
ᨨ
�矄矄

矄鼀
�
 矄�ᨨ 
 �泐




҈ � DU nCv DT Cv ҈ = = = = = Q p nC p DT C p g 䰘 ⏌ � 䰘

㡰 
�  
 �泐
ᨨ

鼀ᨨ矄
ᨨ

�
�矄

␔
├ ᨨ犨
 
Vrms
㦨


Vrms 㡰 He = rms 㡰 H


\
V




 ´

=
= � � ´  ҈ ´
+ ҈´
é ù = ú êëQ M m�x =
û

RT M

M H
M HE

=


҈ =



b

㡰 ␔
├ ᨨ犨
ᨨ

�矄矄
Cp

Cv
㦨
R

� �  R
錐
C p = R
錐
g =

҈� \ C p
Cv =
 
C p + Cv =

� 䰘 䰘 ᨨ
�ᨨ鼀�ᆠ
�矄



Cv = R
錐
C p = R
錐
g =

� � + =
C C \
C p
Cv =
 
p v


ᨨ
鼀ᨨ ᨨᨨ鼀�ᆠ
�矄



Cv =

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EBD_8341 PS-138





Physics

b㡰 
ᨨ
 矄�ᨨ 
├� �ᆠ
 �泐
ᨨ

鼀ᨨ矄
ᨨ
�矄
㦨



҈
�



æ ҈
ö ÷和 꾀 ø è 꾀


㦨
D
꾀
㦨
ç

Q

PV
㦨
㡰


㦨
 ҈


㡰├ 和
㦨D ├ 和

Ӹ矄

 
鼀ᨨᆠ盐矄
ᨨ

�矄
ᆠᨨ�
犨�
ᆠ
ᨨ 
 
 
ᨨᆠ��矄 ᨨ
 
鼀ᨨᆠ盐矄
ᆠ  �
盐
ᨨ
ᆠᨨ�矄�ᨨ


æ ö = ⧤  ᨨ鼀

ᆠᨨ 矄 㡰 ÷ çQ è 꾀 ø

Topic-4 : Miscellaneous (Mixed Concepts) Problems ҈


㦨







 ᆠ

ᨨ�ᨨ
㡰
�矄
ᆠᨨᆠ


㡰 

�矄
矄盐�

ᆠᨨ 矄 
矄矄盐

�.e.

 
ᨨᆠ矄矄 �矄
�矄ᨨ�ᆠ \
Q
㦨
n Cp
Dt

=

� é� ù êë
R úû ´ Dt
㦨


맔


맔


҈
맔
�
㦨




 � æ ö çQ C p =
R ᨨ 鼀ᨨ ᨨ㜠ᨨ鼀�ᆠ �矄 ÷ è ø

ᆠ㡰

҈和

æ nRT ö \ç T = ᆠᨨ 矄 
ᨨ
T V ҈ = ᆠᨨ 矄  è V ÷ø
\
PV
㦨
nRT㡰 �  �� �
 
⼄盐�ᨨ

犨
�

ᨨ
盐鼀
ᨨ
鼀ᨨᆠ盐矄⏌矄ᆠ



T
T T
dT  dV =
ᨨ dT =
dV
V V V

= ҈和 =




�㡰

ᨨ鼀
 
⼄盐�ᨨ

dV
㦨
Vg
dT

g
㦨
ᆠᨨ �ᆠ� 
ᨨ
ᨨ盐鼀
  矄�ᨨ
ᨨ
�矄
㦨


dV V


 dV = V
 T ҈ � DU nCv DT Cv ҈ = = = = = Q p nC p DT C p g 䰘 ⏌ � 䰘

ᨨ鼀
⼄

�㡰

g
㦨
㡰





黨

J㡰

䰘


ᨨ
 
��ᆠ
ᨨᆠ矄矄

TVg҈
㦨
ᆠᨨ 矄 

æVf ö T Þ � =ç ÷ T f è V� ø

g ҈

æ
V ö T Þ � = ç �÷ T f è V� ø � ᨨ
鼀ᨨ ᨨᨨ鼀�ᆠ
�矄
g =  T� T \T f = = �

�  ҈

(

g ҈

)

ᔴ  �
�
�  
 �泐
Du
㦨
nCVDT

 æT  ö æ ö = ҈ ´ R ç �  T� ÷ = ´
ç ÷ ´ ҈

= 黨

J è
ø ø
è

�



鼀ᨨ
矄ᨨ
犨�
盐矄
ᨨ

ᨨ矄��ᨨ
犨�
⼄盐
矄矄盐 ᨨ
� 
矄�矄

 ᨨ

҈
㦨





鼀

҈

鼀






㦨


㦨

㦨
҈ 㝐
㝐҈ 


° 
㝐҈ 

´
= ҈黨
° \ a
㦨
= = = 
+ 䰘㡰 ҈ 㝐


䰘

҈
㦨
Þ





b

ᆠ㡰
ᨨᆠ
㦨
矄矄盐
맔
꾀

҈ =

҈





 

= 꾀 꾀





D  ҈㡰 和 = D D

D ├  和 æ D ö æ ├  ö =ç ÷和 ÷ç b l è l øè b ø

졠�ᨨ
ᆠ㡰
�矄
ᆠᨨᆠ
꾀矄
D
犨�
ᆠ矄
犨�
�鼀
矄ᨨ

ᨨ
鼀ᨨᆠ盐矄
�� �  矄 
 ᨨ盐�
 
盐
ᆠ矄矄
犨�
�鼀
 ᆠ
ᨨ�ᨨ
㡰
�矄
� ᆠᨨᆠ
b

ᆠ

㡰 㡰 꾀矄
犨
├ ᨨ犨

Q
㦨
mc
DT

Þ

෌�

T�
㦨
҈

├

Vf
㦨

V�

ᨨ

T�V�g҈
㦨
TfVf g҈


和

ᨨᆠ
 ᆠ
㦨
矄矄盐
맔

㦨
ᨨ
盐鼀
ᨨ
鼀ᨨᆠ盐矄
맔 b D
├ 和
㦨
l ҈和b
᲼b
㦨
�矄ᆠᨨ盐矄
ᨨᆠ
�� 㡰桌 Þ ҈bl 㦨
D ├

和ᨨ
ᨨ�ᨨ
㡰
�矄
ᆠᨨᆠ


pT

㦨
ᆠᨨ 矄 
�� 㡰




D ├  和 b

dQ dT dQ = mc µ C �




ᨨ

矄ᨨ�ᨨ
µ
ᔴ
ᨨ
dt dT dt

ᔼ
 
 �


ᨨ
҈

⧤
ᨨ鼀
 
�

ᔴ
� ᆠ矄矄
犨� 鼀盐
盐
ᨨ
� 泐
  ᨨ
 


犨 �ᆠ

�矄 矄ᨨ
�矄
犨�
鼀盐

盐
ᨨ
� 泐
㡰 꾀矄
 
盐
ᨨ
ᔴ
�矄
�
�
 
鼀盐
 �



㜠 �

⧤

 

矄ᨨ
�
� ᆠ矄� �
 
鼀盐
ᨨ鼀


㜠 ҈

⧤
�矄
矄矄
 
 

⼄盐�
ᨨ
� ᆠ矄� �
 
鼀盐 ᨨ鼀




㜠
�

⧤
ᆠ㡰 ᨨ鼀
 
�
 
盐
ᨨ
ᔴ
ᨨ矄
ᨨ
ᆠ  �
�
  鼀盐
 �



㜠�

⧤

  ᨨ
 
�矄
ᨨ
ᆠ  �
�

 

ᨨ

矄ᨨ�ᨨ
�
 �矄
 �
㡰 꾀矄
 
盐
ᨨ
ᔴ
� ᆠ矄矄
ᨨ鼀



㜠

⧤

 

ᨨ 
矄ᨨ�ᨨ
� ᆠ矄矄
�
 
 �



㜠

⧤


b

㡰 

㡰



ᨨ 

ᔼ矄ᨨ 鼀 ᨨᆠ矄矄

꾀 




 


 ��
�㡰


ᨨ






 ��
��㡰



␔ᨨ├ᨨ 
㦨
꾀
盐 
P

V
ᆠ盐
 
V㜠 �矄 \
W҈
>
W
㡰 ᨨ
矄盐泐
㜠
��鼀
ᔼ
 
��
ᨨᆠ矄矄

AB
�矄

矄�� 
� 

ᔼ
矄

��
矄ᨨ  

ᨨ矄��
� ᆠ



⼄盐�ᨨ
ᨨ
� 
�矄
P
㦨
a
졔
b


�㡰

WWW.JEEBOOKS.IN

PS-139

Kinetic Theory ␔ 
a
 
b

ᨨ矄��
ᆠᨨ 矄 矄


Q PV = nRT nRT nRT \ =  aV + b \P = V V  aV
bV ᨨ
T
㦨
+ nR nR

ᨨ



��㡰

㦨
-
´


���㡰

�矄
矄 矄

ᨨ
�
鼀矄
ᨨ
T
 
V
\ 

AB
ᆠᨨ鼀矄


ᨨ
ᨨ
ᆠ㡰 ᨨ
P㜠T
��鼀

�鼀� 

�
�㡰
 
��㡰

҈҈


҈



nRT \ P = a + b ᨨ P
 Pb = anRT P

dV




d
T dV
V
㦨


黨


=

҈-

é P
+ H rl g ù \ T = T
ê ú ë P
+  H - y 㡰rl g û




㡰

-
 ´


a 
a +
= nR nR




�矄
��

ᔼ
鼀 矄
T
矄
矄ᨨ鼀
鼀 �鼀盐鼀
盐


҈


b
�矄
 
盐
ᨨ
鼀 �鼀
ᨨ
鼀盐

a




T ᲼ P +  H - y 㡰rl g 桌 � nR ´

V=
᲼ P
+  H - y 㡰rl g 桌 ᲼ P
+ H rl g 桌� nRT
= 

\
盐ᨨ泐 ᆠ泐
ᨨᆠ

DW҈ + DU҈ = DQ҈  
DW
+ DU
= DQ


DQ҈ + DQ
=


 g 桌�

[ P
+ H rl g ] T҈ T
꾀
ᆠᨨ 矄 
矄矄盐

V
µ
T
ᨨ

= V҈ V
T T \
҈ =

\V
㦨
A
맔
h㡰 Ah҈ Ah
T h



´҈ = 鼀 Þ h

㦨

҈ = 

 T҈ ᲼ P
+  H - y 㡰rl

҈



\ T
㦨
�
�
⧤

nRT
rl g

= V rl g =

㡰 꾀矄
 
矄矄盐
ᨨ
�矄矄
�
ᨨ
ᆠᨨ鼀鼀 矄
�矄
 
矄鼀 \
nC p҈ ( 䰘

- T ) = nC p
( T -


)




᲼ P
+  H - y 㡰rl g 桌� [ P
+ H rl g ] �

 䰘 R ( 䰘

- T ) = R (T -


)

Þ
҈

- T = 䰘T -



Þ ҈
T

㦨

黨

\ T
㦨

黨

⧤

�



�T
㦨
䰘T





Þ ҈
T

㦨



꾀泐� �
�矄
犨
ᨨ
 鼀ᨨ泐 鼀�ᆠ矄

é P +  H - y 㡰rl g ù � T = T
ê
ú ë P
+ H rl g û b㡰 Q
PV
㦨
nRT nRT nRT Þ
V = = P P
+  H - y 㡰rl g ␔ 
P
�矄
矄矄盐
ᨨ
 
盐


�泐
ᨨᆠ�ᨨ
�矄  泐
ᨨ鼀
 
ᨨᨨ鼀
和盐矄�盐� �
 
盐
ᨨ
P
 
T
ᨨ鼀
ᨨ
犨
�

꾀矄ᨨ
PAVA = PBVB
RTA = RTB ᨨr TA = TB ᔼ
�ᨨ� �
ᨨ鼀
꾀
ᨨ


 
鼀盐
ᨨ
 
�矄
�矄
� ᆠ矄矄
ᨨ 鼀 �鼀盐鼀

V
㦨
b⏌
a


 
鼀盐
ᆠ矄矄
 
矄ᨨ
ᨨ
ᨨ��� 
盐
㡰 
T

 
� 
鼀盐
ᨨ
 
�矄矄
犨 
⼄盐���盐鼀 �矄
ᆠ �

ᨨ矄
泐
鼀ᨨ ᨨ鼀�ᆠ
�矄

ᆠᨨ 矄 
ᨨ盐鼀 㦨

�� 
泐
�ᨨ鼀�ᆠ
�矄

ᆠᨨ 矄 
矄矄盐

� 䰘 R ( 䰘

- T ) = R (T -


)



� �⏌ 

é P
+ H rl g ù  �









= T
ê ú ë P
+  H - y 㡰rl g û




�㡰

\
nCv҈ ( 䰘

- T ) = nC p
(T -


)

҈



 ᨨ

ᨨ
犨ᨨ├
ᨨ 
㦨
҈


㡰 
ᨨᆠ矄
ᆠ� �
矄�矄
盐ᨨ泐 ᆠ泐
ᨨᆠ
犨 �ᆠ
�矄
  ᨨᆠ
盐
ᨨ
矄矄盐
ᨨ
 
�⼄盐�
 �㡰
ᨨᆠ
ᨨ
��泐
�ᆠ泐
ᨨ犨 犨矄㡰 ��㡰
�矄ᆠᨨ盐矄
ᨨᆠ
�ᆠ泐
ᨨ犨 犨矄㡰 b㡰 꾀矄
 
盐
ᨨ矄
ᨨ
 ᆠ  �
 泐

犨�
 
�⼄盐�
和ᨨ

 
ᨨᆠ矄矄
�矄
��ᆠ
꾀泐� �
�ᨨ

T g P҈ g = ᆠᨨ 矄 

éP ù g T
= T҈ ê ҈ ú ë P
û 
T҈
㦨
T

P҈ = P
+ H rl g

T

㦨
T

P

㦨
P � P = P
+  H - y 㡰rl g

g = 

ᨨ鼀
eq ( ��� )
T =

d
T

R �R ´ ҈䰘� +
´ ´҈
� 㦨
�
�R
졔


�R
㦨

҈


R



҈-g

Pb P
+ ᨨ
T
㦨



�㡰 a nR a nR

�矄
矄 矄

ᨨ
�
鼀矄
ᨨ
T
 
P
\


AB
ᆠᨨ鼀矄


ᨨ
ᨨ
㡰 ��ᨨ
ᨨ

ᨨ �
AB

 aV
bV + nR nR dT 
aV b \ = + dV nR nR b dT =

V = ␔ 
dV
a

- ( DW҈ + DW
) = DU҈ + DU


㦨
nCv҈ ( �
� - 䰘

) + n
Cv
( �
� -


)


�

෌矄
�矄
ᆠᨨ鼀矄矄
犨� ᨨ盐

 ᆠ  �

和ᨨ
ᨨᆠ矄矄
�矄
��ᆠ ᨨ鼀
T҈V҈r҈
㦨
T
V
g҈

æ V
ö \
T '҈
㦨
T

ç ÷ è V҈ ø

g-҈

æ
ö =


ç ÷ èø


⏌�

㦨






⧤

WWW.JEEBOOKS.IN

EBD_8341 PS-140

҈�


Physics

␔ 
ᆠᨨ � 
�矄
矄ᨨ

ᨨᆠ�泐
ᆠ矄矄
泐
v



ᆠ㡰

҈
 ᆠ
├� �ᆠ
 �泐
ᆠ矄矄
泐
MV



dv ҈ = k T ҈⏌
dT
kdT ᨨ
dv = 犨  dT = ҈K
= 

⧤

T dv kdT ҈ ҈ æ dT ö = ´ = ç ÷ ᨨ
v
T k T
è T ø ҈ dT dv \ ´ ҈

= ´ ҈

v
T ҈ æ ҈ ö ҈ ᨨ
奠
ᆠ  �
�
v
㦨
´ ç ÷ ´҈

= =

҈
䰘奠
è 

ø


ᨨ


ᆠ矄
�
├� �ᆠ
 �泐
㦨
ᔼ ᆠ矄
�
�  
 �泐
ᨨ
 
�矄

҈
mv
㦨 nCVDT


mv

҈
m æ ö mv
= ç R ÷ DT \ DT =
M è
ø R ҈




T

鼀盐
ᨨ
 
�矄


�矄 ᆠ

泐

ᨨ鼀
ᨨ �盐鼀
�


҈ 矄
�



�鼀 �



㡰

ᨨ
��ᆠ
ᆠ  �

PV
㦨
ᆠᨨ 矄 

犨
犨ᨨ
矄盐ᆠᆠ矄矄�
ᆠᨨ�矄�ᨨ 矄









㦨

l
㦨

m

Q
l
㦨
҈
鼀㡰

\ V ( dP ) + P gV g ҈ ( dV ) =


�矄 ᆠ
= = ҈



鼀⏌矄 鼀矄
㦨
�鼀 ҈⏌ �



ᨨ
1m

㡰

Q ᔴᨨ鼀矄矄���泐 ( C ) =

҈ \ DC = C
 C҈ gP ҈ ҈ ҈æ ҈ ҈ö ᨨ
DC =  = ç  ÷ gP
gP҈ g è P
P҈ ø

 ├


  -
 ´ ҈

⧤
쟼

㦨
├ = ´ ҈

´ ҈




(

)

㦨

҈

맔
҈

҈

ᆠ㡰

ᨨ鼀
�
�矄
⼄盐�ᨨ

㦨
nRT
㦨


꾀��

P҈ =

鼀

㝐

ᨨ
҈
㦨


 ᆠ
鼀矄矄
ᨨ
�盐鼀
�矄
�
 
ᨨ

鼀
㦨



҈

� 㡰 
 
盐鼀
ᨨ
鼀ᨨ矄
ᨨ
�矄
B
㦨
n


盐鼀
ᨨ
鼀ᨨ矄
ᨨ
�矄
꾀
㦨
҈

nRT ҈黨 Q U = g ҈


m = ҈ \ Um = U A + U B

( nA + nB ) RT g m ҈

=

ᨨ





ᨨ


㦨

n
ᨨ
n
㦨



gRT M n A M A + nB M B ᨨ
鼀� 盐

M
㦨
n A + nB (҈ ´
) + (
´ 
)

= g 鼀ᨨ ҈ M
㦨
҈+
 ҈黨

҈ ´ 

´  ᨨ v =

҈鼀⏌矄 \v = ´ ҈

´ ҈


㡰

和
ᨨ
矄ᨨ盐 
v 㦨


V

(҈ +
) ´

҈ ´ 



ᨨ P҈ =

䰘
䰘黨 V




��㡰




���㡰

҈æ ҈ ҈ö ҈ é҈ ù  ÷=  ҈ú ç g çè P҈ ´ �g P҈ ÷ø gP҈ êë �g û ҈V é ҈ ù ҈ ᨨ
DC = ҈黨 ´ 䰘
䰘黨 êë �҈黨⏌҈ úû (҈ ´

黨
) V (҈ ´ ҈) V ҈V = 㦨
[

҈  ҈] =  ҈黨 ´ 䰘
䰘黨 ҈黨 ´ 䰘
䰘黨 ҈黨 ´
҈
\ DC =

n A RT nB RT + g A ҈ g B ҈

 �n ᨨ ҈ + ҈n = 黨 + ҈�n = +



( n A + nB ) RT

盐� �
盐
ᨨ
P҈

P

�
⼄
�㡰

n ҈+ n ҈ = + ҈黨 � 䰘 æ ö æ ö æ ᨨ
ç ҈ ÷ ç ҈ ÷ ç ҈ ö÷ è ҈ ø è  ø è � ø

҈ (҈ + n )




�㡰

V g g æ �V ö P
æ V҈ ö = ç ÷ = ç ҈ ÷ = �g 꾀��

P҈ è V
ø è V҈ ø ҈黨 ö g æ �� ÷ ᨨ
P
= P҈ ´ � ç 犨  g = è ø ҈

PVM ҈

´ ҈ ´
=
Þ
鼀
㦨



҈

�鼀 \
m = RT
� ⏌  ´ ҈



\

҈ 盐├
鼀ᨨ盐盐矄
( E )

\C =

꾀�
├� �ᆠ
 �泐
ᨨ

ᨨ鼀
ᨨ

鼀ᨨ ᨨᨨ鼀�ᆠ
�盐鼀

�矄
⧤
쟼
㦨
\

gP dP dPV =  ᨨ  = gP dV V dV

ᨨ
盐├
鼀ᨨ盐盐矄
㦨
쟼
㦨
g



 ´
� ⏌  ´

Þ ҈



㦨

\

㦨
҈


⧤ 㡰 鼀矄
㦨
㝐
´ ҈
-

҈䰘


ᨨᆠ�泐
ᨨ
矄ᨨ盐 
v µ T \ v
㦨
ᆠᨨ 矄 
k㡰
T҈⏌


҈



㦨








䰘V
㦨



䰘
맔
҈
�
V


ᨨ
矄矄盐
 
泐
 
鼀� 盐
P
㦨
҈
�
鼀


鼀盐
T
㦨



⧤
錐

ᨨ盐鼀
㦨





鼀 
 

x
�鼀
ᨨ


\
鼀矄矄
ᨨ
꾀
㦨




x㡰g

x 錐



- x 盐鼀
ᨨ
�鼀
鼀ᨨ矄
ᨨ
꾀�ᨨ

n

㦨




盐鼀
ᨨ
�鼀
鼀ᨨ矄
ᨨ
ᨨ

n҈
㦨


盐
ᆠᆠᨨ� �
ᨨ
ᨨ Ἐ矄
犨
ᨨ
�
矄矄盐

n҈RT n
RT + V V x RT x 㡰 RT 

+ ҈
�
㦨


V

V

P
㦨
p҈
졔
p

㦨
ᨨ


WWW.JEEBOOKS.IN

PS-141

Kinetic Theory

҈
� ´

´



Þ 㦨
x
졔





Þ x
㦨



䰘

g

҈
´ 



҈黨


\

nR T - T 㡰 g -҈

´

 


- 

㡰 㦨



黨
맔
҈
�






㡰
҈


졔

� -҈ 

㦨
P҈
V


V҈㡰

졔


Pf
㦨
P鼀
졔


Þ Pf
㦨
҈
�
졔



´ ҈
-

ᔼ 㝐ᨨ ᨨᨨ鼀�ᆠ ෌矄




㦨


맔
҈
�
⏌鼀
꾀鼀
矄矄盐
㦨
҈
�⏌鼀


҈㦨

⧤


҈


k 㦨




⏌鼀

� 
ᨨ盐鼀

Vf 㦨
V҈
졔
xA 㦨




맔
҈

졔


҈
맔


맔
҈

㦨



맔
҈

鼀

PV 꾀泐� �
҈ � =

Pf V f

T�

ᨨ


Tf
㦨


Tf


Þ Tf
㦨


�ᆠ鼀

PV � �


´ ҈
� ´ 

´ ҈
- ´ 

҈
� ´


´ ҈
-

=


⧤
꾀

҈




㡰

\



\

P҈
맔



맔
҈

㦨


맔



맔


P҈
㦨



黨
맔
҈
�
鼀
ᨨ
ᨨᆠ矄矄
҈
®


�矄ᨨ�ᆠ㡰






V҈ V
ᨨ


=



Þ
T
㦨




⧤ =
T҈ T



ᨨ
ᨨᆠ矄矄



®

�b�ᆠ㡰 ҈

é T ù g -҈ é


ù � -҈ V
㦨
V
ê
ú
㦨
҈҈l =

ê ú ë 

û  ë T û � ᲼Q g
㦨

ᨨ
鼀ᨨ ᨨ
ᨨ鼀�ᆠ
�矄桌 

ᨨ犨
PV
㦨
nRT






҈
㦨




꾀




� ´

� ҈
�



鼀ᨨ矄
 矄 
ᨨ鼀
��
鼀盐
矄�
ᨨ
ᨨ犨
鼀盐 矄�
盐
V

V

n

n ᔼ ��泐

T
V
g -҈ = TVg -҈ \

ᔴ

졠


맔

㦨


��㡰 ᨨ
� 
 
� 
ᨨ盐鼀
 
矄矄盐
ᨨ
 
�矄
꾀泐� � P҈V҈
㦨
nRT҈

҈

�ᆠ鼀

ᔴ

P2




� ´

� ´

҈
� é

+


� ù


� ´

� ´

Þ 
맔
ê úû =
҈
� ë +
᲼
Q

㦨
҈ 桌


Þ 
㦨
䰘�


ᆠ鼀




l





盐� �
 �矄
盐
ᨨ
P҈
�
px
㦨
P҈x҈


㡰



l



P1



㦨
҈
졔
�
矄�


㦨
҈
졔


� 和盐矄�盐� �
 �矄
盐
�
���㡰 ҈
맔
҈
㦨
᲼҈
졔


�桌
맔

ᨨ
҈
맔



㦨
᲼҈
졔


�桌
맔



�








x1


ᆠ鼀 㦨


鼀� 矄� 
° � 矄� 
° = = 꾀 꾀 꾀 꾀 ´ �㡰 ´ � 矄� 
° 㦨
�
矄�


ᆠ鼀
ᨨ
� 㦨

꾀

 ҈
҈㡰

x

鼀
�ᆠ




㦨



ᨨᆠ矄矄
ᨨ P

V
��鼀
�矄
矄
矄 ᨨ犨
ᨨ犨
҈㦨


x

x 2㦨

矄矄盐
盐
ᨨ
鼀ᆠ盐泐
ᆠᨨ盐鼀

҈
Q
㦨
P D V + nCv D T + k x
PV ҈ ҈ ҈ Cv DT + kx
㦨
҈
�
᲼



맔
҈
桌
졔
RT҈
� - ҈ ҈
´


´҈
 㦨



졔
´ ´
´ �


졔
´



맔


҈
㦨
䰘




´ 


�㡰

㦨

黨


졔
䰘
䰘


㦨
҈

�



꾀

 

ᨨ
ᆠᨨ矄矄㜠矄ᆠ�ᨨ
ᨨ
 
盐
和� ᆠ
鼀盐
�矄
 
矄鼀

泐� �
ᨨ泐Ἐ矄
犨
�



PV
㦨 ᆠᨨ 矄 
�
犨ᨨ
矄�矄
ᨨ
鼀ᆠ盐
ᆠᨨ盐鼀


맔

맔
꾀㡰
㦨


맔


맔
꾀㡰   
맔

맔
꾀㡰
㦨
҈
맔
 ҈
맔
꾀㡰 \ ҈
맔
 ҈
맔
꾀㡰
㦨


맔


맔
꾀㡰 ᨨ
҈ ҈
㦨





�㡰 犨 


㦨
҈
졔
矄矄盐
盐
ᨨ
鼀ᆠ盐泐
ᆠᨨ盐鼀

Pf V f T�



矄盐�
泐
 

�矄
盐矄
ᨨ
  矄�ᨨ
ᨨ
 
�矄
� ᆠ矄� �
�矄
鼀盐
 
矄ᨨ� �
ᨨ �
 �泐
�
  矄� �







nRT
´

 ´ 

= 㦨





맔
҈
�
⏌鼀
V ҈҈ ´ ҈
-

Q T
㦨
T҈
��

���㡰 ␔ᨨ├
ᨨ 
泐
 
�矄 W
㦨
W҈

졔
W


\ 㝐矄矄
ᨨ
ᨨ
㦨



䰘

g

  鼀矄矄
ᨨ
꾀�ᨨ
㦨







䰘

㦨


黨


g � 
矄矄盐

kx A



´

҈

P
㦨


Q 꾀泐� �
PV
㦨
nRT ᨨ
 
盐 䰘

맔

㦨

맔

䰘 Ἐ
맔

㦨

졔
㡰

맔

䰘 ��� �

犨
� Ἐ + = 䰘


n졔x
졠ᔴ

n㜠x � 泐



졠ᔴ

ᔼ ��泐 � 泐



�㡰

WWW.JEEBOOKS.IN

EBD_8341 PS-142

Physics

ᨨ
�� 
盐 䰘

맔

㦨

맔

䰘 Ἐ
맔

㦨


㡰

맔
� 꾀��
��� �

犨
�

 é


ù P ´ V - P
V
ú
êë 


䰘
û æ黨 ö ҈� 㦨
- ç - ҈÷ P
V
= - ´ RT

㦨
҈

䰘�
RT

è
ø
㦨
-

ᔼ ��泐 � 泐

 Ἐ - � = ´ 䰘

䰘





��㡰




ᨨ鼀
⼄

�㡰
 
��㡰

+ - � = ´




䰘



Þ
㦨





㦨



䰘
Ἐ

´ 䰘
=

䰘� ᆠ鼀 �





\
Ἐ
㦨
= ҈+


䰘



�㡰





�㡰

Þ



 ´ V
g 


V
æ 
ö =ç ÷ V
è

 ø

�⏌

\

n
㦨


҈
� ´



 ´ ҈

 = ҈
黨䰘 ´ ҈

䰘
´ 
´





҈-g

ᨨ


㦨



= 㝐 g - ҈㡰


´

҈

㦨
�


鼀矄҈ 
´ ҈

҈ - ҈㡰 ҈




�
҈
㦨






㦨




鼀鼀
�


錐




䁸 ҈
㦨
 錐



㦨

錐


҈
㦨


錐



㦨





҈

㦨


黨黨


҈҈ 

҈



´


黨黨

= = 䰘黨
鼀鼀�

҈


\



㦨
=

҈

҈- �⏌

æ

 P
ö =ç ÷ è 
P
ø

҈ 㝐




R ҈ Mv

= Cv

盐
Cv 㦨
g -҈
҈ R
\ Mv
=
g -҈





��㡰

꾀��
盐矄� �
PT
⼄盐�ᨨ
ᨨ
�� 
ᆠ 鼀
�



P҈g
Tg
㦨
ᆠᨨ 矄 

æ T҈ ö æ P
ö ç ÷ =ç ÷ T è
ø è P҈ ø



 ´

҈
´ 

= 㦨





鼀⏌矄 㝐 
⏌҈




\





V
= V

䰘
䰘



´ V
´ ´ T
ᨨ鼀
⼄
�㡰
 
��㡰


T҈
㦨

V
´
䰘

䰘 T
= ҈

黨T
ᨨ
T҈
㦨
҈
V҈
㦨

V


V

㦨

V




g

v
㦨


㦨
nCv
D
㦨
҈
맔
Cv
맔
҈
㦨
Cv




V





Þ V

㦨

䰘

盐 V҈
졔
V

㦨

V
\

҈
�
鼀

⧤
쟼

ᨨ
҈
�鼀
鼀ᨨ
ᨨ
ᨨ 泐�


⏌ �




�㡰

��㡰 ⧤� �ᆠ
 �泐
ᨨ
鼀ᨨ�ᨨ
ᨨ
ᨨ 泐�
鼀ᨨᆠ盐矄
犨�
 ᆠᨨ 
� ᨨ

 �泐


ᨨ
 
�� 
ᆠ 鼀

��ᆠ
ᆠᨨ鼀矄矄�ᨨ
ᨨᆠᆠ盐矄 \
P
V
g = P
´

\
n 㦨


ᨨᨨ
鼀
矄⼄盐
ᨨᆠ�泐

P
V
P
´

 = ´ V҈ T

´ T҈

 V҈T
´ V



D Þ D = ´ D = ҈
� ´ ҈ = ҈
� D


맔

鼀
㦨
҈
�




ᨨ
 
 
ᆠ 鼀

Þ T҈
㦨


ᨨᆠ

㦨

맔
꾀
㦨
҈
�
맔
҈
㦨
҈
�


ᨨ犨

鼀ᨨ鼀 盐鼀
ᆠ  �

矄ᆠᨨ  D㡰
㦨

맔

鼀


��㡰 ␔ 

㦨
盐鼀
ᨨ
ᆠᨨ�矄�ᨨ 矄

矄ᆠᨨ 

矄⼄盐
鼀
 ᨨ鼀
�㡰
 
��㡰





���㡰

和盐矄�盐� �
 �矄
盐
ᨨ

�
⼄
�㡰





�㡰

Þ

æ T
ö ç ÷ è T
ø

��㡰

␔ᨨ├
ᨨ 
泐
 
�矄
�
�� 
ᆠ 鼀
��ᆠ
ᨨᆠ矄矄㡰

W
㦨


҈  P
V
- P
V
㡰 ҈- g





ᨨ


T

㦨



�
T







\ ᨨ
矄矄盐
�
 

㦨
䰘黨

졔

�
㦨

҈䰘
鼀鼀
� ᨨ鼀


㦨


␔ 




矄鼀
n µ V 꾀矄
ᨨ盐鼀矄

矄鼀

�



҈ᆠ
ᆠ
ᨨ
ᆠ
泐ᨨ�
 
ᨨ 泐�

和ᨨ ᨨ
矄鼀矄
犨�

⼄盐
盐鼀
ᨨ
鼀ᨨᆠ盐矄

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PS-143

Oscillations

Oscillations

13

Topic-1 : Displacement, Phase, Velocity and Acceleration in S.H.M. ҈





㦨


w 矄�
w

泐
㦨

w ᆠᨨ矄
w

 

㦨
w

㡰

҈



Þ  = 
+ 
泐 +  


( w 矄� w ) + ( w ᆠᨨ矄 w )


Þ=




+ ( w )






 =
w



ᆠ㡰 ᨨ鼀
 
犨ᨨ
鼀盐盐泐
 �ᆠ盐
和

㝐
矄

 
�  ⼄盐�ᨨ
ᨨ
�矄矄ᨨ盐矄
��盐


x
A




+

y
B




-

㦨

ᔴ泐
A ¹ B
 ᆠ
�矄
b㡰 㝐 �鼀盐鼀
ᨨᆠ�泐
�
和㝐

鼀
㦨
w 㝐 �鼀盐鼀
ᆠᆠ�ᨨ
�
和㝐

꾀鼀
㦨
w
犨 

 
w

鼀 �鼀盐鼀
鼀�盐
 
 �盐
⼄盐 ᆠ泐
෌�
 



꾀 鼀 = ҈
 鼀












b㡰 ␔
├ ᨨ犨
 

㦨
w 꾀 -

æ
꾀 ö ᔼ ��泐

㦨
w 꾀 - ç ÷ è  ø




졠
��� �
犨
�





꾀

ᨨᆠ�泐
ᨨ
㡰 


p⏌

꾀
҈ 㦨
꾀




黨


㡰


�鼀

㦨

㡰

꾀


p 




쟼⼄盐���盐鼀
ᨨ�  꾀


p
犨�
矄ᆠ
ᨨ
x҈
t㡰
矄
矄 ᨨ犨
�
 
��盐


꾀

㦨
꾀

é

꾀
ù 䰘꾀
꾀
黨 ê 꾀 - 黨 ú 㦨
꾀Ἐ





\

꾀Ἐ
㦨
 黨 ëê ûú �


ᨨ
= - 
犨 

�矄

ᨨ矄��
ᆠᨨ 矄 
b㡰 犨ᨨ
矄� 盐矄ᨨ�
�矄ᆠ鼀 矄
x҈t㡰
㦨
A
矄�
wt
 
x

t㡰
㦨

f
㦨



␔ 
꾀Ἐ㦨
� 
鼀�盐
෌�


㦨


æ
꾀 ö 꾀
- ç ÷ è  ø

 ᆠ
ᆠᨨ泐
ᨨ
�ᆠ
犨�
ᨨᨨ├
�├
矄
ᆠ㡰
㡰 ᔼ
� 
和

㝐


 
矄ᨨ� �
ᨨᆠ
ᆠ� �
ᨨ
�ᆠ
矄 ᨨ盐 犨泐矄

ᨨᨨ�ᨨ 
ᨨ
 
�矄ᆠ鼀 
ᨨ
 
�ᆠ
  �ᆠ
ᨨ犨矄
 
⼄盐���盐鼀
ᨨ矄��ᨨ
�



µ

졔
f㡰
� �矄
 
鼀矄矄
ᨨ

ᆠᨨ鼀
矄

�矄
�矄
ᨨ矄矄�
犨 
  鼀�盐
ᨨ
 �
B
㦨
A
 
�矄
� �

 矄
�  ᆠ
ᨨ





æ
꾀 ö � 泐


㦨
w 꾀Ἐ - ç ÷ è  ø

 㦨
҈

p 錐 x
t㡰
㦨
a
 
y
t㡰
㦨


w


ö æ p ÷

鼀�盐
꾀
ᆠ

犨�

 矄
�  ᆠ  ø è p ᨨ



ᔼ
�矄
��
 
矄� 盐矄ᨨ�
�矄ᆠ鼀 
xt㡰
㦨
B矄�
wt 







ᆠ㡰 꾀
t
㦨



x
t㡰
㦨


錐 y
t㡰
㦨

x
t㡰
�矄

矄� 盐矄ᨨ�
盐 ᆠ�ᨨ

A
矄�
ç wt +

㝐 �鼀盐鼀
ᆠᆠ�ᨨ
꾀鼀
㦨
w

㦨
�


鼀⏌矄


æ
꾀 ö 꾀Ἐ
- ç ÷ è  ø

q p 㦨

㦨
p 
p


꾀
t
㦨
䰘


�




w
㦨
҈

矄҈ �矄ᆠ鼀 
�矄
��
泐
㦨

矄�
w
졔
p⏌
㡰 



㦨





㦨
� �
㦨

矄�
p⏌
�
㦨

矄�

�
Þ

㦨
�






p 


ᔼ
 泐
ᆠᨨ矄矄
ᆠ
ᨨ 

�鼀

 


xy ᆠᨨ矄 d = 矄�
d AB

x
㦨
A
矄�
at
졔
d㡰 y
㦨
B
矄�
bt
졔
r㡰 


꾀 �
ᆠᨨ
ᨨ
鼀
q
㦨



㦨


졠


㦨
꾀 㡰 
⼄盐�ᨨ
ᨨ
 
和

㝐

x
㦨
a
矄�
wt

Þ

p æ
p ö æ
p ö x = a 矄� ç ÷ ´ t = ҈矄� ç ÷ t
= 矄� t è
ø èT ø
Q a
㦨
҈
ᆠ鼀

T
㦨

s

ᨨ鼀
� 㡰

꾀 ᨨ


ᨨᆠ�泐

v =

dx d é æ p ö ù p æ pö = ê矄� ç ÷ t ú = ᆠᨨ矄 ç ÷ t è
ø dt dt ë è
ø û


\ 꾀ᆠᆠ�ᨨ
a
㦨


d
x dt





æ pö æ pö = - ç ÷ 矄� ç ÷ t è
ø è
ø

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EBD_8341 PS-144

Physics

꾀
t =

a
㦨



= ҈



d
x dt





p
-p
p æ pö = - ç ÷ 矄� ´ = 矄�

è
ø
 ҈


Þ

ᔼ
和㝐
Y
㦨
A
矄�
w졔
j㡰 y҈
㦨
҈

矄�
pt
졔
p⏌
㡰















��㡰

Þ A
=
x
Þ x =

é҈ ù  ᆠᨨ矄p ú 㦨
҈

矄�
pt
졔p⏌㡰 \


y

㦨
�
맔


ê 矄� p + ë
û
A҈


�ᨨ
ᨨ
鼀�盐矄
= =

҈



҈

㦨
҈


҈ A




b

ᆠ㡰


��
⼄盐�ᨨ

x
㦨
A
矄�

wt
졔
B
ᆠᨨ矄

wt
졔
C
矄�
wt
ᆠᨨ矄
wt  �� �
 
⼄盐�ᨨ
ᨨ
和㝐
 
矄� 
 
ᆠᨨ矄� 
盐 ᆠ�ᨨ 矄 矄 ᨨ盐

� 
ᨨ犨


A B C 
矄�
wt 㡰 + 
ᆠᨨ矄
wt 㡰 + 
矄� wt ᆠᨨ矄 wt 㡰


A B C 㦨
᲼҈ - ᆠᨨ矄
wt 桌 + ᲼҈ + ᆠᨨ矄
wt 桌 + ᲼矄�
w t 桌


C 㡰 ᨨ
A
㦨


 
B
㦨



x
㦨
矄� 
wt 㡰



ᨨ
⼄盐�ᨨ
矄 矄
和㝐
㡰 ᔼ
A
㦨
B
 
C
㦨

B
 
x
㦨
B
졔
B
矄�

wt

�矄
�矄

⼄盐�ᨨ
ᨨ
和㝐
ᆠ㡰 A
㦨

B

C
㦨

B錐 \ x
㦨
B
ᆠᨨ矄

wt
졔
B
矄�

wt

犨ᨨ
和㝐矄

矄盐ᨨ矄
ᨨ
��
 ᨨ 
和㝐
⼄盐�ᨨ
㡰 A
㦨
B

C
㦨


\ x
㦨
A


�矄
⼄盐�ᨨ
ᨨ矄
ᨨ
矄 矄
和㝐


ᆠ㡰 꾀泐� �
矄盐ᨨ矄��ᨨ
� ᆠ� y 㦨
y҈
졔
y

졔
y 㦨
a
矄�
wt
졔
a
矄�
wt
졔

� 㡰

졔
a
矄�
wt
졔
黨
㡰 㦨
a᲼矄�
wt
졔
矄�
wt
졔
黨
桌
졔
a
矄�
wt
졔

� 㡰 㦨

a
矄�
wt
졔

� 㡰
ᆠᨨ矄

�
졔
a
矄�
wt
졔

� 㡰

Þ








Topic-2 : Energy in Simple Harmonic Motion ҈


ᆠ㡰

ᨨ鼀
�
⼄盐�ᨨ
ᨨ
和㝐

X = A ᆠᨨ矄 wt ҈㡰

꾀


T
�ᆠ
�矄

鼀
ᨨ矄��ᨨ



\
꾀ᆠᆠ�ᨨ
㦨



ᨨᆠ
㦨



= A ᆠᨨ矄 wt

Þt=

T


㡰 ⧤� �ᆠ
 �泐

├ =

҈ 鼀w
꾀
ᆠᨨ矄
w


҈ 鼀w
꾀
矄�
w
├ p ҈ = ᆠᨨ
w = ᆠᨨ

҈
㡰 =  黨
 ҈
ᆠ㡰 ᨨ �
 �泐
㡰
㦨
kx
҈
҈
⧤� �ᆠ
 �泐
⧤㡰
㦨
kA - kx

꾀ᆠᆠᨨ� �
ᨨ
 
⼄盐矄�ᨨ


㦨
├

҈ ҈ ҈ \ kx
= kA
- kx



A Þ


㦨
꾀


ᨨ

x = ±




b㡰 ᨨ

�ᆠ
 ᆠ盐� �
和㝐 꾀
鼀
ᨨ矄��ᨨ 錐
t
㦨



wt
㦨



y
㦨



V
㦨
Vmax
㦨
aw

҈ mw
a

T p 꾀
 鼀
ᨨ矄��ᨨ


t
㦨


w
㦨


y
㦨
A

V
㦨
Vm�n
㦨





\

K.E. 㦨
KEmax
㦨


\

K.E.
㦨
KEm�n
㦨



⧤� �ᆠ
 �泐
�
SHM

KE
㦨


҈ ù mA
w
ú
û

E矄盐 
㦨
 +

㡰 E矄� �

\







ᨨ �
 �泐

 =

ᔴ泐

矄盐 
鼀ᨨ�ᨨ
�矄
和㝐
ᨨ
鼀�盐
A
㦨

+ ҈ 㡰
a  
� 
�
 矄
泐

�
ᨨ
泐
黨

�
ᨨ
 
�矄


E矄盐  æ A ö
= ç ÷ = 
+ ҈㡰
=  +

㡰 è aø E矄� �

A

+A

\
x
㦨

A
犨 �ᆠ
�矄
ᨨ
ᨨ矄矄� \
҈,

 


ᆠᨨᆠ


㦨

+ ҈ 㡰
a
矄�
wt
졔

� 㡰 㦨
A
矄�
wt
졔

� 㡰

é ë

҈ ҈ k  A
- x
㡰 = kx






A
㦨
鼀�盐
ᨨ
和㝐 x
㦨
�矄ᆠ鼀 
ᨨ鼀
鼀
ᨨ矄��ᨨ





�㡰

쟼 �泐
E
�
和㝐
µ
鼀�盐㡰




Q E = ê

T
�ᆠ
�矄

鼀
ᨨ矄��ᨨ
矄ᨨ
ᨨᆠ�泐
�矄


鼀 �鼀盐鼀
꾀ᆠᆠ�ᨨ
㦨


㡰 ␔ 
⧤쟼
㦨
쟼

\ x
㦨


҈



꾀
t =

㡰

- p
ᆠ鼀 ⏌ s



y

㦨
�
矄�
pt
졔
�  ᆠᨨ矄
pt

҈҈



㡰 꾀
T
�ᆠ
��

 鼀
ᨨ矄��ᨨ
矄ᨨ
ᆠᆠ�ᨨ
�矄 鼀 �鼀盐鼀



s
ᆠᆠ�ᨨ



㦨
�


҈



mw a 
y 㡰


҈ mw
a
ᆠᨨ矄
wt


 ᆠ
�
㡰
ᆠᨨᆠ泐
�ᆠ矄
├� �ᆠ
 �泐
�鼀
�
b㡰 和� ᆠ
矄泐矄鼀
�矄矄�矄
�矄
 �泐
�盐泐

 
 ᆠ 鼀�盐
犨�
矄ᨨ
ᆠ矄矄
犨�
�鼀
ᆠᆠᨨ� �
ᨨ 
㦨


⏌鼀















�㡰 Q 쟼 �泐
ᨨ
��ᨨ
ᨨ
ᨨ

ᨨ
�矄
� ��
盐
쟼
㡰

矄 쟼
µ


Þ

µ
쟼

=


 ҈
-
 
Þ
= =
鼀

҈ ҈


ᨨ鼀
⼄
�㡰



= 
 -  ҈



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PS-145

Oscillations 





҈ =  -  ҈

ᨨ

= ҈

 
=



\




㦨

�
矄ᆠᨨ 矄 ҈
҈ ҈


㡰 ⧤
쟼
㦨
k  A - d 㡰
 

쟼

㦨
kd







é êë

㡰 ෌�

U
x㡰
㦨
K ҈ - e - x




ᆠ㡰

҈ - ᆠᨨ矄
wt ö y
æ
҈ - ᆠᨨ矄
wt 㡰



çQ 矄� wt = ÷ø è

y -y Þ y -
=
ᆠᨨ矄
wt

Þ y = A ᆠᨨ矄
wt y \
꾀鼀�盐 =

Þ y=

꾀
鼀
ᨨ矄��ᨨ

d
㦨



꾀
 鼀
ᨨ矄��ᨨ 矄
d
㦨
꾀

䰘


҈ k ҈ YA =
p m
p mL y = y
矄�
wt f =

ù
\
F
㦨
- dU =
kxe - x
úû dx


ᆠᨨ矄ᨨ � �
u

x
�
ᨨ
 
��
ux㡰
㦨
K ᲼҈

ex 桌
犨 �ᆠ
�矄

 ᨨ  �泐
� ᆠ矄� �
�
ᨨ
u
犨�
x
�矄
矄
矄 ᨨ犨
�
 
��盐
U(x)

꾀 �盐
ᨨᆠ�泐
㦨

w

ky
k
g = mg Þ = m y

꾀矄ᨨ

矄� �
ᆠᨨ 矄 
k = m
w㡰
k g
g ҈
g Þ
w = = Þw= = m y

y

y


ᨨ
⼄盐���盐鼀
ᨨ
鼀矄矄



C

K

B

A x x=0 ᨨ鼀
 
�
�
�矄
ᆠ
 
 
ᨨ �
 �泐
�矄
鼀� �鼀盐鼀
 x
㦨




 ᆠ

x
㦨


�矄
 
矄
ᨨ
矄
⼄盐���盐鼀

ᨨ犨
�
犨 �矄ᆠ
 
�ᆠ
ᨨ鼀
x
㦨


 
ᨨ
�矄ᆠ鼀 矄
 
�ᆠ  矄
ᨨ
��
 
ᨨ矄��ᨨ
x
㦨


犨�

ᨨᆠ
F
㦨







黨



kx ex







ᨨ
矄鼀
盐矄
ᨨ
x
犨

F
µ
x
 ᆠ
ᨨ
矄鼀
�矄ᆠ鼀 
ᨨ鼀
x
㦨


 
鼀ᨨ�ᨨ
�矄
矄�鼀 鼀ᨨ �ᆠ
ᆠ㡰 盐� �
ᨨ 
ᆠᨨ鼀
ᨨ矄ᆠ��ᨨ

 
├� �ᆠ
 �泐
ᨨ �ᆠ
 ᆠ盐� �
和㝐
犨�
ᆠᨨ鼀
鼀 �鼀盐鼀
犨�ᆠ


 ᨨ
 
⼄盐 ᆠ泐
犨�
犨 �ᆠ
�矄
├� �ᆠ
 �泐
ᨨ矄ᆠ�矄 犨�


f


鼀� � �
 �泐
㦨


Þ AἘ =



ᆠ㡰

A


T =
p

l g

␔ 
�鼀鼀矄
ᨨ
�矄ᆠᨨ盐矄
�⼄盐�

g ö ҈� g æ amt = ç g - ÷ = ҈
ø ҈
è l ᨨ犨
T ¢ =
p =
p


b

ᆠ㡰 쟼 �泐
ᨨ
 
ᨨ矄ᆠ�ᨨ

҈
҈

㦨
kA 㦨
´
´ ҈
´ 


҈㡰 = ҈



�


㝐
ᨨ矄��ᨨ

҈ æ ҈
ö ҈
ç kA ÷ø = kA Ἐ
è





A' 㦨
犨
�矄 ᆠ
ᨨ
ᨨᆠ├
ᨨ鼀
鼀
ᨨ矄��ᨨ

҈ ҈ mw
a
= m ´
p
v
a


҈

�
� ´ ´ a

\ a
=



鼀 Þ

黨
㦨
´


´
p ´
p p

쟼 鼀
ᨨ矄��ᨨ

ᆠ㡰

l t = -
⧤ ᆠᨨ矄 q


k=

YA L

Dl ù l
úû

=




҈�

T

=

l


q

b㡰 꾀
矄�ᆠ
犨�
ᆠ


矄

矄� �
 
�矄
矄� � ᆠᨨ 矄 


é F êQ Y = A ë

l

҈� g ҈



ᨨ⼄盐
盐
ᨨ
矄� �
ᨨᆠ


x
㦨
A x
㦨

⧤
쟼

㦨

⧤
쟼

㦨
҈

 E
㦨
҈


 E
㦨
҈


 

쟼

㦨
҈

 

쟼

㦨


 Topic-3 : Time Period, Frequency, Simple Pendulum and Spring Pendulum ҈


҈
kA


꾀
⼄盐���盐鼀
ᨨ矄��ᨨ


ᨨ
 
鼀矄矄
ᨨ
ᨨᆠ├
├矄
ᨨ

矄ᨨ �矄
ᨨ �
 �泐
ᆠᨨ鼀矄




� ෌�

m
㦨




├�

w
㦨


⧤
쟼

㦨


�


쟼

㦨




 p

ᨨ
鼀ᆠ  �ᆠ
 �泐
E
㦨
҈




҈
kx


꾀
鼀
ᨨ矄��ᨨ

ᨨ �
 �泐
=

꾀矄ᨨ


쟼

㦨


҈



ᨨ �
 �泐
ᨨ
矄� �
=



x
㦨
�矄 ᆠ
ᨨ
ᨨᆠ├
ᨨ鼀
鼀
ᨨ矄��ᨨ
k
㦨
矄� �
ᆠᨨ 矄 



 ᆠ


  a
㦨
䁸


쟼

㦨
⧤
쟼

졔

쟼

㦨


�
졔



㡰

㦨


黨


㡰

q

⧤

⼄盐 ᆠ泐
ᨨ
ᨨ矄ᆠ��ᨨ


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EBD_8341 PS-146

Physics

æ ⧤l
ö Þ t=ç ÷ q = -ᔴq è
ø

é ⧤l
ù ê ᔴ = ú
úû êë

\
f2 =


f1 Þ





㡰

⧤l

= ҈
⧤ 㝐l

p 㝐 ҈


҈ ᔴ ҈ =
p ᔼ
p

=

�ᨨ ҈



෌�

y
㦨
� éë矄� p㡰 +  ᆠᨨ矄p㡰 ùû

㡰 꾀ᆠᆠ�ᨨ
盐
ᨨ
��泐
�
㦨


㝐 æ  = ç �  㝐  çè 

㝐矄矄
ᨨ
ᨨ 
ᨨ鼀
ᨨ
矄�

㦨

҈҈





㡰







⼄盐���盐 

㦨
鼀�


҈



鼀

҈


盐矄� �


㦨
w


f
=

҈ C
p ҈

҈ C
ML
҈
p


p
=

├ 鼀

鼀
㦨
҈
├�

C M Mö æ L
ç + ÷
ø è 

꾀矄
⼄盐 ᆠ泐
盐ᆠ矄
泐


奠

=

҈ ├´
҈ = 
p �


ᆠ㡰

�鼀
ᨨ矄⏌�� 

泐
=



├� ҈ µ Dq ´





矄ᆠᨨ 


꾀矄
犨
├ ᨨ犨

�鼀
�ᨨ


=
p

ᆠ㡰

l �

␔ 
��ᨨ 
鼀矄矄
㝐
�矄

 

p
r� 鼀

l + Dl g

TM =
p

␔
;䰘
黨�
矄҈





=

Þ

҈ ├ = ҈ 
p 鼀

q




㦨




q
q
㦨
҈


Þ q 㦨

� ᔴ




f҈ 㦨

⼄盐 ᆠ泐
ᨨ
矄� �

㡰
=

҈ a 

 q㡰 ´









�㡰
҈
= a q 

㡰 ´








��㡰


 q 졠
��� �
犨
�

 = q 



( p r� ) 
㦨


ᆠ㡰

ᆠ㡰

҈
=

鼀� 쟼 
ᨨ泐 
ᨨᆠ
㦨
r꾀 � 

졔

㦨
鼀�
졔
鼀 \

㦨
鼀
㦨
r꾀 �
㦨
p
r�㡰

黨


)

´҈
- ├�

ᔼ
ᨨᆠ├
ᨨ
鼀矄矄
鼀
㦨
҈
├�
�矄
ᆠ 
 
├
㦨

p
ᨨ犨

� �ᆠ
矄� �矄

ᆠ 
�

犨�
鼀矄矄
鼀
㦨

├�

 ᆠ
├⼄
㦨

├





├

ᨨ 盐矄㡰

Þw=





´ ҈



和ᨨ� �
犨
�

矄� �
ᆠᨨ 矄 
⧤
㦨
䰘
҈⏌鼀

෌㝐







(

҈



҈ ├ ´



´ ҈

 = ҈
҈

p ҈

´ ҈
-

Þ  =
 矄



҈ ├ = ҈
҈

p 鼀

犨 
鼀
㦨
鼀矄矄
ᨨ
ᨨ 
ᨨ鼀


ö æ҈ö ҈ ÷÷ =  ç ÷ = èø  ø

 ҈ � 꾀矄ᨨ
µ Þ = =  � � 

�

鼀 㦨

䰘 㝐

b㡰 꾀矄
犨
├ ᨨ犨

⼄盐 ᆠ泐
�
和㝐
㦨


pö æ Þ
y
㦨
҈
矄� ç p + ÷ ø è \
꾀鼀�盐
㦨
҈

ᆠ鼀
p
p

㦨

㦨
矄

�鼀
�ᨨ


㦨
w p  䰘





���㡰

和ᨨ� �
⼄盐�ᨨ 矄
�㡰

��㡰


���㡰

Þ


和ᨨ

⼄盐 ᆠ泐
ᨨ
矄盐� �
ᨨ矄ᆠ��ᨨ 矄










f
=


f҈

㝐 =




�㡰

l + Dl l


⏌


⏌


鼀

鼀




��㡰 㝐 





l + Dl æ 㝐 ö = ÷ è ø l

Þç




㝐� æ 㝐 ö = ҈+ ᨨ

ç è ÷ø 꾀Ӹ \


ù 꾀 ҈ éæ 㝐 ö = êç - ҈ú ÷ Ӹ êè ø úû 㝐� ë

㝐�l ù é êëQ Dl = 꾀Ӹ úû

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PS-147

Oscillations ҈



㡰 ᔼ
矄�鼀
鼀ᨨ �ᆠ
鼀ᨨ�ᨨ

矄� �
ᨨ鼀
矄
꾀
t
㦨




x
㦨
A x
㦨
꾀ᆠᨨ矄w





�㡰 ␔ 
t
㦨
t

x
㦨
A

a ␔ 

㦨

t

x
㦨
A
a ᨨ鼀
⼄盐�ᨨ
�㡰



��㡰 A

a
㦨
Aᆠᨨ矄w t A

a
㦨
A
ᆠᨨ矄
w t



���㡰 꾀矄
ᆠᨨ矄
w t
㦨


ᆠᨨ矄

w t 
҈


�㡰 ᨨ鼀
⼄盐�ᨨ
��㡰

���㡰

 
�㡰

æ P xg ö Mg - ç

g ÷ A = F矄ᨨ� � ç  x - x㡰 ÷ è
ø g æ ö x
÷ = F矄ᨨ� � ᲼ x - x » x 桌 P
A ç҈ g

ç  x - x㡰 ÷
è ø gP Ax F=-
x





\
⼄盐 ᆠ泐
犨�
犨 �ᆠ
�矄ᨨ
 ᆠ盐矄
和㝐


A - a æ A-a ö =
ç ÷ -҈ A è A ø A - a
A
+
a
-
Aa - A
= Þ A A
Þ

A


aA
㦨
A

졔

a



Aa

Þ

a ҈
a

㦨
aA
Þ A
㦨

a
Þ
= A


҈



҈�


ᆠᨨ矄 wt =

A- a ҈ Þ
ᆠᨨ矄 wt =



ᨨ A


Þ
㦨

t b㡰 ᨨ鼀
�
�
�矄
ᆠ
 
犨 


㦨
҈鼀



㦨

矄
꾀矄
犨
├ ᨨ犨





\ ҈



p
p t=  T

 + ҈㡰 ҈ =




 + ҈㡰 ´
p




æ

ö ҈ æ

ö ÷ ´ =ç ÷ è 䰘 ø
è 䰘 ø


�= = 黨

䰘鼀 ⏌ 矄

黨

҈黨


ᨨ


鼀
p 鼀 =
p
ᨨ

w ⧤ ⧤

҈ =
�

ᆠ㡰

҈ [Q 鼀 = ҈├�錐 w =
�  ⏌ 矄 ] ⧤

�
鼀矄 -҈

 ᆠ

�矄
ᨨᆠ
犨�

矄
V
㦨
�

鼀矄҈ 㡰 ᔼ
 
矄� �
ᨨ
矄� �
ᆠᨨ 矄 
k҈
�矄
ᆠᨨ鼀矄矄
泐
x҈
   
ᨨ
矄� �
ᆠᨨ 矄 
k

�矄
ᆠᨨ鼀矄矄
泐
x

 

x҈ =
҈


…��㡰

ᆠ㡰 ᨨ鼀
��盐

ᆠᨨ鼀矄矄�ᨨ
ᨨ
 
矄� �

x
㦨


ì î

�矄ᨨ

L q



L ü qý L
þ

꾀矄ᨨ
ᨨ⼄盐
㦨
Ia
㦨



kL



æ kL
ö ML
a = -ç ÷q ç
÷ ҈
è ø

L⏌


x q




P
Ax
g = PA x
- x 㡰 g




…
�㡰

k
A k
+ k҈

ᨨ⼄盐
㦨
ík 㡰


㝐�
㦨

꾀 …
҈㡰 
�矄ᨨ
�矄
�矄ᆠ
泐
�矄 ᆠ
x

 x
- x 㡰g

\ V =

和ᨨ� �
⼄矄

�㡰


��㡰
犨
�

P
V
g = PV g

P=


V 矄�
�º g

kx   k҈ x҈ = k
x
Þ x
= ҈ ҈ k


Mg = P
A

g P
x


҈=


V sin q g

x҈ + x
= A

ᨨ
⧤
㦨


�
鼀҈



鼀 �鼀盐鼀
ᨨᆠ
 
泐
 
矄泐矄鼀
ᨨ
 
ᨨᨨ 㦨
㝐�
졔
⧤
졔
鼀� 㦨


맔
҈

졔


�
맔
҈


맔
҈


졔
҈
맔
҈

㦨



졔
҈

졔
҈

㦨



 ҈䰘


\







l =  㡰 ´
p � �

Þ

㦨
҈ \

졔
҈
㦨
҈
졔
҈
㦨

㡰 �鼀
ᨨ
�� 
ᨨ
ᨨᆠ�
T
㦨


ᆠ㡰 㝐矄矄
ᨨ
���
ᨨ泐
㝐
㦨


├� 㝐矄矄
ᨨ
矄鼀
ᨨ泐
鼀
㦨
҈
├� 和鼀
鼀矄矄
鼀
㦨
҈
├�㡰
 ᆠ盐矄
和

㝐
ᨨ
 �盐
⼄盐 ᆠ泐
w 㦨

�
⏌矄 꾀鼀�盐

㦨
҈


ᆠ鼀
㦨
҈


맔
҈

꾀矄
犨
├ ᨨ犨


=
p


҈
 

=
p � �

꾀矄
l҈ < l

  ᨨ
҈ <



㦨


 泐
矄
矄犨� �� �
ᨨ� 

和� ᆠ
 �
�鼀
�ᨨ矄 
�  

 
矄犨� �� �
犨�
ᨨ

�
盐 �矄ᨨ
犨泐矄

졠 泐 犨 
盐鼀
ᨨ
ᆠᨨ鼀
ᨨ矄ᆠ��ᨨ 矄
� 
泐

� �

  犨ᨨ
 盐盐鼀矄
犨�
��
��
ᨨ
矄犨� �
ᨨ�  
ᨨ �
 �
 盐盐鼀
ᆠᨨ鼀

ᨨ矄ᆠ��ᨨ
 
矄 ᨨ  �
 盐盐鼀
ᆠᨨ鼀

졔
҈㡰
ᨨ矄ᆠ��ᨨ

ᨨ
盐 �矄ᨨ
矄犨� �� �


p


=
p Þ
� = �


㦨

´ ç

gP
A
MV



҈
 



 
�鼀
�ᨨ
ᨨ
 
犨ᨨ
 盐盐鼀矄

㡰

҈ =
p

ᨨ犨

A

a
㦨
꾀
ᆠᨨ矄wt Þ

gP
A ҈ x
M
㦨

p

҈
p

f =

ᔴ泐�  ᆠᨨ � � � �
�矄

҈ æ
ö çQ I = ML ÷
è ø

q

L⏌


x

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EBD_8341 PS-148

\ \




b㡰 \ ᨨ

Physics

҈
k w = f

㦨

p
p M

dy d
y = v =
kt




 





= a =
k dt dt
ay =
鼀 ⏌ 矄

Q k
㦨
҈
鼀⏌矄

�� 㡰

l l
ᨨ


T҈
㦨

p g g

T҈


\

T



=

g + ay g

\

᲼k桌
㦨





Þ



k = m+m

T҈
T



=

҈

= ҈
�

\

è
m ø

ᨨ


m

æ kA ö kA = mç = è
m ÷ø


Ӹ g  g 矄�


a 㦨
ay g 矄�
a g 矄�
a 矄�
a  g ᆠᨨ矄 a
矄�

a
㦨
ax



g
䰘


a ax = g 矄�
a ᆠᨨ矄
a 㦨
a
ᆠᨨ矄
a ay = g  g 矄�

a = g҈  矄�

a㡰 㦨 g ᆠᨨ矄

a


 ᆠ�
ᆠᆠ�ᨨ
盐
ᨨ
��泐
ᆠ� �
ᨨ
 
ᨨ

㦨


g 矄� a ᆠᨨ矄 a + g ᆠᨨ矄 a

㦨
g
ᆠᨨ矄a
矄�
a + ᆠᨨ矄
a \




T
㦨

p

L geff


㦨

p

L g ᆠᨨ矄 a

kk Ἐ k +kἘ


�㡰

Y, AL

k҈
㦨


k

F YA
㦨
l L

YA L

k




m

m

\

ᨨ

k҈
㦨


\

T
㦨

p

�㡰

YA L

��㡰





��㡰

mk + k҈ 㡰 m
㦨

p keq kk҈

YA ö æ mçk + ÷ mYA + kL㡰 L ø è 㦨

p
㦨

p YA YAk k´ L x 㡰 
A
㦨
L


 

q
㦨
L 㝐ᨨ盐盐矄
ᨨ
����泐

h
㦨
F⏌A
q \ 矄ᨨ� �
ᨨᆠ F
㦨

h
A
q
㦨

h
Lx ᨨ
ᆠᆠ�ᨨ
a
㦨





keq
㦨


ᨨᆠ
㦨
k  �

ᨨᆠ YA
㦨
 � L

ax
+ a
y


Þ

F L Y
㦨
A l

㡰 꾀矄
矄 ᨨ犨
�
 
��盐

ᆠᆠ�ᨨ
ᨨ犨
 
 
a
㦨 gs�na
�矄
 
矄盐ᨨ
ᆠᆠ�ᨨ
�
泐
 
ᨨ矄
�
  ᆠᆠ
鼀




= ML-҈T -


҈

ᨨ
犨�


k
m

�矄
ᆠᆠ�ᨨ
ᨨ
 
ᨨ犨
ᨨᆠ├
�矄
ᨨ�
泐
�ᆠ�ᨨ

\ 㝐 �鼀盐鼀
ᨨᆠ
ᨨ
�ᆠ�ᨨ




L

ᨨᆠ
ᆠᨨ 矄 
ᨨ
矄� �
㦨
k




geff
㦨


ML
T -


҈ ҈ ҈ = + keq k k Ἐ

a鼀 = Aw
= A æ k ö = KA ç ÷






᲼ x 桌

=

b㡰 
盐矄
ᆠᨨ 矄�
 
犨�
矄ᨨ
矄

矄� �
ᨨ
矄� �
ᆠᨨ 矄  k'


 
ᆠ矄
ᆠᨨ鼀矄
 
ᨨ
犨ᨨ
矄� �
ᆠ 
�
矄�矄



⼄盐� 
矄� �
ᆠᨨ 矄 

㝐 �鼀盐鼀
ᆠᆠ�ᨨ
ᨨ
 
矄泐矄鼀
�
和㝐

f鼀 = ma鼀

᲼U 桌

a




㡰 
ᆠᆠ�ᨨ
�矄
矄鼀
ᨨ
ᨨ
P
 
Q
矄
ᨨᆠ├
Q ᨨ矄ᆠ�矄
盐
ᨨ矄
ᨨ
矄�


P

Q㡰
矄泐矄鼀
ᨨ矄ᆠ�矄
犨�  �盐
⼄盐 ᆠ泐
w


矄� �
�矄
矄ᆠ 
泐
A
꾀 �盐
⼄盐 ᆠ泐
ᨨ
 
矄泐矄鼀


w=

෌�


U
x㡰
㦨
k

x


 ᨨ

T
µ


l g + ay

T

㦨

p

㡰

ᨨ犨
�鼀
�ᨨ
鼀泐
 
ᨨ
T
µ
鼀矄矄㡰 x 鼀�盐㡰y
k㡰z \ ᲼M
L
T桌
㦨
᲼M桌x
᲼L桌y
᲼ML҈
T
桌z





㦨
᲼Mx
졔
z
Ly

z T


z桌 쟼⼄盐� �
 
ᨨ犨矄

犨
� 

z
㦨
҈
ᨨ

z
㦨

҈⏌
y

z
㦨



ᨨ

y
㦨
z
㦨

҈⏌
 ᆠ
T
µ
鼀�盐㡰
҈⏌

µ
a҈⏌


෌�


y
㦨
kt


␔
├ ᨨ犨
 
T
㦨

p






�


æ
k ö
÷q = - w q èMø

a
㦨
- ç

x

 q

F hL =x M M

和� ᆠ

a
µ

x
和ᨨ

ᨨ矄ᆠ��ᨨ 矄

和㝐

 ᨨ

�鼀
�ᨨ ᨨ
犨 �ᆠ
�矄
��
泐

㦨
g
ᆠᨨ矄
a ᨨ


T
㦨

p

�矄ᆠ鼀  x
㦨

p ᆠᆠ�ᨨ

a

T
㦨

p

M hL

WWW.JEEBOOKS.IN

PS-149

Oscillations




b㡰 ᔼ
x
�矄
 
�矄ᆠ鼀 
 

矄ᨨ� �
ᨨᆠ
㦨
 
盐 盐矄
졔
 
矄� �
ᨨᆠ \

M w
x = [ rAg + k ] x

é rAg + k ù Þ w
㦨
ê ú ë M û


黨


҈⏌


҈ é rAg + k ù



Þ v
㦨

p êë M úû

҈⏌


ᔴ泐
쟼҈
>
쟼


㡰 ෌�
鼀 �鼀盐鼀
ᨨᆠ�泐

v҈
㦨
v


ᨨ


a҈w҈

㦨
a
w
Q V鼀 �鼀盐鼀
㦨
aw Þ

\





æ ö æ 㝐 ö çQ 
= ç ÷ ҈ ᨨ鼀 ⼄ �㡰 ÷ 㝐 鼀 + è ø è ø
㝐
҈
҈æ 㝐 ö
맔

㦨
ç ÷ ҈ 㝐 + 鼀㡰

è㝐+鼀ø

a҈
맔



p
p = a
´ T҈ T


a҈ T҈ = = a
T


m
p k҈ m
p k


ᔼ 矄  ᨨ盐矄
矄



ᨨ
 
ᆠᨨ鼀� 
鼀矄矄矄 

㦨


=

k
k҈





盐




�盐


 ᔴ
















��㡰

盐


꾀



�盐


쟼⼄盐���盐鼀
ᨨ矄��ᨨ


ᨨᆠ�泐
ᨨ
 
�ᆠ
ᆠᨨ鼀矄


�u

 
�鼀
t

矄� �
⼄盐�ᨨ

㦨
鼀
ᆠᨨ矄
wt

�u

㦨
u

ᆠᨨ矄
wt



├�
ᨨᆠ├
犨�
 ᨨ鼀
和㝐
 
ᆠᨨ�矄�ᨨ

矄ᨨ
矄� �
盐 矄
ᨨ �矄
盐 矄ᆠ 
ᨨ矄��ᨨ
ᨨ
 
�矄
�鼀
 


T m
=p = p
㦨

҈

矄
㦨
p
k


ᨨ
 
�矄
�鼀
㦨

t
㦨


҈

\

㝐 꾀

㦨
꾀 꾀 Þ 㝐+鼀 ҈

㦨


p 

鼀 p + ├ 

㦨


䰘π


鼀
├




A
= A҈

҈ 꾀矄ᨨ
쟼҈
㦨
㝐҈



҈ ҈ 㝐졔鼀㡰


㦨
㝐
졔
鼀㡰



T


鼀 p 鼀
\
t
㦨
├  ├

鼀 é҈ ҈ ҈ ù 鼀 π 鼀 + =π 졔 졔 ├  ├ ├ êë  
úû


�鼀
├
ᨨ
�ᆠ
ᨨ
矄矄
 ᨨ盐�
 
⼄盐���盐鼀 ᨨ矄��ᨨ
矄ᆠᨨ 
�鼀

⧤ 㝐+鼀

M M +m


p 鼀
 ├


�鼀
├
ᨨ
 
鼀 �鼀盐鼀
ᆠᨨ鼀矄矄�ᨨ

㦨
t꾀
졔
t꾀
졔
t꾀ᔴ

㝐 ҈
㦨
㝐
졔
鼀㡰





�㡰 㝐
꾀҈
맔
w҈㡰
㦨
㝐졔鼀㡰
꾀
맔w
㡰

⧤
㦨
㝐졔鼀㡰
꾀 꾀

맔
㝐

t=


�鼀
├
泐
 
�ᆠ
ᨨ
矄矄
 ᨨ盐�
 
⼄盐���盐鼀



b

㡰 ᔴ矄
�㡰


꾀泐� �
� ᆠ�
ᨨ
ᆠᨨ 矄�ᨨ
ᨨ
�  鼀ᨨ鼀 盐鼀


㝐꾀҈
맔


Þ


�鼀
�ᨨ
=
p


´ 
҈

㦨



黨
㦨



黨
鼀 

\

p
p = ´T  t

\

�矄 ᆠ
ᨨ
⼄盐�
矄�ᨨ
犨
 
ᨨᆠ├矄

  쟼
㦨

V = ҈
Þ


㦨


�u


�u


�.e.

 
�ᆠ
ᨨ盐 矄
犨�
 
矄鼀
矄

 ᨨ
  �ᆠ
犨�
盐
ᨨ
�矄
⼄盐���盐鼀
ᨨ矄��ᨨ
犨�
矄
u





v

㦨
鼀⏌矄

v҈
㦨
鼀⏌矄  

҈


e=



v҈ + v


Þ v

졔
v҈
㦨

ᨨ鼀
⼄矄

�㡰
 
��㡰


㦨
v҈t
㦨


鼀+㝐 ⧤

꾀矄ᨨ
꾀

㦨
꾀҈   쟼

㦨
쟼҈ ᔼ
 �矄
ᆠ矄
矄ᨨ

� 矄  ᨨ盐矄
矄

ᨨ
ᨨ
 
ᆠᨨ鼀�  鼀矄矄矄
ᆠ矄矄


㡰

�ᆠ
ᆠᨨ�矄
矄�ᆠ泐
犨�
���
犨
\

ᨨ鼀
鼀ᨨ鼀 盐鼀
ᆠᨨ 矄�ᨨ
� ᆠ�
m҈u҈
졔
m
u

㦨
m҈v҈
졔
m
v


졔
҈
맔


㦨
҈v҈
졔

v





















�㡰 ᔴᨨ�矄�ᨨ
�矄
矄�ᆠ

 ᆠ
e
㦨
҈

t
㦨


㝐҈


҈ 㝐+鼀

ᔴ矄
��㡰



犨
�鼀
�ᨨ


㦨






黨㡰 
ᨨᆠ��矄
ᨨ
҈
├�
 


├�
ᨨᆠ├矄
盐矄
 
ᆠᨨ�矄�ᨨ

v҈
  v

矄ᆠ�泐
盐矄
 
ᆠᨨ�矄�ᨨ

v
v҈ ҈
├�

├�

e
㦨
҈
㦨


鼀+㝐 ⧤


犨
�鼀
�ᨨ


㦨



éπ

㦨

ê

ë




鼀ù ú졔π ├û

鼀 æ
ö � 鼀 鼀
㦨
π ç 졔 ҈÷ 㦨 π ├ è ø  ├ ├

 
和㝐


a
�矄 ᆠ
y
ᨨ
 
鼀
ᨨ矄��ᨨ
ᨨᆠ�泐
ᨨ ᨨᆠ├

v = w a
- y
꾀 
ᆠ � �
ᨨ鼀
矄� �


ᨨ犨 犨
ᆠᆠ�ᨨ
ᨨ
  ᨨᆠ├
㦨
g
\ �� 
� 
泐
 
ᨨᆠ├
㦨
h

WWW.JEEBOOKS.IN

EBD_8341 PS-150

Physics

(

)

w
a
- y
v


ᨨ

h = y + \ h= y+
g
g dh ᨨ
h
ᨨ

鼀 �鼀盐鼀
=

y = y ã
dy
dh w
\ = ҈+ ( -
y ã)
ᨨ

= ҈ -
w y ã dy
g
g

 p 㦨
q

w
y ã g = ҈ ᨨ y ã = g w
和� ᆠ
aw
> g ( �� )

g

ᨨ盐鼀
ᨨ
 
ᨨ
㦨
LS ␔�� 
ᨨ
 
ᨨ
㦨
LS
d҈g  盐矄
ᆠ� �
ᨨ
ᨨ
㦨
LS
d
g







和� ᆠ

d

>
d҈
�� 㡰
\ 
ᨨᆠ
ᆠ� �

 
ᆠ 
ᨨ 鼀矄矄
ᨨ
 
ᨨ

�
ᨨ矄��ᨨ

LS
d
g

LSd҈g㡰

ᨨ⼄盐
ᨨ盐
 �矄
ᨨᆠ
ᨨ盐
P t
㦨
F
맔
r҈
㦨
LSd
g

LSd҈g㡰
맔
PN

L \ t
㦨
LSg
d


d҈㡰
맔
矄� q
犨 
q �矄
矄鼀

矄� q » q \

t
㦨


L
Sg d
- d҈ 㡰q



ᨨ ᨨ

LSd 2g

\ q q

L/2

M

\

LSd 1g P N

=

d2

��㡰

L
Sg d
- d҈ 㡰



Þ Iw

㦨


L
Sg d
- d҈ 㡰





��㡰


鼀ᨨ鼀 
ᨨ
� �
I
ᨨ
 
ᨨ
ᨨ盐
P
I
㦨


҈ ҈ ML
= LSd҈L
 

҈ a
p q ҈
k ⼄盐 ᆠ泐
ᨨ
ᆠ

=
p m ⼄盐 ᆠ泐
ᨨ
ᨨ矄ᆠ��ᨨ
f =

҈
´

҈ ҈ = 矄ᆠ -҈


��㡰 p
p

҈ v鼀

 
ᨨᆠ�泐

 
鼀
ᨨ矄��ᨨ




L L Sg Sd҈ = d
- d҈ 㡰 
d
- d҈ 㡰 g Sg d
- d҈ 㡰 Þ w
㦨
Þ w
㦨
d҈ L
LSd҈ w
´

I a = -
kR
q
犨 
a
㦨
 �盐
ᆠᆠ�ᨨ

æ
k ö mR
㡰
a
㦨


kR
q ᨨ
a = - ç ÷ q è mø

㦨

黨�
맔
҈




盐� �
 �矄
盐
ᨨ
I
�
⼄

��㡰 





졠 㝐

 ᆠ
 �盐
⼄盐 ᆠ泐
ᨨ
ᨨ矄ᆠ��ᨨ


g d
- d҈ 㡰 w
㦨

d҈ L �


턼

\ ᨨ矄矄
�
矄�ᆠ
ᨨ �
 �泐
㦨
��
�
├� �ᆠ
 �泐
é҈ æ pö ù é҈
ù
ê K ç
R ÷ ú =
´ ê mv鼀 ú
ø úû ë
û êë
è K Rp ´ =



p 鼀⏌矄
㦨






鼀⏌矄 \ v鼀 = m  é҈
ù ���㡰 ᨨ
 �泐
㦨

ê mv鼀 ú 㦨
⧤
쟼


鼀
ᨨ矄��ᨨ 㡰 ë
û ҈ é
ù 㦨

ê ´

҈



p㡰 ú ë
û




�㡰

和� ᆠ

t µ q

 ᆠ
鼀ᨨ�ᨨ
�矄
矄�鼀
鼀ᨨ �ᆠ
졠
ᆠᨨ鼀� �
�
犨�
t
㦨
C q

犨
� C
㦨


q

ᔴᨨ鼀矄矄�ᨨ
㦨
Rq
㦨
ᨨ ��ᨨ
㦨
x \ ᨨᆠ
 
泐
ᆠ
矄� �
ᨨ
ᆠ

㦨


kx \ ᨨ
ᨨᆠ
ᨨ
ᆠ

盐
ᨨ
犨ᨨ
矄� �矄
㦨


kx \ 矄ᨨ� �
ᨨ⼄盐
ᨨ盐
ᨨ���
O
㦨



kx㡰R \ t
㦨


kRq㡰R

犨 
q
㦨
꾀 �盐
�矄ᆠ鼀  ᨨ t
㦨


kR
q 和� ᆠ
ᨨ⼄盐
t㡰
�矄
ᨨᨨ�ᨨ 
ᨨ
q

ᆠ

 ᆠ盐矄
 �盐 和㝐
ᨨ盐
 
ᆠ 
O
꾀��

t
㦨


kR
q

\ a > y ã
y ã
ᨨ鼀
鼀
ᨨ矄��ᨨ


a
w
g  ᆠ
yã =
w




q R



ᨨ

\
a >

꾀

�㡰 ෌�


盐
 �
ᨨ
矄� �

l

㦨





p
㦨
pR 和� �
ᆠᨨ 矄 

k
㦨


҈
⏌鼀 㝐矄矄
ᨨ
ᆠ
ᨨᆠ├
꾀
 


m
㦨


҈
├� �盐矄
ᨨ
ᆠ�ᆠ

㦨





鼀 꾀矄
ᨨ
 
矄

�矄ᆠ
泐

 �
q = p ⏌

�
犨� 矄ᆠ
ᨨ
 
�鼀
PQ
ᨨ
 
ᆠ�ᆠ
 
矄
ᨨ鼀
矄

ᔼ 矄盐矄
� ᨨ
ᆠᨨ鼀矄矄�ᨨ
ᨨ
矄� �
�
盐
矄�鼀 
 


⼄盐
ᨨ ��ᨨ
ᨨ
矄� �
�
ᨨ犨
矄�鼀 


�

x

PB
  QA
 ᨨ
x
�
 
��盐


ᨨ

q -q
�






�

턼 ⧤

鼀
�

鼀҈�

꾀
⼄盐���盐鼀


æ ᨨ⼄盐  ⼄ 盐 ᨨ ö æ ᨨ⼄盐   盐 ᨨ ö ç ÷= ç ÷ è �⼄盐� ᨨ  矄�泐 r ø è �⼄盐� ᨨ  矄�泐҈
� r ø m
g
맔
QM
㦨
m҈g
맔
PN

WWW.JEEBOOKS.IN

PS-151

Oscillations m
g R
矄�

�
졔 q 㡰
㦨
m҈gR
矄�

�
 q 㡰

\

VrgR
矄�


�
졔 q 㡰
㦨
҈
�VrgR
矄�

�
 q 㡰

Þ

矄� 
� + q㡰 = ҈
� 矄� 
� - q㡰

Þ

矄�
�° ᆠᨨ矄 q + ᆠᨨ矄
�° 矄� q  = 矄�
�° ᆠᨨ矄 q - ᆠᨨ矄
�° 矄� q


꾀��
�鼀
├
ᨨ

ᨨ鼀
ᔴ
ᨨ




�㡰

和ᨨ� �
犨
�


 q =

t

㦨


\ ᨨ
�鼀
ᨨ
ᨨ矄ᆠ��ᨨ


㦨

t҈
졔
t

졔
t
졔
t

㦨



䰘
�
졔


�
졔
҈

�
졔


�
㦨




꾀ᨨ �鼀泐㡰 



҈ �

㡰 
盐矄
ᨨ犨
�矄ᆠ
 
�⼄盐�矄
�
 �ᆠᨨᆠ├犨�矄

�ᆠ�ᨨ

ᨨ �
 
ᆠ�ᆠ盐鼀  ᆠ
ᨨ
盐
 ᨨ盐�

 �
a
\

ᨨ⼄盐 t
㦨
m
gR
矄�

�
졔
q 졔
a㡰

m҈�
矄�

�

q 
a㡰 㦨
VrgR
矄�

�
졔
q 졔
a㡰

҈
�VrgR

�

q 
a㡰 㦨
VrgR
矄�

q 졔

� 㡰
ᆠᨨ矄
a

졔
VrgR
ᆠᨨ矄

�
졔
q 㡰 矄�
a 
҈
�
VrgR
矄�

�

q㡰
ᆠᨨ矄

a 졔
҈
�
rgR
ᆠᨨ矄
�

q㡰
矄�
a 矄� �
⼄

�㡰
犨
�

m m

1

m

2

2


x҈

 
  矄�ᨨ
�
⼄盐���盐鼀
犨 
ᨨ
m҈
 
m

 矄盐矄 
\

m҈ + m
㡰 g k

m҈
졔
m
㡰g
㦨
kx҈
Þ
x҈
㦨


t = V rgR éëᆠᨨ矄 (
�° + q ) 矄� a + ҈
�ᆠᨨ矄 (
�° - q ) 矄� aùû


x


 
  矄�ᨨ
犨 
ᨨ 泐
m

�矄
 


t
㦨
VrgR
᲼ᆠᨨ矄

�
졔
q㡰
졔
҈
�
ᆠᨨ矄

�

q㡰桌
矄�
a 犨 
a
�矄
矄鼀
�� 㡰 \ 矄�
a
»
a \ t
㦨
VrgR
᲼ᆠᨨ矄

�
졔
q㡰
졔
҈
�
ᆠᨨ矄

�

q㡰桌a 和� ᆠ

t
 
a

ᨨᨨ�ᨨ 

鼀ᨨ�ᨨ
�矄
矄�鼀
鼀ᨨ �ᆠ
㝐ᨨ鼀 
ᨨ
� �
ᨨ盐
O I
㦨
m҈R

졔
m
R

㦨
VrR

졔
҈
�
VrR

㦨


�
VrR


\

ᨨ犨
ᨨ鼀

ᨨ


T
㦨

p

A
㦨
x҈

x

㦨


w
㦨
黨


g




ᆠ鼀

k҈ 꾀

ᔴ

\

t҈
㦨


m


= 





��㡰

M M
=
Þ

Þ

M
㦨
҈


├�
黨 M +
M +


ᆠ㡰 �鼀
ᨨ

 
鼀�盐
�矄
㦨

s 矄� �

A
㦨
A
ekt

A
= Ae e - k ´

A
= Ae e- kt   ҈




T m


=p =p =

䰘
� 矄ᆠ
k




�矄 ᆠ

=

�
㦨
ᨨᆠ�泐 ҈



m


=p = ҈

� 矄ᆠ
k҈ ҈



M +
k

Topic-4 : Damped, Forced Oscillations and Resonance

ᨨ犨


�鼀
├
ᨨ

ᨨ鼀

ᨨ
ᔴ

t
㦨
p






�㡰

��� �
⼄

�㡰
泐
��㡰



ᨨ犨
 
ᨨᆠ├
犨�
ᆠᨨ鼀矄矄
 
 
矄� �
 
 

 
矄� �
��
� 矄
�矄
ᨨ鼀
 �


�鼀
├
ᔴ
ᨨ
꾀

M k M
=
p k


㦨

p

҈




k m


␔ 
鼀矄矄
�矄
� ᆠ矄
泐


├�

k


v

m҈ g k

꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

�
 
鼀矄矄
�矄
� ᆠ矄
泐


├�
 
�ᨨ � ᆠ矄矄
泐
҈
矄

�鼀
�ᨨ
 
矄� �

ᨨ


T m
㦨


犨 
T
㦨

p
k

ᨨ
 �矄
ᨨᆠ矄矄

�鼀
├




��㡰

T
㦨

p

҈ö æ çQ  q = ÷ �ø è

҈


R

m
g k

꾀 �盐
⼄盐 ᆠ泐
犨 
m҈
�矄
鼀ᨨ
ᨨ 泐
m

�矄
 


᲼ᆠᨨ矄
� + q㡰 + ҈
�ᆠᨨ矄
� - q㡰桌V rgR T

㦨

p

ᨨ x҈
㦨





�㡰

ᨨ鼀
⼄矄

�㡰
 
��㡰

鼀�盐
ᨨ
ᨨ矄ᆠ��ᨨ

和� ᆠ
꾀
�矄

矄鼀ᨨᨨ


和ᨨ
ᨨ
ᆠᆠ�ᨨ
ᨨ
�ᨨ
�

 �矄
��ᨨ
矄ᨨ
鼀ᨨ�ᨨ
ᨨ鼀
ᔴ
ᨨ

ᨨ
ᔴ
�矄
盐 � ᨨ鼀
犨�
矄 ҈


鼀⏌矄



鼀矄矄
犨�
矄�├
 
�� 
矄� �

ᆠᨨ鼀矄矄
�


⧤
쟼

ᨨ  
鼀矄矄
犨�
ᆠᨨ 
� ᨨ

쟼

ᨨ
 
矄� �

꾀��
 
矄� � 犨�
盐
ᨨ
�矄
ᨨ鼀
矄�
 泐
ᆠᨨ � �
�矄

쟼

ᨨ
⧤
쟼
ᨨ
 
ᨨᆠ├


t

㦨


kx

㦨
m
g


I C

V r ´

�R
㡰

㦨

p


䰘




=

� ҈










�㡰


��㡰

��� �
�㡰
泐
��㡰
 
矄ᨨ� �

犨
� t
㦨



矄 b㡰
꾀鼀�盐
ᨨ
��ᨨ

�鼀

㦨


�矄
��
泐 꾀
㦨
꾀



҈맔


㦨
҈
맔
꾀

㦨
꾀
矄ᨨ


㦨


�
꾀 =

꾀



WWW.JEEBOOKS.IN

EBD_8341 PS-152

Physics ҈



҈ Þ =  

҈




㦨
҈




;䰘矄 b㡰 ᔼ
�矄
ᆠᨨ�矄�ᨨ
鼀盐
鼀ᨨ鼀 盐鼀
犨�

�鼀
ᨨ
矄泐矄鼀
�
矄ᆠᨨ 
ᆠᨨ�矄�ᨨ
犨 
鼀ᨨ鼀 盐鼀
ᨨ
㝐
졔
鼀㡰
�矄
�
ᨨᨨ矄� �ᆠ�ᨨ
鼀盐
鼀ᨨ鼀 盐鼀
ᨨ
�ᆠ
犨�
鼀├
�矄
鼀ᨨ鼀 盐鼀
ᨨ
졠
҈
ᆠᨨ�矄�ᨨ

鼀 ® 㝐 + ҈
錐 㝐 + ҈鼀 ® 

E҈
㦨


鼀盐 盐 = 鼀盐
㦨
㝐
졔
҈ 鼀㡰
Þ
 = 㝐 + ҈鼀 ҈� 
㦨
w꾀 Þ

盐 = ҈�





�


҈ 䰘� ҈ = ҈� ҈ 

Q 꾀 = 꾀



鼀

E

㦨


-

�㡰

-

…��㡰

꾀
㦨


䰘
黨
꾀
\

a
㦨


䰘
黨 㡰 r r ᆠ㡰 
矄盐 
ᨨ
 矄矄
a t
 
�
a r
ᆠᨨ鼀ᨨ  
ᨨ

ᔼ


r

 㡰 ᔼ
和㝐

ᨨᆠ�泐
ᨨ
�ᆠ矄
�ᨨ矄
ᨨ
ᆠ矄� �
ᨨ鼀 鼀 �鼀盐鼀
盐
ᨨ
ᨨ
矄
 
�ᆠ矄

ᨨ鼀
鼀
ᨨ矄��ᨨ

ᨨ
 鼀
ᨨ矄��ᨨ


 ᨨ
�
 
�鼀
├
ᨨ
 
ᨨ泐
ᨨ
�ᨨ
ᨨ鼀
O
ᨨ A⏌

�矄
T҈
 
ᨨ
�ᨨ
ᨨ鼀
A⏌

ᨨ
A
�矄
T

 
ᨨ�ᨨ盐矄泐
T҈


T


FL æ YA ö ÞF = ç ÷l 
㡰 꾀矄
犨
├ ᨨ犨


Y = è Lø Al

a a
㦨
n

 

㦨
n
 
ᨨ鼀
⼄

㡰 b R

w҈ w҈
E҈

㦨

´ n
㦨
w
E
w
҈҈


\

꾀 �盐
⼄盐 ᆠ泐
w
㦨


n
㦨



K YA
ᨨ
w = m ml

n ´҈
´

黨 ´҈


҈´҈ 黨

E҈ E

㦨
w҈ w




㡰 꾀泐� �

ᨨ⼄盐
t = I a

æl ö 矄� q÷ + Mg l 矄� q㡰 = I A a A è
ø

ᨨ
ᆠ矄
꾀


mg ç

l/


YA æ YA ö Kl = ç ÷ l
Þ K
㦨
L è Lø

ᨨ
҈

=




�㡰




㡰

꾀矄ᨨ



㦨
Kl

\




���㡰

 
ᨨ鼀
⼄

�㡰



ᨨ


x R

w

㦨
n
w҈



黨


E


w


E҈ w҈
a

㦨
E
w

R


 
ᆠᆠ�ᨨ
�矄
矄 
泐
a





p

ᨨ鼀
⼄ 矄

�㡰
 
���㡰


҈�㡰
鼀

ᨨ鼀
⼄ 矄
�㡰
 
��㡰



䰘


҈ mw

R



a w

㦨
b w҈

…
�㡰

꾀 
҈

鼀ᨨ
矄ᆠᨨ 


꾀 = 꾀





��㡰

p鼀
㦨
mv鼀
㦨
mw
R R
㦨
mw
R
Þ mw

㦨
҈ ᨨ鼀
⼄ 矄

��㡰
 
�㡰


犨 

꾀


㦨
鼀 �鼀盐鼀
鼀�盐㡰 꾀ᆠᆠᨨ� �
ᨨ
 
⼄盐矄�ᨨ 矄

 
�
矄ᆠᨨ 




黨꾀
= 꾀

鼀

x

a

ᨨ
矄ᆠᨨ 
鼀ᨨ �ᆠ
ᨨ矄ᆠ�ᨨ
㝐矄矄
㦨
m 꾀 �盐
⼄盐 ᆠ泐
㦨
w
꾀鼀�盐
㦨
R 㝐 �鼀盐鼀
鼀ᨨ鼀 盐鼀

p鼀
㦨
R

ᨨ
 �泐
㦨
E


b㡰 
ᆠ  �
�
�鼀
�ᨨ
ᆠᨨ鼀
ᨨ
 
盐 鼀 ᨨ矄ᆠ�ᨨ
� ᆠ矄矄
泐

奠
ᆠ㡰

b




�㡰

҈ a
㦨
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-

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w҈

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mv鼀
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ᨨ
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q

l/

矄�
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l
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黨
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黨
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PS-153

Oscillations

æl ö ᨨ
ᆠ矄



mg ç 矄� q÷ + Mg l 矄� q㡰 = I B a B


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ø ᨨ鼀
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
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M -


ML T -


L
=T

-dV  x㡰 dx

T⏌A L l⏌ L YAl Þ T
㦨









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L

ᨨ鼀

Y
㦨


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l

x

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ᆠ㡰 矄ᨨ� �
ᨨᆠ

é YAl + x 㡰 YA l ù L L úû ë

㦨


᲼T'  mg桌
㦨

ê



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 


GM e 矄ᆠ
矄
Ve
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犨�
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= ᲼ ML-
T -
桌

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x


=

ML
T -


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T
Þ

ᨨ �  �泐

a=

u




҈



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\




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R
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µ 鼀�盐㡰
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쟼

µ 


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\

和� ᆠ
I A > I B




\ a A < a B
 ᆠ
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⼄錐 
矄錐
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矄錐

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├
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㡰


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矄�� 
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S
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ut
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⼄盐�ᨨ
犨�
F 㦨

mw
x mw

㦨


YA YA
p =
Þ w
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L mL T

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⼄盐 ᆠ泐
f =

YA mL

҈ ҈ YA = T
p mL

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EBD_8341 PS-154

Physics

Waves

14

Topic-1 : Basic of Mechanical Waves Progressive and Stationary Waves ҈






҈ æ gRT ö 錐 C鼀矄
㦨
ç

C鼀矄
µ
÷ è M ø m � é ù ê g mᨨnᨨatᨨm�c = 
R
T 
ᆠᨨ 矄  ú ë û C鼀矄҈ æ m
ö \ = ç C è m ÷ø

㡰 꾀矄
犨
├ ᨨ犨


w ´ rV
V é Bù w êQ K =
V = ú rû V ë DP
҈
҈  鼀= 鼀鼀 » 鼀鼀 Þ S
= » 
҈

rV w ҈ ´ 

´ ҈


b㡰 ෌�


�矄 ᆠ
犨
ᨨ 
ᆠ矄
 
ᨨ 
ᨨ盐�
㦨
҈
�
鼀 l = 
n҈ + ҈㡰
�矄 ᆠ
犨
犨ᨨ
ᆠ矄矄
㦨
�
鼀
= n
l ҈
� 
n҈ + ҈㡰 = Þ n
= ҈
n҈ + � �
n

n҈
 
n


� �
ᔼ n҈ = ҈
n
= � \l = ҈ n҈ =

n
= ҈� \l = ҈⏌  n҈ = 䰘
n
=
� \l = ҈⏌ � ҈ ҈ ҈  ᆠ
ᨨ矄矄�
犨 � 矄



鼀
҈  � b㡰 꾀
t
㦨



x
㦨



y
㦨

f
㦨
p
 㡰 矄� �

b
㦨
҈
矄矄盐
鼀�盐

DP
= aKB = S
KB = S
´










�






䰘


æ I ö ᨨ
҈


㦨
҈

ᨨ�҈
ç -҈
÷


�㡰 è ҈
ø
P = 꾀矄ᨨ
I
㦨



��㡰
pr

pr
졠
矄ᨨ� �
ᨨ
⼄盐�ᨨ 矄

犨
� r
㦨



ᆠ鼀
㡰 ᔴᨨ鼀� �
 
��
⼄盐�ᨨ

泐
㦨
҈
矄� �

졔

㡰
犨�
矄 
⼄盐�ᨨ
泐
㦨

矄� w

├ 㡰 Þ 犨
�矄
鼀ᨨ� �
ᨨ �

㜠 �矄
犨�
矄 w �
=
� 鼀⏌矄ᆠ
= Þ= ⧤
㡰 ෌�

鼀�盐

㦨
҈

ᆠ鼀 犨
ᨨᆠ�泐
㦨


맔
鼀 �鼀盐鼀
�ᆠ
ᨨᆠ�泐 �

wl =
w
p p ᨨ
l
㦨


㦨


맔
҈

㦨



ᆠ鼀 㡰 �ᆠ
ᨨᆠ�泐
vp
�矄

ᨨ
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�矄ᆠ鼀 
ᨨ
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ᨨ鼀
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鼀
ᨨ矄��ᨨ
矄 泐 v p =
pv A
 y
 ævö v p =
p ç ÷ A
 y





èlø

p ˆ
p

 鼀⏌矄 ´

҈ (

҈)  (


�)
=
�


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 
犨
�矄
矄� 盐矄ᨨ�
鼀ᨨ� �
�
ᨨ矄��
㜠 �矄

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
ᨨ
泐㜠 �矄
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ᨨ�ᨨ 矄
ᆠ㡰
 
㡰

盐
ᨨ盐

꾀矄  
犨
鼀ᨨ矄
ᨨ犨
�
ᨨ矄��
㜠�ᆠ�ᨨ

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矄 ᨨ盐 鼀ᨨ
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

�
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ᨨ矄��
Ӹ㜠�ᆠ�ᨨ


㦨


b㡰 和
ᨨ
矄ᨨ盐 
�

�矄


鼀矄


黨


҈



҈

dy = - Aw ᆠᨨ矄  kx - w t 㡰



\ v鼀
㦨
Aw 㡰 v 㦨
dt 㝐 �鼀盐鼀
�ᆠ
ᨨᆠ�泐
ᨨ
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�ᆠ
�
和㝐

v鼀
㦨
Aw xù é b㡰 y
㦨
y

矄�

p ê f t - ú lû ë \ ᨨ


dy é x öù æ = y
ᆠᨨ矄
p ç f t - ÷ ú ´
pf dt êë l øû è é dy ù = y
´
pf êë dt úû 鼀

෌�

㝐 �鼀盐鼀
�ᆠ
ᨨᆠ�泐
㦨

c
㦨


f
맔
T㡰 \

y

맔

pf
㦨

f
맔
l㡰

l
㦨


p y



҈

҈҈


෌�


y =

҈



꾀
t
㦨


犨
�
y
㦨
҈
犨 
x
㦨

҈ 꾀��

y =
҈ + ( x - ҈) 和ᨨ


t
㦨


y
㦨
҈
犨 
x
㦨
҈


犨
盐矄
矄


�矄 ᆠ
ᨨ
҈鼀
�


矄ᆠ
҈ -҈ \ ᨨᆠ�泐
ᨨ
犨
盐矄
v
㦨
=

� 鼀矄
�ᆠ
ᨨᆠ�泐
鼀�盐
㦨
鼀
㦨
Aw
㦨
A
pn �ᆠ
ᆠᆠ�ᨨ
鼀�盐
㦨
a鼀
㦨
Aw

㦨

p
n
A v ҈ T 

㡰 ␔ �
ᨨ
盐矄

l = = ᨨ



µ

f f m ␔ 

㦨
 矄�ᨨ
ᨨ
矄� �

҈
>


\ l҈
>
l



ᨨᆠ��矄
ᨨ
 
犨ᨨ
盐矄矄
ᆠ

ᨨ

矄鼀

鼀�㜠ᨨ� 
矄 ᨨᆠ�泐
� �
ᆠᨨ
⼄盐 �泐
矄
�ᆠ�ᨨ


҈


(҈ + x)


T

矄ᨨ
矄

 泐
ᨨ矄��ᨨ
犨�

矄鼀
ᨨ
ᨨ
盐矄矄
m   ᨨ
�鼀
├
泐
ᨨ
盐矄矄
犨�

矄鼀
�



꾀졠
㦨
졠꾀 

ᆠ

㡰
ᨨ

 
犨

�  矄�泐
�



 �泐
ᆠᨨ矄矄� �

盐 � 


盐 �
�鼀
�矄
ᆠᨨ 矄 


ᨨ� 矄
盐
ᨨ

矄 �ᆠ
犨

�  矄�泐


�矄 ᆠ

ᨨ鼀

ᨨ�  矄ᨨ盐ᆠ ҈ I
µ

r 盐
 
ᨨ
�  矄�y
ᨨ
 
矄 �ᆠ
犨
ᨨ
 
矄 �ᆠ 矄盐 ᆠ
ᆠ 

 
矄ᨨ盐ᆠ
鼀� 矄
ᆠᨨ 矄 


�鼀矄

㦨


҈



ᨨ
� 
矄ᨨ盐ᆠ
I
µ


҈
矄 �ᆠ
犨
�矄
ᨨ
ᨨ盐ᆠ
泐
 
�  r

矄ᨨ盐ᆠ


WWW.JEEBOOKS.IN

PS-155

Waves ҈�
҈





b

ᆠ

㡰
ᔼ
 
犨
鼀ᨨ�ᨨ
y = a ( kx - wt )

y
ᆠ
矄 
ᆠ�ᆠ
 
鼀� �ᆠ
�矄
�
ᆠᨨ鼀� �ᆠ
犨矄
  �矄ᆠ鼀 
 
矄矄盐
�
矄ᨨ盐 
犨矄
b

ᆠ

㡰
ᔴᨨ鼀� �
 
��
⼄盐�ᨨ

y
x

t㡰





㦨

=

+ �㡰
+ � � ù é ҈
ê +  ú + �
û ë ␔�
 
⼄盐�ᨨ
ᨨ
鼀ᨨ� �
盐矄 y
㦨
f
x
졔
t㡰 � -҈

� -҈ 鼀矄 
㦨
鼀矄 =

泐 和ᨨ

 
犨
犨�


�矄 ᆠ

҈

鼀 ᨨ


�
鼀
�


矄ᆠ


































҈䰘


҈


(
x + �t ) + �


= =

҈
鼀 졠 
� \
㝐 �鼀盐鼀
�矄ᆠ鼀 
�矄


҈

鼀


矄 
ᨨ
 
盐矄

x
㦨


 
t
㦨


�矄
矄
矄 ᨨ犨
�
��盐
  �
�矄
矄泐鼀鼀�ᆠ


ᆠ㡰 ᨨ

 矄矄
矄� 盐矄ᨨ�
犨
� �
ᨨ

矄� �
 
鼀 �鼀盐鼀
ᨨᆠ�泐
v鼀
㦨
aw


\l =



҈



҈ -

=








= v


v҈ = v


T҈ v Þ ´
= T
v

(Q V µ T )



´ ҈
T









´ ҈

=

�҈� ´ ҈

N
Þ T

㦨
�
҈�
맔
҈

N 㡰 ෌�

l
㦨



ᆠ鼀

m
㦨


�

A
㦨
҈
鼀鼀


v
㦨
黨

鼀⏌矄
 
Y
㦨 ҈

맔
҈
҈҈
鼀
Þ T
=

�


¶
y

f ( x
t ) = f ( x + l
t )



���㡰 
b
ᆠ
㡰
෌�
y
㦨
҈


矄�


t
졔

x㡰 ᔴᨨ鼀� �
 
��
⼄盐�ᨨ
犨�
 
矄 
犨
⼄盐�ᨨ

y
㦨
a
矄�
wt
졔
k x㡰 꾀鼀�盐
a
㦨
҈

鼀錐
k
㦨


鼀҈ 
꾀 

w
㦨



⏌矄



Þ



pf
㦨



\ f
㦨
 p
p
p = l
=
㦨
p 鼀
k w

和
ᨨ
犨

㦨
= 㦨


鼀⏌矄 k


f x
�
Tx T 黨 = = \ x = =


�
f z 

Tz Tz
b㡰 
ᨨᆠ�泐
ᨨ

 矄矄
犨
�

矄ᆠ 
犨�
�矄 ��
泐

\





�㡰 ¶t
¶x



犨
鼀ᨨ�ᨨ
�矄
ᆠ ᆠ�
泐
 
犨ᨨ
ᆠᨨ ��ᨨ 矄 f ( x
t ) = f ( x
t + T )



��㡰






҈ Tx ҈ Tz
 
f z =
l m
l m

T m ␔ 
T
㦨
 矄�ᨨ
�
 
犨� m
㦨
� 
 矄�泐
ᨨ
犨�

҈ ҈�p ҈ ´ 㦨
҈
�
鼀⏌矄 Þ l
㦨
㦨




鼀

 

㦨

p � �

¶
y

T҈

├�

v=


p
p  = ҈
p = ҈�p
 
l l

ᨨ



l
㦨


ᆠ鼀


g l҈ =
l҈ = ҈

ᆠ鼀 T҈
㦨

�
ᨨᨨ鼀㡰
g ҈ T
㡰 矄� �
f =
l m 鼀矄矄 犨 

T
㦨
 矄�ᨨ
 
m =  � fx =

҈

p


ᆠ㡰
ᨨ
犨
鼀ᨨ�ᨨ

 
�  �
⼄盐�ᨨ
�矄 ¶
y æ w
ö ¶
y é wù = ç ᆠᨨ 矄 

÷


êQ v = k ú ¶t k ¶ x è ø ë û

T T l
=
l҈ M T҈

V = l

҈

鼀



├�





=




p ´ ҈
 ҈
=
p ´ ҈
-
鼀 꾀 
l
㦨
=  n ҈
⏌
p b

ᆠ㡰
෌�
⼄盐�ᨨ


y
㦨
A
矄�
҈

px
졔
҈�
pt
졔
p⏌㡰 ᔴᨨ鼀� �
 �矄
⼄盐�ᨨ
犨�

矄 
⼄盐�ᨨ
ᨨ

犨 � �
�

X
�ᆠ�ᨨ

T


T

㦨

�
 ᨨ㡰


p é
p  ù é
p ù t+ x + fú y
㦨
A
矄�
ê t + x㡰 + f㡰 ú Þ y
㦨
A
矄�
ê l l l ë û ë û

҈黨


ᆠ㡰 矄� �

V = f l

n=




 ҈
= = ҈鼀⏌矄 ҈
҈
aw
㦨
҈




Þ

҈

맔
2pn
㦨
҈
᲼Q w
㦨

pv桌

Þ n
㦨



҈҈

´ ҈

´ ҈

= =



 ҈� �

V҈ V
V = Þ l
=
l҈ l҈ l
V҈ 꾀��
盐矄� �


෌�
鼀 �鼀盐鼀
ᨨᆠ�泐
㦨
\

㡰 

l҈
㦨
҈䰘
ᆠ鼀
 
l

㦨



�
ᆠ鼀

V
㦨


鼀⏌矄

f
㦨
䁸 l =
l
- l҈ 㡰 =
´ 


� - ҈䰘㡰 = ҈� ᆠ鼀 ᨨ犨

ᨨ鼀
v = f l Þ 
= l ´ ҈� ´ ҈
-



㦨








꾀
x =

t =

y =

Topic-2 : Vibration of String and Organ Pipe

矄� �

v =

T mv
´l ÞT = m I

T DL ⏌ L
A Tl mv
´ l
´ ҈
 ´ 黨

DL = = = =  ´ ҈

m YA l Ӹ꾀㡰 ҈
´ ҈
҈҈ ´ ҈

㦨




鼀鼀

꾀��
ᨨ鼀

Y =





b㡰 ␔
矄
 㦨

m

犨 
ᆠ
�矄

矄

㦨



\

=

㝐� m

和�鼀�泐
犨 
 
ᆠ
�矄
鼀ᨨ� �
犨�
ᆠᆠ�ᨨ



WWW.JEEBOOKS.IN

EBD_8341 PS-156

Physics

(

㝐 �
졔 



� 㦨

)

鼀 ´ ´�

��㡰 l ᨨ鼀
҈㡰
 

㡰






































 = � l 

҈


m


Þ


� =



  矄�ᨨ =

�
졔 
�





�
졔 

æ

� ö
㦨҈졔 Þ �
졔 
㦨 �
졔 �
맔 çè ҈졔 ÷ø 㦨






�
�  㦨� 㦨














᲼犨 �ᆠ
�矄
ᆠᨨ矄矄
ᨨ
�⏌�桌


䰘


b㡰

-҈⏌
 = � 

黨


҈



҈


꾀 l =  Þ l =

l 㦨
҈

鼀
 Q


㦨



ᆠ鼀
㦨



鼀
�� 㡰 矄� �

㦨
l

 l⏌


҈�


꾀

Þ f =

҈



=

犨 
m =

T mù   m = ú lû m

Tl T Y Dl Þ = ADl A l

҈
l

y Dl lr





�㡰

盐� �
 
盐
ᨨ
l


Dl
r  
y
�
⼄

�㡰

犨
�
l


҈
 ´ 䰘 

ᨨ


f
» ҈䰘



 㡰 ᨨ
 �
ᨨ
 
犨�


㦨
҈҈

ᆠ鼀

҈







㦨
҈







҈



 


 
 � 矄
ᨨ
 
 
矄 f =

҈

Q


r

㦨

r҈

꾀矄
µ

҈ 

҈ ҈ ҈

= ҈



 ҈ 


\

҈







㦨


\

æ ҈
ö ҈ = ç ´ ҈҈
÷ = 䰘
ᆠ鼀 è ҈
+
+  ø




�㡰

鼀 鼀矄矄 ᨨ 矄� � =   � ᨨ 矄� �

é êQ v = ë

Dl 㦨



҈
l  r 㦨
䰘
䰘
맔
҈

├�⏌鼀
�� 㡰 y
㦨




맔
҈
҈҈
⏌鼀

�� 㡰

㡰 ␔
├ ᨨ犨
 
ᨨᆠ�泐
�
矄� �
�矄
��
泐

m

T Ar

l 㦨
҈
�
鼀






l p
r 꾀矄




 
l

矄鼀
ᨨ
ᨨ
 
犨�矄

҈
㦨

҈ ҈ = 



v ҈ T ҈ = =
l
l m
l

꾀矄ᨨ

Y =

=


Þ



l


盐 鼀 
⼄盐 ᆠ泐
�
ᆠ矄
㡰
�矄   = = f '
㦨

l ⏌
㡰
l ᆠ㡰  �
ᨨ
�
㦨

�

ᆠ鼀
㦨



�鼀 ⼄盐 ᆠ泐
ᨨ
ᨨ矄ᆠ��ᨨ 矄
ᨨ
�
ᆠᨨ盐鼀
�
ᆠᨨ矄
ᨨ�
�
�矄 ��
泐

n - ҈㡰u 
n - ҈㡰u f = £ ҈
�
Þ
f =
L
L 
- ҈㡰 ´ 

£ ҈
�
Þ

n

҈


҈

�
»

Þ


� ´
b㡰 盐 鼀 
⼄盐 ᆠ泐


f =

 �

� ´ ҈
 Þ = = l ҈

㦨





맔
҈


; �
⧤ b㡰 ҈
㦨




®
和鼀 
®
和鼀 l
®
和鼀 ⼄盐 ᆠ泐
ᨨ
��ᨨ

҈  =
r҈ r


㡰


盐 鼀 

⼄盐 ᆠ泐
�
ᆠ矄
㡰
�矄
=

҈



Ӹ 黨

䰘 ´ ҈
҈
= r

䰘 ´ ҈
 
㦨
�

�
맔
҈

鼀⏌矄ᆠ 和� ᆠ
ᨨ
�矄
ᆠ鼀

鼀�
盐 鼀 
犨
矄 
�矄
矄 ᨨᨨ犨

l 㡰

(
 - ҈)  =

=

(
 - ҈) ҈
l 犨 
҈
㦨
盐 鼀 
⼄盐 ᆠ泐
ᨨ
��ᨨ 㡰  ᆠ







㦨



҈㡰
맔
҈�

Þ 
㦨
䰘
҈
»
䰘 \ 盐鼀
ᨨ
ᨨ
ᨨ 矄
㦨
ᨨ

ᨨ
鼀ᨨ
ᨨ
��ᨨ 㡰

҈ 㦨
䰘

҈
㦨

㡰 ᔼ
矄ᨨ�矄

ᨨᆠ�泐
ᨨ
犨



l =
=

� ҈


f

b㡰 l

l VA u r = B = B =
= A Þ l =
l Þ P = ҈ A B lB VB u A rA q
㡰 ᔼ

ᆠᨨ矄
�
��ᨨ
�

鼀ᨨ
 
⼄盐 ᆠ泐
ᨨ

=

҈҈





= �´ \  =



= 

鼀⏌矄ᆠ 




=

  ᆠ㡰 꾀矄
 
鼀盐矄

ᨨ

ᨨ
 矄
 

 
ᨨ� 
ᨨ
  犨�
A
 
B
矄ᨨ

��ᨨ







Þ


꾀矄ᨨ
f҈ =



l

\ ò -҈⏌
 - � ò 

l
=
ᨨ
l = m


ᨨᆠ�泐

v
㦨
f l =


´



l

WWW.JEEBOOKS.IN

PS-157

Waves æ
ö 
= ç ´ ҈҈
÷ =

ᆠ鼀 è ҈黨 ø

҈䰘


҈




҈


æ  ö    = ç ´ ҈҈
÷ = ҈
ᆠ鼀 è ҈黨 ø  ᆠ
 
��矄
矄 ᨨ盐

ᆠ


䰘

ᆠ鼀
 

䰘

졔



㦨
黨
ᆠ鼀
ᨨ鼀
ᨨ 
 
b㡰 ᔴᨨ 矄�� �
 
 
ᆠᨨᆠ�ᨨ
᲼e
㦨



D 犨  
㦨
�鼀桌 é v ù f = n ê
l + e ú ᨨ
�矄
矄ᨨ  ᆠ

n
㦨
҈ )û ë ( v v \
f
㦨

( l +

D ) Þ l =
f 

D æ 
´ ҈

ö ᨨ
l = ç ÷ 

 ´
= ҈�

ᆠ鼀 è
´ �҈
ø 㡰 ⼄盐 ᆠ泐
ᨨ


鼀ᨨ �ᆠ
ᨨ
矄� �
㦨
盐 鼀  ⼄盐 ᆠ泐
ᨨ盐ᆠ
�
 
�

㡰 ⼄盐 ᆠ泐
ᨨ


鼀ᨨ �ᆠ
ᨨ
ᨨ
�
v v f҈
㦨
= l l ⼄盐 ᆠ泐
ᨨ

鼀ᨨ �ᆠ
ᨨ
ᆠᨨ矄
�


v nv = l
l 
n
�矄

ᨨ
盐鼀

ᨨ鼀
⼄

�㡰
 
��㡰





��㡰

f

㦨


f

㦨






�㡰

n f҈


ᨨ
�矄
矄ᨨ  ᆠ

n
㦨
�
� f҈
ᔼ

矄ᨨ  ᆠ
盐


\
f

㦨





ᆠ㡰








l҈ 졔

l
졔



�
鼀

ᨨ
�矄
矄ᨨ  ᆠ

l l҈ + e =
盐 
㦨
nl
\ 
㦨
n l
+ e㡰 





鼀

æv ö v
ç ҈ ÷ =
è
l҈ ø Al
é ҈ Tù v \
´ê ú= ë
l҈ m û
l


҈黨







҈ �


\ = Þ m
㦨





├�
鼀҈

� m
´


 ᆠ

 
鼀矄矄
ᨨ
 
矄� �

m҈
㦨
m
l҈








































㦨





맔


�
├�
㦨
҈
� 㡰 ᨨ
鼀� 
 
矄
ᨨ
矄ᨨ盐  盐矄� �

矄ᨨ  ᆠ
ᆠᨨ盐鼀

ᨨ �矄 ᨨ
 
盐 � �
ᨨ├

├
�

�ᆠ  

꾀矄
矄 ᨨ犨
�
 
��盐

  � �矄
ᨨ
 
盐 � �
ᨨ├

├
�
 �ᆠ
 


㡰

⼄盐 ᆠ泐
ᨨ
�矄
鼀ᨨ �ᆠ
�
꾀
㦨
f҈
㦨


⼄盐 ᆠ泐
ᨨ
矄ᆠᨨ 
鼀ᨨ �ᆠ
�
ᔴ
㦨
f




㦨
n

 l҈ 졔
e㡰

\

l
+ e =





l



v  l҈ + e =
n

Þ l
+ e =




�㡰

Þ





��㡰

和盐ᆠ� �
�㡰
 
��㡰


 =
n l
- l҈ 㡰
\
D
㦨

n  Dl
+ Dl҈ 㡰 㦨


맔
�҈

맔


҈
졔



҈㡰
ᆠ鼀⏌矄
㦨






ᆠ鼀⏌矄  ᆠ

鼀 �鼀盐鼀
ᨨ矄矄�
ᨨ
�
矄

Dv
㦨






ᆠ鼀⏌矄 b㡰 ⼄盐 ᆠ泐
ᨨ
�矄
ᨨᨨ 
�
ᆠᨨ矄
ᨨ�
�


v P
l҈ r҈ ⼄盐 ᆠ泐
ᨨ
�矄
ᨨᨨ 
�
ᨨ
ᨨ�
�


n
㦨


҈ P l
r


v
㦨
vἘ nἘ
㦨


҈ T
l m

v P ҈ =
l҈ r҈ l


҈ T
l m

f҈
㦨
f

�� 㡰

҈ T҈ ҈ T
=
l m l m ᨨ
҈
㦨





�㡰 쟼⼄盐� �
ᨨ⼄盐矄
盐
ᨨ
T ҈
 
T

ᨨ盐
O
ᨨ
ᨨ�ᨨ  ⼄盐���盐鼀
꾀 \
T҈x
㦨
T

L

x㡰 ᨨ
 矄�ᨨ 
⼄盐���盐鼀
T҈
졔
T

㦨
鼀�




҈ ᨨ鼀
�

�㡰
 
���㡰
mg
mg 졠 T҈
㦨

 


㦨
  � � 㜠 ᨨ犨
ᨨ鼀
⼄

��㡰
\


mg L
mg ´x= ( L  x) Þ
x = L  x \ x =
� � �

ᨨ
矄ᆠᨨ 
矄ᨨ  ᆠ






P r


r
l҈ ҈ r


\

l
=

㡰

盐 鼀 
⼄盐 ᆠ泐

f

=

P T
l m

� 黨g  Mg = Þ M
㦨

�
├�
l Ҙ
l Ҙ 꾀矄
⼄ 盐 ᆠ泐
�矄
ᆠᨨ矄ᨨ 矄
ᨨ
� 
 
 
鼀ᨨ �ᆠ P
㦨
�
 
P
㦨

矄ᆠ�泐
҈ æ T ö ҈ æ T ö 㡰 n҈
㦨
ç ÷
 
n

㦨

l ç
÷
l è
pr
rø è pr rø 㦨



�


n
㦨


鼀

\

v ҈ = l l

T l


᲼犨 

㦨
 �
ᨨ
矄� �桌 m


҈ n҈ =
´ =҈ n



é ù 鼀矄矄 r ´ A ´  � = r Aú êQ m =  � =  � ë û

WWW.JEEBOOKS.IN

EBD_8341 PS-158






Physics

㡰 ᨨ
bᨨh

 
ᨨ
ᨨ�
� 盐 鼀 
⼄盐 ᆠ泐

ᨨ
ᨨ
ᨨ�
�
l

c


�㡰
l ᨨ
ᨨ 
 
ᆠᨨ矄
ᨨ�
� ᨨ
 �
鼀ᨨ �ᆠ
⼄盐 ᆠ泐 æ cö n

㦨

çè ÷ø



��㡰
l




���㡰 ෌�

n


n҈
㦨
҈

ᨨ鼀
⼄

�㡰
 
��㡰
n
⏌
 = =
Þ v҈ = v



�㡰  n҈ ҈⏌

꾀��

矄ᨨ� �
⼄

���㡰


�㡰
犨
�



n҈
㦨





ᆠ㡰 ᔴᨨ鼀� �
 
��
⼄盐�ᨨ

y
x

t㡰
㦨





ᆠᨨ矄




n2
㦨


n҈ =


䰘







pö æ çè �
pt + ÷ø
ᆠᨨ矄
҈
px㡰
y
x

t㡰
㦨
A
ᆠᨨ矄
wt
졔
p⏌
㡰
ᆠᨨ矄
k x ᔼ
k x
㦨
p⏌



ᨨ
ᨨᆠᆠ盐矄錐 \ ҈
p x
㦨
p⏌

Þ x
㦨



�
鼀 ᔼ
k x
㦨
p


 � ᨨ
ᨨᆠᆠ盐矄 Þ ҈
px
㦨
p
Þ x
㦨


҈
鼀 꾀矄ᨨ
矄
ᨨ
犨 w �
p = � 鼀⏌矄
 
l
㦨

p⏌k

㦨

p⏌҈
p
㦨




鼀 v
㦨

= k ҈
p 㡰 In a�r


T
㦨
鼀�
㦨
rVg T ҈ rVg =
\ f
㦨




�㡰 m
l m 犨 
 
ᨨᆠ
�矄

�鼀鼀矄
�
犨
T
㦨
mg

盐 盐矄 V Vg 㦨
Vrg

rw g = 
r - rw 㡰

Vg 
r - rw 㡰 ҈ ҈ Vgr 
r - rw 㡰
\ f Ἐ
㦨





㦨

l m
l m
r
r - rw fἘ æ
r - rw ö =


f Ἐ = f ç f
r è
r ø÷

l

Q 







҈



n1
㦨

ᨨ
�矄
鼀ᨨ �ᆠ
l҈ l1

  l ҈ =



Þ



҈ =
l҈
l
l

ᨨ
盐
ᆠᨨ矄


 

n҈ = n





\




\

l =
l

lἘ = l  盐 鼀 
鼀ᨨ㡰
c c \ n
㦨
= = �҈
 l
l   nἘ
㦨



\

l Ἐ =
l �� 㡰

c c æ cö = =
ç ÷
㦨


맔
�҈

㦨
҈




è
l ø l Ἐ
l

ᨨ
ᆠᨨ矄
ᨨ�
�
æ vö f = n ç ÷ 犨 
n = ҈

�





è
҈ø \
l =

nv
f

ᨨ
n
㦨
҈

l҈ =

(҈)( 
)

´ ҈

ᆠ鼀 = ҈

� ᆠ鼀
´


ᨨ
n
㦨


l = l҈ = 黨
䰘� ᆠ鼀 



ᨨ
n
㦨
�

l� = �l҈ = ҈�


� ᆠ鼀 �㡰 
�矄 ᆠ
犨
犨ᨨ
矄盐ᆠᆠ矄矄�
ᨨ矄



� ´


T ⏌m v l = = D
㦨
=

f
f �


҈⏌



黨


2

l = l  盐 鼀 
鼀ᨨ㡰
ᨨ
ᨨ
盐

҈⏌


é
r - ҈ ù 㦨


ê ú  ë
r û ᆠ㡰 和�ᨨ 泐
犨
�矄
ᨨ盐ᆠ
犨 
犨ᨨ
犨矄

�

ᨨᨨ矄�
�ᆠ�ᨨ
ᨨ犨

y
㦨
a
ᆠᨨ矄
k x

wt㡰

a
ᆠᨨ矄
k x
졔
wt㡰 \ y
㦨

a
矄�
kx
矄�
wt
�矄
⼄盐�ᨨ
ᨨ
矄�ᨨ 泐
犨
犨 �ᆠ ��矄

ᨨ

x
㦨


y
㦨

a
ᆠᨨ矄
kx

wt㡰
ᆠ

ᨨ

矄
 �
�ᆠ�ᨨ 矄

ᨨ
ᨨᨨ矄�
l ᆠ㡰 ␔ 
盐
犨矄
ᨨ

ᨨ
 矄

 

= l
v v \ f
㦨
= l
l lἘ l 犨 
盐
�矄

�
�
犨

 

=

Þ lἘ
㦨

l

v v = f \

f
Ἐ
㦨
= l Ἐ
l ᆠ㡰 ᨨ
ᆠᨨ矄
ᨨ�
�


㡰


ᨨ
 �
鼀ᨨ �ᆠ
l


҈

´ ҈



=

҈
= �ᆠ鼀


�


㡰 ෌�
 �矄泐
ᨨ
犨�

s = 黨 ´ ҈
-
├�
ᆠ鼀 Ӹᨨ盐 �Ἐ矄
鼀ᨨ盐盐矄
ᨨ
犨�

Y
㦨
黨
맔
҈
҈

鼀
和�
㦨


黨
맔
҈

和矄矄 T ⏌ A Y= = 和� 和�

T \ = Y ´ 和�
㦨
黨
맔
҈
黨
맔


黨
맔
҈

A 꾀矄ᨨ

鼀矄矄
ᨨ
犨�

m = Al s m 㝐矄矄

盐 �
 �

m = = As J 盐 鼀 
⼄盐 ᆠ泐
�
 
矄� �

f =



=

҈ T ҈ T =
l m
l sA ҈ 黨 ´ ҈
黨 ´

黨 ´ ҈
-

´҈ 黨 ´ ҈


҈ ҈
黨 ´ ҈
黨 -
- = ´ 䰘
= � 






ᨨ



㡰 
l҈
㦨
� ��
 �
ᨨ
� l

㦨
犨
 �
ᨨ
� 
㦨
和
ᨨ
盐 � �
ᨨ├




=





ᔼ
ᆠᨨ矄
ᨨ�
�

=


l҈

WWW.JEEBOOKS.IN

PS-159

Waves æ  ␔ 
盐 � �
ᨨ├
�矄
鼀ᨨ
Ἐ = = ç è  - 

Þ

 æ  ö  ç ÷=
l҈ è  -  ø
l


Þ

w
㦨
҈


鼀矄҈
├  ᆠ
ᨨ�ᨨ 矄
㡰

ᆠ㡰


㡰
矄�矄 泐
 
ᨨ
ᆠᨨ ��ᨨ 矄



b

ᆠ㡰 泐
㦨
᲼


҈
矄�




㡰桌
ᆠᨨ矄




㡰

᲼෌� 桌

ö  ÷= ø
l


和
ᨨ
犨

v =

 -  l
=  l҈

㝐

 - 

-

l
l -l -҈ = -҈ Þ
҈ =   l҈ l҈ ᆠ �
ᆠ  �
⼄盐�
�
 
 �
ᨨ
 
� l
- l҈ -
´ ҈

= ´ ҈

= -



�奠 

l҈  ᆠ

矄鼀矄
盐
ᨨ
ᆠ �
ᆠ  �
⼄盐�
�
 
 � ᨨ
�
�矄




�奠 ᔼ
��盐
㡰
ᨨ
 矄
ᨨ
 
盐 �
�
Þ

䰘






l = l


Þ l
㦨

l
c c \



f
㦨
= l
l ᔼ
��盐
㡰
犨 

ᨨ
  盐
�ᆠ泐
�
�

l 犨
lἘ l = Þ l Ἐ =
l

c c = f \

f
Ἐ
㦨
= (a) l Ἐ
l ҈ T 盐 鼀 
⼄盐 ᆠ泐
f
㦨

ᨨ
f
µ T
l m



l/2






(b)

f µ 犨�� 
ᨨ
鼀矄矄
( w ) f Ἐ µ 犨

盐 盐矄
( F )

u ( x
y ) =
 x = L
y = L [Q 矄� p =

矄�
p =
]

fἘ wF w F = ᨨ

f Ἐ = f f w w 和盐矄�盐� �
 
盐矄

犨
� \


f Ἐ =


黨







( �

䰘) g - (



䰘�) (҈
 ) g ( �

䰘) g

졠�ᨨ
㡰

u ( x
y ) =
 y =

y = L [Q 矄�
=

矄� p =
] u ( x
y ) ¹
 x =

x = L [Q ᆠᨨ矄
= ҈
ᆠᨨ矄
p = ҈]

=


Hz

和 ᨨ矄
�矄 ᆠ
ᨨ鼀
 
犨

犨 �ᆠ
�
�ᆠ矄


� =

҈
�鼀 鼀 �鼀盐鼀
鼀�盐
ᨨ
��ᨨ
㦨
l⏌

㦨

c 
=

� 鼀 Q c
㦨
vl \

l
㦨
= v




b

ᆠ㡰
꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

 
 �
ᨨ
 
�
ᆠᨨ盐鼀
�矄 �
泐
ᆠ  �� �
 

ᨨ
犨
�
 
矄ᨨ  ᆠ
盐
l 矄ᨨ


n
졔
҈㡰
㦨
�

䰘
졔
e


�㡰
l  

n
졔
㡰
㦨


黨
졔
e


��㡰
��� �
⼄
�㡰
泐
��㡰 ᔼ
n
㦨
҈

Þ \

l ⏌
�

䰘 + e = �l ⏌


黨 + e


e
㦨
҈




p
p =

҈鼀 㦨





\
l =



l l ҈ =


� 鼀  �
ᨨ
矄� �

l
㦨
� ´ = � ´





鼀�ᨨ� 
㝐
�矄

 � ᨨ
 
矄
 
鼀 �鼀盐鼀 �矄ᆠ鼀 
㦨



҈
鼀  ω⏌├


盐 鼀 
⼄盐 ᆠ泐

v
㦨
=
l
l


= =


´


� ´




b
ᆠ㡰
和 � �
犨矄

ᨨ盐ᆠ
泐
犨ᨨ
� �ᆠ
犨矄 矄盐ᨨ矄� �
犨 �
� �
�
ᨨᨨ矄�
�ᆠ�ᨨ
b
ᆠ㡰

�矄
ᨨ
 

 y ᆠ鼀

矄ᨨ
�矄
�矄ᆠ鼀 矄 ᨨ �
 
x
 
泐
 矄
犨� Q
L
L
㡰 R 

L
㡰 � ��盐泐

ᨨ

  �矄
졠�ᨨ
㡰

u ( x
y ) =
 x =L
y =L P x u ( x
y ) ¹
 x =

y =




㡰 L


㡰 졠�ᨨ
㡰

u ( x
y ) =
 x =

y =
[Q 矄�
=
] u ( x
y ) =
 x = L
y = L [Q 矄� p =
] 졠�ᨨ
ᆠ㡰

u ( x
y ) =
 x =

y =
[Q 矄�
=
] ᨨ鼀
 
��
⼄盐�ᨨ

├
㦨



�







æ vö 

b

㡰
ᨨ
ᆠᨨ矄
ᨨ�
�

v
㦨
n
ç ÷ 犨 

㦨
҈



�

䰘



è
l ø v v �v \ n
㦨








㦨

















l
l
l ҈

ᆠ㡰
⼄盐 ᆠ泐
n =
ᨨ
�矄
鼀ᨨ
ᨨ
��ᨨ


l 鼀 ᨨ
�n
ᨨ

鼀 �鼀盐鼀

�l
矄 ᨨ盐

鼀� �鼀盐鼀
和� �㜠҈

=

҈




m

和� �㜠


=

҈




=

m


和� �㜠
 =

҈




=

m 

和� �㜠


=

҈





=
m


\

맔



黨
졔
e
㦨
�
맔
�

䰘
졔
�e

l
㦨
�

䰘
졔


黨
㦨
�҈




\
e
㦨


黨
ᆠ鼀 Þ l
㦨





ᆠ鼀
㦨






鼀

꾀矄ᨨ
矄
ᨨ
矄ᨨ盐 


㦨
vl
㦨
�


맔






鼀矄҈
㦨




鼀矄҈
҈


ᆠ

㡰


矄 ᨨ盐


�矄ᆠ鼀 
ᨨ

x
㦨


 

�矄ᆠ鼀   � ᨨ

x
㦨

鼀

 ᨨ

y
㦨



x
㦨


 
y = ± A

x
㦨

鼀



䰘


b㡰
꾀矄
n =

n
l
鼀 

\ =
l 鼀 







m 和� �㜠҈

= l


WWW.JEEBOOKS.IN

EBD_8341 PS-160

Physics ҈ ҈



dE
㦨
dm㡰 A w =  µdx 㡰 a 矄� k x 㡰 
pn㡰







㦨

p
Ҙn
a

矄�

kx
dx Q

㦨
nl  
T ⏌Ҙ éQ  = T ⏌ m ù \ n
㦨
Þ n

㦨

=
ë û l
l l




和� �㜠


=

æö


ç ÷ 

(
m ) è
ø

( )




=






æ�ö


ç ÷ 

( m )  è
ø 和� �㜠
 = =
҈
�


\




和� �㜠


=




æ䰘ö


ç ÷ 

(
m ) è
ø

(҈
)


\

=
҈


l


E
㦨
ò dE = ò
p Ҙ




꾀㜠
錐
㜠
矄錐
ᔴ㜠⼄
矄錐
㜠⼄
 꾀㡰
�
ᆠᨨ矄

ᨨ 
  ␔矄
ᨨ盐ᆠ

ᨨ ��盐� 

= L















�҈












lf


\ l
㦨


 㡰 �
ᨨ

bᨨh
 矄 犨矄
ᨨ盐ᆠ

ᨨ ��盐� 

L

lf

�



= L


















\ l
㦨



lf



黨


vA  = ᨨr ᨨ
vB


= L




















lf



\ l
㦨


㡰 和ᆠh
犨
ᆠ鼀

bᨨh
 矄
 

鼀�
ᨨ�  ␔矄
ᨨ盐ᆠ

 矄矄
�
盐

ᨨ


l
㦨





























æ � ⏌  ö æ ҈
ö ÷ç ÷ 㦨
çè 䰘 ⏌ �ø è 黨 ø

lf

黨
aw =
Þ w
㦨


⏌矄



�㡰  aw 和盐矄�盐� �
 
盐
ᨨ
w
�
⼄

�㡰

犨
�  =

҈鼀 a
㦨




㡰 

p
p w 
 = = = = ᨨ犨

k
㦨




�㡰  ⏌ n 


l ᨨ犨
盐� �
 
盐
ᨨ
a

w
 
k
犨
�
犨 ᨨ鼀  y
㦨


҈
矄�
᲼
t
᠀
x
᠀
f桌



矄� �
�矄
�
盐 鼀 
鼀ᨨ
  ᨨ l =


p
p p =
㦨
l
l l 꾀鼀�盐
ᨨ
��ᨨ


�矄 ᆠ
x
ᨨ鼀
x
㦨


A
㦨
a
矄�
k
x 㝐ᆠ  �ᆠ
 �泐

x
ᨨ
 �
dx
�矄 ᨨ
l
㦨

l

和� ᆠ

k
㦨


gA gB

g A RTA
lB g B RTB MB

=




(Q l A = lB )

MB  M g æ ҈
ö = ( 矄TA = TB ) Þ A = A ç ÷ MA
MB gB è 黨 ø

� 䰘ö æ çè g A =  
g B = ÷ø  �

M A æ
� ö æ ҈
ö


=ç ÷ç ÷ = M B è
҈ ø è 黨 ø ҈
黨 㡰 �ᨨ
ᨨ
盐 鼀 
⼄盐 ᆠ泐
�
�
A
 
�
�
B, 犨 �ᆠ
�矄
ᨨ犨
ᆠᨨ矄

ᨨ
 矄


矄 
犨 ᨨ鼀
ᨨ

 矄矄
鼀ᨨ �ᆠ
�矄盐 ᆠ y
㦨

a
矄�
wt
᠀
kx
᠀
f㡰 ෌�
鼀
㦨
aw
㦨

鼀⏌矄



�㡰 A鼀
㦨

aw

㦨
黨

鼀⏌矄




��㡰 ᨨᆠ�泐
ᨨ
犨
v
㦨



鼀⏌矄



���㡰 ��� �
⼄

��㡰
泐
�㡰

\

f A v A ⏌
l A v A = = f B vB ⏌
lB vB g A RTA MA g A MB =
㦨
g B RTB gB MA MB




�



p Ta
l ᔼ
 
盐 鼀 
鼀ᨨ v l l +


r 㡰 = =
Þ
v
㦨

f
l +


r 㡰

f 





r
㦨
 㜠ᆠᨨᆠ�ᨨ
�
盐 r
㦨
�盐矄
ᨨ
盐㡰
㦨


�
ᆠ鼀 l
㦨
 �
ᨨ
盐㡰
㦨
҈

ᆠ鼀 f
㦨
⼄盐 ᆠ泐
ᨨ
盐 � �
ᨨ├㡰
㦨




 \
ᨨᆠ�泐
ᨨ
矄ᨨ盐 
�
�

v
= 
鼀⏌矄
㡰
⼄盐 ᆠ泐
ᨨ
矄ᆠᨨ 
鼀ᨨ �ᆠ
�
�
A
ᨨ

ᨨ
 矄 㦨
⼄盐 ᆠ泐
ᨨ
 �
鼀ᨨ �ᆠ
�
�
B
ᆠᨨ矄

ᨨ 
 

æ vA ö æ vB ö \

ç
l ÷ =  ç
l ÷ è Aø è Bø

ᆠ㡰 和ᆠh
犨�
ᆠ鼀

bᨨh
 矄 ␔矄
ᨨ盐ᆠ

 矄矄
�
盐


lf

T ⏌Ҙ

æ p x ö a 矄� ç ÷ dx è l ø
l


ᨨ

E 㦨


L

lf

T ⏌ Ҙ

ìæ p ö ü a 矄� íç ÷ xý dx
l
îè l ø þ

ᨨ
 �泐
ᨨ
 
矄� �

dE
㦨

p
Ҙ


( 矄
l A = lB ) (Q TA = TB )

M B ҈
黨 � 䰘 和盐矄�盐� �
M =




g A =   
g B =  A

fA
� ҈
黨  = ´ =
҈



fB 和
ᨨ
矄ᨨ盐 

v
㦨



鼀⏌矄

l


 
 �
ᨨ
�
ᆠᨨ盐鼀
ᆠᨨ矄ᨨ � �
ᨨ
  盐 鼀 
⼄盐 ᆠ泐



v v 

=
҈

�
Þ

l 㦨
= =


鼀


l


҈

�㡰

҈

�㡰 \

l


�


WWW.JEEBOOKS.IN

PS-161




鼀

Waves




鼀




鼀




鼀

҈

鼀



鼀



鼀

҈

鼀

l

矄矄盐


ᨨ㡰 �矄ᆠ鼀   � ᨨ x㦨 矄矄盐
��ᨨ

�矄
ᨨ

㡰 꾀
 泐
ᨨ矄��ᨨ
x

 
矄矄盐 DP
㦨
DP

ᆠᨨ矄
kx
ᆠᨨ矄
wt

ᔼ
ᆠᨨ矄
�
ᨨ 泐
ᨨ
鼀ᨨ �ᆠ矄

ᨨ� 

ᨨ犨

 l҈
l

l 
l

ᆠ


 
 � 矄
ᆠᨨ矄ᨨ � �
ᨨ
 
 鼀ᨨ �ᆠ

�
鼀ᨨ �ᆠ

䰘
鼀ᨨ �ᆠ
ᆠ




鼀�盐
A
㦨
DP

ᆠᨨ矄
kx
㦨
DP

ᆠᨨ矄


҈� ҈� L = 鼀 鼀�
ᨨ� 㡰 㦨


´ ҈

é
p ҈� ù DP ´ ú=
꾀鼀�盐
㦨
DP

ᆠᨨ矄
ê 
⏌


㡰 
û
ë ᆠ㡰 꾀
ᨨ
 
ᨨ
�

矄矄盐
�矄
犨泐矄
矄鼀
�


⼄盐
ᨨ

æ v ö 䰘ç ÷ 㦨

҈

�
Þ l  㦨




鼀錐 è
l  ø ��


\

æ - dH ö a
gH
㦨
꾀

ç è dt ÷ø 和ᨨ


ᨨ

ᨨ
犨

�
 
�


æ - dH ö a
gH çè ÷= dt ø A 和盐矄�盐� �
 
盐矄

犨
�
´ ҈
´ H

- dH 㦨
҈
҈҈
맔
҈

㡰 H dt 犨
�矄
犨ᨨ
矄ᨨ  ᆠ矄

 
犨

矄
ᨨ鼀 


鼀
ᨨ




鼀


Þ

�



\

dH

Þ

ò



Þ


᲼


- 

桌 = - ҈
҈ ´ ҈
-
㡰
t

ᨨ
㡰

ᨨ
矄ᆠᨨ 
ᨨᨨ 
ᨨ
 
ᆠᨨ矄
�

H




= -҈
҈҈ ´ ҈
-
dt

dH H

t

= - ҈
҈҈ ´ ҈
-
㡰ò dt


t »
 s

æ vö æ 
ö
ᨨ




= � ç è
L÷ø è
L ÷ø � ´ 
҈� Þ L
㦨 Þ

= 鼀

´


҈


⼄盐 ᆠ泐

f = P ç

鼀
矄矄盐

Q DP =

P鼀 = P鼀� = P
㡰 꾀
 
ᆠᨨ矄
 
ᨨ
� 㝐 �鼀盐鼀
矄矄盐

鼀
㦨
P

졔
DP
㝐� �鼀盐鼀
矄矄盐

鼀�
㦨
P


DP
px ᆠᨨ矄 ( 黨
pt ) �㡰 y
㦨


矄�
҈� æ p xö 
鼀�盐

A
㦨


矄�
ç ÷ è ҈� ø At x
㦨
�
ᆠ鼀 꾀鼀�盐
ᨨ
鼀 �鼀盐鼀
�矄ᆠ鼀 
æ p ´ �ö A
㦨

矄� ç 㦨


맔






㦨




ᆠ鼀 è ҈� ÷ø ��㡰 ᨨ矄

 
ᨨ矄��ᨨ
犨 
A
㦨



æ px ö x
㦨
҈�


矄� ç ÷ =
= 矄� np


\ è ҈� ø 犨  n
㦨



҈


x
㦨
҈�
ᆠ鼀



ᆠ鼀




ᆠ鼀





���㡰 꾀
x
㦨
䰘
�
ᆠ鼀

t
㦨



�
ᆠ鼀
ᨨᆠ�泐
ᨨ
�ᆠ dy æ p xö =
矄� ç v
㦨

᲼
黨
p
矄�
黨

pt㡰桌 è ҈� ÷ø dt æ p ´ 䰘
� ö v
㦨

矄� ç
᲼
黨
p
矄�
黨

p
맔



�㡰桌 è ҈� ÷ø æ pö 㦨

矄� ç ÷ ᲼- 黨
p 矄� 

p㡰桌
㦨

è
ø æ p xö �㡰 y
㦨

矄� ç ÷ ᆠᨨ矄᲼黨
p t 桌 è ҈� ø é ù æ p xö 㦨

ê
矄� ç ᆠᨨ矄 黨
p t 㡰ú è ҈� ÷ø ë û é æ p xö p xöù æ 㦨

ê矄� ç 黨
p t + ÷ø - 矄� çè 黨
p t ÷ è ҈� ҈� ø úû ë p xö p xö æ æ 㦨

矄� ç 黨
p t + ÷ -
矄� çè 黨
p t ÷ è ҈� ø ҈� ø 㦨
y҈
졔
y
 ᆠ
ᆠᨨ鼀ᨨ  矄
犨矄 p xö y҈
㦨


矄� æç 黨
p t + ÷ è ҈� ø

ᨨᆠ�泐
ᨨ
 盐

v
㦨

gH ᔴᨨ � 盐�泐
⼄盐�ᨨ

҈
 


��矄


- dH 
҈
´ ҈
-
= dt ҈


´ ҈
-


p x l

ᨨ x
=

æ v ö ç ÷ 㦨

҈

�
Þ l҈ 㦨
҈


鼀錐 è
l҈ ø æ v ö �ç ÷ 㦨

҈

�
Þ l
㦨




鼀 è
l
ø

æ v ö 黨ç ÷ 㦨

҈

�
Þ l
㦨



鼀 è
l
ø ᨨ
�� 矄
ᨨ
犨









㡰
鼀





҈

㡰
鼀







㡰
鼀
 







㡰
鼀
 ᆠ
�� 矄
ᨨ
犨





鼀





鼀

҈


鼀
 




鼀

A
 


 

ᨨ
ᆠᨨ矄矄㜠矄ᆠ�ᨨ 矄
ᨨ
 
�
 
ᨨ 矄ᆠ�泐



A
㦨
p


맔
҈

㡰

㦨
҈



맔
҈

鼀
  a
㦨
p
҈
㡰

㦨

҈

맔
҈


鼀


矄矄盐
 � ᨨ㡰 ᨨ
�矄ᆠ鼀 
ᨨ 㝐ᔼ
졠ᔼ x㦨졠 矄矄盐
��ᨨ

x=x �矄
鼀
ᨨ
鼀�

�



p xö æ  

y

㦨



矄�
ç 黨
p t ÷ è ҈� ø

盐
ᨨ

ᨨ
 矄  �
ᨨ
盐
l
㦨



ᆠ鼀
㦨




鼀
盐 鼀 
⼄盐 ᆠ泐
f
㦨



ᨨᆠ�泐
ᨨ
矄ᨨ盐 
�
�
v
㦨



鼀⏌矄  

v
㦨




\



㦨

l +


D㡰




+


´ D㡰

WWW.JEEBOOKS.IN

EBD_8341 PS-162

Physics






졔




D
㦨


�


Þ



D
㦨








Þ D
㦨
´ ҈

ᆠ鼀
㦨


ᆠ鼀
�鼀
ᨨ
盐㡰



盐
ᆠᨨ矄

ᨨ 
 

졠矄 O D q

 

= ⼄盐 ᆠ泐

v
㦨
»
҈


l +

D㡰




+

 ´


㡰

Topic-3 : Beats, Interference and Superposition of Waves ҈









㡰 ᆠ㡰 
⼄盐 ᆠ泐 㦨
�  ᆠ
�
⼄盐 ᆠ�矄
ᨨ
犨ᨨ
犨矄 㦨
҈҈

黨
㦨


 㡰 㡰 ᨨ
⼄盐 ᆠ�矄
ᨨ
盐 � �
ᨨ├

n
᠀
� ҈ ⼄盐 ᆠ泐
ᨨ
矄ᨨ ᨨ鼀
犨�

n µ l n + � ҈

\ = Þ 黨� n + �㡰 = ҈

 n - �㡰 n - � 黨� ᨨ
黨�
n
졔

䰘�
㦨
҈


n

�


ᨨ

�
n
㦨
黨䰘�

黨䰘� = ҈黨�  � 㡰 ␔�
� ᆠ矄
�
 矄�ᨨ

 
⼄盐 ᆠ泐
ᨨ盐ᆠ
泐
矄� � � ᆠ矄矄

꾀矄
 
矄⏌矄ᆠ
ᆠ矄矄
  ᨨ
⼄盐 ᆠ泐
ᨨ
盐 � � ᨨ├ ævö æ 

ö f = ç ÷ +
= ç ÷ +
= 

 è
l ø è
´

䰘� ø ᨨ
n =

�






b㡰 ᨨ

矄� �
⼄盐 ᆠ泐
f =

和ᨨ盐ᆠ V

ᔼ ��泐
q
犨�

矄矄
矄ᨨ
ᆠᨨ矄
q
鼀ᨨ
\
vᨨ矄
鼀ᨨ

 
�
犨�
ᆠ矄






æ v + vc ö æ v + vc ö f
= ç ÷ f҈ = ç v - v ÷ f
è v ø c ø è é 
� + vc ù ҈

� + vc Þ


= ê = ú


Þ v 
� ҈҈ 
� - vc cû ë

Þ vc = �
├鼀⏌  


҈ F
l m

�鼀
�ᨨ

T =
l



� ´ ҈
-

㦨


҈
矄ᆠ
҈




�鼀
⼄盐�
ᨨ
ᆠᨨ 矄盐ᆠ�
�   ᆠ

⼄盐
ᨨ
 
�鼀 �ᨨ
ᨨ

犨
盐矄

 ᆠ
D
㦨


҈矄 ætö b㡰 ෌�
⼄盐�ᨨ

y
㦨


ᆠᨨ矄

çè ÷ø
矄�
҈



t㡰


\
T 㦨

´




䰘






=

+ 
+
´
´  ´ ᆠᨨ矄

A҈
+ A

+
A҈ A
ᆠᨨ矄 f

p = ҈
+ 黨 +
= �


Topic-4 : Musical Sound and Doppler’s Effect ҈






t æ ö
㦨


ç
ᆠᨨ矄
矄� ҈


t ÷ è ø
ᨨ

y
㦨


᲼ᆠᨨ矄
t
졔
҈桌
矄�
҈


t


҈㡰 ᨨ

y
㦨


ᆠᨨ矄
t
矄�
҈


t
졔


矄�
҈


t



㡰 ᨨ

y
㦨
矄�
҈

҈t
졔
矄�
黨黨黨t
졔


矄�
҈


t


㡰  ᆠ

ᨨ
 
��
�ᨨ�ᆠ
鼀ᨨ�ᨨ

 
�   
鼀ᨨ �ᆠ 鼀ᨨ�ᨨ 矄

矄盐ᨨ矄
�㡰 
 
 矄
�  ᆠ
犨
 
犨ᨨ
犨矄
f
㦨
p
矄盐 
鼀�盐

A =

b㡰 ⼄盐 ᆠ泐

泐
 
ᨨ矄

vᨨ矄

æ ö v矄ᨨ盐  =ç ÷ v
v q  ᆠᨨ矄 è 矄ᨨ盐  ø

㡰 ᨨ鼀
 
ᨨἘ矄
 ᆠ
ᨨ
矄ᨨ盐 

⼄盐 ᆠ泐

 犨 
fw = ×f


�㡰 
- v 

v
㦨
矄
ᨨ
盐矄
f
㦨
ᆠ盐
⼄盐 ᆠ泐
ᨨ
矄ᨨ盐ᆠ ⼄盐 ᆠ泐

 
 ᆠ�ᨨ
ᨨ鼀
犨
f'㡰
�矄


+ v 
+ v × fw = ×f 

- v 
+ v Þ
黨
= ×



- v 
´ 䰘 Þv= »
�

鼀⏌矄 㦨 黨҈
├鼀⏌矄 黨҈ ᆠ㡰 ᨨ鼀
ᨨ矄
 ᆠ

⼄盐 ᆠ泐
ᨨ
矄ᨨ盐 

f҈㡰
犨 

矄ᨨ盐ᆠ
�矄
ᨨᆠ � � c f҈ = f
cv 

c
㦨
ᨨᆠ�泐
ᨨ
矄ᨨ盐  v
㦨
ᨨᆠ�泐
ᨨ
矄ᨨ盐ᆠ ⼄盐 ᆠ泐
ᨨ
矄ᨨ盐 


f
㡰
犨 
矄ᨨ盐ᆠ
�矄
ᆠ� � c f
= f
c+v 
⼄盐 ᆠ泐
㦨
f҈

f
҈ ù é ҈ Þ
= f҈  f
= f
c ê  ë c  v c + v úû
v = f
c é v
ù c
ê҈ 
ú ëê c ûú ᨨ c>>
v c
c �
҈ Þ v= = = = 鼀⏌矄 f
҈




f


f Ἐ=

m F ҈
-
鼀矄矄 =

� ´ ҈
-
F
㦨
҈





m =
㦨



 �

\

æ v ö 㡰 
f҈

 
⼄盐 ᆠ泐

泐
犨

f҈ = ç ÷ f
è v - vc ø 

v
㦨
ᨨᆠ�泐
ᨨ
矄ᨨ盐 

vc
㦨
ᨨᆠ�泐
ᨨ
ᔴ
f

㦨
ᆠ盐
⼄盐 ᆠ泐
ᨨ
ᆠ
ᨨ


f


 
⼄盐 ᆠ泐

泐
�
 
 ᆠ�ᨨ
ᨨ鼀
犨


�


ᆠ㡰 f҈
㦨
f

v - v
v + v

 
f

㦨
f v v

盐
⼄盐 ᆠ泐



v
f


f҈
㦨
f ´ v

S҈

L

u

S


WWW.JEEBOOKS.IN

PS-163

Waves
u 
\ u
㦨


�
鼀⏌矄
b㡰 ⼄盐 ᆠ泐
ᨨ
矄ᨨ盐 
矄ᨨ盐ᆠ

㡰
㦨
�


 ␔ 
ᨨ矄
�矄
鼀ᨨ� �
犨泐
ᨨ鼀
 
矄ᨨ盐ᆠ æ n - n Ἐ
ö 꾀 
⼄盐 ᆠ泐
҈ =


=
ç



�㡰 ÷ ç n ÷ è ø 꾀 
犨 
ᨨ矄
�矄
鼀ᨨ� �
ᨨ犨矄
 
矄ᨨ盐ᆠ æ n - n³
ö
= �
=
ç ÷



��㡰 ç n ÷ è ø ᨨ鼀
⼄盐�ᨨ
�㡰 æ 

-  Ἐ
ö


= �

ç ÷ ç 

÷ è ø 


㦨
҈

鼀⏌矄 ᨨ鼀
⼄盐�ᨨ
��㡰 æ  ἘἘ ö �
= �

ç҈ +
÷









\

㦨
҈

鼀⏌矄 ç
 ÷ø è v - v
㡰 f '
㦨
f v + vs ᨨ





䰘


é v ù 

f
= f ê Hz ú= f´ + v v 

sû ë

҈

㦨



´

҈



҈


�矄  ᔴ矄
҈


␔ 
矄ᨨ盐ᆠ
�矄
鼀ᨨ� �
ᨨ犨矄
矄�ᨨ 泐
�矄 



-




\
f
㦨


�




+

b㡰 꾀ᆠᆠᨨ� �
ᨨ
ᨨ矄
 ᆠ

犨 
矄ᨨ盐ᆠ
�矄
鼀ᨨ� �
盐 ᨨ矄

矄 ᨨ




= f





é V ù  㦨

ê ú Þ  㦨

ëV  Vs û

é 

ù ê ú ë 

 
û

é 

ù  


㦨

ê ú ë 

҈䰘 û

҈



 

҈䰘 
 ҈黨 = 㦨 ᨨ
҈ =


 



҈
㡰 ⼄盐 ᆠ泐
ᨨ
 
矄ᨨ盐 
ᨨ盐ᆠ
泐
ᨨ
盐


\

黨


æ v ö
´ 
f =
ç ÷ = =


Hz è
l ø
´

� �
� m⏌s = ҈
黨 꾀矄
 
矄ᨨ盐ᆠ
�矄
鼀ᨨ� �
ᨨ犨矄
 
ᨨ矄
  ᨨ

ᆠᆠᨨ� � ᨨ
ᨨἘ矄
 ᆠ
\
⼄盐 ᆠ泐
ᆠ
泐
ᨨ矄
é
� ù é v + v
ù ê 黨 + 
ú
Ἐ
㦨
ê ú f 㦨
ê ú


ë v û ê 
ú ë û
黨黨� ´



ᨨ


Ἐ
㦨


�
��
;



 㦨
黨 ´ 
㡰 ␔
├ ᨨ犨
 
 
 
⼄盐 ᆠ泐 æ  - 
ö Ἐ= ç ÷
ᨨ鼀
ᨨἘ矄
 ᆠ è  - 矄 ø 犨 


㦨
矄
㦨


鼀⏌矄

ᨨᆠ�泐
ᨨ
ᨨ矄
 
矄ᨨ盐ᆠ 和
ᨨ
矄ᨨ盐 

㦨


鼀⏌矄 ᨨᆠ�泐
ᨨ
ᨨ矄



㦨
҈
´

҈




졔 
 �


㦨






㜠 
Q
⼄盐 ᆠ泐
ᨨ
犨 �矄
 㡰
㦨
�



é v ù 

Hz 㡰 f҈ = f ê ú= f´ 

ë v - vs û

æ f
ö æ 

ö - ҈ ÷ ´ ҈

; ҈
奠 ç - ҈ ÷ ´ ҈

= ç f è 

ø è ҈ ø 㡰 ෌�

⼄盐 ᆠ泐
ᨨ
矄ᨨ盐 
ᨨ盐ᆠ
泐
矄�
f
㦨




 和
ᨨ
ᨨ矄

u
㦨


鼀⏌矄 ᨨᆠ�泐
ᨨ
矄ᨨ盐 

v
㦨



鼀⏌矄 ᨨ

ᨨ
矄


矄ᆠᨨ 
㦨
䁸 ᨨ

ᨨ
 
犨矄
ᆠ�
泐
 
ᨨ矄

矄ᆠᨨ 
㦨
졔

l \ ᨨ

ᨨ
矄⏌
矄ᆠ æ
æ
ö
Vö
´
㦨
- ç - ÷ =
㦨


çQ l = ÷ f ø l è lø l è



´
´


= ҈
㦨



㦨
�


 ᆠ㡰  꾀

鼀⏌矄 ᔴ

҈�


҈



æ  ö æ 

ö  
⼄盐 ᆠ泐
h Ἐ = h ç ÷ = �

ç ÷
㦨
�


   è 矄 ø è 
ø ᔴ矄





␔ 
矄ᨨ盐ᆠ
�矄
鼀ᨨ� �
犨泐
ᨨ鼀
 
矄�ᨨ 泐
�矄  æ  ö æ 

ö h³ = h ç ÷

= �

ç ÷
㦨

黨

 è 

ø è  + 矄 ø ᔼ
ᆠ矄
҈
盐鼀
ᨨ
矄

�矄


 
�
ᆠ矄


盐鼀
ᨨ
矄
 �矄
҈

  ᨨ
⼄盐 ᆠ泐
ᨨ
 
矄ᨨ盐ᆠ


㦨
�҈

 㡰 꾀 
⼄盐 ᆠ泐
ᨨ
 
矄�

泐
 
ᆠ
�

f¢ é v + vᨨ ù 㦨
f ê ú ë v - vs û 

㦨



鼀⏌矄
�� 㡰 � vᨨ
㦨
v矄
㦨

´
㦨
҈

鼀⏌矄 ҈
 é 

+ ҈
ù ᨨ

f ¢ =
ê ú =
´ ҈ »

� ├ ë 

- ҈
û b㡰 f҈㦨
⼄盐 ᆠ泐
ᨨ
 
ᨨ�ᆠ
ᆠ

泐
鼀ᨨᨨᆠ泐ᆠ�矄
f

㦨
⼄盐 ᆠ泐
ᨨ
 
矄�

泐
鼀ᨨᨨᆠ泐ᆠ�矄

- v 
+ v ´ ҈䰘
錐
f

㦨
´ ҈
�錐 f҈
㦨

-


Q f҈

f

㦨


\ v
㦨



鼀⏌矄

æ v + vö b㡰 矄� �
 
ᨨ鼀盐
f' = f ç A è v ø÷ v A + v �
� VB + V
v =

\
B =
=
 
� � v V v A

҈䰘




҈㡰

꾀 
⼄盐 ᆠ泐

泐
ᨨ矄
盐
ᨨ
矄ᨨ盐ᆠ
和


é  + 
ù é 
+ ҈
ù







v҈ = v ê ú = ҈

ê 
ú = ҈

´  = ҈



  ë û ë û



鼀矄҈ 䰘 和




ᆠᨨ矄
䰘

\ Ἐ㦨

҈҈


҈

҈

鼀⏌矄

� ҈

ᆠᨨ矄
�

WWW.JEEBOOKS.IN

EBD_8341 PS-164

Physics

꾀 
⼄盐 ᆠ泐

泐
ᨨ矄
盐
ᨨ
矄ᨨ盐ᆠ
和

æ v + v҈ ö f҈ = f
ç è v  v҈ ÷ø




é  + 
ù é 
+ ҈
ᆠᨨ矄 �° ù v
= v ê ú = ҈

ê ë 
- 
ᆠᨨ矄 䰘° úû ë  - 矄 û

҈
鼀

꾀 
⼄盐 ᆠ泐

�졠 盐
ᨨ
矄ᨨ盐ᆠ


B

A

q

鼀矄

q

q

v é ù nB = n ê úû + ҈ᆠᨨ矄 q v ë \

҈
ᆠᨨ矄


q






é

v

ù

é

v

ù


⼄盐 ᆠ泐

vb = n êë v -
ᆠᨨ矄 q úû - n êë v + ᆠᨨ矄 q úû

҈ ҈ ù é = n vê ë v -
ᆠᨨ矄 q v + ᆠᨨ矄 q úû ҈ ҈ é ù = ҈

´ 
ê � �ú 
+ ú ê 
-
´ ҈ ҈ û ë ҈ ҈ é ù = ҈

´ 
´ ҈ ê ë 
´ ҈ - ҈

´ ҈ + � úû ҈ ù é ҈ = ҈

´ 
´ ҈ ê » �  ë





黨� úû ҈黨



㡰

 -  l
 æ  ö  Þ = ç ÷= l҈ 
l҈ è  -  ø
l
l
l
- l҈ -

 - 

Þ -҈ = -҈ Þ =   l҈ l҈

ᆠ �
ᆠ  �
⼄盐�
�
 
 �
ᨨ
 
�

-
l
- l҈ ´҈

= ´҈

= -



�奠 l҈ 



ᆠ





ᨨ 㦨�鼀矄㜠҈

和ᨨ盐ᆠ 

㦨

黨




⼄盐 ᆠ泐
ᨨ
矄ᨨ盐 
 ᆠ
ᨨ鼀
 
犨

é  + 
ù é 

+ � ù
;





ú
Þ
vἘ
㦨




ê ë 

- � úû ë  - s û

nἘ
㦨
n
ê

矄盐 
ᨨ
鼀�盐

꾀


p p é ù = I
ê矄� O + 矄� + 矄�

+ 矄� p ú   ë û é  ù  = I
ê + ú =  I


ë û

\ IR
㦨
AR

㦨
I
\ n 㦨
 䰘㡰 �矄泐

ᆠ
犨�


矄

ᨨ矄
犨 �ᆠ
�矄 ᨨᆠ � �
 
矄ᨨ盐ᆠ


�
犨�


矄

矄ᨨ盐ᆠ

犨 �ᆠ �矄
鼀ᨨ� �
�
 
�ᆠ�ᨨ
ᨨ
矄ᨨ盐 
⼄盐 ᆠ泐
ᨨ
矄ᨨ盐 
 ᆠ
泐
 
ᆠ
ᔴ҈

矄㦨�鼀矄҈  
�鼀�
ᨨ
 
盐矄
ᨨ鼀
泐
 
犨
 矄
矄
矄ᨨ盐ᆠ㡰

é
v ù é
´
ù
黨
´
= u
ê c ú =
黨
ê ú = 

=
 ë 

û ë v  vc û


҈


 ᆠ

矄鼀矄
盐
ᨨ
ᆠ �
ᆠ  �
⼄盐�
�
 
 � ᨨ
�
�矄




�奠


ᨨ矄
 
矄ᨨ盐ᆠ

鼀ᨨ� �
ᨨ犨矄
ᆠ
ᨨ 


�鼀� ᨨ
 
矄ᨨ盐ᆠ
矄矄
矄
矄ᨨ盐ᆠ
ᨨ
 ᆠ
矄ᨨ盐 
 
�
 矄
矄

ᨨ矄㡰

vc
㦨


鼀矄҈

é v + vc  v + vc ù é v + vc  v + vc ù = u
ê ú = u
ê ú v  vc v  vc ë û ë û

㡰

ö  ÷=
l
ø

Þ

⼄盐 ᆠ泐
ᨨ
 ᆠ
矄ᨨ盐 
矄
ᨨ矄

 
矄ᨨ盐ᆠ






æ

  

è

æ v + vc ö v ÷ø






l҈

␔ 
盐 � �
ᨨ├
�矄
鼀ᨨ
Ἐ = ç

u҈
㦨
u

çè

\

⼄盐 ᆠ泐
㦨
v


v







ᨨ



㡰 
l҈
㦨
� ��
 �
ᨨ
� l

㦨
犨
 �
ᨨ
� 
㦨
和
ᨨ
盐 � �
ᨨ├ ᔼ
ᆠᨨ矄
ᨨ�
�

=

⼄盐 ᆠ泐
ᨨ矄

ᆠ

æ v ö æ v + vc ö u

㦨
u҈
ç ÷ = v
ç ÷









è v  vc ø è v  vc ø

ᔴ



( v҈  v
) æ ҈

ö \ç f
= f
è ҈

÷ø v v  ҈


´ ҈

= = ҈
黨
鼀矄 ҈ = 䰘├鼀 ҈ ᨨ
( v҈  v
) =







҈
鼀矄 ҈ B

�
鼀

�
鼀




v

v҈㡰
㦨
v

v
㡰
»
v 矄
v҈
 
v


泐
泐
矄矄
 
v


҈

鼀 和�ᨨ 泐㡰

҈

矄

æ v + v
ö f
= f
ç ÷ è v  v
ø é v + v҈ v + v
ù æ ҈

ö  ÷ø f
= f
ê ú \
f҈  f
= èç ҈

ë v  v҈ v  v
û
v ( v҈  v
) æ ҈

ö f ÷ø f
= ᨨ
çè ҈

( v  v҈ )( v  v
)


\

⼄盐 ᆠ泐

v
㦨



҈
҈҈
䰘



҈




㦨


҈
�
 b »

҈
 �


㡰 꾀 
⼄盐 ᆠ泐

졠
盐
ᨨ 졠矄⏌�矄  和�ᨨ 泐 矄ᨨ盐ᆠ


A

v é ù nA = n ê













ᆠᨨ矄
q ë v -
ᆠᨨ矄 q úû

҈

꾀 
⼄盐 ᆠ泐
ᨨ
矄ᨨ盐 
 ᆠ
泐
 
ᆠ
ᔴ


é 
+ ҈
´


ù é 
ù \ v
= ҈

ê = ҈

ê ú ú = ҈҈
䰘

 ë 
- 
´


û ë 

û

҈



ᔴ҈






\ ⼄盐 ᆠ泐
ᨨ
矄 㦨 nἘ

n
㦨









㦨



矄

�
 
矄ᨨ盐 
ᆠ 矄
 
ᨨ矄  
� �
 ᆠ
ᨨ鼀

矄�ᨨ 泐 矄盐 ᆠ
 
 
鼀�盐鼀
�矄
矄ᨨ 졠矄 ҈ 矄�ᨨ 泐

 
�鼀�
ᨨ
 
矄ᨨ盐ᆠ
犨� 和 和 ᆠᨨ鼀
 
矄ᨨ盐ᆠ
ᨨ
 ᆠ
矄ᨨ盐 


盐矄
�
ᨨ
 
ᆠ矄矄

ᨨ 
矄ᨨ盐  ᆠᨨ鼀� �
�ᆠ泐
ᨨ鼀
 
矄ᨨ盐ᆠ
   
ᨨ 
ᆠᨨ鼀� �
 
 ᆠ�ᨨ
犨� 
 
矄鼀
 
⼄盐 ᆠ泐
␔  ᆠ

ᨨ
矄
犨�




WWW.JEEBOOKS.IN

PS-165

Waves
�


矄

�  矄�泐
ᨨ
ᨨ盐 矄矄
ᨨ
矄ᨨ盐 


I
㦨


҈ r V w
A



æ v + vm ö f ἘἘ = ç ÷f è v  vb ø

꾀矄ᨨ
�  矄�泐
�矄
矄
�矄 ᆠ
ᨨ鼀
 
ᨨ� 
矄ᨨ盐ᆠ
矄

҈







䰘


 ᆠ

⼄盐 ᆠ泐

泐
鼀ᨨᨨ�矄


I
µ

r 꾀矄
ᨨ 
ᨨ
 
鼀矄

ᆠ  �� �
�
ᆠ矄
ᨨ

ᆠ
�� 
ᨨ 
ᆠ
泐

矄ᨨ
 
�  矄�泐
犨�
鼀�
 
矄鼀
꾀
® ⼄錐

® 錐
ᔴ
®  꾀㡰 �ᆠ ⼄
⼄盐 ᆠ泐 㡰 ⼄盐�泐 
犨 ᨨ鼀 ᔴ㡰 ᨨ盐 矄矄 
�  矄�泐 泐
ᨨἘ矄
ᨨ鼀盐

æ v + vm ö æ v + vm ö fb = f ἘἘ f Ἐ = ç ÷ f ç ÷f è v + vb ø è v  vb ø ᨨ

fb =


vb ( v + vm ) v
 vb


é  + 鼀 ± 
ù ú ë  +  鼀 ± 矄 û



和��
ᆠᨨ  �ᨨ
ᨨ
Vm
�矄
矄
ᨨᨨ犨矄

ᔼ
鼀�盐鼀
�矄
鼀ᨨ� �
ᨨ鼀
s
ᨨ
O
 
졔

 
�ᆠ
矄
和�鼀�泐
v

 
vs

ᨨ矄��
�
 矄

�ᆠ
ᨨ鼀
S
ᨨ
O  
�ᆠ
矄
㡰

 ᆠ 犨

鼀

nἘ
㦨
n
ê

B
= r

ᨨᆠ�泐
ᨨ
矄ᨨ盐 
�
犨
v
㦨







´ ҈
黨

졠
黨


和


犨 �矄
犨 �ᆠ
�矄
鼀�� �
矄ᨨ盐 
�矄
� �
ᨨ
�

ᆠ�ᆠ



ᨨ矄
�矄

�
�矄 ᆠ
ᨨ鼀
 
犨 �矄
�



졠 矄㦨w㦨
鼀⏌矄

҈


꾀

㦨
҈

�
鼀⏌矄

҈
鼀⏌矄

㦨 ҈
�鼀


鼀⏌矄

S   矄鼀�㡰

�  ␔

졠

O  ᆠ�㡰

w㦨

㜠҈

ᨨ㦨


⼄盐 ᆠ泐
ᨨ
矄ᨨ盐 
�
犨 v
㦨


v犨 ҈

� = = ҈
�   T犨 ҈


� ´҈


  㦨w 矄 㦨
鼀⏌矄

෌�

r
㦨
҈
�
鼀
 

w
㦨



矄҈ 和
ᨨ
矄ᨨ盐ᆠ

vs
㦨
rw 㦨
҈
�
맔



㦨


鼀矄҈ ␔ 
 
矄ᨨ盐ᆠ
�矄

 
ᨨ矄��ᨨ
A

 
 
⼄盐 ᆠ泐
 泐
 
ᨨ矄
犨�

鼀 �鼀盐鼀

⼄盐 ᆠ泐
�矄
�   
ᨨ
鼀�盐鼀
\
v犨
㦨
v� 鼀
㦨
졔


鼀⏌矄錐



㦨

錐

矄
㦨
҈

鼀⏌矄

é v + vm  v
ù ú ë v + vm  vs û

꾀泐� �
ᨨἘ矄
ᨨ鼀盐

vἘ
㦨
v ê \

(

)

 ù =


é 
ù =


 êë 
- 
úû ú

ë  - s û

é

nἘ
㦨
n ê

� é ҈

� +
-
ù \
nἘ
㦨
҈


䰘
맔
҈
�
 nἘ
㦨
҈ ´ ҈
ê ë҈

� +
- ҈
úû


␔ 
 
矄ᨨ盐ᆠ
�矄

 
ᨨ矄��ᨨ
B

 
 
⼄盐 ᆠ泐
 泐
 
ᨨ矄
犨�

鼀� �鼀盐鼀

b㡰 ᔼ
�

ᨨᆠ�泐
ᨨ
矄ᨨ盐 

g RT
㦨
㦨
M

é 
ù  ù ú

=


ê 
+ 
ú =


 ë û ë  + 矄 û é

(҈

) ´ (

҈) ´
黨 = 

鼀⏌矄

nἘἘ
㦨
n ê





´ ҈


 ᆠ
 
 �
ᨨ
⼄盐 ᆠ�矄

泐
 
ᨨ矄 㦨






ᨨ







vm
㦨
�鼀矄㜠҈ S





O ҈
鼀⏌矄

= ҈
 ⏌ 矄

 ᆠ�㡰

æ v  v m  v
ö ÷ è v  vm  vs ø

꾀泐� �
ᨨἘ矄
ᨨ鼀盐
vἘ
㦨
v ç \




(

�

nἘ
㦨
҈ ´ ҈



 �盐
⼄盐 ᆠ泐
ᨨ
 
ᆠᨨ

w
㦨

pn =
p ´



- � -
ù
㦨
҈


맔
҈

 ) éêë 

- � - ҈
úû

� p


 �盐
⼄盐 ᆠ泐
ᨨ
 
ᆠᨨ
 
 
矄ᨨ盐ᆠ
ᨨ
矄ᨨ盐  
⼄盐
A w㦨҈
⏌矄 ҈

�


鼀ᨨᨨ�矄
矄
 

⼄盐 ᆠ泐
矄

ᆠ�矄
犨ᨨ
�   ⼄盐 ᆠ�矄
ᨨ 
�ᆠ泐
ᨨ鼀
 
矄ᨨ盐 
矄ᨨ盐ᆠ
ᨨ
 

f'
  ᨨ 
 ᆠ
ᨨ鼀
 
犨
꾀 
⼄盐 ᆠ泐
犨 
�ᆠ
矄ᨨ盐 
ᨨ
 
�矄

泐
  鼀ᨨᨨ�矄

æ v + vm ö f Ἐ= ç ÷f è v + vb ø 꾀��
 
⼄盐 ᆠ泐
犨 

ᨨ
�ᆠ
 
 ᆠ 矄ᨨ盐 
泐
 
鼀ᨨᨨ�矄


鼀

A
A

B


鼀 C

D

A

Þ ␔ 
 
ᆠᨨ
�矄

C
鼀ᨨ� �
ᨨ犨矄
D

 
矄ᨨ盐ᆠ
�矄 
A҈
鼀ᨨ� �
 犨矄

ᔼ
�矄
�
 �矄
矄�盐�ᨨ
 
 
⼄盐 ᆠ泐 
�矄
鼀� �鼀盐鼀

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EBD_8341 PS-166

Physics

é  - 
ù 

-

㡰 =
�䰘
  n鼀�
㦨
n
ê ú = 

´ + + 
㡰   

矄 ë û Q


㦨
꾀w
㦨


맔
҈

㦨



鼀⏌矄 矄
㦨
w
㦨

맔
҈

㦨


鼀⏌矄㡰 꾀��
犨 
 
ᆠᨨ
�矄

C
鼀ᨨ� �
ᨨ犨矄
B

 
矄ᨨ盐ᆠ
�矄 
A
鼀ᨨ� �
�� 犨

ᔼ
�矄
�
 �矄
矄�盐�ᨨ
 
 
⼄盐 ᆠ泐 
�矄
鼀 �鼀盐鼀


v

㦨
�
鼀⏌矄

é  + 
ù 

+

㡰 =


䰘  ú = 

´   

- 
㡰 ë 矄û

矄

犨

é c - 
ù nἘ
㦨
n ê ú ë c - s û

矄




s

졔

�
 泐

�ᆠ
ᨨ鼀
矄ᨨ盐ᆠ
ᨨ
 
ᨨ矄
  

�
 泐

�ᆠ
ᨨ鼀
ᨨ矄
ᨨ
矄ᨨ盐ᆠ


é 
- -�㡰 ù ú =


 ë 
- � û

nἘ
㦨

�
ê




҈ 졠矄 ᨨ
ᨨ├
҈
鼀ᨨ� �
ᨨ犨矄
ᨨ矄


\
⼄盐 ᆠ泐
㦨 vἘ

v
㦨






�

㦨



v f Ἐ= f ( v  vs )
犨 
vs
㦨
ᨨᆠ�泐
ᨨ
ᨨ├


Topic-5 : Miscellaneous (Mixed Concepts) Problems

ᨨ
ᨨ├


鼀ᨨ� �
犨泐
ᨨ鼀
ᨨ矄


f ἘἘ =

v f v + vs

犨 
v
㦨
ᨨᆠ�泐
ᨨ
矄ᨨ盐 


҈


 =
l



n =

g

㝐 ´ ҈


-

´

҈ Þ
v
㦨
vl
㦨
n
맔

l
l

v



㡰
맔



맔
l
㦨


䰘
鼀⏌矄
ᨨ



�
鼀⏌矄 ᲼\
l
㦨


�

졔




�桌 ᨨ
鼀ᨨ ᨨ鼀�ᆠ
�矄
g
㦨҈

䰘

vf vf é ( v + vs )  ( v  vs ) ù 
ᨨ
 = vf ê ú v  vs v + vs êë ( v  vs )( v + vs ) úû vf ´
vs vf ´
vs ᨨ
 = ᨨ
 = 矄
vs



v

v
v v s v
v  ´ 

v = = = ҈
�鼀⏌矄 ᨨ
vs = ᨨ
s
vf
f
´ 



ᨨ
 =

\
 =

g

-

㝐 ´ ҈


= ҈

g ´

҈
㝐

҈
҈
=


㝐 㝐 䰘 ᨨ
Nᨨ
㝐
㦨



\
 =


´
㦨




鼀矄҈ ҈
҈䰘 㦨



鼀矄҈ ᨨ
꾀�ᨨ
㝐
㦨


\

㦨



´ 





㦨
҈
䰘 ´

\
和
ᨨ
盐 � �
ᨨ├
㦨
҈
�
鼀⏌矄



ᆠ鼀㦨l


ᨨ
�ᨨ鼀�ᆠ
�矄
g
=
҈



l⏌


l҈㦨
�ᆠ鼀 l

㡰 Þ

\

⼄盐 ᆠ泐
㦨
f'  f''
㦨






ᔼ 和ᨨ盐ᆠ㡰

和 졠矄㡰

n鼀
㦨
n ê ҈


v

㦨
�
鼀⏌矄

 = ҈









҈
҈
= �黨
´ 㝐 㝐 黨 㦨
҈

䰘
鼀矄҈ ҈
 \
 = �黨
´ 㦨
�

鼀矄҈ �

ᨨ
졠 y�
M
㦨


\
 = �黨
´ 泐
ᆠ矄� �
 
 矄�ᨨ
�
 
矄� �

⼄盐 ᆠ泐
�矄
ᆠ矄� �
\
�矄
ᨨᨨ 
⼄盐 ᆠ泐
ᨨ
矄� �

盐 鼀 
⼄盐 ᆠ泐
ᨨ ᆠᨨ矄
�
㦨



æ v҈ ö æ v
ö é v
ù ÷ç ÷ =
ᨨ v҈ = l҈ ê
+ ú \


ç
l
û è
l҈ ø è
l
ø ë 和盐矄�盐� �
 
盐

犨




ù é = �
m ⏌ s v҈ =


� ê
+ ë
´


úû ᨨ犨

v҈ =




T \ T = mv҈
m

 æ m ö
æ

� ´҈
ö ÷ �
㡰
=
䰘


 ᨨ

T
= ç ÷ v҈ = ç ç


� ÷ èl ø è ø

ᨨ
N�ᨨ�
M
㦨






b㡰 
x

 
 
ᆠᨨᆠ�ᨨ


l


ᨨ


l =
l҈ + x㡰
l
l
+ x 㡰 =

ᨨ

l = l
+ x㡰
    

= = \ n҈ 㦨


\


n

㦨
l҈
l҈ + x 㡰 l

l
+ x 㡰 l҈ + x =



҈鼀㦨l҈

l҈ +x



�鼀㦨l


l
+x

ᔼ
 
矄ᨨ盐 
ᆠ 矄
 
ᨨ矄
 
� �
 ᆠ
ᨨ鼀
 矄�ᨨ 泐
矄盐 ᆠ
 
 
鼀�盐鼀
�矄
矄ᨨ
矄�ᨨ 泐

 
�鼀� ᨨ
 
矄ᨨ盐ᆠ
ᔼ㡰
�
 
 ᆠ� �
矄盐 ᆠ
犨�
ᆠᨨ鼀
 
矄ᨨ盐ᆠ ᨨ
 
 ᆠ
矄ᨨ盐 


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PS-167

Waves ෌�

n҈
㦨
n
\ 






�


  = ᨨ

l҈ + x 㡰
l
+ x㡰

\
x
㦨




�
鼀

v dn æ -҈
v ö 
Vc 矄� a = - c 矄� a ´ ç
÷= dt v è x 矄ᆠ
a ø x
矄ᆠ
a 
vc 矄� a dn =

vc 矄�  a

꾀
a
㦨
黨
\
dt = ҈
ᆠᨨ a 㡰
矄ᆠ
a

ᆠ㡰 꾀矄ᨨ
w
㦨

pn
Þ
w µ
n E �鼀 
=
҈


꾀鼀�盐
㦨
A
 
n҈
㦨
n E �鼀 
=




꾀鼀�盐
㦨
A
 
n

㦨

n

E
A
n

= =


\ E

㦨

E҈ E҈ A
n҈


dn = max dt

b㡰 
矄
ᨨ
ᆠ
盐矄
�矄

ᆠ鼀⏌矄
 
� ��泐
犨ᨨ
盐矄矄 


ᆠ鼀

 
鼀ᨨ� �
ᨨ犨矄
ᆠ
ᨨ 






꾀 
犨ᨨ
矄ᆠᨨ 矄
盐矄矄
犨�
ᨨ
ᆠ
ᨨ 
꾀ᆠᆠᨨ� �
ᨨ
矄盐ᨨ矄��ᨨ
� ᆠ�
 
矄� �
犨�
ᨨ
  泐
�矄ᨨ�ᨨ
 
犨�

矄�� 
 ᆠ
 
犨�

ᨨ

쟼


ᨨ
 �泐
犨�

ᨨ 泐
├� �ᆠ


黨


ᨨᆠ�泐
ᨨ
矄ᨨ盐 
泐

矄ᆠ 
矄� �

㦨


矄ᨨ  ᆠ
�矄
矄ᨨ鼀犨 
矄 ᨨ
 


 +e=

和
ᨨ
矄ᨨ盐 
犨矄
�
犨
�矄
�
 
�
�

矄ᨨ
犨  
矄

鼀�盐鼀
 
 ᆠ
矄ᨨ盐 
犨
犨�

 ᆠ � ᨨ
犨
犨泐
ᨨ鼀
 
ᨨ鼀


b

ᆠ㡰

é v + vc ᆠᨨ矄 q ù é v + vc ᆠᨨ矄 q ù n P = V  V鼀 㡰 ê ú = ҈
҈ - ҈҈
êë úû v v ë û n Q = ( vN  vM ) = ҈
҈ - ҈҈
=  é v + vc ᆠᨨ矄 q ù é v - vc ᆠᨨ矄 q ù n R = ( vN  vM ) ê ú = ҈
҈  ҈҈
㡰 êë úû v v ë û \
n P + nR =
nQ  vᆠᆠᨨ矄
q q vᆠ 꾀 a Vx ᆠ ҈
҈ ҈҈
 ҈
턼  㝐

ᔼ
� 
犨 
 
ᆠ
�矄
矄矄� �
 ᨨ盐�
꾀

é v + vc ᆠᨨ矄 a ù n = ê úû v ë \


…�㡰

dn é v 矄� a ù d n = - ê c


�矄
鼀
犨 
矄�
a
㦨
҈ da ë v úû d a

�





a
㦨黨


턼㡰

d n vc da =  矄� a㡰 …
��㡰 dt v dt ҈
dx ҈

da =-
꾀矄ᨨ

 a =
\
矄ᆠ a dt x x dt

ᨨ鼀
⼄

�㡰


҈ 


ᨨ
 
犨 �
ᨨ
 
矄ᨨ盐 
�
�
盐
ᨨ  
ᆠᨨᆠ�ᨨ
㡰


TἘ ҈
�x T  = \





Ἐ
㦨
v 㦨
v
㦨
҈



 T x Ἐ TἘ

䰘


b


㡰 ␔ 

矄ᨨ盐 
盐矄
�矄
 ᆠ
ᨨ鼀
ᨨ
 
ᨨ

�


 矄 ᆠ  �矄
泐
҈



꾀
��
矄矄盐
盐矄
�



ᆠᨨ鼀矄矄�ᨨ
�矄
 ᆠ 矄

ᨨ犨
矄矄盐
盐矄
�



 ᆠ�ᨨ
␔ 
矄ᨨ盐 
盐矄
�矄
 ᆠ
 ᨨ盐�

���
ᨨ盐 泐
ᆠᨨ矄  
ᨨ

�㡰

ᨨ
 矄
ᆠ  �
ᨨᆠᆠ盐矄
矄ᨨ

��
矄矄盐
盐矄 �矄
 ᆠ
矄

��
矄矄盐
盐矄
e 

㡰


 �
ᨨ
 
�
ᆠᨨ盐鼀

 
�矄

T m

ᨨ鼀
ᨨᨨ├Ἐ矄
犨

 矄�ᨨ
�
矄� �
T
µ   矄�ᨨ
x





…���㡰

ᨨ鼀
⼄

��㡰


���㡰

쟼 �泐
E
µ
A
n

犨 
A
㦨
鼀�盐
 
n
㦨
⼄盐 ᆠ泐


㡰

da -҈
v =
dt x 矄ᆠ
a

҈



҈҈


҈



l


Þ

 =

l -e


 ᆠ

矄ᆠᨨ 
矄ᨨ  ᆠ
 
 �
ᨨ
�
ᆠᨨ盐鼀
�矄
鼀ᨨ
矄 ᆠᨨ鼀
ᨨ
�矄
矄ᨨ  ᆠ

ᨨ犨

ᨨ �
 
 �
ᨨ
�
ᆠᨨ盐鼀
鼀ᨨ
�矄
 
矄ᨨ�ᨨ
ᨨ
 �泐
 
矄矄
�矄
 
�  矄�泐
ᨨ 矄ᨨ盐 

b㡰 
矄
ᨨ
矄ᨨ盐 
 矄
ᨨ
 
鼀
ᨨ
  ᆠ
ᨨ  
ᨨ矄
和꾀
㦨



졔



㦨



鼀⏌矄  
和
㦨






㦨
҈

鼀⏌矄 㡰 
�矄
ᨨ
�
鼀ᨨ�ᨨ
犨
矄ᨨ盐ᆠ
 
ᨨ矄
ᨨ  
矄矄 �矄
�
�
꾀

和� ᆠ

 
矄矄 �矄
�
�
A
 鼀ᨨ� �
犨�

ᨨᆠ�泐
ᨨ



鼀⏌矄
  ᨨ
 
�矄�盐�ᨨ
ᨨ
矄ᨨ盐  �  矄�泐
ᨨ
 
犨 �矄
泐
 
矄矄 �矄
�
�
A
�矄
盐 � ᨨ鼀
㡰 ᨨ
 
矄矄 �矄
�
�


矄ᨨ盐ᆠ
�矄
ᨨᆠ � �
犨� ᨨᆠ�泐



鼀⏌矄
 
ᨨ矄
�矄
ᆠ� �
犨�
ᨨᆠ�泐


鼀⏌矄

é v - v
ù ҈ é 

- 
ù n Ἐ = n҈ ê ú
=


ê 

-

ú =


´ 
v v ë û ë sû é v - v
ù ҈ n ἘἘ = n
ê ú = ҈҈

´ 
ë v - vs û \

n ἘἘ- n Ἐ 㦨
҈҈






㡰
맔


҈ ҈ = 

´ = ҈





҈



㡰 ҈




ᆠ㡰 ҈�



㡰 
��
⼄盐�ᨨ 矄
y҈
㦨
A
ᆠᨨ矄


�
px

҈


pt㡰
  y

㦨
A
ᆠᨨ矄





px

黨

pt㡰
矄 矄
犨ᨨ
ᨨ�矄矄�
犨 � �
�
 
矄鼀
�ᆠ�ᨨ
ᨨ �
x㜠 �矄
犨�
矄�� 
�  ᆠ �
 
⼄盐 ᆠ泐
ᔴᨨ鼀� �
�
犨�
 
矄 
⼄盐�ᨨ

y
㦨
A
ᆠᨨ矄
kx

wt㡰

犨
� w҈
㦨
҈


p
Þ
pf҈
㦨
҈


p
Þ f҈
㦨
�


  K҈
㦨


�
p
Þ


p =

�p
Þ l҈
㦨


鼀 l҈

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EBD_8341 PS-168

Physics

␔
ᨨᆠ�泐
㦨
l҈f҈
㦨




鼀⏌矄

᲼꾀 �泐
盐矄
v
㦨
w

㦨
黨

p Þ
pf

㦨
黨

p
Þ f

㦨





 ᨨ

⼄盐 ᆠ泐
㦨
f҈

f

㦨



 

w 桌 K





p =



p Þ l
㦨
K

㦨





p
Þ

l



␔
ᨨᆠ�泐
㦨
맔


㦨




鼀⏌矄



v
㦨


ᨨ


é

ì  ax + bt 㡰 + ax - bt 㡰 ü ý
î þ ë ì  ax + bt 㡰 - ax - bt 㡰 ü ù 矄� í ý ú




A
ᆠᨨ矄
ax
졔
bt㡰桌
î þû Þ y
㦨

҈


A
矄�
ax
矄�
bt
졔




A
ᆠᨨ矄
ax
졔
bt㡰












=



A ê -

矄� í

�鼀
� 
犨
犨ᨨ
矄盐ᆠᆠ矄矄�
鼀 �鼀矄
�矄

p
맔
҈

矄ᆠ ��㡰 
ᆠᨨ
ᆠ
ᆠ
�
矄盐 
�  矄�泐
>

A
\ 矄盐 
鼀�盐
>

A

犨 

҈


A
矄�
ax
矄�
bt㡰
�矄
 
⼄盐�ᨨ
ᨨ

矄 � �
犨
 



A
ᆠᨨ矄
ax
졔
bt㡰
�矄
 
⼄盐�ᨨ
ᨨ
� �
犨



犨
�矄
� �
�
x
�ᆠ�ᨨ
꾀 � ᨨ矄

 
ᨨ� 矄
犨 
 
鼀�盐
�矄
鼀 �鼀盐鼀



A ᆠᨨ矄 p  n҈ - n
㡰 t ³
A ҈ ᨨ
ᆠᨨ矄 p  n҈ - n
㡰 t ³ ᨨ

҈ é  w - w
㡰t ù ᆠᨨ矄 ê ҈ ú³

ë û ᨨ


é w҈ - w
㡰t ù


ᆠᨨ
�矄
�

犨 
 
盐矄
ᨨ
ᆠᨨ矄 ê ú
�矄
ë û ҈ 犨


 


\


p p w҈ - w
㡰t
�矄
犨

 





dy 㦨

Ab
矄�
ax
졔
bt㡰





Ab
矄�
ax 
bt㡰 dt




㦨

Ab
᲼矄�
ax
졔
bt㡰
졔




矄�
ax

bt㡰桌


㦨

Ab
᲼矄�
axᆠᨨ矄
bt
졔
ᆠᨨ矄
ax
矄�
bt 졔




矄�
ax
ᆠᨨ矄
bt





ᆠᨨ矄
ax
矄�
bt桌 v
㦨

Ab
᲼҈


矄�
ax
ᆠᨨ矄
bt
졔




ᆠᨨ矄
ax
矄�
bt桌


鼀 �鼀盐鼀
ᨨᆠ�泐
犨�
ᨨᆠᆠ盐
犨 
矄�
ax
㦨
҈
  ᆠᨨ矄
bt
㦨
҈
盐 
 矄
ᆠᨨ ��ᨨ
ᆠᨨ矄
ax
㦨


 
矄�
bt
㦨

\ 
v鼀

㦨
҈


Ab 꾀矄ᨨ


v鼀�

㦨

㡰 y
㦨
᲼A
ᆠᨨ矄
ax
졔
bt㡰桌

᲼



A
ᆠᨨ矄
ax

bt㡰桌


㦨
᲼



A
ᆠᨨ矄
ax
졔
bt㡰
졔




A
ᆠᨨ矄
ax
졔
bt㡰桌

᲼



A
ᆠᨨ矄
ax

bt㡰桌












㦨
᲼



A
ᆠᨨ矄
ax
졔
bt㡰





A
ᆠᨨ矄
ax 
bt㡰桌
졔
᲼



A
ᆠᨨ矄
ax
졔
bt㡰桌

w҈ - w
㡰
np t = np


ᨨ
t
㦨

w҈ - w



p
p
np
n - ҈㡰p 矄 =

ᨨ

w҈ - w
w҈ - w
w҈ - w
҈


b
p

b㡰 
犨
�矄
 ᆠ
泐

ᨨ矄ᆠ

�
犨�
矄盐 

 矄 �  ᆠ
ᨨ
p


�  矄�泐
ᨨ
 
 ᆠ
犨
�矄





�鼀矄 ᨨ
 
� ᆠ� 
犨
ᔼ  矄�泐
ᨨ
ᨨ��� 
犨
I
µ
꾀
ᔼ  矄�泐
ᨨ
 ᆠ
犨
I Ἐ
㦨





I Þ I Ἐ
µ
AἘ


Þ




I
µ
AἘ

Þ




A

µ
AἘ


Þ 꾀Ἐ
µ



A 和ᨨ
 
⼄盐�ᨨ
ᨨ
 ᆠ
犨 y
㦨



A
ᆠᨨ矄
ax 
bt
졔
p㡰
㦨





A
ᆠᨨ矄
ax

bt㡰 ᆠ㡰 
矄盐 
犨
⼄盐�ᨨ

y
㦨
y�
졔
y



㦨
A
ᆠᨨ矄
ax
졔
bt㡰
졔
᲼




A
ᆠᨨ矄
ax

bt㡰桌 �ᆠ
ᨨᆠ�泐

㦨

A
矄�
p
n҈
졔
n
㡰
t
ᆠᨨ矄
p
n҈

n
㡰
t 犨 
 
鼀�盐
AἘ
㦨

A
ᆠᨨ矄
p
n҈

n
㡰
t ᨨ犨

�  矄�泐
µ
꾀鼀�盐㡰
Þ I
µ
AἘ
I
µ

A

ᆠᨨ矄
p
n҈

n
㡰
t ᔼ  矄�泐
犨�

鼀 �鼀盐鼀
犨 

ᆠᨨ矄
p
n҈

n
㡰
t
㦨
҈ ᨨ
ᆠᨨ矄
p
v҈

v
㡰
t
㦨
҈ ᨨ
p
n҈

n
㡰
t
㦨
np

�鼀
� 
犨
犨ᨨ
矄盐ᆠᆠ矄矄�
鼀 �鼀

㡰 ᔴᨨ鼀
 
��
⼄盐�ᨨ
y҈
㦨
A
ᆠᨨ矄
ax
졔
bt㡰
犨�
  矄 
⼄盐�ᨨ
ᨨ

 
ᨨ�矄矄�
犨


\
⼄盐 ᆠ泐
ᨨ
� ᆠ� 
犨
n
㦨



pn҈ +
pn
㡰t 
pn҈ +
pn
㡰t ᆠᨨ矄



\

p p p p -

㦨

㦨
´ ҈
矄
w҈ - w

w҈ - w
㡰
w҈ - w
㡰

꾀矄ᨨ


pn
㦨
b

\
ᔼ
҈
矄ᆠᨨ 

y҈
졔
y

㦨



x
㦨


ᨨ
黨

�鼀矄 
 
犨ᨨ
�ᨨ
犨矄

矄 
泐
 
⼄盐�ᨨ 矄 y҈
㦨
A
矄�

pn҈t y

㦨
A
矄�

pn
t 和� ᆠ

A҈
㦨
A

㦨
A  
ᆠᨨ
�矄

x
㦨




⼄盐�ᨨ
ᨨ
矄盐 
犨
ᆠᆠᨨ� �
ᨨ
矄盐ᨨ矄��ᨨ

� ᆠ� y
㦨
y҈
졔
y

㦨
A
矄�

pn҈t
졔
A
矄�

pn
t 㦨
A
᲼矄�


pn҈t
졔
矄�

pn
t桌

Þ


�鼀
�

㦨
t҈

t



p
p æ
p ö = a


Þ l
㦨
x +
pn t÷ Þ l a è l ø


n + ҈ 㦨
ᆠᨨ矄
᲼
n
졔
҈㡰p 
黨

pt Þ t
㦨
҈黨
҈ ҈黨҈ = 黨�

犨 
t
㦨



n
㦨
-
 
犨 
t
㦨
҈


n
㦨




㦨
A
맔


矄�


\

y
㦨
A
ᆠᨨ矄
ç

꾀
x
㦨


y҈
졔
y

㦨
A
ᆠᨨ矄
҈

pt㡰
졔
A
ᆠᨨ矄
黨

pt㡰
㦨

Þ ᆠᨨ矄
҈


pt
㦨

ᆠᨨ矄
黨

pt
㦨
ᆠᨨ矄

黨

pt㡰

҈



t҈
㦨



㦨
҈䰘


p p
 
t

㦨
w҈ - w

w҈ - w
㡰

\

é

n

pù

�


矄�
ax
㦨
҈
㦨
矄�
ê np + -҈㡰
úû ë

é ë

ᨨ

ax = ê np + ( ҈) ҈



n

pù
úû

ᨨ


é

x
㦨
ê n +

êë

 -҈㡰n ù p ú
úû a

�㡰


⼄盐 ᆠ泐
ᨨ
 
犨 �矄

泐
ᨨ矄
ᨨ
 
�

é

v + vw ù é ҈


+

ù = �黨黨  ú
㦨
�


ê v v v + ë҈


+

-

úû ë w sû

nἘ
㦨
n ê

��㡰 
ᆠ ᨨ
ᨨ鼀
 
�
�矄

泐
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�

B
犨 �ᆠ
�矄
 
�矄 ᆠ
x
ᨨ鼀
 
�
矄ᨨ盐 
ᨨ盐ᆠ
泐
�


�矄 ᆠ
҈

WWW.JEEBOOKS.IN

PS-169

Waves ├鼀
ᨨ鼀
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
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ᆠ
ᨨ鼀
A
ᨨ
B

k  x - y㡰
n + ҈ = p


n + ҈㡰 p 犨 

㦨



졔
҈

졔


ᆠ
x

y㡰
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x 꾀

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҈ x + t

㦨
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+

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n

㦨


é v - vw 㡰 + vs ù ú ë v - vw 㡰 - v
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-

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ê ú ë ҈


-

㡰 -

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nἘἘ
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Al r ù T é ê 犨  m = 鼀矄矄  盐 �
 � = l = Ar ú M ë û YAaDq Y aDq =
= Ar r

v=


㦨


҈


\
v
=




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wt ù ᆠᨨ矄 ê úû

ë


矄盐 
�  矄�泐
犨�

ᨨ
犨 

k  x - y㡰 k  x - y㡰 =



Þ


ᆠᨨ矄 =




黨 ´ ҈



҈ 鼀ᨨ �ᆠ


㦨

䰘

鼀⏌矄



鼀ᨨ �ᆠ





n + ҈ 
n + ҈㡰 p p


Þ
x
㦨
犨 
n
㦨



҈









k

Þ z
㦨

A
ᆠᨨ矄


҈
 ´ ҈
҈҈ ´ ҈
䰘 ´ ҈
-� ´



꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

 
� �鼀矄
A
 
B

矄
� ᨨ ��ᨨ 矄
ᨨ
矄鼀
⼄盐 ᆠ泐

� �鼀
C
�矄

ᨨ

和ᨨ


A  
B
 � ᨨ矄

ᨨ鼀



n + ҈㡰 p


��㡰 
 矄矄
犨
z҈
� �
�
졔
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 
z
� � �

졔
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㦨
A
ᆠᨨ矄
k x

wt㡰 z
㦨
A
ᆠᨨ矄
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犨
z
㦨
z҈
졔
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㦨
A
ᆠᨨ矄
k x

wt㡰
졔
A
ᆠᨨ矄
ky

wt㡰


A ᆠᨨ矄

҈
´ ҈
҈҈ ´ ҈
-
´ ҈

҈ ´ ҈
-� ´


㦨
҈҈

´҈

҈ F A Dl  
Dl
㦨
l a Dq 㡰 矄� �
y
㦨
l \ F = YAaDq 和
ᨨ
 矄矄
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犨
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矄 � �
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 ᆠ
ᨨᨨ犨� �
犨ᨨ
⼄盐�ᨨ 矄
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ᨨ盐ᆠ
矄 � �
犨 z҈
㦨
A
ᆠᨨ矄
k x

wt㡰 z

㦨
A
ᆠᨨ矄
k x
졔
wt㡰


矄盐 
犨
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泐
z
㦨
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졔
z
Þ z
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A
ᆠᨨ矄
k x

wt㡰
졔
A
ᆠᨨ矄
k x
졔
wt㡰 㦨

A
ᆠᨨ矄
k x
ᆠᨨ矄
wt


矄盐 
�  矄�泐
犨�

ᨨ
犨 


A
ᆠᨨ矄
k x
㦨



Þ ᆠᨨ矄
k x
㦨
ᆠᨨ矄


҈ T ҈ YA a Dq =
l m
l m





㦨


҈- x ҈ x = +

҈


+

҈


-



Þ x
㦨


黨�
├鼀


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
泐
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�

Y
㦨


⼄盐 ᆠ泐
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ᨨ
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tAH
㦨
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├
泐
矄ᨨ盐 
ᨨ鼀
A
ᨨ
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犨�
ᨨᆠ�泐 ҈



졔


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㦨
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泐
矄ᨨ盐 
ᨨ鼀
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ᨨ
B
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҈






ᨨ鼀
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�㡰
 
��㡰 t҈
㦨
t


Þ

矄矄矄 꾀




\
T
㦨
YA
a Dq = = 矄� Dl l a꾀Dq Dl = a Dq 㡰 (Q
Dl = la Dq Þ l 矄� �


졠
ᨨ ᔴ

꾀 � ᨨ 꾀

�鼀

v҈ v ´ p҈ =
´ p

Þ

l
l

꾀 � ᨨ 

�鼀

p҈ v҈  = = ᨨ

҈҈p҈
㦨
p
p
v
҈҈

�




 �
鼀ᨨ �ᆠ
�
AC
�矄
⼄盐
ᨨ
҈҈
鼀ᨨ �ᆠ
�
CB
ᨨ犨

 
盐 鼀 
⼄盐 ᆠ泐
�
AC 㦨


v҈ ҈҈

= = ��

l
´

�

 
 
盐 鼀 
⼄盐 ᆠ泐
�
CB 㦨


v


= = ��

l
´

�

\
⼄盐 ᆠ泐
�
AC
㦨

맔
��

㦨
҈
�

  
⼄盐 ᆠ泐
�
CB
㦨
҈҈
맔
҈�

㦨
҈
�



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EBD_8341 PS-170

Physics

Electric Charges and Fields

15

Þ
F 
㦨

F
ᆠᨨ矄q

Topic-1 : Electric Charges and Coulomb's Law

Fnet =



҈


æ y
+ a
ö ç ÷ è ø q æ ö
kq ç ÷ y
Fnet =
è
ø ⏌
y + a 㡰



㡰  졔턼

⼄

⏌


 =

ᨨᆠ
盐
ᨨ
ᆠ �
⼄


졔턼

⏌


 =

ᨨᆠ
盐
ᨨ
ᆠ �
졔
턼



⧤턼턼


⧤턼⼄

æö ç ÷ è
ø

Þ








ᨨ
⼄盐���盐鼀


r r  +  =




b㡰

Þ

├턼턼

├턼⼄

+


㦨


(  ⏌
)
꾀 턼 = ò r = ò
 
⏌ (
p
 ) 


æ ç  
⏌ =
p꾀 ò   =
p 꾀 ç
ç 
è  턼
㦨

p꾀҈
⏌㡰


⏌








\⼄ = -

㡰

q

a

a

Þ F
矄�
q

q

x

y + a


Q
y



a㡰






和ᨨ

F
µ
y

㡰 q l

 ⼄
= 鼀�
p Î

´ 鼀�

矄�
q ⼄ ⏌
矄ᨨ
 q » 矄� = l ⼄

⼄
l  Þ =  ᆠ


l =
p Î
鼀�
p Î

´ 鼀�

q

ᆠᨨ矄
q

⼄



鼀�

҈⏌

䰘






æ ⼄
l ö ÷÷ \
= çç è
p Î
鼀� ø ҈⏌

 ᨨ

µ
l


ᆠ �
㦨

㡰 ␔ 

ᨨ矄��
ᨨ� 
ᆠ �
�矄 ᆠ
ᨨ盐矄�

ᆠᨨ 盐ᆠ� �
矄 

 +  �鼀 
ᨨ
ᆠ �
├矄
ᆠ
ᨨ

+ +  
矄盐 ᆠ
矄
矄 ᨨ犨
�
 
��盐

盐 +


ᆠ �
ᨨ
 
矄 
�矄
ᨨ
矄
ᨨ 졔⼄ + ᆠ �
矄
 
ᨨ
 
 
矄 
b㡰 ᔴ � q
矄
ᨨ

��
ᨨ
⼄盐���盐鼀

҈ ҈
҈ 졔턼
҈ ⼄ l


 

졔턼 l

ᔴᨨ 矄�� �
⼄盐���盐鼀
ᨨ

ᆠ �
q
졔
҈ F҈
㦨
F҈


KQ ´ Q 
l㡰 黨





=

KQ ( - q ) l






\
q
㦨 -

Q


æ ҈ q
ö çç ÷
÷ è
p Î
L ø ᨨᆠ
ᨨ
⼄㡰



盐
ᨨ
ᆠ �


犨� �
ᆠ ᆠ
ᨨ盐
泐
ᨨᆠ
ᨨ
⼄㡰
盐
ᨨ ᆠ  �
ᨨ
꾀

和�鼀�泐
ᨨᆠ
ᨨ
⼄
盐 ᨨ
ᆠ �

쟼
犨�
�
ᆠ ᆠ
ᨨ盐
盐
ᨨ ᆠ �
ᨨ

和ᨨ
 

ᨨᆠ
犨�

ᆠ盐矄
ᨨ
ᆠ �
⼄
ᨨ
ᔴ
ᨨ 泐

�


F
矄�
q

F ᆠᨨ矄
q

y


 q =

ö ÷ æ  ö 
⏌  ҈) ÷ =
p꾀 ç 
÷ (  è ø ÷ ø


F

a

×

鼀� = ᆠᨨ矄 q



ᆠ㡰 ᔼ 矄�
 
ᆠ�泐

ᆠ �
�矄
ᨨ
\ Q҈
㦨


 
s1 㦨




�矄
ᨨ
 ᆠ
ᨨ
ᨨ� 
ᆠ �矄
졔Q

Q
 
� 盐ᆠ
ᆠ � ᨨ
�


矄盐 ᆠ
ᨨ
 
ᨨ盐
矄盐 ᆠ
\ Q

㦨


 
s2 㦨

y

kq
y




ᔼ
⼄盐���盐鼀

 = 矄� q

턼


æ ö ç ÷   ҈ = ᨨ� ç ÷
ç҈ 턼 ÷
p꾀 ø è 㡰 和 矄
꾀
 

ᆠ泐
⼄盐
ᆠ �
矄泐
Ἐ⼄Ἐ ├⼄⼄ \ ᨨᆠ
犨
 鼀

= 
⼄ ␔ 
꾀
 
ᔴ

ᨨ盐ᆠ 

ᆠ �
ᨨ
ᨨ
⼄ 꾀 = ⼄ ᔴ =



犨 

 
ᔴ

ᨨ盐ᆠ 

ᆠ �
ᨨ
 ⼄ +⼄ ⼄ ⼄ =
=

ᨨ犨

 
ᨨᆠ
犨
ᆠ �
⼄꾀
 
⼄ ⼄ ⼄ ├⼄ 꾀 ⼄  ├ ´
´
 ├⼄
 Ἐ = = = =






F �


æqö
kq ç ÷ è
ø

F=

҈ q

犨ᨨ
 
ᆠ�
�
盐

p Î
L


WWW.JEEBOOKS.IN

PS-171

Electric Charges and Fields

҈



 ҈

°


Y”

KQ




҈҈


FR
㦨

F
ᆠᨨ矄 q

㡰

ᨨ 泐
ᆠᨨ矄�ᆠ
ᨨᆠ
ᨨ
盐矄�ᨨ
�矄
ᆠ� �
L
犨 �ᆠ
犨�
├
 
犨ᨨ
矄
矄

矄
ᨨ矄矄�


 �
犨

 
犨ᨨ
矄� �矄
犨�

҈





 矄�ᨨ
�
ᆠ
矄� �
犨�

⼄盐
ᨨ
 
ᆠᨨ矄�ᆠ
ᨨᆠ ᨨ
盐矄�ᨨ

졠 ҈

턼 턼 

턼҈
㦨
턼

㦨
턼
 

㦨





㦨


맔


㦨


\

҈
pe


(R

Qx


)


⏌


+x

=

҈Q ´
R
ù⏌



pe
é R + (
R) ë û


=


QR


pe
� � R

҈�


ᨨᆠ
㦨

qE

-
QR

҈



҈


�.e.


ᨨᆠ
�矄
ᨨ
ᨨᨨ�ᨨ 
ᨨ
�矄ᆠ鼀 

pe
� �R
 ᆠ
ᆠ �
�ᆠ
犨�
ᨨ
 ᆠ盐
矄�鼀
鼀ᨨ �ᆠ
鼀ᨨ�ᨨ
T盐


鼀�ᆠ
矄 
犨 �ᆠ
�矄
��泐
ᆠ �
�� 矄
ᆠᨨ 矄  
 ᆠ
�矄
鼀矄矄
� ᆠ矄矄



鼀�ᆠ
矄 
犨 �ᆠ
�矄
ᨨ矄��泐
ᆠ �
ᨨ矄矄
ᆠᨨ 矄  
 ᆠ
�矄
鼀矄矄
ᆠ矄矄


ᆠ㡰


ᆠ�ᆠ ᨨᆠ
ᨨ
 泐
矄 
�矄 q
矄矄
盐
泐
ᔴᨨ盐ᨨ鼀
犨  
ᨨᆠ
盐
ᨨ
ᨨ 

矄 
ᨨ
 ᨨ 
鼀�

ᆠᨨ矄
q
 
矄鼀
q
ᔼ
⼄盐���盐鼀

矄�


q
졠


ᆠᨨ矄q
㦨
鼀� 鼀�  

矄�
q
㦨















\  q =


�㡰 鼀� 꾀矄
ᨨᆠ
犨
犨ᨨ
ᆠ �
ᨨ�矄
ᆠᆠᨨ� �
ᨨ
ᆠᨨ盐ᨨ鼀Ἐ矄
犨 ᨨ矄
ᨨ
 
盐ᨨ
 
鼀�盐鼀

 ᆠ
ᨨᆠ
犨
 鼀 鼀�
矄鼀
 ᆠ
 
�鼀鼀矄
�
�ᆠ�ᆠ
鼀�盐鼀 꾀矄
��
ᨨ
ᆠ  �
�
 �
q


e

æ r ö  
Ἐ ᆠᨨ矄 q= 鼀� ç ҈ -  ÷ è r矄 ø \  q = e r 


��㡰 鼀�҈ -  㡰 r矄 ᨨ犨
⼄盐� �
⼄

�㡰


��㡰 r ҈


҈ ҈-  = Þ҈=
҈ r矄 e  r矄 \ r矄
㦨




├�⏌鼀  ᆠ
ᨨ�ᨨ
ᆠ㡰
�矄
ᆠᨨᆠ

Ἐ 矄� q=

꾀 
 矄�ᨨ

Ἐ =

҈



x
+ a
㡰

㜠q

´

F
矄� q






㡰 F

x

x
+ a








k Q q x


a


㡰 턼

q q




ᆠᨨ矄
q
F 㜠q

 x
+ a
㡰 ⏌




㜠
a㡰

F 矄� q





Q´Q Q
҈ ҈ T
㦨
´ = ´
p e

L㡰

p e

L
矄 쟼ᆠ�ᆠ
�
ᨨ �
 �矄
ᨨ
� �
E=

kQq



= e 
҈

 ᆠ
ᨨ�ᨨ
㡰
�矄
� ᆠᨨᆠ
㡰 
盐矄
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 
ᨨ矄��
ᆠ �
Q


�矄 ᆠ
x
ᨨ鼀  
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
ᨨᆠ
犨
Q
 
q

F

졠
矄ᨨ� �
 �矄 ᨨᆠ
F
矄� q ᆠ ᆠ矄
ᨨ盐


矄盐 
ᨨᆠ

҈



和� ᆠ
FR
�矄
ᨨ
ᨨᨨ�ᨨ 
ᨨ
x

 
鼀ᨨ�ᨨ
�矄
ᨨ
矄�鼀
鼀ᨨ �ᆠ



ᆠ �
Q
犨�
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  ᨨ���
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 
鼀ᨨ
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�
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x㜠 �矄

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矄�鼀
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㡰 ᔼ
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Q


Q
 
Q



ᨨ矄��

 
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犨�
 ᨨ �
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y㜠�ᆠ�ᨨ
矄
ᆠᨨ鼀ᨨ  矄
ᨨ
ᨨᆠ矄
ᨨ �
x㜠 �矄
ᆠ ᆠ ᨨ盐
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Q

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x㜠�ᆠ�ᨨ
矄
ᆠᨨ鼀ᨨ  矄
ᨨ
ᨨ矄
ᨨ � y㜠 �矄
ᆠ ᆠ
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ᨨ �
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ᔼ
Q ҈

Q
ᨨ矄��
 
Q


Q

��
ᆠᨨ鼀ᨨ  矄
ᨨ
ᨨᆠ矄 ᨨ �
y㜠 �矄
ᆠ
ᆠ
ᨨ盐
ᆠ
ᨨ 

和ᨨ

ᨨᆠ
ᨨ
ᆠ �
q
ᨨ � x㜠 �矄
쟼ᆠ
�ᆠ
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⼄盐���盐鼀
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ᨨ
矄� �
T

犨�� � �

矄盐 
ᆠᨨ矄�ᆠ
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F
ᨨ
  犨ᨨ
ᨨ 
ᆠ �矄

ᨨᆠ
F
 
鼀�

 �ᆠ盐


矄ᨨ� �
T
�
 
�ᆠ�ᨨ
ᨨ
mg
 
F
 

泐� �
  ᆠᨨ ��ᨨ
ᨨ
⼄盐���盐鼀
T
ᆠᨨ矄 q 㦨
mg錐 T
矄� q
㦨
F F \  q
㦨




�㡰 mg

F
㦨
FCA + FCB +
FCA FCB ᆠᨨ矄 a

\




+ FCA +
FCA ´ F
㦨
FCA

F
㦨
FCA =  ´

kq


҈






��㡰 ur ur
 
a
㦨



桌  F CB  =  F CA  
T
鼀├

 �
q 犨�
 
�ᆠ



㡰
- 


҈�㡰
㦨



҈䰘
鼀 OC
㦨
 \ OM
㦨


黨黨黨䰘 OC


҈䰘 = \
 q
㦨





���㡰 OM

黨黨黨䰘 ᨨ鼀
⼄

�㡰

��㡰
 
���㡰

CA㡰





҈䰘  ´ 黨 ´ ҈
黨 ´ ⼄
=

黨黨黨䰘 


㡰
´ ҈
- ´ 黨



\
q
㦨

҈

맔
҈
黨
ᔴ


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EBD_8341 PS-172

෌�


㝐矄矄
ᨨ
�ᆠ

m
㦨


黨
맔
҈

├� ᔴ �
ᨨ
�ᆠ
q
㦨

҈


ᔴ Q + + r
㦨
҈
鼀錐 + + Q
㦨
҈
�
ᔴ + r +


ᆠ�ᆠ
�
盐

盐 � ᨨ鼀泐 + + ᆠ �
� �
ᨨ
�盐矄
r


ᨨ�  + �矄 
x
ᨨ鼀
�矄
ᆠ 
ᨨ
�矄
 �矄 x

Qx

E
㦨
k





























r
+ x
㡰 ⏌
\ ᨨᆠ
ᨨ
 
��
ᆠ � q
F
㦨
qE

\ ᨨ


- kQq

´x ᨨ
 r
+ x
㡰 ⏌
Qq A
㦨

k
´x
m  r + x
㡰 ⏌


F
㦨


ᨨ x





r


A = -

kQq 

+

+ + + +

҈䰘


Physics

+

+ + + +


ᨨᆠ
ᨨ
ᆠ �
q
├

A
盐
ᨨ
ᆠ �矄
├

B
 
C F҈
㦨

F
ᆠᨨ矄



æ q
ö F҈
㦨
 ´ ç 黨 ´҈
黨 ÷ ç a
÷ø è


ᨨᆠ
ᨨ
q
盐
ᨨ
ᆠ �

q㡰
├

D q
ö 黨 æ ´ ç 黨 ´ ҈
黨 ´ ÷ 
a ⏌ 㡰

çè a
÷ø ᔴ泐
 
犨ᨨ
ᨨᆠ矄

ᨨ
⼄盐

꾀矄ᨨ
矄
F

>
F҈
 
ᆠ �矄 犨�
鼀ᨨ
ᨨ犨矄
 
ᆠ 


q

F

㦨
黨
맔
҈
黨


Fixed ring

-k Q q

mA
㦨


 r
+ x
㡰 ⏌


w

㦨


\


p = T

kQq mr



´x

kQq mr 

B

kQq mr 

é 㦨
 ê K ëê

é

黨 ´ ҈
- ´ ҈ ù T
㦨


맔

҈
ê 黨 -� -
ú ëê 黨 ´ ҈
´ ҈
´ ҈
ûú



㦨





᲼


҈桌҈⏌

㦨





᲼

҈桌
㦨





矄 ᔼ
 
ᆠ �矄
q҈
 
q


矄�鼀�
�
盐

 
ᆠ�ᆠ
ᨨᆠ qq F = ҈


犨�

盐矄�
 
�矄
ᨨᨨ矄�
ᨨ
 矄�ᨨ
ᨨᆠ
T' l 꾀

⼄҈

P

a   㡰






҈黨


q
a


+K

q

C

é q
ù
 q´q ù +  êK ´ ú
ú 黨 
a ⏌ 㡰
ûú a ûú ëê

Topic-2 : Electric Field and Field Lines ҈


ᆠ㡰 쟼ᆠ�ᆠ
�
盐
ᆠ �
Q


E
= 쟼ᆠ�ᆠ
�
盐
ᆠ �
Q҈

E҈ = Q҈ A





mg

ᨨ
⼄盐���盐鼀
ᨨ


P mg ᆠᨨ矄


㦨
T

F㡰
ᆠᨨ矄a


�㡰 N҈
㦨
mg ᆠᨨ矄

졔
T

F㡰
矄� a


��㡰 和�鼀�泐

ᨨ
 
⼄盐���盐鼀
ᨨ


Q mg 矄�



㦨
T

F㡰
矄� a


���㡰 N

㦨
mg ᆠᨨ矄



졔


㡰ᆠᨨ矄a


�㡰 �㡰

��� �
⼄盐�ᨨ
���㡰
泐
�㡰

犨
�






㦨
 a �


a 㦨





㡰 ᨨ犨
��㡰
ᨨ鼀
⼄盐�ᨨ
���㡰
T

F
㦨
mg ᨨ

㦨

졔
mg
㦨
q҈q
⏌l
㡰
졔
mg


�㡰 ���㡰

ᨨ鼀
⼄盐�ᨨ
��㡰
N҈
㦨
mg ᆠᨨ矄


졔


㡰
矄�


N҈
㦨
mg ᆠᨨ矄


졔
mg
矄�


\
N҈
㦨
mg


��㡰 ᨨ鼀
⼄盐�ᨨ
�㡰
N

㦨
mg ᆠᨨ矄



졔
mg ᆠᨨ矄



㦨
mg


���㡰 꾀��
�
 
ᆠ ᨨ
�矄
ᆠ盐

 
T
ᆠᨨ鼀矄
ᨨ

 ᆠ
ᨨ鼀 ⼄盐�ᨨ
�㡰 q҈q

㦨
mgl




� 㡰

�矄
矄 ᨨ犨矄
 
q҈
 
q

鼀盐矄

盐 �├
ᆠ �矄
ᨨ
矄
ᨨ 鼀�
矄�ᨨ 泐
�㡰 
 
ᆠ �矄
ᆠ
ᨨ
盐
q

ᆠ

 
ᆠᨨ 矄 ᨨ

⼄盐�
� �
 

ᨨ盐
ᆠ �
Q
�矄
ᆠ

  ᆠ 
ᨨ
 
� �
矄
矄 ᨨ犨
�
 
��盐


kQ
x



kQ҈ x҈


x҈

ᔴ mg

F

Aq

a








  㡰





q
q
q
㦨

K
맔

+   K

㦨

+ 㡰 K
a a a

�矄
�矄
 
鼀ᨨ盐 
ᨨ
犨ᨨ├

ᨨ
鼀ᨨ
 
ᆠ �矄
ᨨ
� � �泐



Q ⼄


黨


a

F

q
Q 黨
q  q
=Q

맔
K
´
㦨
´ K ´
Þ


黨 a a


ᆠ �
턼
矄 ᨨ盐

��
ᨨ �
 �泐
ᨨ
 
矄泐矄鼀

mr 
Þ

T
㦨

p kQq

҈

q


F ᆠᨨ矄 
a

��㡰 ᨨ
ᆠ �矄
ᨨ
鼀�
矄�ᨨ 泐

҈⏌


҈




a  D Q

a

´x




ᨨ w
㦨


=

A q

r  ᆠ 鼀ᨨ�ᨨ
�矄
矄�鼀
鼀ᨨ �ᆠ
�
盐
ᔴᨨ鼀� �
 
ᨨ
⼄盐�ᨨ
犨�
A = -w
x \

q


q E


x


q E 

ᨨ鼀
��盐
E x  q =
= ҈ Þ E҈ x






q

黨
q

O

B

Q


E҈

kQ
x = ҈ kQ҈ x
´
x҈
Q x x Q x Q x Þ
҈
= ҈ Þ
=

ᨨ

҈ = ҈
x
Q҈ x҈ Q
x
Q҈ x
ᆠ㡰 ᨨ
矄 �ᆠ
矄  ҈ Q E= �
r ³ R 㡰
pe
r




㦨

�
r


R㡰 ᨨᆠ
ᨨ
ᆠ �
�
ᆠ�
�

F
㦨
qE  ᨨ
r


R㡰 \F =
҈ Qq F=  ᨨ
r  R㡰
pe
r
x



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PS-173

Electric Charges and Fields 


ᆠ㡰 
ᆠ�ᆠ
�
ᨨ盐ᆠ
盐
ᨨ
盐 � ᨨ鼀泐
ᆠ �
� � �  
�矄
盐 � ᨨ鼀

和ᨨ
ᨨ�ᨨ
㡰
 
㡰

犨ᨨ �
꾀 
졔
ᆠ �
 矄�泐
s졔
�矄
���
�
鼀� �盐
矄ᨨ
�矄
� ᨨ �
Y
�ᆠ�ᨨ
犨�

���
 
�
ᨨ

ᆠ �
 矄�泐
s �
X
�ᆠ�ᨨ

 ᆠ
ᨨ�ᨨ
ᆠ㡰
�矄
ᆠᨨᆠ

s E҈ ER ER 
s E

E A ҈ ´ �
æ 黨 ö 黨
҈
= =ç ÷= ´ =
´ ҈䰘 è ҈䰘 ø ҈䰘

EB ᆠ㡰





쟼ᆠ�ᆠ
�

E = E
҈ - x
㡰

b㡰 矄� � v


u

㦨

aS


�㡰 

u
㦨



s
㦨
x 꾀矄ᨨ

Fᆠ�ᆠ
㦨
ma

\
ᨨᆠ

F = qE = qE
҈ - x
㡰

Þ
qE
㦨
ma
Þ a =

E E
E
҈

E

E

ᆠ㡰 ෌�


꾀矄ᨨ

F = ma = mv

dv dx

dv ö æ çèQ a = v ÷ø dx

v
= 黨


qE ҈ - x
㡰dx Þ v dv =
m ᔼ �� �
ᨨ
矄�矄
犨
�
v




qE
҈ - x
㡰 dx 黨x ö v
qE
æ x Þ = ç ÷=
ò m m çè
 ÷ø
x

ax =






ᆠᆠ�ᨨ
ᨨ �ᆠ
�矄
ᆠᨨ 矄 

� �� ᨨᆠ�泐
�矄
ᨨ
  ᨨ 
�矄
矄�� 
� 
ay = g

泐

 쟼


졔s 
졔s



r s  ᆠᨨ矄

° xˆ  矄�

° yˆ 㡰
 
E
=
e


s æ ҈  ö yˆ ÷ çç  xˆ 
e
è

÷ø 쟼ᆠ�ᆠ
�
�
 
��ᨨ
矄 ᨨ犨
�
��盐
㡰


E m

r r r æ s é ҈ ö ù EP = E҈ + E
= ê  xˆ + çç҈  ÷ yˆ ú
e
ëê

÷ø úû è


a=

æ
E ö
çè ÷ +g mø

҈



Rö æ b㡰 쟼ᆠ�ᆠ
�

꾀
ç R Ἐ = ÷
ø è q E A
ds = e




A 3R




r s ( R ⏌
) æ sR ö =ç ÷


Þ E A = e
è
e
ø æ Rö e
×
p ç ÷ è
ø 쟼ᆠ�ᆠ
�矄

�B

r Þ EA =



=

r F = qE�ˆ + mgˆ


æ Rö r´ p ç ÷  è
ø

쟼҈

㡰

ᨨ鼀
��盐
r s E҈ = yˆ
e


Þx=








qE
x m





 a 㡰 
ᨨᆠ
ᆠ� �
ᨨ

 
�ᆠ


)

qE qE Þ a= m m 和盐矄�盐� �
 
盐矄
�
�㡰
犨
�

dv \ mv = qE
҈ - x
㡰 dx

Þ ò v dv =

r r E 鼀盐矄

 �
ᨨ
p \
Eˆ
�矄

ᨨ
�ˆ +  ˆ -
kˆ

(

졔
s

졔
s

E

�


r r

和� ᆠ
r × p =


䰘


҈





r ҈䰘 æ sR ö sR sR Þ EB = Þ EB = ç ÷ e
�
e
�
è e
ø







æRö
k ´ r´ pR k ´ r´ p ç ÷ r  è
ø  EB =
R
 æ Rö ç ÷ è
ø

 r sR æ ҈ ö ( s )
p æ R ö Þ EB = -ç ÷ ç ÷ e
è
pe
ø æ R ö
 è
ø ç ÷ è
ø

⏌


r xˆ ù s éæ ö ᨨ
EP = êç҈  ÷ yˆ  ú
e
ëêçè
÷ø
ûú ᆠ㡰 ᔼ
 

�ᆠ�ᨨ

Fx
㦨
qE Þ ma x
㦨
qE E q Þ ax =
m ᨨ
矄
ᨨ

ᨨ盐


v

+ vx
= 
v
㡰
Þ v x =  v
= a x t Þ v
=
+ ®

®

®

®

®

qE
t Þ t= m

v
m E
q

҈҈
㡰 E =  E + E 㡰 +  E + E 㡰 ҈

ᨨ
E
㦨

E
ᆠᨨ矄
a


E
ᆠᨨ矄
b –q d q d d q d –q

E3 a D

E1

WWW.JEEBOOKS.IN

EBD_8341 PS-174

㦨


Physics
kq





D + d 㡰

D

´

D
+ d



kqD

=





⏌


D + d 㡰 ᨨ







Eµ

D 

-

-


kq


 D + 
d 㡰




D

´

D
+ 
d 㡰



kqD


᲼ D + 
d 㡰
桌⏌


ᆠ㡰 �
� 矄
ᨨ��� 
 �ᆠ盐
ᨨ鼀
ᨨ矄��
ᆠ �  
鼀� 
 �ᆠ盐

��
ᆠ �

盐 
 �矄
矄泐矄鼀 ᆠ


矄

ᆠ�ᆠ
�ᨨ
҈

㡰 쟼ᆠ�ᆠ
�
盐
ᨨ
ᆠᨨ鼀
�矄ᆠ
R
㦨

a㡰


�矄 ᆠ x
 
ᨨ
�矄
 �矄 ҈�


é ù x s ê҈  ú
E҈ =
e
êë R
+ x
úû s é h ù é 
x
㦨h ù ҈ =
e
êë
a úû ëê  
R =
a ûú E҈ =

҈

µ

D D
҈

㡰 꾀
⼄盐���盐鼀
矄盐 
ᨨᆠ
ᨨ
ᨨ
鼀盐矄

ᨨ

矄ᨨ T
ᆠᨨ矄
q
㦨
mg





�㡰 T
矄�
q
㦨
qE





��㡰 Y 和ᨨ� �
�㡰
 
��㡰
犨
�  q =

qE mg



-


� ´ ҈
´





҈ =  q = -

´ ҈
´ ҈


҈


T

X q mg ᲼

q
㦨
�
맔
҈


ᔴ
쟼
㦨





⏌鼀

鼀
㦨


맔
҈

├�桌

 ᆠ

c =

rö

҈ Q · 錐
pÎ
R
Q⏌
҈
Q ҈ E = E
= · ·
pÎ
R
錐

pÎ
R


҈



쟼Ἐ =

\쟼Ἐ =


é R R ù é R ù =
pr
ê - ú =
pr
ê ú
úû êë  êë ҈
úû

=

pr
R 

æ pr R ö E

pr = ç
÷ è  Î
ø \

쟼ᆠ�ᆠ
�
ᨨ盐矄�
 


E =

s
Î


r
R  ҈
Î
r


ö ÷ ÷ ø

æ ç҈ ç
+
è

ö ÷ ÷ ø

s æ
- ö
ç ÷÷
= 쟼
Î
çè
ø ҈



ö æ çè 쟼 - 쟼÷ø ´ ҈

҈


҈
奠 ; 䰘

䰘奠 = = 쟼 ҈
㡰 Wh
h
ᆠ�ᆠ
�
�矄
ᨨ

ᔼ
⼄盐���盐鼀

ᨨᆠ
盐
ᨨ
ᆠ�ᆠ
�
㦨
犨��  qE
㦨
mg Þ qE
㦨





æ ç҈ ç
+
è

\
奠
盐ᆠ�ᨨ
�
 
盐
ᨨ
ᆠ�ᆠ
�

҈黨




s
Î


ᨨ鼀
⼄盐矄�ᨨ


㦨

�盐矄
ᨨ
�矄ᆠ㡰

ò
çè҈ - R ÷ø r dr
=
pr
ò
r
dr - rR dr R

s �� 㡰
Î


쟼ᆠ�ᆠ
�

ᨨ �
 
 �矄

 泐
�矄 ᆠ

ᨨ鼀
 
ᆠ 
ᨨ  
�矄ᆠ

éé  ù R é
ù R ù
ù é  ê ê r ú - ê r ú ú =
pr ê R - R ú =
pr





êê  ú ê
R ú ú êë 
R úû ë ë û
ë û
û

q=

ᔴ泐
E

>
E҈
>
E 㡰 쟼ᆠ�ᆠ
�
�  矄�泐

 
ᆠ 
ᨨ
 
�矄ᆠ


쟼=

Q
dv
㦨

p
㡰


s
ae


E҈ =

҈䰘
ᆠ㡰

ò

Ræ

a ᨨ

h ù s é hù s é ҈  ҈ 
㦨
sh
e
êë
a úû
e
êë a úû
e
a

q�n = dq = rdv


=
pr



a

s æ hö ç҈  ÷
e
è a ø 쟼ᆠ�ᆠ
�
盐
ᨨ
��
�矄ᆠ E
㦨
E҈

E


)

rö æ = r
ç҈ - ÷
pr
dr è Rø

ù ú úû

E
=

)

(

é h ê҈ 
êë
a + h


和�鼀�泐

ᆠ�ᆠ
�
盐 ᨨ
�矄ᆠ
R
㦨
a㡰

q

æ҈ö Þ  -҈ ç ÷ è
ø b㡰 쟼ᆠ�ᆠ
�
ᨨ
 
 �矄
ᨨ

� �
ᨨ
�盐矄



�矄 ᆠ
ᨨ鼀
 
ᆠ 
├턼 쟼= ⏌

+
쟼 =
ᔴᨨ ��ᨨ

ᨨ
鼀 �鼀盐鼀
ᆠ�ᆠ
�
 é ù d ê kQh ú Þ ú =0 dh ê

⏌
ê R +h ú ë û 泐
盐矄� �
 
ᆠᨨ ᆠ
ᨨ
鼀 �鼀
 
鼀� �鼀
犨
�

=
rö æ 㡰 ᔴ �
 矄�泐

r = r
ç ҈ - ÷ è Rø dq
㦨
rdv

(

҈



q qE

s
e




pR rg pR rg


\ q =  E




�㡰

Wh
h
ᆠ�ᆠ
�
�矄
矄犨�ᆠh
ᨨ ␔�� 
ᨨ
 
ᨨ
㦨
�矄ᆠᨨ盐矄


ᨨᆠ
ᨨ
 
ᨨ 鼀�
㦨

phRvt

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PS-175

Electric Charges and Fields


 pR rg =
phRvt

\ R = 

黨hvt
rg

é s
s $
s s ù $ + + ú  - k 㡰 㦨
- e k
ë
e

e

e
û

㦨
ê




��㡰




é 黨hvt ù
rg ´ ᨨ鼀
⼄

�㡰


��㡰
q = p ê  ë
rg úû E






ᆠ �矄
ᨨ
 
矄盐 ᆠ矄
ᨨ
 
鼀�ᆠ
矄 矄


矄 ᨨ犨
�
 
��盐
ᆠ �
 矄�泐
σ
㦨
⼄盐矄�ᨨ

Q҈
pR =




=

졔

Q҈ + Q

p 
R 㡰


p R 㡰




졔




Q҈ + Q
Þ Q
= Q҈


졔

졔

Q҈+ Q
㡰 졔





졔

 –Q҈





졔

졔 Q҈ 

R

졔








졔



졔

졔



졔

Q҈+ Q
+ Q 





졔



졔


�


졔



졔

졔


R



졔

R 

졔



 졔

Q҈ + Q
+ Q = Q҈ 黨



졔

  졔





䰘


졔 졔

b㡰 
 
ᆠ �

盐 �
ᨨ盐鼀 
s
 
O

 
ᆠ 
ᨨ

盐 � ᨨ鼀矄 ᆠ �
矄ᨨ�
矄 


盐矄
ᆠᨨ 矄� 
盐 � ᨨ鼀泐
ᆠ �
矄 
ᨨ
�� ᆠ �
 矄�泐
s
� �
�矄
ᆠ 
 OἘ

꾀矄ᨨ

OOἘ

⼄盐
ᨨ
a


r҈

ur s uur E
= r
Ἐ \ 
ᨨ
ᆠ�ᆠ
� e
ur ur ur s E = E҈ + E
= a e
 ᆠ
ᆠ�ᆠ
�
犨�


� �
盐
犨 �ᆠ
犨�

盐 � ᨨ鼀
ᆠ㡰 
�ᆠ�ᨨ
ᨨ
ᆠ�ᆠ
�矄

ᆠᆠᨨ� �
ᨨ
 
ᆠ � ᨨ
 
矄 矄
�



ᨨ �
��
z㜠 �矄
E1 E3 Z=a

E2

2 -2s


ᨨ
ᆠ�ᆠ
�

ᨨ� 
P

ur ur ur ur E = E҈ + E
+ E  㦨
E҈
 - k$ 㡰 + E
 - k$ 㡰 + E  - k$ 㡰

3 -s

E

+ R

E E

S




\
E
㦨



rR pR
r ҈ pR r ⏌

rR = 

㦨

e ´
´ e
黨
҈

pe
R
pe

R


\
k
㦨


㡰 
 


矄 �ᆠ
矄 
ᨨ
�盐矄
x
 
 �ᆠ├ 矄矄
dx
ᨨ盐鼀
ᨨ
 �矄
矄 


㦨


px
dx㡰

ᔴ �
 ᆠᨨ矄
�
 �矄
矄 
dq = 
px
㡰 dx ´ kx a ᨨ
dq =
pkx
+ a dx
ᨨ
r
=
R


ᨨ
ᆠ �
 ᆠᨨ矄
�
  矄 
ᨨ
�盐矄
R R

Q=

\

ò


E҈ =


pk x
+ a dx =
pk

dx x

R + a + a


쟼ᆠ�ᆠ
�

r
㦨

R

҈
pkR + a ҈
pk ҈+ a = R

pe
 + a 㡰 R
pe
 + a

ᨨ
r
=
R⏌




ᨨ
ᆠ �
 ᆠᨨ矄
�
 
矄 
ᨨ
�盐矄
R⏌

R⏌


Z = –a

O

E

Q ҈
pe
(
R )


lᆠ泐� 
㦨
pR r

꾀矄ᨨ
 
ᆠ�ᆠ
�
盐
ᨨ
矄鼀
矄 

P


pe
(
R )



 졠¢






Z = 3a

g

졠

ur s ur E҈ = r҈ e


1 -s

r

+T

pR  r
æ Rö 턼矄 
㦨
p ç ÷ r =  è
ø



盐矄
ᆠᨨ 矄�

�泐
ᨨ� 
P
�
 
矄鼀
矄 

 ᆠ�ᆠ
�
盐
ᨨ
ᆠ �
ᨨ
��
矄 
盐矄� �
෌盐矄矄Ἐ矄
 ᨨ鼀






-



r҈ 

E U

㡰 
ᆠ�ᆠ
� 矄
ᨨ
ᨨᆠ
ᆠ

ᨨ
 
 
鼀�ᆠ
矄 
矄 쟼
㦨


� 矄�
 
矄ᨨ�
鼀�ᆠ
矄 

꾀矄ᨨ

 
ᆠ�ᆠ
� 矄
ᨨ ᨨᆠ

ᨨ
 
鼀�ᆠ
矄盐 ᆠ
ᨨ鼀泐
ᆠ㡰 쟼ᆠ�ᆠ
�
泐犨 
� 矄�
 
鼀�ᆠ
ᨨ�ᨨ
�


ᆠᨨ 盐ᆠᨨ
ᨨ
矄 
�矄
ᨨ
쟼ᆠ�ᆠ
�
� 矄

犨泐矄
ᨨ鼀
ᨨ

矄盐 ᆠ
ᆠ㡰 쟼ᆠ�ᆠ
� 矄
ᨨ
ᨨᆠ
ᨨ
ᨨ
ᨨ鼀
ᆠᨨ矄
ᨨᨨ矄

㡰 
鼀� �盐
ᨨ
 
ᆠ�ᆠ
�

 
ᨨ� 
P
犨 �ᆠ
�矄 

�矄 ᆠ

R
ᨨ鼀
 
 �矄
ᨨ
 
ᆠ泐�  쟼
㦨
쟼ᨨ

쟼ᆠ�泐 㦨


Þ

Q҈

Q

㦨
Q Þ

Q҈

Q҈
㦨
�Q҈
㦨
Q \
Q҈
Q

Q = ҈

�
҈


E

ur ur E
졔 E 㦨

E


졔

 Q҈+ Q
㡰

 졔

Q҈ + Q
+ Q

\
Q҈ =

ᆠ �
 
矄
  졔

r


和�鼀�泐
쟼ᆠ�ᆠ
�
盐
ᨨ
Q ᨨ
O
�矄
ᆠ ᆠ
泐
ᆠ�ᆠ
� 盐
ᨨ
T
 
O



ᆠ�ᆠ
�
盐
ᨨ
R
ᨨ
O
�矄 �
 
矄鼀
�ᆠ�ᨨ
矄
 
ᨨ
U ᨨ
O

 ᆠ
 

ᆠ�ᆠ
�

ᨨ

q
㦨
䰘


맔
҈
҈黨
ᔴ b㡰

Kq

쟼ᆠ�ᆠ
�
盐
ᨨ
P
ᨨ
O
�矄
ᆠ ᆠ
泐
ᆠ�ᆠ
�
盐
ᨨ
S ᨨ
O
P+Q



é 黨 ´ ҈

´ ҈
-� ´
´ ҈
- ù
黨

´ 黨

´ 䰘
= ´pê ú ´ 
´ 黨

´ 黨


҈p ´ ҈
� ë û




ur

쟼ᆠ�ᆠ
�

 E =

ᆠ㡰

QἘ =

\

ò



pk x
+ a dx =


pk  R ⏌
㡰+ a + a


ᆠ�ᆠ
�

r = R⏌


E
=

҈
pk  R ⏌
㡰+ a ҈
pk æ R ö = ç ÷

pe
 + a  R ⏌
㡰
pe
 + a è
ø

҈+ a

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EBD_8341 PS-176

Physics

҈ E
҈ ҈+ a ҈
pk æ R ö ҈ ҈
pk ҈+ a = ´ R çè ÷ø 
pe
 + a 㡰

pe
 + a


Q
E
= \


盐 矄� q =

�鼀
ᨨ
�� 
҈ = 泐

=

Þ ҈
졔
a
㦨

\
a
㦨


黨


⼄E꾀㡰
和� ᆠ
ᆠᨨ矄�ᆠ
�
�矄

ᆠᨨ 矄�
� \ WPQ 졔 WQR 졔 WRS㡰
졔
WSP
㦨

ᨨ WPS
졔
WSP
㦨




ᨨ
WPS
㦨
WSP \

(

=

)





҈









(



�鼀



E


B
�
� A

E


KP R

P O


⧤ 

⧤ 

�

�ᆠ�ᨨ
ᨨᨨ矄�
ᨨ
 
�ᨨ


  
ᆠ�ᆠ
� 矄 ᨨ盐
ᆠ ᆠ
ᨨ盐
 �矄
�

�



EB
㦨

®

r - ⧤


ᆠ�ᆠ
�

꾀

q

E


ᨨ
 
ᆠ�ᆠ
�ᨨ


ᆠ�ᆠ
�

盐
ᨨ
�ᨨ
�矄


®

⼄
㦨
鼀
 

㦨
⼄쟼
\泐
= 쟼 泐
㦨





맔
҈
҈
鼀 é ⼄ ù ҈
ᔴ├� -҈ ҈
�� 㡰 ú Q = êë 鼀 û

쟼




 ´҈
҈


�ᆠ
ᨨ鼀
졠
ᨨ
꾀㡰
矄
ᨨ� 
꾀
�矄
ᨨ
 
 �
� 
ᨨ
ᆠ�ᆠ �ᨨ


  
ᆠ�ᆠ
�
쟼

矄 ᨨ盐
矄ᨨ

�
 
�ᆠ�ᨨ

ᨨ
졠
ᨨ
꾀
ᨨ犨
ᆠᨨ 矄�� �
ᨨ� 

犨 �ᆠ
�矄

ᨨ� 
ᨨ
 
⼄盐�ᨨ�
� 

 �


㦨
�
鼀

盐 =
҈
´҈

鼀矄 -҈




ᨨ犨
ᆠᨨ 矄�� �
ᨨ� 
꾀

 
ᆠ�ᆠ
�
盐
ᨨ
�ᨨ



盐 
쟼
=

Þ

q
㦨



ᨨ

҈


Þ q
㦨


ᨨ



�ᆠ
�矄
 
�

�
q
㦨


ᨨ
q
㦨




)

ᆠ�ᆠ
�矄
⼄盐�ᨨ �

꾀矄ᨨ KP/R ᆠ�ᆠ
�
�矄
ᨨ鼀
ᨨ
矄盐ᆠ
 E ⼄ � 
 
�矄
 
�ᆠ�ᨨ
ᨨ ᆠ�ᆠ
�
�矄
� 
�
ᨨ  
鼀� �盐
ᨨ
ᆠ�ᆠ
� 矄 ᨨ盐

ᨨ

ᨨ� 矄
ᨨ
  ᆠ�ᆠ





 Þ 矄�
q =



´
҈
´҈

´

P
ˆ ˆ � + 
 


ᔼ ��
矄
ᨨ
ᆠ �
�ᆠ

盐 =
҈
´ ҈
鼀⏌矄

盐
矄�
q

´ ҈
҈
矄�
q Þ�=  �

㦨 泐


 ´ ҈
҈





 ´ ҈
҈


b

㡰 ෌�
�ᨨ
鼀ᨨ鼀 
ᨨ
ᆠ�ᆠ �ᨨ
P =

\ WPS
㦨

WSP
㦨

qEa \
␔ᨨ├
ᨨ 
泐
�
ᨨ �
ᨨᆠ矄矄
PQRS
㦨
qEa 矄 ᨨ

�ᆠ
ᨨ
鼀ᨨ
�
ᆠ�ᆠ盐
鼀ᨨ�ᨨ

犨


ᆠ � ᨨᆠ
犨 �ᆠ
�矄
ᨨ
�
T盐


ᆠ�ᆠ
�
ᨨ盐ᆠ
犨
 


ᆠᆠ�ᨨ �矄
盐 � ᨨ鼀


ᨨᆠ
ᆠ� �
ᨨ
ᆠ �
�ᆠ
ᆠ
�


ᆠ�ᆠ
�
�矄
��
泐
F
㦨
qE
 ᆠ

 
ᨨᆠ矄
ᨨ
 
犨ᨨ
ᨨᨨ 矄

� �ᆠ
� �
⼄盐 ᆠ �
 
盐 � ᨨ鼀
ᆠ�ᆠ
�
T盐 ␔ 
��
 �泐
X㜠泐
鼀
矄












�
犨�
├ ᨨᆠ├
ᨨ盐
ᆠᨨ 矄
ᨨ鼀
  矄鼀
鼀

盐
ᨨ
 ᨨᨨᆠ�ᆠ  ᆠ

鼀├� �
�
ᨨ矄��泐
ᆠ �
  
ᆠᨨ矄�ᆠ
ᨨᆠ
�
 
�ᆠ�ᨨ
ᨨ ᆠ�ᆠ
�
ᆠ矄

 ᨨ
 

犨� 
 ᆠ
�
 
�ᆠ�ᨨ
ᨨ
ᆠ�ᆠ
�
r -҈ b

ᆠ㡰
෌�

ᆠ�ᆠ
�

쟼 泐 = -


 ᔴ

҈


� m矄  ᨨ q=

° 㡰


ur

WSP = qEa





� m矄  ᨨ q = 
° 㡰



盐 矄� q =

�鼀
ᨨ
�� 

= 泐

r uur ␔ᨨ├
ᨨ 
㦨
qE
SP (\ F = qE ) r r WSP = qE�ˆ ( rP  rS ) (Q E ҈҈´  �矄 )

WSP  qE�ˆ a�ˆ + bˆ


´
҈
´ ҈

´

쟼꾀 =

r
⧤




�㡰

R ®

®

®

®

+ 쟼

㦨
-
쟼
+ 쟼
= - 쟼
 
҈ ´
꾀��
ᨨ鼀
⼄

�㡰
쟼

㦨

pÎ

 R  㡰 \ �




㦨

p Î
쟼


҈⏌

é 
ù \
㦨
ê ú
쟼 pÎ

û ë

㡰 
盐矄
ᆠᨨ 矄�

ᨨ� 
㝐
� 矄�
 
ᆠ�泐
犨 
ᆠ�ᆠ
� 矄
ᨨ

ᆠᆠ盐

꾀矄矄盐鼀
 
ᆠ�泐
ᨨ
ᆠᨨ �
矄�鼀�
ᆠ � �矄�盐�ᨨ
ᨨ
ᨨ矄��
 
��
ᆠ �
矄
 
矄
ᨨ
矄 
쟼ᆠ�ᆠ
�

M
盐
ᨨ
盐 � ᨨ鼀泐
�矄�盐
ᆠ �
ᨨ
 
犨 ᨨ 矄 
ᨨ
�盐矄
R҈

r r r E҈ = r e























쟼ᆠ�ᆠ
�

㝐
盐
ᨨ
 

��
ᆠ �
�矄�盐�ᨨ
�
  ᆠ�泐

ur r uuur E
= MP e

㝐 

  졠

R҈

\ 
ᨨ
ᆠ�ᆠ
�

㝐
� 矄�  
ᆠ�泐


ur ur ur r r r uuur E = E҈ + E
= r+ MP e e

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PS-177

Electric Charges and Fields

ur r r r r r r uuur r a  r éQ r + MP = a ù E= r+ ë û e e ur r r \ E= a e ur  ᆠ
� 矄�
 
ᆠ�泐
E
�矄
盐 � ᨨ鼀
 
ᨨ
�矄
鼀� �盐 r  
�ᆠ�ᨨ
 
ᨨ
a


(

\

)

r






ᆠ㡰

҈

쟼ᆠ�ᆠ
�
쟼҈
盐
ᨨ
矄鼀
矄 
�꾀



҈ E҈ =
pe


 꾀

Q

E҈
㦨


pÎ
r



졔Q 졔 졔 졔

r


졔졔 졔졔 졔졔

r


E

㦨


҈


l

pÎ
r


҈ 턼 s = Þ 턼 =
ps



pÎ


Î


㡰

쟼ᆠ�ᆠ
�
E

盐
ᨨ
���
矄 
�B




 p
R r R r R ҈  ( )
E
= =

pe
e
(
R ) E҈ = E




쟼

r
㡰
㦨
쟼r
㡰

҈
l s l = Þ 
=
pÎ


Î
sp ᆠ㡰

= Q

쟼҈

⏌
㡰
㦨


҈
턼
p Î





´
l
҈ ҈
l ´ =
p Î


p Î


E҈r
㡰
㦨
E
r
㡰


\ Q
㦨

l
r


黨


é ҈
l ù ҈
l ú= ë
pÎ

û
pÎ



꾀 

E

r
⏌
㡰
㦨

ê

҈ 㡰 E

r
⏌
㡰
㦨

p Î

s
s =
E
r
⏌
㡰
㦨

Î
Î
䰘



 p
R r ҈  ( )
E
=
pe
( �R )
E҈
졔
E

㦨



҈
pe
\



 pR r҈ p (
R ) r
҈   + =


pe

R ( ) ( � R )


r҈ 
= r

�

r҈ R r R =

Î
´   Î


Þ

r҈ =
r




㡰 ᔴ泐
ᨨ鼀
��盐
ᆠ�ᆠ
�
� 矄

ᨨ�� � �
ᨨ鼀
Q҈
  鼀� � �
ᨨ
Q

\ Q҈
�矄
ᨨ矄��
 
Q


�矄
��
盐鼀
ᨨ
ᆠ�ᆠ
�
� 矄
ᨨ��� � �
ᨨ鼀
Q҈
�矄
鼀ᨨ
 

鼀� � �

Q

\
Q҈
>
Q

和� ᆠ


� �
�矄 ᆠ
ᨨ
 
�� 
ᨨ
Q


 
ᆠ�ᆠ
�
�矄
ᨨ



ᆠ�ᆠ
�
ᆠ
泐
Q



�ᆠ盐
ᨨ� 
犨�
ᆠ ᆠ
ᨨ盐  
ᆠ�ᆠ
�
ᆠ
泐
Q҈

盐


� �
�矄 ᆠ
ᨨ
 
  ᨨ
Q҈
 
ᆠ�ᆠ
�
�矄
ᨨ 㜠ᨨ


ᆠ㡰


ᆠ�ᆠ
�
� 矄�
 
矄 

҈ Qq r

pe
R 

E

ᨨ

E
µ
r
졠盐矄�
 
矄  R


r


¥㡰
E
㦨
ᨨ

E
µ


҈ Q

pe
r


A

E


҈

B r

r


O r=R  ᆠ

E
� ᆠ矄矄
ᨨ
r


R
 
ᆠ矄矄
ᨨ
R





¥


b

㡰 쟼ᆠ�ᆠ
�
E҈
盐
ᨨ
矄鼀
矄 
ἘAἘ

Q

쟼ᆠ�ᆠ
�
E

盐
ᨨ
���
矄 
ἘBἘ

Q

\

r


R㡰
E
㦨



l
l ҈
l = = 
⏌

p Î

pÎ

l
= ´ Î
p



 pR r҈ ҈  E҈ =
pe
(
R )


졔 졔 졔 졔 졔
졔 졔 졔 졔 졔 졔 졔 ® ® 쟼
 쟼҈ 졔 졔 졔 졔 졔 졔 졔 졔
졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 r


s E
㦨
2Î



쟼҈
r
㡰
㦨
쟼r
㡰

\


졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔
졔 졔 r҈

턼

E҈r
㡰
㦨
E
r
㡰
㦨
Er
㡰
෌� 㡰 㡰


r҈ ´ pR r pR r҈ R ҈  ´ ҈ = 㦨


p Î 
Î 
R㡰

´






b㡰

ᨨ

ᨨ� 
ᆠ �
쟼
㦨



쟼
µ

҈ d

kp




d



�




 
ᨨ

�ᨨ

a


k l r

҈

쟼
㦨

�



쟼
µ  d d ᨨ

� � �
ᨨ �
� 
ᆠ � 쟼
㦨


kQ


k l ҈
�



쟼
µ d d


k l r





졔l l ᨨ
犨ᨨ
� � �
犨�矄
ᆠ泐� �
盐 � ᨨ鼀
� 
ᆠ �
 矄�泐


쟼
㦨


l

l
k l
k l l
k l ´ = ᆠᨨ矄 a
㦨




r d
+ l
d +l d +l

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EBD_8341 PS-178

Physics

ᨨ

E µ

҈ d


Q
l s
) e


rr  =  
쟼

盐
 ᨨ盐�


ᆠ矄
㦨


␔ᨨ├
ᨨ 
泐
ᆠ�ᆠ
� W
㦨
qv
㦨
( q
E ) a ᆠᨨ矄
�o

\

W
㦨



q
s҈ - s
㡰 a
e


Topic-3 : Electric Dipole, Electric Flux and Gauss's Law ҈


b㡰 
v

 
矄
ᨨ
�ᨨ
矄� �
 �泐
ᆠᨨ 矄�ᨨ

K� + U � = K f + U f Þ
-


k × p҈ r

҈ ҈ p
ᆠᨨ矄 ҈

°㡰 = mv
+ mv
+





qE md


㦨

쟼
ᆠᨨ矄
q ᲼␔ q
㦨
 �
犨
�ᨨ
 
 �ᆠ盐
ᨨ
 
�桌 㦨


҈

黨㡰
҈
㡰
ᆠᨨ矄

� 㦨



䰘
䰘
맔
҈



 㦨

䰘
맔
҈

䰘 b㡰 ␔ 
ᆠ盐
�矄
ᨨ
矄�

 
ᨨ� 
ᆠ �
턼
�矄

 
ᆠ  ᨨ
 
ᆠ盐
 
 
ᨨ
ᆠ�ᆠ
盐
盐
ᨨ
 �矄
ᆠ �
犨� 矄矄
 泐
 ᨨ盐�
 
矄�
ᆠ矄
ᨨ
 
ᆠ盐

和ᨨ

 
ᆠ�ᆠ 盐
 ᨨ盐�
ᨨ 
ᆠ
犨�

⼄盐
ᨨ
҈⏌

ᨨ
 
ᨨ
ᆠ�ᆠ 盐
盐
ᨨ
 �矄
ᆠ �


\

ᆠ�ᆠ
�

E
㦨
E҈

E




=

b㡰











㦨
q

E
맔





�


ᔼ
 
ᨨ盐鼀
ᆠ �
 矄�泐
�矄
ᆠᨨ 矄 
 
쟼
µ






㦨


Q ҈ Q ÞE=
r³R
pe
r
e


b㡰 t
㦨

PE
矄�
q ᨨ Ia
㦨

PE q㡰

a=

 ö æ 4π 
  \
⼄
㦨
çè  - ÷ø


ᨨ



㦨 ò
π   ò

S

E

㦨


q҈l e


E
맔

pr

㦨




ᔴ �
�
 
矄 
鼀  ⼄
㦨

맔









®

Ñò E
d A = ᨨ



和盐 ᆠ
ᆠ �
 矄�泐
 矄
ᨨ 泐
盐
ᨨ
Q

꾀矄ᨨ

®





ᆠ㡰

㡰 b㡰

p ҈ a
p Î
ma

\ 䰘


턼 Îᨨ

盐
 ᨨ盐�
҈
ᆠ

㦨


턼
Îᨨ

㡰 T
㦨
쟼
矄�
q
ᨨ⼄盐
 � ᆠ
泐
 
�ᨨ
�

ᆠ�ᆠ r r r �

T = P ´ E

r p
㦨
p
ᆠᨨ矄q �ˆ 졔
p 矄�
q ˆ r r E҈ = E� r r r T҈ = p ´ E҈
㦨
p
ᆠᨨ矄
q �ˆ
졔
p
矄�
q ˆ 㡰
맔
E �ˆ 㡰 t kˆ 㦨
pE
矄� q
 kˆ 㡰


�㡰 r ˆ E
=  E҈  r T
= p ᆠᨨ矄 q�ˆ + p 矄� qˆ 㡰 ´  E҈ ˆ


��㡰 tkˆ =  pE҈ ᆠᨨ矄 qkˆ

ᨨ鼀
⼄ 矄

�㡰
 
��㡰 pE
矄� q
㦨
 pE
ᆠᨨ矄q  q 㦨


\

q
㦨




WWW.JEEBOOKS.IN

PS-179

Electric Charges and Fields



黨


(c)



盐
� ├
犨�
ᆠᨨ矄
矄盐 ᆠ矄
和҈

和


和


和



҈



҈ 
⼄㡰 ᨨ
矄盐 ᆠ
和҈

f҈ = e
҈ ҈ ⼄ + ⼄ + ⼄ - ⼄㡰 =
⼄ ᨨ
矄盐 ᆠ
和


f
= e
e
҈ ҈ ⼄ + ⼄㡰 = 
⼄㡰 ᨨ
矄盐 ᆠ
和

f = e
e
҈ ҈ 
⼄ -
⼄ -
⼄㡰 = 
⼄㡰 ᨨ
矄盐 ᆠ
和


f
= e
e
 ᆠ

f҈
㦨
f

㦨
f
㦨
f

�



ᆠ�ᆠ
盐
�矄
矄鼀
ᨨ
 矄盐 ᆠ矄
⧤
�
鼀� 

 
ᆠ�ᆠ
�
盐
ᨨ

ᆠ �
ᨨ盐矄�
和
  和
㡰

 
෌盐矄矄�
矄盐 ᆠ
ᆠᨨ �盐矄
ᨨ

盐
 ᨨ盐�
  矄盐 ᆠ

ᆠ盐矄
矄
鼀 泐
� 矄
盐
ᨨ
 
ᆠ �
 
 
矄盐 ᆠ 矄

�
ᆠ㡰 꾀泐� �
෌盐矄矄矄
犨 r uur Q Q +
pAr
-
pAa
Ñò S E × ds = Î
\ E
맔

pr

㦨
Î


dr dV Q
㦨
r
pr
r=

෌盐矄矄�鼀 矄盐 ᆠ



r

A
Q
㦨
ò
pr dr
㦨

pA᲼r


a
桌 r a

E
㦨


ù ҈ é 턼 -
p꾀
+
p꾀 ú ê

p Î
ëê  ûú

ᨨ
E
ᨨ

�   
ᨨ
�r Q


pAa

㦨

\

A
㦨


҈


҈



ᆠ㡰

҈҈


҈
҈


p Î

ur ᆠ㡰 ෌�
E = E
xˆ

졔⼄

� 
ᆠ � 졔졠 q


⏌
 ⏌


Nm
C f҈

f҈҈㦨





⼄

f҈҈ =



ur
A 㦨a´
a =
a

E

a
㡰 ´

҈


҈







㡰 
盐矄
ᆠᨨ 矄�

� �
鼀 
ᨨ
�盐矄

 
 �ᆠ├ 矄矄
 和盐 ᆠ
ᆠ �
 矄�泐
ᨨ
�矄ᆠ
ᨨ
�盐矄



ö æ s 㡰 = s
ç҈ - ÷ è ø

 ö æ s矄㡰 = s
ç ҈ - ÷ è ø

ᔴ �
ᨨ
�矄ᆠ
鼀 











㡰 쟼 q 꾀 




㡰

2

泐



f
=



ö æ ⼄ = s
ç҈ - ÷
p è ø

 ö æ ⼄ = s
ç҈ - ÷

p è ø ᨨ犨
ᨨ鼀
෌盐矄矄Ἐ矄
 ᨨ鼀

쟼ᆠ�ᆠ
盐

 ᨨ盐�

�
矄 �ᆠ 矄盐 ᆠ
 
 ᆠᨨ矄矄
 
ᆠ �
�矄ᆠ
ᆠᨨ鼀泐


q
㦨

�





㡰

q

\ f҈҈ = é
´ �ˆ   y
+ ҈㡰 ˆ ù

�ˆ = ҈
x ë û

ur




㡰

ᆠ泐� 

r uur ᨨ
矄盐 ᆠ
BCGF
fn = ò E
dA 


Q 꾀 �
犨
쟼
 
꾀

턼
㦨

� 㡰

҈ a⏌
= a ⏌


\ f҈ = ò E
A ᆠᨨ矄 黨
° =


�


ᆠ�ᆠ
�
E
ᆠ矄
ᨨ �
졔
x �ᆠ�ᨨ
 
�矄

ᆠᨨ 矄 


 ᨨ
 
ᆠ�ᆠ
盐
 ᨨ盐�
 
矄 
ᨨ�ᨨ
犨 ᨨ矄 

㦨
E
a


䰘C

\
쟼ᆠ�ᆠ
盐
矄矄� �
 ᨨ盐� 꾀 ᔴ ᆠ泐� �ᆠ
矄盐 ᆠ
�



ᨨ


lL  �
㦨


 ᆠ


㦨


e
҈�


㡰 r r 盐
ᨨ
ᆠ�ᆠ
�
E
 ᨨ盐�
 泐

A
�矄
 � 
矄 f = ò E
A ᆠᨨ矄 q 

q
㦨
 �
犨
ᆠ�ᆠ
�
 

ᆠᨨ
ᨨ

矄盐 ᆠ ᨨ
矄盐 ᆠ
ABCD
꾀 �

q
㦨
黨




r r f = E
A = EA ᆠᨨ矄 q= E

a
㡰 ᆠᨨ矄

�
㦨


ᨨ鼀
��盐
 q
㦨


( ␔  L =  � ᨨ ᆠ泐�  )

=

⼄

C

\ q
㦨

쟼ᆠ�ᆠ
盐
 ᨨ盐�
  ᆠᨨ鼀
ᆠ泐� 
泐

















෌盐矄矄矄
 ᨨ鼀 q �n lL 㦨 f
ᆠ泐� 
㦨

e
e


졔⼄

҈

X
C

㡰 
盐
 ᨨ盐�
 
෌盐矄矄�
矄盐 ᆠ
�矄
盐
ᨨ
 
ᆠ �矄 � 矄�
 
෌盐矄矄�
矄盐 ᆠ

盐
 
ᆠ�ᆠ
�
ᨨ
 
෌盐矄矄�

矄盐 ᆠ
�矄
 
ᆠᨨ
矄盐鼀
ᨨ
ᆠ�ᆠ
�矄
盐
ᨨ

 
ᆠ �矄 �.e.

⼄҈

⼄҈
 
⼄



҈


㡰

Q


pa
ᨨᆠ
ᨨ
� ᆠ�ᨨ

Z

q -
C f = �n = e
e


  턼

Y

㡰 ᨨ鼀
 
��盐
ᨨ ᆠ �
 ᆠᨨ矄
�
 
ᆠ盐�ᆠ 矄盐 ᆠ
q�
㦨
C
졔

C

䰘C
㦨 
C

꾀ᆠᆠᨨ� �
ᨨ
෌盐矄矄Ἐ矄  ᨨ鼀
 
ᆠ�ᆠ
盐  ᨨ盐�
 
ᆠ盐

ò ⼄ = e


æ

 ö

ò s
çè҈ - ÷ø
p




e


WWW.JEEBOOKS.IN

EBD_8341 PS-180

Physics

쟼ᆠ�ᆠ
盐
 ᨨ盐�
 ᨨ 
矄 �ᆠ
矄盐 ᆠ
ᨨ
�盐矄
⏌


҈黨


⏌


ö æ s
ç ҈ - ÷
p ø ò ⼄ =
è f= e
e


ò

҈䰘


æ 
ö

s

p ò ç  - ÷  ç ÷ f
ø
è
 = 
=



\ = = ⏌
æ f �


ö s

p ò ç  - ÷  ç ÷ 
 ´

ø
è 

b

ᆠ㡰 R 㡰 ᨨ
h
>

R
 
r
㦨


Q
e



矄ᨨ�
 �
矄盐 
泐
 

矄盐 ᆠ

P

R �  = = 䰘° R � é
ù Q q�n = Q [҈ - ᆠᨨ矄䰘°] = Q ê҈ - ú = ë �û �

R R �


ᨨ鼀
෌盐矄矄Ἐ矄
 ᨨ鼀
Q
㦨


㡰

҈ ö æ =
p (҈  ᆠᨨ矄 q ) =
p (҈  ᆠᨨ矄
�° ) =
p ç҈ ÷ è
ø \
盐
矄矄� �
 ᨨ盐�
ᆠ盐
矄盐 ᆠ

q�

e


Q \

f = � Î
ᨨ
h
>

R
 
r
>
R






졔Q h


R

f=

R

q�n Q = Î
Î


r

ᆠ㡰

ᨨ
h




R

㡰


 R
 
r
㦨
R � � q�n =
Î


q

R


R �
= =


Þ q = �°
矄� q = R �






é  ù
Q q�n = Q [҈ - ᆠᨨ矄 q ] = Q ê҈ - ú = � ë �û \

f =

q
 �矄
ᆠᆠᨨ� �
ᨨ
෌盐矄矄Ἐ矄 Î


 ᨨ鼀
졠�ᨨ
㡰
 
㡰

  
ᨨ
 
ᆠ盐盐
犨 �ᆠ
�矄 �  

ᨨ� 矄
A
 
B
ᆠ盐矄
ᆠ�泐
�矄
��ᆠ
ᆠ㡰 쟼ᆠ�ᆠ
�


�矄 ᆠ r
ᨨ鼀
 
 �矄
ᨨ
ᆠ泐� 











+ + l + + E
㦨
+ + l
pe


҈ \


E
µ

l㦨
ᆠ �

盐 �
 � 㡰 


Q � Î




b㡰
꾀ᆠᆠᨨ� �
ᨨ
෌盐矄矄Ἐ矄
犨
쟼ᆠ�ᆠ
盐

f

҈ ö æ
p ç҈ ÷ ҈ ö
QἘ è Q æ
ø ==҈÷ e

p
e
çè
ø 

ᆠ

㡰 盐
ᨨ
矄泐鼀鼀泐

 

ᆠ�ᆠ
盐
矄矄� �
 ᨨ盐� a x =+

a a

z 㦨
+
�矄
矄鼀
x
㦨


꾀ᆠᆠᨨ� �
ᨨ
෌盐矄矄Ἐ矄
 ᨨ鼀


ᆠ�ᆠ
盐
 q�

ᆠ�ᆠ
盐
ᆠᨨ矄矄� �
 ᨨ盐�
 泐
ᆠᨨ矄
矄盐 ᆠ
f
㦨
e
- q + q - q q = 㦨

e
e
ᆠ

㡰


ᨨ �
ᨨ

ᨨ� 矄
�
ᨨ
 
ᆠᨨ 盐ᆠᨨ
�矄
矄鼀

盐矄 ᨨ �

A
㦨
ᨨ �

B

ᨨ
ᆠ�ᆠ
盐
 ᨨ盐�
ᆠ�泐
=


ᨨ
h
>

R
 
r
>
R �
R �

҈




҈


f=


R R � �


� q

 


矄� q =

q


졔턼


ᆠᨨ鼀ᨨ  
ᨨ ᆠ�ᆠ
�
㜠  �ᆠ盐
ᨨ
 
 쟼 쟼ᆠᨨ矄
q 矄盐 ᆠ
�矄
E
ᆠᨨ矄
q

E
矄
犨
矄
q
ᆠ  �矄
ᨨ
�  
ᨨ� 
ᨨ
 

矄盐 ᆠ



ᨨ
盐
 ᨨ盐�
 
ᆠ盐
 

矄盐 ᆠ
矄 ᨨ盐

矄矄

�






㡰 
ᆠ�ᆠ盐鼀㜠  ᆠ
ᨨ
 
 矄盐 ᆠ
�矄


⼄盐�ᨨ �
ᆠ盐矄  
ᆠ�ᆠ盐鼀  ᆠ
�矄 ⼄盐��矄 
ᨨ鼀
졔
턼

-

+ r

+

+

+

-

Topic-4 : Miscellaneous Problems z

l

҈ ҈ 㦨

⼄
㦨

᲼l
맔



矄�


桌 Î
� Î


꾀

l R 㦨
Î
꾀
㦨

矄�



ᨨ
꾀ᔴ
㦨


矄�


꾀矄ᨨ

ᆠ�ᆠ
�
�矄
 �ᆠ盐
ᨨ
  犨�
  ᨨ
�矄
㜠ᆠᨨ鼀
ᨨ   犨�

ᨨ


 ᔴ

҈


㡰


⼄盐 ᆠ泐
ᨨ
和㝐
 ᨨ鼀
泐
犨ᨨᨨ
ᨨᆠ├

쟼





҈



n

㦨


҈
p

k
m

졔
턼

�矄
盐
ᨨ
⼄盐 ᆠ泐
犨�
鼀�
盐 ᆠ  �
ᆠ盐矄
犨 

ᆠ�ᆠ
�
�矄
矄犨�ᆠ 
ᨨ

 
盐
ᨨ
├
 
鼀
�矄
ᨨ
 ᆠ QE  
鼀
ᨨ矄��ᨨ
ᨨ
和㝐
矄 � 矄
ᨨ犨矄
�� 
泐
l
㦨
k 盐
ᨨ
ᨨᆠ
ᆠ� �
F
㦨
qE


WWW.JEEBOOKS.IN

PS-181

Electric Charges and Fields



㡰 
ᆠᨨ矄�ᆠ
ᨨᆠ

盐 �

ᆠᨨ矄�ᆠ
矄矄盐 

ᨨ� 
ᨨ
 
矄盐 ᆠ
ᨨ

盐 � ᨨ鼀泐
ᆠ �




s
҈ e
E
=

Î
s
´ p
\

ᨨᆠ
ᨨ

鼀�矄 �ᆠ
矄 
F
㦨

Î
s
R
ᨨ

F µ e
矄  =




é kq
҈ ù ᆠᨨ =
+
ê
´ ú =
ûú
a ëê a 和盐 ᆠ
 矄�ᨨ
ᨨᆠ
F矄
㦨
ga






ᔼ
⼄盐���盐鼀


q




a


q


æ q
ö a k = ç ÷ \ ç g ÷ è ø 
҈
㡰 y




├⼄

l Fnet
㦨
pe x
�



f
µ
x





├⼄







´ ᆠᨨ 矄  㦨 ga

l
x \
Fnet
㦨

pe


d x ␔ 
x



d
 

´ ᆠᨨ 矄 

a










├⼄ 

҈⏌ 





x

l éd + x  d + xù \
Fnet
㦨

pe
êë d
- x
úû

㡰 
ᆠᨨ矄�ᆠ
ᨨᆠ
ᨨ
ᨨ 
ᆠ �
盐
ᨨ
鼀� � �
  ᆠ �矄


kq






\
N

㦨


䰘


p













q x





㡰 l 졔 x0


⧤ d 鼀 ᔼ
�ᨨ
矄

⼄盐���盐鼀

⧤

⼄
㦨
⧤ 졠��� 
⼄盐 ᆠ泐

=









�㡰

l +
㡰 ␔ 
�矄ᆠ
ᨨ犨矄
�� 
泐
 �

 =


⧤ l +
+ 㡰

× ⼄ 
⧤


졔
㡰 

鼀 =

x ù
⧤⼄ é ҈+ ú

⧤
x

졔
x㡰  ê x
û l  l + x
㡰 ë 


⧤⼄ é x ù = ҈ú

⧤
x

졔
x㡰 ê x
û l + l + x
㡰 ë

=


⧤⼄ l + x
㡰




⧤ =

é  x
ù x + ҈ú
⧤x


⧤x = - ⧤ ê l + x l + x

ë û

-
⧤ 鼀 ҈
⧤ ҈ ⧤ 犨
⼄盐 ᆠ泐

Ἐ = =
=
p 鼀 p 鼀 \ d 㦨
p
㦨

҈
矄 쟼ᆠᨨ矄�ᆠ
ᨨᆠ
�矄
ᆠᨨ 矄�
�
盐

 ᆠ
犨ᨨ├
ᨨ 
�矄 
�   
꾀矄
ἘlἘ
�矄
��

鼀
㦨

⧤

Þ  =

�


ᆠ㡰 
ᨨᆠ
ᨨ
ᆠ �
q
犨 
�
�矄
��

矄鼀
�矄ᆠ鼀  x l l Fnet
㦨
F҈

F
㦨 l l ҈ ҈  졔q F

pe
d  x
pe
d + x F҈





x d

x d


d � 
ᆠ �
҈

l

� 
ᆠ �



l

�矄ᆠ鼀 
  ᨨ 
q  
ᆠ �
졔q
犨�
 ᆠ盐
和㝐
F
ᔼ
ᆠ矄
ᨨ
ᆠ �
q㡰
F

> F҈ F ҈
�.e.

ᆠ �

q
�矄 盐 矄

  ᨨ
  ᆠ �
q
ᆠᨨ � 盐矄
ᨨ 鼀ᨨ
�
 
�ᆠ�ᨨ
ᨨ
�矄 � 
ᆠ �
҈ � 
ᆠ �

�矄ᆠ鼀 
v 㡰 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ
盐 
 
� 盐 ᆠ
ᨨ
ᆠᨨ盐ᨨ鼀 
⼄ �
ᨨ
ᆠ �
졔턼

ᆠ �
q
�矄 鼀ᨨ� �
ᨨ盐 
�
�

��ᆠ 졔Q ᨨ�
 
 �矄
矄�盐�ᨨ
�矄
矄 ᨨ犨

�
 
��盐
犨 �ᆠ
�矄
矄�鼀�
ᨨ
  
ᨨ� �
ᨨ盐 
矄盐


�矄 ᆠ
ᨨ
 q
ᨨ鼀
졔
Q
�矄 ᆠ  �� �

  ᨨ

ᨨᆠ
犨
 
ᆠ �矄
犨�
ᆠ  �

 矄
ᨨ
 
ᆠ �

q
犨�

�
犨 
 
ᆠ �
�矄
 ᨨ
졔
Q
矄
ᆠᨨ鼀
ᨨ
犨 
�
�矄


 ᆠ


 
 �盐
ᨨᆠ�泐 ᨨ
ᆠ �

q
�矄
ᨨ
ᆠᨨ 矄 


�ᆠ�ᨨ
ᨨ
 
ᨨᆠ�泐
ᆠ  �矄 ᆠᨨ � 盐ᨨ盐矄泐

  ᨨ

� 
鼀ᨨ鼀 盐鼀
�矄
ᨨ
ᆠᨨ 矄 

  �盐
鼀ᨨ鼀 盐鼀
ᨨ
ᆠ �
 q㡰
ᨨ盐
Q
�矄
ᆠᨨ 矄 
ᆠ盐矄  
ᨨ⼄盐
ᨨ盐
ᆠ �
졔
Q
�矄
ᨨ
y 

ᆠ㡰 
Q

 
ᆠ �
ᨨ

 
� �

 
��
ᆠ �
 x Q q
�矄
矄
ᨨ鼀
ᨨ� 
P






㡰


ᆠ�ᆠ
�

P E 盐
ᨨ
 
ᆠ �
� �
犨�
 q O z ᨨ �
ᨨ矄��
z㜠 �矄
 
�矄 P





㡰 鼀� �盐
犨�
 R  Qz
҈
E㦨
pe
 R
+ z

㡰⏌


 ᨨ

ᨨᆠ
ᨨ
ᆠ �
P
犨�

ᨨ犨矄
ᆠ 
矄
矄 ᨨ犨
 
�矄
鼀� �盐
�矄 ҈ Qq
Fe
㦨
qE
㦨







҈㡰
pe
 R
+ Z

㡰⏌
和�鼀�泐

犨 
�
ᆠᨨ矄矄矄
 
ᨨ���

 
ᨨᆠ
�矄
��
ᨨ犨矄 ᆠ 
O


盐矄
 
鼀ᨨ�ᨨ
ᨨ
 
�ᆠ
�矄
�ᨨ�ᆠ
ᨨ

盐矄
ᨨ
Z
泐� �
犨


 
¥
和ᆠᨨ 泐
�
Z




R

R

졔 Z

㡰⏌

» R

҈ Qq

Z ᲼ ᨨ鼀
⼄盐�ᨨ
҈桌
pe
R 
�.e.
 
矄ᨨ� �
ᨨᆠ
Fe
µ 
Z


 ᆠ
 
鼀ᨨ�ᨨ
ᨨ
 
�ᆠ 犨�

矄�鼀
鼀ᨨ �ᆠ


��
矄��
�鼀�矄
 
  ᨨᆠ
�矄
ᨨ犨矄
�矄
鼀
ᨨ矄��ᨨ 㡰
Fe
㦨


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EBD_8341 PS-182

Physics

Electrostatic Potential and Capacitance

16

ᔴ �
ᨨ
 
�


矄 

Q҈ = s
pr
ᔴ �
ᨨ
 
ᨨ盐
矄 

Q
= s
pR


Topic-1 : Electrostatic Potential and Equipotential Surface ҈


\
ᨨ
ᆠ �

Q = s
p r
+ R
㡰

KQ  ᆠ㡰 ᨨ �

 
ᆠ 

VC = QQ  =
\VC =


Þs=

ᨨ �

 
ᆠᨨ鼀鼀ᨨ
ᆠ 


æ ҈ ö KQ҈ KQ
ç 犨  K = ÷ +
pe
ø r R è K s
pr
K s
pR
+ = K s
p  r + R 㡰




= r R KQ
pr + R㡰 ҈  r + R 㡰Q =




=




pe
r
+ R
㡰
pr + R 㡰 VC =


E

ᆠ�ᆠ
�
ᨨ盐ᆠ
泐
ᆠ
ᆠ �

 
ᆠ 

 

矄盐 
ᆠ�ᆠ
�
犨�

EC
㦨



矄� ᆠ
⼄盐
ᆠ�ᆠ
� ᆠᨨ矄

ᆠ� �

⼄盐
 �
矄ᨨ
 �
矄盐 
�矄
⼄盐
ᨨ
ᨨ

E

䰘



E

䰘


䰘



E

�


䰘






㡰 ᨨ
ᆠ �
Q҈ + Q
= Q Ἐ҈ + Q Ἐ



鼀� �鼀盐鼀
ᨨᆠ�泐

㡰
矄 ᨨ盐
盐矄
矄盐 �ᆠ� 
ᨨ
ᆠ
  ᨨ� 
ᆠ �

 
ᆠ 

  ᨨ

é ⧤⼄ ⧤⼄ ù ҈ 鼀

= ⼄ ᲼ᔴ -  桌 = ⼄ ê
ë  � úû
⧤⼄

҈ ⼄


= = ҈�鼀 ҈�
pe
鼀

KQ Ἐ҈ KQ Ἐ
=
㦨
҈



㦨
黨
Ҙᔴ R
R   Q Ἐ҈ =
Q Ἐ
Þ
Q Ἐ
+ Q Ἐ
= 黨mC \ Q Ἐ҈ =
mC
 
Q Ἐ
=  mC

kQ kQ VA = ҈ +
R
R

R
R

æ

kQ҈ kQ
+
R
R











ç 

├
㦨


è

ᨨ �
�  ᆠ


DV = VA - VB =



҈

㡰 ␔

��
犨ᨨ
鼀�ᆠ
ᨨᨨ犨
矄 矄
ᨨ
��
R
 

R � �
ᆠ �矄
Q҈
 
Q

矄ᆠ�泐
ᨨ �
ᨨ
 
矄盐 ᆠ
ᨨ
�

 Q
Q҈ 矄 

A㡰

VB =





B

ᨨ犨
犨 

ᆠ �


턼
�矄 ��
ᨨ
ᨨᨨ犨
矄  쟼ᆠ�ᆠ
�
犨

矄盐 ᆠ
鼀�
盐 ᆠ  �


҈ ö ÷
pe
ø

졔턼 턼

졔턼

쟼

졔턼 턼

턼

r KQ E=
r

矄

�
� 矄�
 
ᨨᨨ犨
矄 �ᆠ
矄 
㦨

\
ᨨ �
�  ᆠ
犨
 鼀
鼀�
盐 ᆠ  �

s

ò

r r

�


E
dr = V 䰘


C r

㡰 ␔ 
ᆠ �
턼
�矄
ᨨ
�


矄ᨨ�
ᆠᨨ 盐ᆠ� �
矄  쟼ᆠ�ᆠ
�
犨
矄 �ᆠ
矄盐 ᆠ

r KQ r r E =
Sᨨò E
dr = V g�ven r

A

㡰 
s

 
矄盐 ᆠ
ᆠ �
 矄�泐
ᨨ
 
矄 矄


R

ö

æ
⼄
ç ÷ 鼀 è ҈� ´
pe
 ø

Þ 
=

Q  kQ҈  × = × ҈ ҈
p Î
R
R

s


+ 


l =
+ 
= �

\ V҈ = V
=

ᨨ �
ᨨ
 
矄盐 ᆠ
ᨨ ᨨ盐
矄 

B㡰

⧤⼄


㦨
 
㦨




E




= ҈
mC - mC = 黨mC

犨ᨨ
�矄ᨨ
ᆠᨨ 盐ᆠ� �
矄 矄
S҈
 
S


ᨨ犨
ᆠᨨ

ᆠ
泐 
ᆠᨨ 盐ᆠ� �
犨�





ᆠ㡰 ᨨ �

 泐
ᨨ� 
ᨨ
 
ᆠ �
� �

 =

䰘

E

Q


p  r
+ R
㡰

uur

ᆠ㡰 ෌�

E = ( Ax + B ) �ˆ ᨨ E
㦨


x
졔
҈


ò

矄� �
V = Edx


犨
 ҈

V


V҈
㦨



ò (

x + ҈
) dx

-�


㦨

҈



ᨨ V҈

V

㦨
҈




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PS-183

Electrostatic Potential and Capacitance

黨


b㡰 쟼ᆠ�ᆠ
ᨨ �
�矄
ᆠᨨ 矄 
� 矄�

ᆠ �
矄 �ᆠ
矄 
㡰 


�矄 ᆠ
� 
ᨨ鼀
ᨨ� 


ᨨ
 
�ᨨ矄
ᨨ盐ᆠ 矄鼀
ᨨ �


⼄


⼄


⼄
⼄ \ =
( + )

( )

Þ


= + Þ =



҈




҈

 ᨨ
�矄 ᆠ
ᨨ鼀
꾀

犨 �ᆠ
ᨨ
ᨨ
 鼀
ᨨ盐ᆠ
  矄鼀
ᨨ �

= ҈




҈

+ =



҈

b㡰 
Q

 
ᨨ
ᆠ �
ᨨ
 
矄盐 ᆠ
ᨨ
 
矄 �ᆠ 矄 


矄盐 ᆠ
ᆠ �
 矄�泐
�矄


҈


Q

꾀
ᨨ
�矄
ᨨ犨
ᆠ盐
pR


ᨨ

a
p
㡰


ᆠ �
ᨨ
 �矄
ᨨ
矄泐
q


⧤⼄ 꾀 ⧤⼄  ⧤⼄ᔴ + +   ᆠ

 =

҈
p Î


 =

s Î


s ᔴ s  s  꾀

é s
p
s
p
s
pᆠ
ù + ê ú  ᆠ ûú ëê 

 ᆠ

é 
- 
ù + ᆠú ê úû ëê 

ᆠ㡰 ᨨ �
�� 
�矄
��
泐
D
㦨
쟼




㦨
쟼
鼀 㡰 D
㦨
쟼
ᆠᨨ矄
q
㦨




맔
ᆠᨨ矄



㦨



 ᆠ

鼀 �鼀盐鼀
ᨨ �


ᨨ� 
ᨨ
 
矄  㦨
�
黨




-  = ҈
    -  S
= = ҈


b㡰 S҈ =

ᆠᨨ 矄 


쟼㡰

\

矄ᨨ盐ᆠ
�矄

� � �泐
�
ᨨ
ᆠᨨ 盐ᆠ� �
 �ᆠ├

ᨨ  �ᆠ├ 矄矄


鼀


q Q =
\
q
㦨
a턼 Q



a 
pR 㡰
pR


r×꾀 µ  e
r
㦨 ҈


ᨨ
   £ ҈鼀
dv 㡰 꾀矄
犨
├ ᨨ犨

쟼
㦨
 dx \҈
 ×҈
꾀 =


ᆠ�ᆠ
ᨨ �
盐
ᨨ

盐 � ᨨ鼀泐
ᆠ �
 �
矄 �ᆠ 矄 

r
>
R㡰
V
=

 =

KQ R ҈



ᨨ �

 
ᨨ� 

㦨

鼀

泐
㦨

鼀
�矄
��
泐

V

R⏌
P

ò dV = 

M A R




B R ҈�


ᨨ犨

ᨨ �

KQ K aQ KQ 꾀
ᨨ� 
A
VA = = ҈ -
a 㡰 R R
R VP ҈ - a = \ V A ҈ -
a

҈



ò


�dx + 
dy 㡰




ᨨ
矄ᨨ� �
犨
�


㦨

҈҈

ᨨ
ᆠ㡰 ᨨ �
�  ᆠ
犨
 泐
犨ᨨ
ᨨ� 矄
�

ᆠ�ᆠ �
�矄
��
r uur泐

㦨
 E × dx VA

KQ K aQ KQ 꾀
ᆠ 


VP = = ҈ - a 㡰 R R R = V
҈ - a 㡰 = V
- aV










ò

dV = - ò 
x
dx

VO




V A - VO = -᲼҈
x  桌

= -

⏌ ᔴ

ᆠ㡰

졠

ᆠᨨ矄q q ᔴ 矄� q ⼄ x  q

⼄

Eᆠ�ᆠ
�

KQ

R㡰
KQ K aQ㡰 KQ = - aV
( EB ) �  = 
R 㡰

R 㡰
R
K aQ㡰 aV ( EP )� �� =

 
( EP ) �  = R
= R
⧤턼 b㡰 ᨨ �
ᨨ盐矄�
 
矄 

ᨨ盐矄� = 

( EB )� �� =

҈҈


犨 

�矄
�矄 ᆠ
ᨨ
ᨨ� 
ᨨ鼀
 
ᆠ 
ᨨ
矄  ᨨ �
� 矄�
 
矄 

� 矄� 犨 
� 
�矄
�盐矄
ᨨ
 
矄 

⧤턼 =

鼀� ᔼ
⼄盐���盐鼀


 = 矄� q 鼀�
㦨
ᆠᨨ矄 q

 q =

 ⼄
= = 鼀�
p Î

´ 鼀� \


⼄

p Î
 q 鼀�

쟼ᆠ�ᆠ
ᨨ �

 
ᆠ 
ᨨ
 
� 

= ҈䰘


├⼄ ├⼄ + =
├鼀� ⏌  q ⏌
⏌


㡰 꾀ᆠᆠᨨ� �
ᨨ
෌盐矄矄Ἐ矄
 ᨨ鼀

ᆠ�ᆠ
�
犨
犨ᨨ

l

�矄
ᆠ�ᆠ
�
犨�
ᨨ盐ᆠ

ᨨ �
pe
urr r �  ᆠ矄

dV
㦨

E
d r

␔ 

ᆠ �
 矄�泐
�矄
��
ᨨ
 

ᆠ泐� 矄
E =

�


ᆠ泐� 

 
ᨨ �
ᨨ

�矄
矄盐 ᆠ
�矄
�  

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EBD_8341 PS-184

҈



Physics

ᨨ鼀
 
ᨨ
 
ᨨ盐
ᆠ泐� 

�矄
�矄
ᆠ盐矄
 
ᨨ � ᆠ矄矄
犨�
�矄 ᆠ
ᨨ

ᆠ �
ᆠᨨ 盐ᆠ� �

ᆠ泐� 
犨 

 
ᨨ� 
ᨨ
ᆠᨨ 矄��ᨨ
�矄
ᨨ盐矄�
 
ᆠ泐� 
盐
犨 

ᆠ �
 矄�泐
�矄
��
ᨨ
 
ᨨ盐
ᆠ泐� 

�
犨� ᆠ  �
�矄
ᨨ �
泐
 
矄鼀
鼀ᨨ盐 
矄
 
ᨨ
 
�

 ᆠ泐� 

 ᆠ
ᨨ
ᨨ �
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犨�

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犨

 
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b㡰 꾀矄
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鼀ᨨ
ᨨ �
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ᨨ
ᆠ�ᆠ
�
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ᨨ � ᆠ矄矄
\ VA
>
VB
 
VA
㦨
VC
ᨨ
⼄盐�ᨨ �
矄盐 ᆠ







dv dt qEdt q 矄� ҈


t �ˆ dt Þ

㦨

㦨
m m  Q
쟼
㦨
矄�
҈


�ˆ �� 㡰


㡰

v

\ ò dv =


泐




҈㡰















҈


㡰 



\

㡰 
ᨨ �
�  ᆠ
ᆠᨨ矄矄
ᆠᆠ�ᨨ
矄
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矄犨�ᆠ
S
�矄
ᆠᨨ矄

␔�
 
ᆠᨨ矄� �
ᨨ
矄犨�ᆠ S
 
S ҈
 
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ᆠ �
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C

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 泐
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 
��
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 ᨨ


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 
C
犨�




 




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
ᨨ���
犨�

��
泐


�









㦨


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҈

⏌鼀 = 


҈

p⏌ w

҈ ҈


´ 

᲼ᆠᨨ矄҈


t 桌
p ⏌ w ҈




é ë






p ù  ᆠᨨ矄
ú
㦨
᲼҈҈桌
㦨


鼀矄҈ ҈


û

d 
x
㡰 -dV = -
x




Þ Ex
㦨
dx dx


ᆠ�ᆠ
�

ᨨ� 
҈





㡰
 
x
㦨
҈㡰 Ex 㦨



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ᨨ
ᆠ�ᆠ
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
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B
ᆠ盐矄

B  
⼄盐�ᨨ �
矄盐 ᆠ矄

ᆠᨨ矄矄
ᆠ

㡰 쟼ᆠᨨ矄�ᆠ
�

P
�矄 P

r ᔴ҈

ur ur ur r uuuur (  r) uuuur E p = E҈ + E
= C҈ P + C
P e
e
r uuuur uuuur r uuuur uuuur r uuuuur = C҈ P  C
P = C҈ P + PC
= C҈C
e
e
e


b

ᆠ㡰uuur uuur uuur  E D  uuur uuur  E A  uuur =  EB  =  EC  = =  E E  =  EF  = K 





(






\

)

(

졔
q F 졔

q A


㦨



҈
鼀

E 
q

(

)

҈ ҈
졔
⧤
졔
⧤㡰
맔

㦨

⧤




q D

O

鼀� 
q

졔
q B

C

EB 졔 EE


꾀矄
⼄
㦨

 
鼀 = p r  \
맔
҈


맔
҈
҈黨
맔



맔
҈




p 

맔
҈
䰘㡰
맔
黨


맔
҈


´҈
-䰘 㡰 ´ 黨

´҈
Þ = p´
\
㦨


´ ҈

´ ҈

´ ҈
-҈黨

)

EO
㦨
EA
졔
ED 졔
EF
졔 EC㡰
ᆠᨨ矄


º
졔
EB 졔 EC㡰
ᆠᨨ矄


º

㦨

⧤
졔

⧤
졔
⧤
졔
⧤㡰
맔


和犨�ᆠ

–r ᔴ



҈

쟼⼄

=


㦨


 ᆠ
鼀 �鼀盐鼀
矄
� 
泐
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�ᆠ
෌�
ᆠ�ᆠ
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V
㦨

x

ᨨ矄

 ᨨ

ᆠ�ᆠ
ᨨ �
ᆠ  �矄
ᨨ 泐
ᨨ �
x㜠 �矄


␔ 
鼀� 
ᨨᆠ�泐
�矄
ᆠ � ⼄쟼
㦨
鼀�


㦨






T
p ù é 矄� ҈


t dt ê 鼀
矄 �矄  =
w´
úû ë


㦨
 êᆠᨨ矄҈


´

␔
├ ᨨ犨
Ex
㦨



䰘


㡰 
ᨨ �
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矄 
�矄
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ᨨ �
ᨨ
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

Q
ᆠ �
ᨨ
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矄盐 ᆠ

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ᨨ �
ᨨ
  矄盐 ᆠ
ᆠ  �矄
泐
 
矄鼀
鼀ᨨ盐 
矄
 
� 矄�

 ᨨ
  ᨨ �
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 
矄鼀
v㡰
b㡰 
ᆠ�ᆠ
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
 
矄盐 ᆠ
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
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ᨨ
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矄鼀
矄
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ᆠ 
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矄 
   泐
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 
矄 
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
矄盐 ᆠ
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矄 
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 ᆠ 
ᨨ
 
矄 
㦨
҈
 
㡰 

ᨨ

ᨨ
ᆠᨨ 矄
矄 
�
 
ᨨ�
ᨨ 쟼ᆠ�ᆠ
�

쟼 =

ò

æQ m = ҈
kg
q = ҈C
E
= ҈NC ҈ ö ç ÷ ç and w = ҈
 矄
҈ ��

÷ è ø

ù ҈ ҈ ҈ q é҈ + +



ú V

㦨
ê
pe
ë x

x
x

x
û q  
㡰 V

㦨

pe
x

҈


p⏌w



쟼

҈黨


q m

q é ᆠᨨ矄҈


t ù \ 
㦨
 ê m ë ҈


úû


ᔴ





㡰 꾀

Q

㦨
鼀
\
⼄쟼
㦨
m



º O



º

EA

ED

EF 졔 EC



WWW.JEEBOOKS.IN

PS-185

Electrostatic Potential and Capacitance

( )

쟼ᆠ�ᆠ
ᨨ �

O


\ pR =
pa
t Þ R = a
t 

҈ VO
㦨

᲼
q
졔
q
졔
q

q

q


q桌
㦨


p e
L


黨






҈ QA ҈ QB \
= =V
p Î
RA
p Î
RB Q
RA
>
RB
\
QA
>
AB


sA s
 
EB = B Î
Î
E A s A RB = = < ҈
\
쟼


쟼 꾀
 EB s B RA V


졔
졔
졔
졔
졔

졔
졔
졔

졔

졔
졔
졔
졔
졔
졔
졔
졔


R


R


r


졔
졔
졔
졔
졔
졔
졔 졔 졔


ᨨ �
ᨨ
 
�⼄盐�
盐
V =

㡰 
P


ᨨ� 
�
 
X㜠Y
 
犨�
ᆠᨨᨨ� 矄
x, y㡰
 犨 �ᆠ
 
ᨨ �
盐
ᨨ
ᆠ �矄


Q
ᨨᆠ
 a


㡰
 
졔
Q
ᨨᆠ

졔



㡰

ᨨ
Y

Px
y
㡰

y

y A 㜠
Q X' 㜠a

㡰 O x a 졔x㡰

Y'

K 
Q 㡰


V 㦨

a

�a

黨a

x

a

B 졔Q C X a

㡰 �a

㡰 a㜠x㡰




=

K +Q㡰 a - x㡰
+ y


K  -
Q㡰 K +Q㡰 ҈ +
犨 
k
㦨
x + a  x - a㡰
pe

ù é ҈ \ V
x㡰
㦨
KQ
ê
ᨨ

x

>
a ë x - a x + a úû V
x㡰
㦨



졔
졔
졔
졔
졔

졔
졔
졔




Q Q = C
pe
R



ᨨ �

M

㡰 ᨨ
r


R


 
ᨨ �

�矄 E ᆠᨨ 矄 
 
 
ᆠ�ᆠ
�  矄�泐 �矄
ᨨ

졠�ᨨ盐矄泐

  ᆠᨨ矄�ᆠ
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�矄
ᨨ
ᨨ
r

R

R
r b㡰 

�矄
盐矄
矄

  ᆠ

ᨨ
ᨨ �
�
ᆠ�ᆠ ᆠ�ᆠ盐�矄
ᨨ
ᆠ�ᆠ
盐ᨨ矄矄
ᨨ
矄 �ᆠ
ᆠᆠ�ᨨ

ᆠᆠ� ᆠ
ᔴ
㦨

pe
R
 
ᆠ�ᆠ ᨨ �
V =



҈

�矄
�矄
 
⼄盐�ᨨ
ᨨ

ᆠ�ᆠ
犨�
ᆠ 

�a


㡰
 
�盐矄

a
b㡰 ᨨ
x

a 
盐矄
ᆠᨨ 矄�

ᨨ� 
�泐㡰
M


�矄 ᆠ
x
ᨨ鼀
  ᨨ���
ᨨ
x㜠 �矄


RB RA

㡰 E A =








æ aö =Vç ÷ è t ø

Þ
a - x㡰
+ y
= a + x㡰
+ y
Þ x

�a㡰

졔
y㡰

㦨

a㡰


A



b

㡰


��
V㜠r
�
�矄
ᨨ
ᆠ � ᆠᨨ 盐ᆠ� �
矄 
ᨨ
�盐矄

㡰 

犨 ᨨ
ᆠ �
Q
犨� 
 ᆠᨨ矄
�

矄 
ᨨ �鼀

R

㡰 쟼ᆠ�ᆠ
�
E 㦨
ᨨ
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矄 

 ᆠ
ᆠ�ᆠ
� �矄
�矄ᆠᨨ � 盐

r
㦨
R

ᆠ㡰 ᔴ  �矄
�

 
E
 ᆠᨨ � 盐ᨨ盐矄泐
矄 
ᨨ
r
>
R


졠�ᨨ
ᆠ㡰
�矄
� ᆠᨨᆠ






(a t )

҈

a + x㡰 + y

QA

҈


æ Va ö Kç ÷ è kø

Kq V҈ = = R

\


R
R


pÎ
꾀V
pR A Q s = B´ A= B´ ᆠ㡰 꾀矄
A = 㦨

Q sB B R QB R

p Î
RBV A



ᨨ �
ᨨ
 
ᨨ

ᨨ �


ᨨ� 矄
ᨨ
 
� 

�矄
矄鼀
ᨨ
ᨨ
� 
和
PR �矄
 �ᆠ盐
�矄ᆠᨨ
 
⼄盐ᨨ�
� 㡰
ᨨ
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ᆠ�ᆠ �ᨨ矄
AB

FE
 
BC

 ᨨ
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ᆠ�ᆠ
ᨨ �
犨�
 ᨨ

 泐
ᨨ� 
ᨨ
PR
ᆠ

㡰 㡰 ෌盐矄矄Ἐ矄
犨
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�
ᨨ 泐
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쟼
µ 
㡰 ෌盐矄矄Ἐ矄
犨
ᆠ

ᨨ

盐矄
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
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�ᨨ
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ᆠᨨᆠ
矄
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ᔼ
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 ᆠ
 泐
ᆠ

ᨨ

ᨨ
㡰 WAB
㦨
q
DV㡰
㦨҈
VB

VA㡰 ᨨ

WAB
㦨
VB

VA 

b

ᆠ

㡰 㡰 쟼ᆠ�ᆠ
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� 矄�

矄 �ᆠ
鼀�ᆠ
矄 
犨�
ᆠ �
ᨨ

 
矄盐 ᆠ
�矄
犨泐矄
ᨨ
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EA
� 矄�
㦨

㡰 ␔ 
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矄 矄

ᆠᨨ

ᆠ
犨�

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鼀
犨�
 

ᆠ�ᆠ
ᨨ �矄
犨�

⼄盐

�.e.

VA
㦨
VB
㦨
V




pRB

҈

和�鼀�泐

ᨨ






x



a


ù é ҈ ë a - x a + x úû

V
x㡰
㦨
KQ
ê

和� ᆠ
ᆠ�ᆠ
ᨨ
ᨨ
ᨨ �
ᆠ盐矄
 
x㜠 �矄

a

㡰
 
黨a


㡰  ᆠ

V
x㡰
㦨



x
㦨
a


x
㦨
黨a 掸 ᨨ鼀
 
ᨨ
 矄矄�ᨨ 矄 V
x㡰
®
¥

x
®
a
 
V
x㡰
®

¥

x
®


a 掸 V
x㡰
®


矄
x
® ±
¥ 掸

V
x㡰
�矄


҈ �
� 
x

盐 ᆠ�ᨨ
x㡰
ᨨ
 
犨 ᨨ
x �矄
矄
矄 ᨨ犨
ᨨ犨
V

a a

a

黨a

x

kq Va Þq= a K

␔  q 㦨
ᆠ �
ᨨ
 
�⼄盐�
盐

 
ᆠᨨ矄� �
 
�盐矄
ᨨ
ᨨ
ᆠᨨ鼀矄
R
盐
 ᨨ盐鼀
鼀� 矄
ᆠᨨ 矄 

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EBD_8341 PS-186

Physics \
ᨨ
ᨨ �

 
ᆠᨨ鼀鼀ᨨ
ᆠ 
�졠

ᆠ㡰 ᨨ鼀
 �泐
ᆠᨨ 矄�ᨨ
� ᆠ� 
v

 
矄
ᨨ
�ᆠ ⧤
쟼

졔

쟼
㡰ᆠ 
㦨
⧤
쟼

졔

쟼
㡰ᆠ�ᆠ盐鼀  ᆠ

V
㦨
V�
졔
V

㦨


é Qq
Qq ù ҈
é Qq
Qq ù = mv + K ê

졔
K
ê ú
a
a
ë û ë
a ҈
a úû ҈
KQq


\ v
㦨
mv =

a �


KQq =
ma

҈ æ Qq ö ç ÷
pe
è
ma ø

҈


KQR


R +r




㦨


kq A kqB kqC + + a b c


q





kq A kqB kqC s + + ᨨ

VB
㦨
b b c e


é a
ù ê - b + cú êë b úû


矄��
矄 ᨨ犨矄
 
 �泐
ᨨ
 
ᆠ �
�矄
ᆠ矄� �
ᆠ㡰 꾀ᆠᆠᨨ� �
ᨨ
犨ᨨ├
 �泐
 ᨨ鼀

��
�
├� �ᆠ
 �泐
�矄 ⼄盐
ᨨ
犨ᨨ├
ᨨ 
�
�矄ᆠ鼀 
ᨨ
ᆠ �


\

vµ

m




v
= =

҈
v ҈ é ù ê ҈  - q 㡰Q ú q- q㡰 qQ ê ú 㡰 U = + + dö æ dö ú
p Î
ê d æ çè D + ÷ø çè D - ÷ø ú êë

û
é ù
q qQd ҈ = -
ú

ᔼ� ᨨ� �
d ê
p Î
ëê d D ûú






b㡰 
ᆠᨨ矄�ᆠ
 �泐
ᨨ
 
矄泐矄鼀


pR


qR

㦨
Qr


qr


æ q = ᆠ �
ᨨ
�


矄  ö \ q
㦨













çè Q 
q 㦨
ᆠ �
ᨨ
ᨨ盐
矄  ÷ø R +r

é ⼄
턼⼄ 턼⼄ ù  = ⧤ê + + =
  
úú ëê û

Qr


ᨨ �

O
盐
ᨨ
�


矄 

턼

é ҈ ù Þ ⼄ = -턼 ê҈ + ú





 ë û

æ ö q = K ç Qr ÷


ç r è R + r ÷ø r Qr


V�
㦨
K

҈

\

qö æ çè 和盐 ᆠ ᆠ �  矄�泐
s = ÷ø A

V�
㦨
K

҈
mv = q DV


㝐矄矄
ᨨ
泐ᨨ�
�ᨨ
鼀
㦨
҈ 㝐矄矄
ᨨ
�盐鼀
�ᨨ
鼀
㦨



Q-q

ᨨ
qR

㦨
Q

q㡰
r





ᨨ


Þ턼=

R
+ r


ᨨ �

O
盐
ᨨ
ᨨ盐
矄 
ù é Q - q㡰 = K êQ - Qr ú V

㦨
K

R
R + r
ûú ëê R

-⼄

졔҈

졠


㡰 턼


�




ù
é
KQR K ( QR ) K ëQR + Qr - Qr û =
㦨
㦨


R
+ r
R R + r 㡰 R R
+ r


(

d



DV
㦨
ᨨ �
�  ᆠ
犨
犨ᨨ
ᨨ矄��ᨨ 矄
ᨨ
ᆠ �
q
ᨨ
矄鼀
q
 
DV.

ᨨ ac

bc
졔
c

㦨
a


b

졔
c




ᨨ
c
㦨
a
졔 b ෌�
矄盐 ᆠ
ᆠ �
 矄��矄

⼄盐



pr


d⏌



q
æ k
q k
q ö Du = q ç =÷ è d ⏌
d ⏌
ø pe
d

kq A kq B kqC + + c c c


ù é s a -b +c ᨨ

VC
㦨
ê ú e
ëê c ûú 㡰 Q VA = VC
�� 㡰 s s é a
- b
+ c
ù a - b + c 㡰 = ê ú c e
e
ëê ûú

=

R
+ r


X

q

O

d⏌


꾀 
VC
㦨


q

KQ R + r 㡰

q

s ᲼ a - b + c桌 e


和�鼀�泐

VB
㦨


\

+

Y

҈
pa
҈ 
pb
㡰 ҈ 
pc
㡰 ´s´ s + ´s 㦨
a b c
pe

pe

pe






R +r




㡰 ᔴ  �
�
ᨨ �
 �泐

Du = qV f - V� 㡰 ᨨ �
ᨨ
 q
�矄
矄鼀
矄
� ��
 
� 
ᨨ� 
ᨨ
 




k s 
p a
㡰 k ( -s ) 
pb 㡰 k s 
pc
㡰 + + 㦨
a b c

㦨





Topic-2 : Electric Potential Energy and Work Done in Carrying a Charge

ᨨ �

 泐
矄 
�矄
盐
ᆠ �矄 
矄 
꾀


 
ᔴ
ᨨ �
ᨨ
矄 
A VA 㦨


KQr

) (

)

졔⼄




졔⼄

턼



졔

㡰

b㡰 졠




㡰 턼


턼





㡰

ᨨ �

ᨨ���

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PS-187

Electrostatic Potential and Capacitance

⧤턼 ⧤턼 ⧤턼 ⧤턼 + + +





҈ ö æ  
ᨨ �

¥
㦨


㦨⧤턼
ç ҈+ ÷ �ø è


㦨


q

� 
 �泐

\
␔ᨨ├
⼄盐�
ᨨ
盐

� 
ᆠ �
턼

ᨨ���
␔
㦨
턼

ᆠ㡰 
犨ᨨ├
ᨨ 
�
鼀ᨨ� �

ᆠ �
ᨨ �

⼄盐�ᨨ � 矄盐 ᆠ
�矄
犨泐矄
ᨨ



�ᆠ�ᨨ
ᨨ
ᆠ�ᆠ
�
�矄
 �ᆠ盐
ᨨ
 
⼄盐�ᨨ � 矄盐 ᆠ
ᨨ
� 矄


䰘


ᆠ㡰 㡰 
ᆠ�ᆠ
�
盐
ᨨ
ᆠ �


æ 
q ö çè ÷  ø

꾀矄ᨨ


º

⼄  

҈





҈҈


ᔴ 꾀 㦨矄�


º
Þ
ᔴ
㦨 㦨
 꾀


é( ⼄ ⏌ )(
⏌ ) ù é ( ⼄⏌)( -
⼄ ⏌ ) ù é( ⼄⏌)( -
⼄ ⏌ ) ù ú+⧤ê ú +⧤ê ú
 ë û ë û ë û ├⼄
é ҈
ù -
㦨 ú
¹

黨 êë
û
ê

ᆠ㡰 㝐� �盐
ᨨ
ᨨᆠ
犨

 
ᔴ

(

)

҈



é +q ⏌  ù é +q ⏌  ù é -
q ⏌  ù 졔
⧤
ê
졔
⧤
ê 㦨

ú ú ë û ë û ë úû

)

)

)

Qq q
Qq + + =
a a a


ᨨ

Q
+ q
+ Q =
\ Q
㦨
-

q

+҈

=-


q


+


b㡰 㡰
ᨨ

d


x


d

 
ᆠ�ᆠ
�
�矄
ᨨ �
졔
x㜠 �矄
ᨨ

ᨨ 
ᨨ� 矄

쟼
�矄
ᨨ �
��
x㜠 �矄
泐 쟼  ᆠ

쟼
犨�
ᨨ

矄鼀
�ᆠ�ᨨ

졔q㡰 ᨨ �
 �

x㜠 �矄
























 쟼 㡰 쟼ᆠ�ᆠ
�

P


ᨨ� 
ᨨ

쟼q㡰 泐㜠 �矄
�矄

ᨨ
x㜠 �矄

ᔼ
�矄
ᨨ ᨨ �
x㜠 �矄
�矄
ᆠ㡰 쟼ᆠ�ᆠ
ᨨ �
 q 졔q x ᨨ���
㦨
ᨨ 
d


㡰 



㡰 d


㡰 ᨨ
犨ᨨ├
矄
ᨨ

ᨨ 
�
� �� �

矄
ᆠ �
ᨨ鼀
� � �泐
ᨨ  
ᨨ���
㡰 
�ᨨ
鼀ᨨ鼀 
�矄
�ᆠ
ᨨ鼀

q
ᆠ �
ᨨ
졔q
ᆠ �
ᨨ �
��
�ᆠ�ᨨ

ᨨ
x㜠 �矄
\
�ᨨ
鼀ᨨ鼀 
㦨



qd
ᨨ �
x㜠 �矄
b㡰 
犨ᨨ├
ᨨ 
�
鼀ᨨ� �

ᆠ �
q
ᨨ鼀
ᆠ 
ᨨ
ᨨ 
� � ᨨ
ᨨ 
㦨
ᆠ  �
�
ᨨ �
 �泐 Q҈

㡰 ᨨ �
졠㦨
⧤
ê



(

\



⼄
҈ (
⼄ ⏌ )( ⼄ ⏌ )

㦨

4π Îᨨ �
π Îᨨ


ax � 泐

b㡰 和� ᆠ

ᆠᨨ矄�ᆠ
 �泐
㦨



졠

㡰 ᨨ �
 �泐
ᨨ
 
矄泐矄鼀
⧤

㦨

Qq
pe


(



졠
犨�
�
ᆠ ᆠ


ᆠ�ᆠ
�

O
盐
ᨨ
ᆠ �

q ҈
q ⏌ 
㦨

4π Îᨨ R

π Îᨨ
꾀ᔴ -
⼄ ᔴ ᔼ
D
꾀ᔴ 㦨

矄�

º 꾀 

º 꾀 꾀 ⼄ Þ꾀ᔴ㦨 㦨



x a졔x


(

q q

꾀
 
ᆠ �

  

쟼=





q


Qqa ҈ ù+ ҈ q = é ҈ êë a + x + a - x úû
pe

a
pe a
- x

é ù
Qq ê ҈ a ú U - Uf = \
�
pe
ê a a
- x
ú ëê ûú é


ù
Qq ê a - x - a ú -
Qqx
= =
pe
ê a a
- x
ú
pe a
ëê ûú
犨 
x
҈桌 犨
ᆠ �
㦨
KCV
犨
ᨨ �
 �泐 㦨


V
d

҈ KCV




E
㦨



矄鼀
ᆠᆠ� ᆠ
ᨨ
 
ᨨ
ᨨ�ᨨ


V
d

�㡰  ᨨ
盐 ᆠ �
ᆠᆠ�ᨨ
矄 ᔴ �
ᨨ
ᆠᆠ�ᨨ
A qA
㦨


맔
҈


맔
҈


㦨

맔
҈

ᔴ












ᔴ �
ᨨ
ᆠᆠ�ᨨ B
qB
㦨



맔
҈


맔
҈


㦨



맔
҈

ᔴ ␔ 
盐 ᆠ �
ᆠᆠ�ᨨ
矄

 
ᆠ �
ᨨ
ᆠᆠ�ᨨ
A

C
 
B 
q҈

q

 
q
矄ᆠ�泐
泐
ᆠ �
ᆠᨨ 矄�ᨨ


҈ ҈ ҈ = + dC dC҈ dC


犨 
dC҈
㦨
ᆠᆠ� ᆠ
ᨨ
ᆠᆠ�ᨨ
犨�
�ᆠ�ᆠ
K҈ dC

㦨
ᆠᆠ� ᆠ
ᨨ
ᆠᆠ�ᨨ
犨�
�ᆠ�ᆠ
K

+ + +

A

m q҈

҈



+ + +

K҈ bdx 㡰 e
dἘ d é xù d'
㦨
d

x = d ê҈ - ú l ë lû

dC҈
㦨


B


m q
҈



\

dC҈
㦨


K҈ A e
( dx) K҈b ( dx) e
K҈bl ( dx) e

= d l - x 㡰 d l - x 㡰 é xù d ê҈ - ú ë lû

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PS-197

Electrostatic Potential and Capacitance

和�鼀�泐

dC

㦨


K
e
bdx 㡰 K
e
bdx = xd d -dἘ d -d + l

㦨


㦨


K
e
b
l
dx K
e
Adx = xd xd \

҈ d l - x 㡰 xd = + dC K҈ Ae
( dx ) K
Ae
( dx )

Þ

K҈K
A e
dx = dC K
ld + d  K҈ - K
㡰 x

l

dx

ò
K
ld + d  K҈ - K
㡰 x

CA
Ἐ
㦨


l

é ᨨ�᲼ K
ld + d  K҈ - K
㡰 x桌 ù ú d  K҈ - K
㡰 ë û


�



Ae




´


� ´ ҈
-҈
= 㦨




맔
҈
黨
F d


� ´ ҈
-


q
㦨
CAV
㦨


맔
҈
黨
맔
҈҈

㦨




맔
҈
䰘
C \

K҈ K
Ae
K ᨨ� ҈ d  K҈ - K
㡰 K


W
㦨


ù 


´ ҈
-䰘 㡰
é ҈ ҈ ê -黨 -黨 ú

´ ҈
û ë


´ ҈


ᨨ

W
㦨





맔
҈
�


e
e r AB d


 
矄泐矄鼀
矄
矄 ᨨ犨
�
 
��盐矄

�矄
⼄盐� 
ᨨ
犨ᨨ ᆠᆠ�ᨨ矄
�

ᨨ 
犨�
�
 
ᨨ 
犨�
ᨨ�
犨�
�ᆠ�ᆠ
C
㦨
C҈
졔
C


���㡰 ᔴᆠ� ᆠ
ᨨ
B
㦨


k e
 x ´ ҈㡰 e
᲼҈ - x 㡰 ´ ҈㡰 + 㦨




d d

㦨


e
᲼kx + ҈ - x桌 C
㦨
d

ᔴ �
ᨨ
A

qA
㦨




맔
҈
䰘
C
�矄
�矄�盐
� ᨨ
犨ᨨ
矄
\ q҈
졔
q

㦨




맔
҈
䰘
C 꾀矄ᨨ
 
ᨨ �
�  ᆠ
ᆠᨨ矄矄
A
㦨



ᆠᨨ矄矄
B





�㡰

x
㦨
矄�
ᨨ
 
�ᆠ�ᆠ
㦨
҈m
 
d
㦨



҈m

和ᨨ
I =

\

A ᨨ犨

ᆠᆠ� ᆠ CA
㦨
C҈
졔
C





´ ҈
-䰘 㦨
҈


맔
҈
䰘
C ҈




ᨨ
 �泐
矄ᨨ

㦨


㦨


+

+ - A⏌
A⏌
-




´ ҈
-黨 CA
q
㦨





q

q

㦨
CB ҈

´ ҈
-黨


Þ q҈
㦨




맔
҈
䰘
C

A ᆠᆠ�ᨨ矄
�


ᨨ 
� �
�ᆠ�ᆠ
矄
 
  
A  ᨨ 
ᨨ 
�矄
犨� ᨨ盐
�ᆠ�ᆠ
矄
 


+


㦨
҈


맔
҈
黨
F






q

졔
q

㦨




맔
҈
䰘

Þ q

㦨


�㡰 
ᆠᆠ�ᨨ
A
犨�
�ᆠ�ᆠ
矄
ᆠ

ᆠᨨ 矄�
矄
犨ᨨ

҈҈





� ´ ҈
-


Þ q҈
㦨





� ´ ҈
҈
´ �

[҈҈  ҈] ´



҈ =



� ´ ҈
黨 A


҈

+ + +




� ´ ҈
-҈
´ 黨 ´





q҈ q =
C Ἐ A CB

e V \ q = CV =
ëé҈ + x ( K  ҈)ûù


҈ dq e
V dx = [ K  ҈] \
ᔴ盐 

I = dt


҈ dt

�䰘


҈ ҈ C V

㦨

맔


맔
҈
黨
맔
҈҈
㡰

A


q
q
W
㦨

C A Ἐ
C A

㦨
K҈K
Ae

ê

ᨨ

C
㦨


᲼҈ + 黨桌
㦨


맔҈
黨
F

㦨
҈

҈
맔
҈
�
 ��㡰 ␔ �
├� �
ᨨ盐
 
�ᆠ�ᆠ

 
ᆠ �
ᨨ
 
ᆠᆠ�ᨨ 
鼀� 矄
 
矄鼀
盐
ᆠᆠ� ᆠ
ᨨ
꾀
犨�
�
ᆠ  �
 ᆠ
 �泐
犨�
ᆠ  �

�矄
ᆠ  �
�
 �泐
�矄
⼄盐
ᨨ
犨ᨨ├ ᨨ 


ò
K
ld + d  K҈ - K
㡰 x

㦨
K҈K
Ae




´


� ´ ҈
-


UA
㦨

K҈ K
Ae
dx

l




´


� ´ ҈
-҈


쟼 �泐
矄ᨨ
�
 �矄
ᆠᆠ�ᨨ
犨 
ᆠᨨ

ᆠ
ᨨ
V
㦨
҈҈

ᨨ 泐


 ᆠ
ᆠᆠ� ᆠ
ᨨ
 
犨 ᨨ
ᆠᆠ�ᨨ
C
㦨


 A ⏌
㡰 e
 A ⏌
㡰 e
er A e
+ = ᲼҈ + e r 桌 d d
d




´


´ ҈
-҈

´


´ ҈
-黨

+

q҈
q
+

C Ἐ A
CB

҈

´ ҈

´ ҈
-҈

´ ҈

´ ҈
-


㦨




맔
҈
�
졔


黨
맔
҈
�
㦨
҈
҈
맔
҈
�
 �



ᨨ �
�  ᆠ
ᆠᨨ矄矄
ᆠ
ᆠᆠ�ᨨ
�矄
V
矄
ᨨ
ᆠᆠ�ᨨ矄 
�



ᨨ
 �泐
㦨
 �泐
�
A
졔
 �泐
�
B

WWW.JEEBOOKS.IN

EBD_8341 PS-198

㦨


Physics

҈ ҈ CV
+ CV

㦨

CV







�㡰

␔ 
 
矄犨�ᆠ
ᨨ 
 

�ᆠ�ᆠ
�矄
� 矄
犨
  矄
ᨨ
ᆠᆠ�ᨨ矄

 
犨
ᆠᆠ� ᆠ
�矄
ᔴ
盐
ᨨ � �  ᆠ
ᆠᨨ矄矄
�
�矄
V 쟼 �泐
�
A
㦨


҈  C 㡰V



= CV




쟼 �泐
�
B
㦨


CV
҈ q
҈ CV 㡰
= ´
㦨

KC
C




ᆠ �
ᨨ
ᆠᆠ�ᨨ
B
鼀� 矄
矄鼀
犨 
矄犨�ᆠ
�矄
ᨨ 㡰
ᨨ
 �泐
㦨
 �泐
�
A
졔
 �泐
�
B \

ᨨ
 �泐
㦨


 ҈ � CV
+ CV
= CV








��㡰

ᨨ  �泐 � ��泐  CV
= = �

ᨨ  �泐 � 泐 � CV


\ \

V=

쟼ᆠ�ᆠ
�
盐
ᨨ
 
�


鼀ᨨ矄
ᨨ矄��
 
��
ᆠ �矄

é

qù ú
ᨨ犨矄
 
ë r
û


M
E҈ =
ê k

é q ù E
=
ê k
ᨨ犨矄
�� 

쟼ᆠ�ᆠ
�
盐
ᨨ
ú ëê 
r 㡰 ûú  
ᨨ盐鼀ᨨ矄
ᨨ矄��
 
��
ᆠ �矄

M

ᆠ㡰 꾀ᆠᆠᨨ� �
ᨨ
 
⼄盐矄�ᨨ


ᨨᨨ
�矄
矄

矄
鼀�㜠 犨泐
犨
 
犨ᨨ
矄
 
�矄
ᨨ盐 
ᨨ
鼀ᨨ


�

矄ᨨ


ᨨᆠ
�矄


�
ᨨ鼀 �ᆠ
 
犨ᨨ
ᨨᆠ矄
ᆠ� �
ᨨ
  qE 졔 ᨨᨨ
盐矄
 
 
矄

矄 q 㦨

� 矄 ᨨ犨
�
 
��盐


































qE
㦨
mg

æV ö q ç ÷ = mg èdø

 졔  졔  졔 㝐 턼

쟼ᆠ�ᆠ
�
盐
ᨨ
 
 
ᨨ矄��
 
��
ᆠ �矄

M

Topic-4 : Miscellaneous (Mixed Concepts) Problems ҈



ᆠ�ᆠ
ᨨ �
盐
ᨨ

 
ᆠ �矄

M �矄
ᨨ
�




㦨



㝐 ␔ 
 
矄泐矄鼀
ᨨ
ᆠ �矄
�矄
ᨨ
ᨨ盐
� 
PM

 
 ᆠ盐 
犨�

ᨨ
 ᆠ
鼀� �ᆠ
�
B
㦨



M. ␔ 
B
㦨


 
m
㦨

⼄㡰 㝐
�矄
 
鼀�㜠ᨨ� 
犨
 
犨ᨨ
�

鼀ᨨ矄
ᆠ �矄


v

mg

mgd ҈

䰘 ´ ҈
-
䰘 ´ ҈
´҈
-
= ҈
-黨 
㦨
-҈黨 q ҈

´ ҈


é r r r r q ù E =
ê k
ᨨ犨矄
 

�



E҈ + E
+ E = d
ú
ëê r 㡰 ûú


ᆠ�ᆠ
ᨨ �
盐
ᨨ
 
ᆠ �矄

M

é +q -q q q q q ù + - + =
ë r r
r
r r r úû

V =kê


ᆠ盐 
盐
ᨨ
 
�

鼀ᨨ矄
ᨨ矄��
 
��
ᆠ �矄
�矄 ᨨ

和�鼀�泐
 

ᆠ盐 
盐
ᨨ
ᨨ 
ᆠ �矄
�
�矄
�矄 ᨨ

 ᨨ

B
㦨


\m
㦨

㡰

졔

 ᆠ

�矄
泐
҈
맔
҈
黨




ᨨ
ᆠ�ᆠ
盐
 ᨨ盐�
ᆠ�泐
=






ᆠ

㡰 
ᨨ �
ᨨ

ᨨ� 矄
�
ᨨ
 
ᆠᨨ 盐ᆠᨨ
�矄
矄鼀


盐矄
ᨨ �

A
㦨
ᨨ �

B

q
 �矄
ᆠᆠᨨ� �
ᨨ
෌盐矄矄Ἐ矄 Î


 ᨨ鼀
졠�ᨨ
㡰
 
㡰

  
ᨨ
 
ᆠ盐盐
犨 �ᆠ
�矄 �  

ᨨ� 矄
A
 
B
ᆠ盐矄
ᆠ�泐
�矄
��ᆠ
㡰 
턼
�矄
 �ᆠ盐
ᨨ
 
 
ᨨ
 
 �ᨨ


졔





졔

㝐 졔

턼



ᔴ泐
ᆠ�ᆠ
�

 
ᆠ 
M
ᨨ
 
 �ᨨ
盐
ᨨ
  ᆠ �矄

 
ᆠᨨ 矄
ᨨ
�盐
 �ᨨ
�矄
ᨨ


쟼

쟼 쟼졔 ҈

҈

쟼졔 ҈

쟼졔

 E+  =  E- 
쟼

�


E
㦨



M

泐
矄泐鼀鼀泐㡰

턼

 㝐 

졔



졔



ᆠ�ᆠ
�
盐
ᨨ
��
ᆠ �矄
쟼�


㦨



和�鼀�泐  

ᆠ�ᆠ
�
盐
ᨨ
ᨨ矄��
ᆠ �矄
쟼ᨨ盐
㦨


\
쟼
㦨




ᆠ�ᆠ
ᨨ �
盐
ᨨ
��
ᆠ �矄

M
�矄
�  
ᨨ鼀  
ᆠ�ᆠ
ᨨ �
盐
ᨨ
ᨨ矄��
ᆠ �矄

M
\

¹



㝐
␔ 
 
矄泐矄鼀
ᨨ
ᆠ �矄
ᨨ

犨
�

ᆠ盐 
I҈
盐
ᨨ

��
ᆠ �矄
 
 ᨨ 
ᆠ盐 
I
盐
ᨨ
ᨨ矄��
ᆠ �矄



鼀� �盐
ᨨ
B

M
盐
ᨨ
 
ᆠ盐 矄
�矄
�  
\
B
¹


 
m ¹



ᆠ�ᆠ
�
盐
ᨨ
�  
ᆠ �矄
ᆠ ᆠ
ᨨ盐
�
�矄
\
쟼
㦨

矄㡰


a





졔

a

M

P 

졔



Q


ᆠ�ᆠ
�

M
盐
ᨨ

 
ᆠ �矄
�矄
ᨨ
ᆠ盐矄


ᨨ �

M
盐
ᨨ
 
ᆠ �矄

WWW.JEEBOOKS.IN

PS-199

Electrostatic Potential and Capacitance

é +q q æ q öù V = kê + -
¹
ç �a ÷ ú a⏌
a⏌
ê ú ç ÷ è
ø úû ëê



㡰

 졔

졔

㝐

҈





���㡰 


᲼Q 턼
µ

桌

ᨨ鼀
⼄

�㡰

��㡰
 
���㡰

␔ 
 
犨 ᨨ
矄泐矄鼀
�矄
矄
� ᨨ
ᨨ�ᨨ
犨�

ᆠᨨ 矄   �盐
ᨨᆠ�泐
ᨨ盐
 
� 
PQ
犨
�
 
ᨨᨨ矄
�
犨 �ᆠ ᆠ盐 
�矄
ᨨ犨� �



鼀� �ᆠ
�
盐
ᨨ
 矄
ᆠ盐 矄
ᨨ盐ᆠ

矄盐  鼀� �ᆠ
�

M
犨 �ᆠ
�矄
ᨨ
⼄盐
ᨨ
ᨨ

 ᨨ

 鼀� �ᆠ
�ᨨ
鼀ᨨ鼀 
犨�

ᨨ盐ᆠ
\
B
¹


 
m
¹



졔


µ

\

ᔼ
µ







㦨
ᔼ
µ 

҈ 



P

 泐
ᨨ� 


�矄 ᆠ
x
ᨨ鼀
 
ᨨ���

꾀矄
矄 ᨨ犨
�

 
��盐

 

犨ᨨ
ᨨᆠ矄
ᨨ
盐矄�ᨨ
ᆠ� �
盐
ᨨ
犨ᨨ ᆠ �矄
ᨨ
졔


鼀ᔴ



ᨨᆠ
�矄

F
ᆠᨨ矄
a
ᨨ犨矄
�� 
和�鼀�泐
 

犨ᨨ
ᨨᆠ矄
ᨨ
ᆠ�ᨨ
盐
ᨨ
犨ᨨ
ᆠ �矄
ᨨ 
҈
鼀ᔴ



ᨨᆠ
盐
ᨨ
 矄
ᨨᆠ
�矄

FἘ
ᆠᨨ矄
b
ᨨ犨矄
 




턼






쟼
¹


矄
 
犨�



ᆠ�ᆠ
�
盐
ᨨ
 
 �鼀  ᨨ
ᆠ �矄

M
ᨨ犨矄
 
�� 
矄�
쟼ᆠ�ᆠ
ᨨ �
V
㦨



M
盐
ᨨ
矄泐鼀鼀�ᆠ
 �鼀 
ᨨ ᨨ矄��
 
��
ᆠ �矄

ᆠ盐 
�矄
ᨨ
犨 
 
矄泐矄鼀
ᨨ
ᆠ �矄
ᨨ矄
ᨨ盐 PQ

盐
ᨨ
矄泐鼀鼀�ᆠ
 �鼀 
ᨨ
ᆠ �矄

\
B
㦨


 
m
㦨
㡰 ᔼ
ᨨᆠ矄矄
҈

犨ᨨ├
ᨨ 
泐
泐

犨
㦨
⼄
맔

㦨
ᔴᨨ
맔
ᨨ
㦨 ᔴᨨ


҈
쟼 �泐
矄ᨨ
�
 
泐
쟼ᔴ㦨
ᔴ



K ´
´ ҈
-
´

҈ ´ ҈
-



�矄矄�
ᆠᨨ矄矄
矄�矄 ᆠ
쟼
㦨
␔

쟼ᔴ
㦨
ᔴ


�






V ö ù V ö ҈ éæ Vᨨ ö æ
V æ -
÷ + ç ᨨ - ᨨ ÷ + çVᨨ - ᨨ ÷ ú C êç ø è  è
ëè  ø  ø û





䰘


҈ CVᨨ



㡰 꾀�
ᆠ盐 

ᔼ

µ 
턼
µ



턼 

x
+

´





䰘


x

 x
+



Þ x
㦨
᠀


�


�
m ᨨᆠ
�矄
盐矄�
ᨨ犨矄
졔
ve
x㜠 �矄
 
ᨨ
x





´ 黨 ´ ҈
黨 ´
´ ҈
-
�
䰘 +



é
ë

-

҈ù


´ 黨 ´ ҈
黨 ´ ҈
-
�  +




㦨


맔
黨
맔
҈
黨
맔
҈


ê - ú =

䰘 ´ ҈




û

⧤� �ᆠ
 �泐
�矄
⼄盐�
ᨨ
ᨨᆠᨨ鼀
 
ᨨᆠ
ᨨ
盐矄�ᨨ




�㡰


��㡰

҈


F qE ö ҈ 턼쟼
҈ æ 턼 ´

ö
æ  㦨
ç
㦨
÷ ÷ ´  çèQ a = = m mø
鼀
è 鼀 ø ᨨ鼀

和
㦨
盐
졔

æ
ö ç x +
÷ è ø

x

´

� m
ᨨᆠ
�矄
ᆠ�
ᨨ犨矄
��
x㜠 �矄

� 쟼ᆠ�ᆠ
ᨨ �
ᨨ
 
ᨨ盐
ᆠ �矄

x
㦨
V
㦨


ᆠ㡰 꾀 
ᆠᨨ�� �
 
ᨨ


 

犨�
��
�� ᆠ �
 
�

泐
 
ᨨ

 
ᨨ盐 ᆠ
ᆠ├
ᨨ
  ᨨᨨ鼀


盐

ᨨ
ᨨ
 ᆠ盐
矄�鼀
鼀ᨨ �ᆠ
鼀ᨨ�ᨨ

矄
ᨨᆠ
ᨨ
�
µ
x




K ´ ҈ ´ ҈
-
´

҈ ´ ҈
-


꾀
x
>


\
ᨨ
 �泐
�矄矄�
ᆠᨨ矄矄
 
矄�矄 ᆠ


ᨨ

Ed
㦨


㦨


҈ ᔴ  � 㡰



҈
㦨
ᔴ  � 㡰


ED =

æ

䰘 ö ç x +
÷ è ø

҈ ҈ ᔴ

= ᔴ





\




쟼ᔴ
㦨
쟼 ᆠ㡰 
�
 


 
� ��
 
� 
ᨨ�
�
ᆠ
矄
ᨨ ᨨᆠ矄矄





쟼 �泐
�矄矄�
㦨
␔泐

D 㦨
ᔴ
� 㡰 

ᨨ

ᨨᆠ
ᨨ
ᆠᨨ鼀
ᨨ

F
ᆠᨨ矄
a
㦨

F'
ᆠᨨ矄
b

ᨨ鼀
µ
ᨨ
x
㦨

�




犨ᨨ├
ᨨ 
�
 �矄
ᨨᆠ矄矄
W
㦨
q
V㡰 \ W
㦨


҈
맔
҈


맔


䰘
맔
҈


㦨


䰘
맔
҈

 泐
 �泐
ᆠᨨ 矄�ᨨ


҈ mV

㦨


䰘
맔
҈



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EBD_8341 PS-200

Þ

Physics

҈ ´
´ ҈
-
V

=

䰘 ´ ҈
- Þ V

㦨

鼀⏌矄


K.E. at the ᨨr�g�n ᨨ �

ᨨ���

Vx 㦨


㦨


҈




´ 黨 ´ ҈
黨 ´
´ ҈
-

䰘


-


´ 黨 ´ ҈
黨 ´ ҈
-


㦨




맔
҈

꾀��
泐
 �泐
ᆠᨨ 矄�ᨨ




ë

r

黨


졠
 �
ᆠᨨ ᆠ


é R

sa
e


ᔼ
෌





Sn = ᨨ


a 

qs  a
+ x
- x㡰
e




\ U 鼀� =  鼀� ᨨ鼀
⼄盐�ᨨ
��㡰

�
犨

Ux㡰
 
x

ᨨ

�矄
矄
矄 ᨨ犨

ᨨ犨


҈

QR é æ R ö ê҈  ç ÷ r êë è R + r ø

nù

ú úû

qn
qn
=
C
(
pe
R )








��㡰

U( A) = mgH +
mg é a
+ H
- H ù ëê ûú

H
㦨


) 犨 
r �矄
ᆠᨨ鼀鼀ᨨ
�ᨨ


é QR ìï æ R ö n üï ù ҈ ҈ ᨨ

U n = ´ ´ê í҈  ç ÷ ýú

pe
R ê r ï è R + r ø ï ú î þû ë 㡰 �鼀�� �
盐
ᨨ
 �泐
矄
n ® ¥

盐矄
ᆠᆠ盐
qn
犨 
n
 矄
ᨨ
¥
a ᨨ
෌



S ¥ = 犨  r҈ = ᆠᨨ鼀鼀ᨨ
�ᨨ ҈ - r҈ QR é QR ҈ ù ᨨ q¥ = \ q¥ = R ú R+r ê r ê҈ ú ë R+r û

q¥ ҈ æ QR ö \ U¥ = =ç ´ ÷
C è r ø
´ (
pe
) ´ ( R )

b㡰 ᨨ
ᨨ �
 �泐
ᨨ
 
�ᆠ

�� 
H U
x㡰
㦨
mgx
졔
qV
x㡰



(҈  r҈ )

qn =

Un =

mgH
㦨

mg
᲼a - { a + H 㡰 - H 晤桌

= mg é
a
+ H
- H
ù êë úû dU ᨨ
⼄盐���盐鼀


=
dH a

(

a ҈  r҈n

쟼ᆠᨨ矄�ᆠ
 �泐
ᨨ
S

 
n
矄盐ᆠ
ᆠᨨ ᆠ矄




㦨
mgx
졔

mg
᲼ a
+ x
㡰 - x桌

û

n
é R æ R ö æ R ö ù +ç + +





qn = Q ê ÷ ç ÷ ú R+r è R+r ø è R + r ø ûú ëê

盐� �
 �矄
盐
�
⼄盐�ᨨ
�㡰


㦨
mgx
졔


 æ R ö ù

ë

s mgH
㦨
q
᲼a

{ a
+ H
㡰 - H 晤桌



�㡰
e


e
g = Þ
sq =
mg ᨨ鼀
 
��
�ᨨ
m s
e


H
㦨





졠

ᆠᨨᆠ


␔ 
 
�ᆠ
�矄
ᨨ
ᨨ �
 
 �矄
ᨨ
�矄

 �泐
�矄 ᆠᨨ 矄

ᨨ矄矄
ᨨ
���ᨨ 
ᨨ �
 �泐
㦨
��
�
ᆠ�ᆠ ᨨ �
 �泐 mgH
㦨
qDV
㦨
q
᲼V
㡰

V H㡰桌

ᨨ

æ R ö

q
㦨
Q
ê R + r + çè R + r ÷ø + çè R + r ÷ø ú ê ú

s ᲼ a
+ H
- H 桌
e





=

æ R ö ᨨ rq
= RQ  Rq
+ RQ ç è R + r ÷ø éæ R ö æ R ö
ù ᨨ
q

㦨
Q ê ç ÷ +ç ÷ ú êë è R + r ø è R + r ø úû

꾀
x
㦨
H

꾀 
VO㡰
㦨


kq҈ ҈ æ R ö 犨  k = ᨨ q҈ = Q ç ÷ R
pe
è R+r ø

kq
r R ᨨ rq
= RQ  Rq
+ Rq҈

s ᲼ a
+ x
- x桌
e


V
H㡰
㦨



=

k éëQ  ( q
 q҈ ) ùû

⧤
쟼

㦨


҈
맔
҈


᲼

䰘
맔
҈







맔
҈

桌 㦨


҈
맔
҈


맔



맔
҈


㦨

맔
҈


 㡰 ෌�
a
㦨
�盐矄
ᨨ
�矄ᆠ

s
㦨
矄盐 ᆠ
ᆠ �
 矄�泐
q⏌m
㦨

e
g⏌s ᨨ �
盐
ᨨ

ᆠ �
�矄ᆠ

 泐
 �
ᨨ� 
矄�盐

 �矄 ᆠ
x
ᨨ鼀
O V
x㡰
㦨


�矄
矄
⼄盐���盐鼀
ᨨ矄��ᨨ


졠
矄ᆠᨨ 
ᆠᨨ ᆠ

矄�鼀�泐


û






㡰 
S҈
ᆠᨨ �
ᆠ �
Q
 ᨨ
ᆠᨨ ᆠ
犨�
S

졠
ᆠᨨ ᆠ



ᆠ �
q҈
矄 � 
ᨨ鼀
S҈
ᨨ
S



k (Q  q҈ )

V - Vx =
ù ⧤
쟼

㦨
q
é x= � ê ú \

a

 ᆠ
H =





���㡰

ᨨ ҈҈


U¥ =

Q
R


r
´
´
pe
R

ᨨ U ¥ =

Q
R
(
pe
) r


㡰 꾀
ᆠ �
�矄
盐 � ᨨ鼀泐
�矄�盐
ᨨ

ᨨ 㜠ᆠᨨ 盐ᆠ� � 矄 �ᆠ
ᨨ盐鼀
ᨨ
�盐矄
R


盐矄
ᆠᨨ 矄�

矄 
ᨨ
 
 �ᆠ├ 矄矄 dx


�矄 ᆠ
x
ᨨ鼀
 
ᆠ 
ᨨ

矄  ᨨ盐鼀
ᨨ
 
矄 
㦨



é
ù p ê x + dx 㡰 - p x  ú  ë  û

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PS-201

Electrostatic Potential and Capacitance

ᆠ㡰 ᔼ
 
矄鼀
ᆠ �
ᨨ
Q
ᆠᨨ ᨨ鼀
�矄
��
ᨨ

矄 �ᆠ ᆠᨨ 盐ᆠᨨ
ᨨ
 
矄鼀
�盐矄

 
C =
pe
R  
 �泐


  㦨
p éë x + dx㡰 - x ùû   ù
 é æ dx ö p ê ҈ + x 㦨
çè ÷ø - ҈ú x  ëê ûú 㦨


E= ҈




 é dx ù p x ê҈ + - ҈ú  x ë û

=

҈



r

 
ᆠ �

盐 �
ᨨ盐鼀
ᨨ
 
矄  \ ᔴ �
ᨨ
 
矄 
㦨
dq
㦨

px
rdx



�㡰 ᨨ �

 
矄盐 ᆠ
ᨨ
 
矄 
ᨨ
�盐矄
x

҈ 㦨
´
pe



r ´ px   x

㝐� �鼀盐鼀
�鼀
ᨨ
 
�ᨨ
ᨨ
��
ᨨ �
 
�ᆠ�ᨨ
ᨨ ᆠ�ᆠ
� t
㦨


 dx px ´
㦨

px
dx x 

҈ Q
҈ Q
=

pe
R
C

T
p ML p ML
= =


qE

qE

쟼ᆠ
�ᆠ
犨�

�
⼄盐���盐鼀
盐 
 
ᆠ
ᨨ
 
ᨨᆠ  矄�ᨨ
ᨨ
矄� �
T

犨�� � �

矄盐 
ᆠᨨ矄�ᆠ
ᨨᆠ
F
ᨨ  
犨ᨨ
ᨨ 
ᆠ �矄

ᨨᆠ
F
 
鼀�

 �ᆠ盐


qù é êëQ V = k r úû

\
ᨨ �

 
矄盐 ᆠ
ᨨ
 
矄 
ᨨ
�盐矄

x
㦨


rx
e


␔ᨨ├
ᨨ 
�
� �� �
 
ᆠ �
dq
ᨨ
 
矄 
ᨨ
�盐矄 x
dW

rx
rx
㦨
맔
dq Þ
dW
㦨
맔

p x

r
dx e
e


\
␔ᨨ├
ᨨ 
�
ᆠᆠ盐鼀盐� �
 
ᆠ �
Q
ᨨ

矄 �ᆠ ᨨ盐鼀
ᨨ
�盐矄
R R
pr



pr
x dx = W
㦨
ò
e e




R

é x� ù
pr
R� ê ú = e
� ëê � ûú






p æ Q ö R� Q
= 㦨
ç ÷ e
è
⏌ pR ø �

pe
R

矄ᨨ� �
T
�
 
�ᆠ�ᨨ
ᨨ
mg
 
F
 

泐� �
  ᆠᨨ ��ᨨ
ᨨ
⼄盐���盐鼀


F \  q
㦨
mg

Q
KQ
҈ =
犨 
K
㦨
� ´ 
pe
㡰 R �R
pe
҈ ᆠ� �
ᨨ
⧤
泐
G
 
Q

泐
M


pe


\ g
㦨


E
㦨


GM �R

\

gR




Þ
G
㦨
M R






E =




F
㦨
FCA =




´

҈


kq






��㡰

CA㡰
ur ur  F CB  =  F CA 
 
a
㦨



桌


T
鼀├

 �
q 犨�
 
�ᆠ OC
㦨
\





㡰
- 


҈�㡰

㦨



҈䰘
鼀 

OM
㦨


黨黨黨䰘

\
 q
㦨


 gR M   ҈ 㦨 gMR = ´ ҈
´

� ´ ҈
� M R � � = ҈
� ´҈



\




F
㦨
FCA + FCA +
FCA ´




GM





�㡰



F
㦨
FCA + FCB +
FCA FCB ᆠᨨ矄 a

�矄
犨ᨨ├
ᨨ 
�矄
⼄盐
ᨨ
 
 �泐
矄ᨨ
�
 
矄泐矄鼀
b㡰 
ᆠᨨ矄ᨨ � �
 �泐

ᨨ
ᆠᨨ鼀泐
�矄矄矄鼀  
 

�� 矄
 
���ᨨ 
盐
E
㦨


T
矄� q
㦨
F

T
ᆠᨨ矄 q 㦨
mg錐

OC


҈䰘 = OM

黨黨黨䰘






���㡰

ᨨ鼀
⼄

�㡰

��㡰
 
���㡰




҈䰘  ´ 黨 ´ ҈
黨 ´ ⼄
=

黨黨黨䰘 


㡰
´ ҈
- ´ 黨

\
q
㦨

҈

맔
҈
黨
ᔴ


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EBD_8341 PS-202

Physics

Current Electricity

17

Topic-1 : Electric Current, Drift of Electrons, Ohm's Law, Resistance and Resistivity ҈






҈
�

黨 ´ ҈


´҈

´ ҈
-҈黨 ´ � ´ ҈
-
҈



ᆠ㡰 꾀鼀鼀


ᔼ
矄�矄
ᆠᨨ

ᆠ�ᨨ

 
矄鼀
ᆠ盐 
ᨨ犨矄  ᨨ盐�

 
ᆠᨨ鼀ᨨ  矄

ᔼ
�鼀矄

鼀矄盐� �
 
ᆠ盐  ᨨ犨� �
 ᨨ盐�
 
ᆠ�ᆠ盐�
 
 ᆠ

�
�矄
ᆠᨨ

ᆠ
�
矄�矄
ᨨ鼀


꾀
ᨨ鼀
鼀矄盐矄
ᨨ�
ᆠ  �
犨
犨ᨨ ᨨ� 矄
�

ᆠ�ᆠ盐�

和ᨨ
犨

ᨨ
ᆠ
 
ᨨ鼀
�
 犨�
 
ᆠ�ᆠ盐�
ᆠᨨ鼀ᨨ  
b㡰 rM = 黨
´ ҈
-


r A =


� ´ ҈
-
rC = ҈
䰘

´ ҈
-
rT = �

� ´ ҈
-
\rM > rT > r A > rC






㡰 㡰

 쟼

҈҈


Þ m=

쟼 쟼꾀 ᔼr㡰 = Þ 쟼=  ᔼ 꾀

  꾀
= = 쟼 ᔼr

(

҈
҈ ´҈
 ´ p ´ � ´҈


)




҈



dR =

� ´ ҈
䰘 ´҈




px

䰘






dx
px

b



Þ

R=

r é -҈ ù
p êë x úû a

æ r ö æ҈ ҈ö R = ç ÷
ç - ÷ è
p ø è a b ø m ᆠ㡰 r =
ne t 黨
҈ ´ ҈
-҈ =

� ´҈


´ ҈

´҈
-҈黨 㡰
´
� ´ ҈
-҈� 㦨
҈


W㜠鼀 㡰 盐鼀


�矄
矄矄ᨨᆠ�
犨�
 

ᆠᨨᨨ盐

�矄
ᆠᨨᨨ盐
�矄 ᆠ
泐
�
Q
ᔴᨨᨨ盐
ᆠᨨ
��盐
ᨨ
�
�矄
� \

犨
矄�矄 ᆠ
㦨
�


W 㡰 ᔴ泐

ᨨ鼀
� ᔴ盐 

ᔼ
㦨


黨


㡰

෌�

҈
㦨
҈


W

Ἐ
㦨
⏌




㦨
䁸

ᨨ


a =
´ ҈


r

-


°ᔴ - ҈

rl



Q

꾀
맔
 �
㦨
ᨨ盐鼀 꾀

꾀


⼄ =
  =
矄
᲼和� ᆠ
⼄
�矄
ᆠᨨ 矄 桌 

矄� �

ᔼ
㦨

꾀

\�  矄   =

҈

꾀

µ

҈

ᆠ㡰

쟼ᆠ�ᆠ
�


�矄 ᆠ
r
ᨨ鼀
� 
ᆠ �




ᨨ µ

҈









Q


꾀
㦨
p


꾀 

= ҈
Þ
= ҈
´ ҈

= ҈


W

矄�矄 ᆠ
ᨨ
犨
犨�
҈










b





㡰

 ᆠ

=

r㡰dx㡰

ò

a

҈
҈

´ �



和� ᆠ

r ®
ᆠᨨ 矄 


 ®
ᆠᨨ 矄 


R = dR

ò dR = rò

a=

矄�矄��泐
ᨨ
犨�

=

鼀 m = ҈

矄 㡰




= = ҈҈
W ᔼ




㦨
҈



�� 㡰
鼀盐
ᆠᨨ �ᆠ� 
ᨨ
矄�矄 ᆠ

a
㦨
䁸 盐矄� �

㦨


҈
졔
a㡰 ҈҈

㦨
҈


҈
졔
a�

㡰




�


矄�矄 ᆠ
 
鼀盐
� ᆠ矄矄
泐
�

ᔴ
�




ᆠ㡰

 =

[ ␔ 
 = � 
ᨨᆠ�泐 ]

 
矄�矄��泐
( r ) =

㡰 ᔴᨨᨨ盐
ᆠᨨ
ᨨ
ᆠᨨ
矄�矄ᨨ 





졠

Ӹ

෌

盐



෌

␔





҈













�









䰘






黨 矄�矄 ᆠ


㦨
꾀
맔
ᔴ
᠀
  矄
꾀
 


 
�矄
犨ᨨ
矄�� � �ᆠ 
��盐矄
ᨨ
矄�矄 ᆠ 
 
ᔴ
� �ᆠ矄
 
ᆠ�鼀
鼀盐��
ᨨ
 
盐鼀
ᨨ
ᨨ矄  
ᨨᨨ犨
꾀
 
 
 

�矄
ᨨ ᆠ

෌ᨨ
㦨
᠀
�奠
和�
㦨
᠀
҈

奠
ᨨ
ᆠᨨᨨ盐
㦨
᠀


奠

= � ´ ҈

± �奠 = �
├W ± �奠

␔ 
�
㦨



V
㦨
e
㦨
҈
�
ᨨ ᔴ �
鼀ᨨ��泐

m㡰 =


㦨





鼀鼀矄҈

҈


dV l =
per dr l dV =  dr
per E=

l
㦨
� 
ᆠ �
 矄�泐
ᨨ
犨�㡰

ᔴ盐 
 ᨨ盐�
 
鼀 
矄 

l dr lsl  dV 
per I= = = dr ҈ dr e ´ s
prl Q = r

l  ҈ dr 㡰 \  = r = 꾀
prl s
prl

�矄
ᆠ盐 
�矄
�泐
ᨨ盐犨矄


d  lsl \ ( ll ) = Þ dt e  ( s ⏌ e )t

l

ò

l


t

dl æ sö =  ç ÷ dt è eø l

ò


Þ l = l
e lsl ls I \ J= = =
prl
perl
per

WWW.JEEBOOKS.IN

PS-203

Current Electricity

æ l
s ö  ( s ⏌ e )t e Þ J = J
e ( s ⏌ e ) t
per ÷ø l b㡰 V = IR = neAvd 㡰r A  \ r
㦨
  


҈


ᨨ

J = ç è

҈



和ᨨ
ᨨ
�
㦨







�


´ ҈






´ ҈

´ ҈


-҈黨





´

� ´ ҈
-
´

҈

ᆠ㡰

�
㦨
neAVd
 
Vd µ

E
෌� 㡰

ᨨ

� µ E �

µ
E �

µ
V  ᆠ
�
ᆠ㡰
ᆠᨨᆠ泐
��ᆠ矄
 
V㜠I
�
ᨨ

犨�
鼀 ᨨ
矄盐ᆠ
泐
ᨨ
鼀�
҈



\ R=

҈䰘


L

 ᆠ R
�矄
�   
ᨨ
L  
� 矄泐
ᨨᨨ�ᨨ 
ᨨ
t
ᆠ㡰 꾀
ᨨ鼀
�矄
鼀
泐
ᆠᨨ

ᆠ� �

��
矄�矄 ᆠ
R҈
�

矄�矄
犨�
 
� ᨨ鼀
G҈


R҈

꾀
鼀鼀
�矄
鼀
泐
ᆠᨨ

ᆠ� �

ᨨ犨
矄�矄 ᆠ
R

�
 犨�
 
� ᨨ鼀
G






ò






G


҈

pr æ ö l ç
÷ è ҈ ø

҈黨











I V


҈
s

ᔼ

ᆠᨨ 盐ᆠ� �
矄ᨨ�

 
ᆠᨨ�矄�ᨨ 矄
ᆠᨨ鼀
鼀ᨨ
⼄盐 
犨� � ᆠ矄
ᨨ
鼀盐


R


҈



æ ö 
 l ç
÷  è ҈ ø Þ ᔼ=
= pr

 ᆠ
ᨨ�ᨨ
㡰
�矄
ᆠᨨᆠ
ᆠ㡰 
ᆠᨨ 盐ᆠ��泐
s㡰
ᨨ

矄鼀�ᆠᨨ 盐ᆠᨨ
� ᆠ矄矄
犨� � ᆠ矄
�
鼀盐
�


 
矄�矄��泐
f㡰
ᆠ矄矄
犨� � ᆠ矄
�
鼀盐
矄
P
㦨


ᨨ鼀
�矄
ᆠᨨ

ᆠ
�

犨�
 
矄
矄�矄ᨨ
RT
�

ᆠ�ᆠ盐�
꾀鼀鼀
�矄
ᆠᨨ

ᆠ
�
矄�矄
犨�
 
矄
矄�矄ᨨ
RT
�
ᆠ�ᆠ盐�
꾀
�
ᨨ�
矄ᨨ盐ᆠ
V
�矄
ᆠᨨ

ᆠ
�
矄�矄
犨�
 
矄 矄�矄ᨨ
RT
�
ᆠ�ᆠ盐�
㡰 ᔼ
ᆠ�ᆠ
ᆠ�ᆠ盐�
鼀鼀
�矄
ᆠᨨ

ᆠ
�
矄�矄
犨� 矄�矄 ᆠ
 
ᨨ鼀

犨�
 

矄�矄 ᆠ
ᔼ
ᨨ 鼀Ἐ矄
犨

犨
ᆠ ᆠ├
V
㦨
IR 泐
泐� �

矄�矄 ᆠ
ᨨ
  ᆠ�ᆠ盐�
㡰 ᔴᨨ
�矄

鼀
犨 矄
෌鼀 �盐鼀
�矄
和鼀�㜠ᆠᨨ 盐ᆠᨨ
矄�矄 ᆠ
ᨨ
鼀
ᆠ矄矄
 
矄鼀�㜠ᆠᨨ 盐ᆠᨨ
� ᆠ矄矄
犨� ᆠ矄
�
鼀盐
㡰 矄�矄 ᆠ
� ᆠ矄矄
犨�
鼀盐
矄 
졔


҈
졔
µ
㡰 R҈
㦨
R

҈
졔
a t҈㡰

Þ
҈
㦨


᲼҈
졔




҈
�
맔

䰘桌 R

㦨
R

҈
졔
a t
㡰

Þ


㦨
R

᲼҈
졔




҈
�
맔 t
桌 和ᨨ� �
犨
� T

㦨

�
ᔴ
㦨

�
ᔴ
졔

䰘 ᔴ
㦨
҈҈
䰘
⧤

T2

 ҈  æ
ö Þ = l ç ÷ pr è ҈ ø rp

r r 鼀
鼀
 = ⼄쟼 Þ 쟼 =  ⼄ R҈ r 鼀ᔼ
쟼=




꾀
⼄ I ᔼ ö æ Q �  ᨨᆠ�泐
 = çè ÷ 

꾀 ø r uur D = ò 쟼
 ᨨ

D µ 


\D µ ᔼ


t

G҈

҈ 矄ᨨ
ᨨ
�

 ᆠ
ᨨ�ᨨ
㡰
�矄
ᆠᨨᆠ
꾀 

ᨨ
ᆠ�ᆠ盐
鼀ᨨ�ᨨ
ᨨ
ᆠᨨ

D
ᨨ矄 ᔼ


矄盐 ᆠ

�� 
ᨨ �
矄ᨨ
 
ᆠ�ᆠ
�
ᨨ矄 �泐
ᨨ盐犨
R
和ᨨ
ᨨ�ᨨ
㡰
�矄
ᆠᨨᆠ
DV

L

rL r = L´t t

҈ = 

矄�矄 ᆠ


=

l ᆠ㡰 ␔
├ ᨨ犨
矄�矄 ᆠ

R = r a  l
㦨
L
 
a
㦨
L
맔
t

T1 I

和ᨨ㡰T



和ᨨ㡰T҈
\
 矄�矄 ᆠ㡰T

>  矄�矄 ᆠ㡰T ҈ V ᨨ

鼀�ᆠ
犨�

矄�矄 ᆠ
� ᆠ矄矄
犨�
鼀盐
\
T

>
T҈ 矄

盐
ᨨ
 鼀
 �泐

ᆠᨨ 矄
�

ᆠᨨ 盐ᆠᨨ


  
 鼀
ᨨᆠ��矄


ᆠᨨ 矄

鼀ᨨ�ᨨ
�
 ᨨ鼀
�ᆠ�ᨨ 矄 
�
 
矄 ᆠ
ᨨ
ᨨ �
�  ᆠ


ᆠ

㡰
矄�矄 ᆠ
ᨨ
鼀 泐
矄�矄

\ò

㦨
҈


맔
҈
�
W鼀 ҈�


I ҈ = V R

Þ 矄�矄 ᆠ
R =



㦨
ᨨ �
�  ᆠ 
㦨
 �
ᨨ
犨�


㦨
ᨨ

ᨨ
ᆠᨨ 矄

盐 �
ᨨ盐鼀
ᨨ
ᆠᨨ 盐ᆠᨨ

㦨
ᨨ

ᨨ
ᆠᨨ 矄 ᆠ� �
 
盐
ᨨ
ᨨ
鼀矄
犨
�
矄�矄��泐

r=

T盐

ᨨ鼀
 
��
ᨨ�

ᆠ盐 
�


�


l A V 꾀 
V = Al Þ A = (V = ᨨ盐鼀 ᨨ 犨� ) l rl rl
\R = = V ⏌l V

r
 
V

ᆠᨨ 矄  \
R
µ l DR ´ ҈

=
\
ᆠ �
ᆠ  �
�
矄�矄 ᆠ
R 奠ᆠ  �
�
l㡰
㦨



҈奠㡰
㦨



奠 꾀矄
犨
├ ᨨ犨

矄�矄 ᆠ

R = r

Topic-2 : Combination of Resistances ҈


b㡰

ᨨ鼀
ᆠ�ᆠ盐�
��鼀


҈ ҈ ҈
= + Þ R҈ = R҈ ҈
� ҈ ҈ ҈
= + Þ R
= R

 �

WWW.JEEBOOKS.IN

EBD_8341 PS-204

Physics

꾀

ᔴ





䰘


R = R҈ + R
=

�=




ᆠ㡰 ᆠ㡰



+ =
W � �






´ ҈
� = W ҈









⼄ 㦨





ᔴ

ᔴ



ᔴ

㡰 ᔼ
矄泐
矄

ᨨ犨
ᨨ
ᆠ盐 
 ᨨ盐�
ᆠᆠ�ᨨ
犨�
 ᨨ
ᔴ盐 
 ᨨ盐�
 
ᆠ�ᆠ盐�


E r + r


Vc =



R
㦨


맔

㦨
҈

W R⼄
㦨


ᔴ

꾀



ᨨ �
�  ᆠ
 ᨨ盐�
ᆠᆠ�ᨨ

␔ 
 �
ᆠᨨ鼀矄
ᨨ盐
�矄
矄�矄 ᆠ
ᆠᨨ鼀矄

R Ҙ l
㡰



�=


� � � v

= = ҈
A

\ I BC = - = =
A � � � R






ᔴ

黨


æ E Q = E - �r 㦨
E - ç C è r + r


ö r
÷ r
\
Q = CE r + r
ø

b㡰 
ᨨ �
�  ᆠ
�
ᆠ
ᨨᨨ
�矄
ᨨ
\
ᨨ
ᆠ盐 
犨�
ᨨ犨
ᨨ
ᆠ盐 
�
ᆠ
矄�矄 ᆠ
�矄
ᨨ
b㡰

꾀矄
矄�矄 ᆠ
ᨨ
犨�

R
㦨


òA l

rFe ´ lFe ҈
䰘 ´ �
´҈

� = = ´҈

RFe
㦨


AFe
´ ҈


ᆠ㡰 R

R

 
R�

�
矄�矄
矄ᨨ
 �
⼄盐�  R
㦨



졔
�
졔

�
㦨
�

W

�矄
�矄

犨�
R


 
矄ᨨ

矄�矄 ᆠ
ᨨ
 
ᆠ�ᆠ盐�

r Al ´ l Al

䰘 ´ ҈

´ �
´ ҈


䰘 ´ �
= =
맔
҈
� RAl
㦨
AAl 
黨 
㡰 ´ ҈


�

㦨



맔
҈

꾀矄
ᨨ �
�  ᆠ
ᆠᨨ矄矄
ᨨ
矄�矄ᨨ矄
�矄
矄鼀

矄ᨨ
 泐
 �

ᆠᨨ鼀� �ᨨ


\ RPQ
㦨
҈



矄�矄 ᆠ
犨

 
턼

� ´  æ  ö
= � 턼
㦨
 P ç + ÷ = è 
ø � ҈҈ + 


҈

æ ҈
´ �
ö W ÷ + ҈� + 
=  è ҈
+ �
ø e ҈� 黨 = = A 和ᨨ
� = R⼄ (҈

⏌ ) 


R⼄
= ç

�


b㡰

RFe ´ RAl ҈

� ´҈

´

 ´ ҈

҈
䰘� = = mW

RFe + RAl ҈


´҈



矄�矄 ᆠ
犨
턼
 



´   
턼 = P  + 㡰 =
 =  

 +  ҈҈


㡰

矄�矄 ᆠ
犨

 


矄�矄 ᆠ


µ l
矄ᨨ
矄�矄 ᆠ
ᨨ
ᆠ
矄�
ᨨ
 
⼄盐� � �
㦨


W 矄�矄 ᆠ
⼄
犨
 泐
犨ᨨ
�ᆠ矄

҈ ҈ ҈ 㦨 졔 Þ ⼄
=
W ⼄ ҈





㡰

矄�矄
㦨
҈
졔


졔






졔


҈ ҈ ҈ ҈ = + +






+  ҈


  ´   æ ö 
 =   = P ç + ÷ =  è
ø  +   ҈҈ 
҈҈


 ᆠ

�
�矄
ᆠ
 
턼
�矄
鼀 �鼀盐鼀 㡰 
ᆠ盐 
�

W
矄�矄ᨨ
㦨
ᨨ
ᆠ盐矄
�
�矄
ᨨ


ᨨ  泐
ᆠᨨ矄
ᨨᨨ
҈


�W



 ҈
W


W

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PS-205

Current Electricity ෌�
矄�矄 ᆠ
ᨨ
ᆠ
矄�矄ᨨ
Ἐ Ἐ

ᆠ㡰

RPQ ҈


҈







㡰

�
 R
RQR = R   RPR = R = ҈҈ ҈҈ ҈҈

\
턼
�矄
鼀 �鼀盐鼀
㡰 
ᆠ�ᆠ盐�
ᨨ
 
ᨨ犨
 
 �矄
POQ 矄泐鼀鼀�ᆠ  
矄 矄
 ᆠ
犨 矄ᨨ 
��

 ᆠ
 
ᆠ  矄�矄 ᆠ


�矄
�  ᆠ�

R


R


R


R

r

r


R


R


R r


矄�矄 ᆠ
ᨨ

W
�矄
�

犨�
R

�
 ᆠ� 矄�矄 ᆠ

r

P

Q
R


R

P

2R

Q

҈ ҈ ҈ = + R Ἐ 
R 
R RἘ = 
+ R

 ᨨ
 
⼄盐� 
ᆠ�ᆠ盐�
�矄
犨
矄
ᨨᨨ犨矄


r + r +
R ҈ ҈ ҈ + +

㦨

Rr RPQ
R
R
r
Rr ᨨ
RPQ
㦨
R+r

҈

\

҈



ᆠ㡰

=

꾀矄ᨨ


㦨
ᔼ

R Ἐ = ҈� =

V

㦨
㦨


҈
꾀鼀
ᔴ盐 
�
 
ᆠ�ᆠ盐�


I = R

҈㡰

҈䰘


ᔼ
 
��盐
㡰
ᨨ
ᆠ盐 
ᨨ犨矄
 ᨨ盐�

W
矄�矄 ᆠ


W


W


W


W


W



�

҈
W ҈
W

 ᆠ 犨 矄ᨨ  �� ҈ ҈ ҈ = + ⼄ 



W

\ ⼄ =

´
=
W +


I
㦨


҈



æ Rö ç ÷
꾀
 
 
矄�矄 ᆠ矄

R


R
 
R

�

矄 è
ø

矄 ᨨ犨
�
��盐


R ҈ ҈ ҈ ҈
= + + = ᨨ

RAB
㦨

R AB
R
R R R
R

 ᆠ



R

A


´ ҈
黨 = W \
R⼄㦨





B

R


W
W


W

 ᆠ 犨 矄ᨨ  ��

҈黨




㡰
I

2k W

҈
W

R



W

㡰

 ᨨ
 
ᆠ盐 
 ᨨ盐�
 
矄�矄ᨨ
R
㦨

W

V

�

� = = = ҈A 黨

� R
+


R҈




V

҈
W


k W

I=

矄� �
�㡰



 =
� 꾀 ҈

W

꾀��
 
⼄盐� 
矄�矄 ᆠ
ᨨ
 ᆠ
犨 
矄ᨨ 
�� ��
㡰
ᨨ
ᆠ盐 
 ᨨ盐�
҈
W
矄�矄 ᆠ




�


R 
+ R

 ᆠ
ᨨ鼀
� � V
㦨
E

IR 㦨
�㡰


�꾀㡰




W㡰



㦨
�

�
㦨



㡰

ᔼ



´

R Ἐ+
�

Þ


졔
R
㦨

R Þ R
㦨


W 

㡰


�
 
ᆠ�ᆠ盐�


�矄
ᆠ
ᨨ
�
  ᨨ


鼀 ᨨ
 
ᆠ�ᆠ盐�
㦨


�㡰
㦨



V ᔼ  
矄�矄 ᆠ
ᨨ
ᆠ
ᆠ

r
㦨



W

矄�矄 ᆠ
�
 
ᆠ�ᆠ盐�
㦨






W㡰

㦨
҈


W \ ᔴ盐 
ᨨ犨� �
 ᨨ盐�
 
ᆠ�ᆠ盐�


҈W
W

ᔼ

Þ ҈
=

Þ R Ἐ+
� =

Þ R
㦨
҈�

ᔼ
 
��
ᆠ�ᆠ盐�


黨
҈ ҈ ҈ = +
= Þ
R⼄
㦨



W R⼄ 



´



҈�





�㡰

Rp =

I RL 㦨
҈
�
kW

҈



R҈




V Rp

R
´ RL
´ ҈
� 黨 = = kW R
+ RL
+ ҈
� 䰘
�

\
Rᨨ
㦨
R҈
졔
Rp
㦨


졔
() I =

黨
㦨



kW 䰘
�

V

= 鼀꾀 = 䰘
� 鼀꾀 R 



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EBD_8341 PS-206

Physics

I
㦨
IR

҈

\

æ RL ö æ ҈
� ö IR
= ç ÷I = ç ÷ ´ 䰘
� = ҈
� mA è ҈
� +
ø è RL + R
ø

Þ R 졔


졔

R
㦨
R

졔

R Þ
R


R



㦨

Þ R



R
졔
R



㦨


Þ R R


㡰
졔
҈
R


㡰
㦨

Þ ᲼R
졔
҈桌
᲼R



桌
㦨



Þ R
㦨

W


\
IR 㦨
IR

IR
㦨
䰘
�

҈
�
㦨

鼀꾀 

㡰

҈




ᨨ �
�  ᆠ
ᆠᨨ矄矄
ᨨ ��㡰

VRL = ( IRL ) RL =
´҈
� = 黨V ᆠ㡰




\

҈




㡰








␔ 
R҈
 
R


� ᆠ  �

 

R҈P =


kW 䰘

6W

6W

3W

6W

6W

3W 3W

3W

6W

꾀


W


W

6W

3W


W


W

D

3W

3W 2W

Topic-3 : Kirchhoff ’s Laws, Cells, Thermo e.m.f. & Electrolysis ҈


ᆠ㡰

�W

�


A

B

D


W

҈W
W ᨨ¥

E �҈

҈



W
W

�
�

C




W


W

F

矄� �
⧤�ᆠ ᨨ Ἐ矄
ᨨᨨ
犨
�
ᨨᨨ
ABCD


´R
R+


C ҈W

6W

҈
W

 
W
W
W ᨨ �
�  ᆠ
犨
A
 
B VA

VB
㦨
黨

҈
졔

㡰
㦨

V
㦨

�҈ \
�҈
㦨


�
꾀 \
�

㦨
҈

�҈
㦨


�
꾀 和�鼀�泐

ᨨ �
�  ᆠ
犨
ᔴ
 
 VC

VD
㦨
VA

VB㡰

�


졔

㡰 㦨




�

㦨







�㡰
㦨

V
㦨

� \
�
㦨



�꾀 \ �

㦨
�


�
㦨


�




�
㦨



�꾀 �㡰 ᔼ
ᆠ矄
ᨨ

� � �

ᨨ
矄�矄 ᆠ矄

 
 ᆠ� 矄�矄 ᆠ

鼀� 矄
矄鼀
犨 
ᨨ 
� �ᆠ
�鼀
�矄

ᨨ 鼀ᨨ
ᨨ鼀
�
쟼 ᆠ�
矄�矄 ᆠ
犨
ᨨ� 矄
C
 
D
 R  

 
ᆠ�ᆠ盐�
ᆠ

犨
矄
矄 ᨨ犨

쟼 ᆠ�
矄�矄 ᆠ
犨
A  
B

A ҈W

3W

6W

3W

�҈ 졔 �


�


�҈

Req
㦨
҈
졔


3W

6W

 ᆠ
矄�矄 ᆠ
犨
A
 
B

RAB
㦨

W

 ҈ = 

 ᨨ
ᨨ犨 黨 

ᔴ

�


黨

3W

3W

 黨 = = ҈꾀 㦨
ᆠ盐 
 ᨨ盐�
W
矄�矄ᨨ 黨 W

3W

Þ

6W

3W

3W

Þ

3W 3W

3W

V ᨨ
P
µ
V

犨�
ᆠ矄
泐

ᆠᨨ
ᨨ
黨
R

҈꾀

B

6W

3W

6W

3W

6W

㡰 

矄�矄 ᆠ
ᨨ
 
ᆠ�ᆠ盐�
㦨
黨W
\
ᔴ盐 
犨
ᨨ鼀
 
泐





Þ

3W





҈


6W

3W

�



ᨨ犨
ᨨ �
ᆠᨨ鼀矄


�
㦨


3W 3W

3W

æ
⏌䰘 ö =

ç = V è
+
䰘 ÷ø

�矄矄�

㦨



㦨
꾀


 = ҈
�
꾀 꾀


3W 3W

3W

ᨨ犨



ᆠᨨ矄矄
RL Ἐ VRL

�CD 㦨
�EF 㦨


6W

R
RL
´ ҈
�
= = kW R
+ RL 
� 䰘

RἘᨨ
㦨


졔
R ἘP
㦨


졔


RAB㦨
҈W
졔
҈W
㦨

W


\

IAB
㦨


�CD 㦨
�EF 矄
矄�矄 ᆠ矄
RCD 㦨
REF

�ᨨ
ᨨ
ᨨ犨矄
�矄矄�
�
R҈
 
R



PR҈ ( I R ) R҈ ( 䰘
�) ´

� = = =

 PR
҈
� ´
( ) I R (R )








´R
㦨
R+


҈
졔


-��
- ҈
�҈ + �
㡰 -
�
+

=
Þ - ҈
�҈ - ҈䰘�
+

=





�㡰

矄� �
⧤�ᆠ ᨨ Ἐ矄
ᨨᨨ
犨
�
ᨨᨨ
BEFC

Þ -҈
+
�҈ + ҈
�҈ + �
㡰 =
Þ ҈
�҈ + ҈
�
+ ҈
=





��㡰

㝐盐�泐� �
⼄盐�ᨨ
�㡰
泐
҈


犨


҈
�҈ + ҈䰘�
=

㡰 ´ ҈
Þ ҈

�҈ - ҈䰘
�
=







���㡰

㝐盐�泐� �
⼄盐�ᨨ
��㡰
泐
҈䰘

犨


҈
�҈ + ҈
�
= ҈
㡰 ´ ҈䰘 Þ

�҈ - ҈䰘
�
= ҈䰘





�㡰

졠
矄ᨨ� �
⼄盐�ᨨ 矄
���㡰
 
�㡰

犨
� A


B

-҈
�҈ = 
Þ �҈ = -

1W C

R

2W D

F


= -


҈䰘 ҈


�҈
�矄
��
�
鼀 矄
ᆠ盐 
ᨨ犨矄
ᨨ鼀
ᨨ矄��
ᨨ
�� 鼀� 


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PS-207

Current Electricity



㡰




D

E ҈꾀

�

B





ᨨ犨
�
ᨨ




W

҈꾀
㦨
�҈ �




�

㦨

꾀



A

æ e ö P
㦨
� R
㦨
ç ÷ R
è R+r ø dP =

ᨨ

鼀 �鼀盐鼀

dR

F




C
꾀 ҈ 
盐矄
矄矄盐鼀
 
ᨨ �

A
㦨
VA
㦨

矄� �
⧤�ᆠ ᨨ Ἐ矄
盐 ᆠ�ᨨ
盐

C

犨
�

�҈ + � = �
҈꾀 + � =

꾀 Þ � =
A

ᨨ

ᨨ犨
盐矄� �
⧤�ᆠ ᨨ Ἐ矄
ᨨᨨ
犨
ᨨ �
ACDB

VA + ҈ + � 
㡰 -
= VB Þ VA + ҈ + � ҈㡰 -
= VB
Þ VB - VA =  -
= ҈ ᨨ 


䰘


ᆠ㡰 
⼄盐� 
ᆠ�ᆠ盐�
ᆠ

犨
矄
W

A

�


� ҈
W

C

꾀

쟼҈ 쟼
쟼  ҈
 + + + + ҈ 
 ҈ ҈ ҈
= ᔴ = = ҈ ҈ ҈ ҈ ҈ ҈ = =
 + + + +  ҈ 
 ҈ ҈ ҈

ᆠ㡰

ᔴ盐 
矄矄� �
 ᨨ盐�
矄�矄 ᆠ
҈



W


W






d éæ e ö ù êç ÷ Rú =
dR êè R + r ø ú ë û

ᨨ R
㦨
r b㡰 ෌�

쟼҈
㦨
҈

쟼

㦨



쟼
㦨


҈
㦨
҈W


㦨
҈W
 

㦨
҈W


W
W

æ e ö �=ç ÷ èR+rø

㡰

 ҈
= =

�꾀
 

�

㦨

҈

҈
 ҈W  b㡰

҈

W  和 ҈ 턼
 ҈
W

�҈ =



 ᨨ�
ᆠᨨ矄矄
AC
㦨


 矄�矄 ᆠ
RAC
㦨


졔


㦨


W

�҈ =



b㡰

黨


V
= = ҈ 꾀鼀 RAC
+


ᨨ
 
��
ᆠ�ᆠ盐�

矄� �
⧤�ᆠ ᨨ 矄
犨

P
犨
�

 - ҈
 - ҈  -
+ + =
҈
҈
᲼
ᨨ �



턼


㦨


 


㦨


�=

   ҈
҈
Þ + + = + + ҈
҈
҈
҈
҈
+ � + ҈

+ ҈ æ ҈
ö 䰘 Þ =
Þ ç ÷ = ҈

è ҈
ø
䰘 ´҈
䰘
Þ= = = ҈҈
�
ᨨ ҈
´




+


ᨨ犨
ᨨ�
ᆠᨨ矄矄
꾀

�´
=

 ´
=

+


Þ
黨
㦨


졔








҈ Þ= W
�


㡰

҈



㡰 ᨨ鼀
⧤ 


졔
ᔼ҈
졔
҈
ᔼ�

ᔼ
㡰
㦨






꾀泐� �

ᆠᨨ鼀� �ᨨ
ᨨ
�矄

E҈ E
E + + ҈+҈
҈ +҈ ҈ ҈ ҈ + + ҈ +҈
҈+҈


+ +


= �´
҈ ҈ ҈  + +


҈
= 㦨

ᨨ 


W 黨 ҈W

W

q


W



㦨

ᔼ҈
졔
ᔼ҈

ᔼ

錐


ᔼ҈

ᔼ

㦨

















҈㡰 
黨
졔

ᔼ


ᔼ҈

ᔼ
㡰
졔
ᔼ

㦨


ᔼ҈
졔

ᔼ

㦨
黨






㡰 졠
矄ᨨ� �
҈㡰
 

㡰 ᔼ҈
㦨


҈꾀

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EBD_8341 PS-208

Physics

�ᆠ�ᨨ
턼
ᨨ


矄� ᆠ
ᔼ҈
>
ᔼ

ᆠ㡰 꾀矄
ᨨ
ᆠ盐 
ᨨ犨矄
 ᨨ盐�
鼀
EB
 


㦨

 쟼
㦨

 
㦨

 꾀
㦨
� 和ᨨ

ᨨ �
�  ᆠ
犨
 
ᨨ� 矄
A
 
D VA

VD
㦨
�

҈҈


҈



b㡰

ᔼ

�m

W

ᔼ


ᔼ҈


R


R

黨W

҈





ᔼ
矄泐
矄
ᆠᆠ�ᨨ
�矄
盐泐
ᆠ �
 ᆠ
ᨨ
ᆠ盐 
犨� ᨨ犨
 ᨨ盐�
� 


泐
矄�鼀泐 � �
 
ᆠ�ᆠ盐�

ᨨ
鼀 �鼀盐鼀
ᨨ犨

R  
㦨
R�  
R
㦨


W \ R
㦨


W �㡰 
�

 
ᆠ盐 
ᨨ犨� �
�
 
ᆠ�ᆠ盐�

꾀泐� � ⧤�ᆠ ᨨ 矄
犨
⧤
�
 
ᨨᨨ
ᔴ쟼 ᔴ 


�

� 졔


㦨

Þ
�
㦨
 \
�
㦨
҈
꾀
v ᨨ犨
ᨨ
盐
ᨨ
ᨨ犨
 ҈W ᔴ �

꾀



졔
҈
맔
҈
㦨


\

҈





҈



盐矄
ᆠ盐 

ᔼ = ҈


b㡰



W �҈ A

æ V 㡰 V ö t÷ ´ t çQ H = R R ø è


҈҈
W

黨V
´ t

Q 盐
Q
㦨
mcDT㡰 R











�㡰

ᔴ盐 
 ᨨ盐�
AB

�҈ =



=




+



ᔴ盐 
 ᨨ盐�
AD

�
=



҈ = 黨
+ ҈҈
�

矄� �
⧤
�
꾀
ᨨᨨ

 NV 㡰
㦨
´ t (Q R µ l )
R

VB + �҈ 

㡰 - �
黨
㡰 = VD ҈
Þ VB - VD = 黨
㡰 - 

㡰 Þ VB - VD = ҈
- ҈
=
 � ҈


꾀矄ᨨ
HἘ
㦨
m'cDT
㦨

m㡰
CDT

N
V
´t
R






��㡰

҈䰘







㡰
W

��� �
⼄

�㡰
泐
��㡰

A

C

mcDT 黨V
´ t ⏌ R ҈ 黨´
=

=

ᨨ

mc DT N V t ⏌
R
N


�


ᨨ








N

㦨
҈

맔






\


N
㦨

b㡰 
⼄盐� 
ᆠ�ᆠ盐�矄

矄 ᨨ犨
�
 
��盐


҈



C

�
黨
W


RἘ

 
矄�矄 ᆠ
ᨨ
 
犨�
ᨨ
ᨨ盐
 �
�
l \ ᨨ犨
 �泐
矄
�
㜠矄ᆠᨨ 矄
'



鼀ᆠD
㦨


D

B


W
W

�҈


W

�


F

E �


W

�

҈



R

R

쟼

D


꾀矄ᨨ

H
㦨
mcD
㦨


�



W


R

 
矄�矄 ᆠ
ᨨ
犨�


쟼 �泐
矄
�
t
矄ᆠᨨ 

H
㦨



W

v

B


㡰






=
㦨

ᨨ

ᔼ

㦨

㦨



䰘
꾀   + 黨 ҈


B 

꾀


㦨
�

 ᆠ
矄盐 
ᨨ �
�  ᆠ
ᆠᨨ矄矄
矄�矄 ᆠ矄
犨�









A

ᨨ

꾀





맔
҈
㦨


黨W








R 4W E

\
꾀


㦨
�


W

R

҈ ҈ ҈ R 㡰
R㡰 = + =
R

Þ
Req = Req R
R R㡰 + 
R㡰

҈�









R

E 4W

� 


� 



� 
҈

R


R

꾀矄
ᆠᆠ�ᨨ
�矄
盐泐
ᆠ �
ᨨ
ᆠ盐 
犨�
ᨨ犨
 ᨨ盐�
�


R E


W


R

R R


R


W


R

R


R

҈꾀


R

A

҈꾀 E
W  ᆠ ␔ 
矄ᨨ  ��


ᆠ�ᆠ盐�
矄 矄
 ᆠ
␔ 矄ᨨ 
��

 ᆠ
ᨨ ᆠ盐 
犨�
ᨨ犨
ᆠᨨ矄矄



W
矄�矄 ᆠ


W
W

E
꾀


꾀

҈



B
W

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PS-209

Current Electricity ␔

 
ᆠ盐 
�矄�盐�ᨨ
矄
矄 ᨨ犨
�
 
��盐


҈
V  VP =
V ᨨ VP =
V

æ
´
ö +
쟼⼄盐� 
矄�矄 ᆠ

⼄
㦨
ç è
+
÷ø

VA  VQ = I ´
W = (҈A)(
W)

҈
V  VQ =
V

҈
҈
´  = =  꾀鼀
҈
+


VQ = ҈
V 
V =
V

�҈
㦨


꾀
 
�

= ҈
꾀 VAEB
㦨
҈
맔


졔

맔


㦨


 ҈

矄

ᔼ

ᆠᨨ泐�ᆠ
矄ᨨ盐�ᨨ

 
ᆠ�ᆠ
ᆠ盐 
�矄
鼀� 泐 盐
ᨨ
 
鼀ᨨ鼀 
ᨨ
�ᨨ 矄
ᨨ
ᆠᨨ 矄
҈黨


b

㡰 ᨨ
ᆠ盐 
�矄
ᨨ犨� �
 ᨨ盐�
矄�矄 ᆠ
R

꾀泐� �
⧤
�
ᨨᨨ

\
� =

҈ + 



















�㡰 ҈ + 


V  VS =
V Þ VS =
V

\
Vs


VQ
҈



졔


I


I҈㡰
졔
҈
졔
҈
맔
I




졔

I

㦨



Þ
I҈

�I

㦨

҈ A

1V

�

R

ᔼ



W

҈



矄�矄 ᆠ
ᨨ
 
盐
鼀


I


R⼄


I҈
㦨

⏌҈
꾀
 
I

㦨
�㡰

҈
V \ I҈ = = A
W æ ҈
ö I
= ç ´  =
A è
+ ҈
÷ø I = I҈  I
= ҈A

ᨨ �
�  ᆠ
ᆠᨨ矄矄
A
 
P


VA  VP = I
´
W = (
A)(
W)

ᨨ �
�  ᆠ
犨
B
 
D



é�
ù - ú ´
= VD


\
VB  BD
㦨
 ҈ ҈ ҈ ë û

VB
졔
ê


W






(
W )(҈
W ) =
W RR = ҈
= R҈ + R

W + ҈
W





��㡰

� 꾀 ҈




ᆠᨨ矄矄
G 㦨
 -




ᆠᨨ矄矄 H
㦨
҈
졔
҈
맔


҈
W

쟼⼄盐� 
矄�矄 ᆠ
ᨨ
  ᆠ�ᆠ盐�


ᔼ҈

ᔼ
㜠ᔼ҈

Þ I҈

I

㦨
҈ 和ᨨ� �
⼄

�㡰
 
��㡰

犨
�

��㡰

R҈ =
W +
W +
W =
W

I҈

3W

҈ H


黨 - ҈

҈  ´= = ҈ ҈ ҈

᲼Q  
ᆠ
�矄
�
�矄ᆠ �� �
鼀ᨨ桌


矄�矄 ᆠ
ᨨ
 
ᨨ犨
鼀

R
=
W +
W +
W = ҈
W

1W

D


W



1W

C G ᔼ҈  꾀泐� �
⧤�ᆠ ᨨ Ἐ矄
犨
⧤
�
ᨨᨨ
BCDB

犨
� 

I


I҈㡰
졔
҈
졔
I҈


졔
I҈
㦨



F

W

B

E

F

V


A \ ҈ ҈
졔
҈ 
㦨
҈ ҈
졔

҈ Þ ҈ 
㦨

҈ ᔼ
҈
㦨


 
҈
㦨

㦨

ᔼ
҈
㦨


 
҈
㦨

㦨


ᔼ
҈
㦨



 


㦨
҈ ᔼ

҈
㦨


 

㦨

҈
㦨






b

ᆠ

㡰 矄�矄 ᆠ
ᨨ
鼀
PQ
 
ST
ᆠᨨ鼀矄
�  ᆠ�
矄
P


Q
  S


T


 
矄鼀
ᨨ �


⼄盐� 
ᆠ�ᆠ盐�
�矄
矄 矄 ᨨ犨
�
 
��盐

W  和
W
W I


I҈




ᔼ


2W

E

B

҈ ҈ + 
= ҈ ҈ + 

턼





�㡰

R


 ҈ - � ҈ =

\
� =




��㡰 ҈




꾀泐� �
⧤�ᆠ ᨨ Ἐ矄
犨
⧤
�
ᨨᨨ
BDAB

R҈

V҈

꾀泐� �
⧤
�
ᨨᨨ
ᔴ쟼












ᨨ鼀
⼄

�㡰


��㡰


VP  VS = (
A)(
W ) =
V

D �

C

҈ - � ҈ + 
- �  =


ᨨ �
�  ᆠ
ᆠᨨ矄矄
P
 
S


2W


ᆠ盐 

� =

ᨨ �
�  ᆠ
ᆠᨨ矄矄
A
 
Q


I҈

I

҈


�㡰


҈黨 =  ҈ ҈ ᲼Q ᆠ
�矄
�
ᆠ �� �
鼀ᨨ桌

和� ᆠ
ᨨ
ᆠ盐 
�矄
├
ᨨ鼀
 
泐

E҈ E
E + + r҈ r
r SE ⏌ r = \ VAB
㦨
E⼄
㦨
҈ ҈ ҈ ҈ S + + r r҈ r
r

( ҈ +
҈ + ҈҈ ) =
V 㦨
҈ ( ҈ ) + ( ҈҈ ) + ( ҈҈ ) ᔼ  
矄�矄 ᆠ
ᨨ
⼄盐� 
泐
㦨


�҈ =

҈ W 

VB  V A + E҈ 
+  = = ҈A r҈ ҈

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EBD_8341 PS-210

Physics

�
=

VB  V A + E

+
= =
r
҈

  � =

VB  VA + E 
+ ҈ = = ҈A r ҈





b㡰

�


㡰

��㡰 ᔼ
r

�矄
矄 ᨨ㜠ᆠ�ᆠ盐�
 
矄�矄 ᆠ
ᨨ
 �矄
 ᆠ
�

ᨨ
r҈ Q P I҈ E҈ A

R U

E


I
I

I҈졔I
졔I

T

r

E

R

҈ = �
҈ =

B

S

꾀泐� �
⧤�ᆠ ᨨ Ἐ矄
犨
�
PQRUP
矄� �
ᨨ鼀
P 鼀ᨨ� � ᆠᨨᆠ├犨�矄 I҈r҈

E҈
졔
E

㦨



ᨨ

I҈


졔


㦨


\
I҈
㦨
҈꾀 꾀泐� �
⧤�ᆠ ᨨ Ἐ矄
犨
�
URSTU 矄� �
ᨨ鼀
U 鼀ᨨ� � ᆠᨨᆠ├犨�矄 
E

졔
E

Ir
㦨



ᨨ



졔
҈

I
㦨



\
I
㦨
҈꾀 


矄��
ᨨ
I
� �ᆠ矄
 
 
�ᆠ�ᨨ
ᨨ
ᆠ盐 
�

 ᆠ
UTSR �矄
ᨨᨨ矄�
ᨨ
 
矄矄盐鼀
꾀泐� �
⧤�ᆠ ᨨ Ἐ矄
犨
�

AURBA 矄� �
ᨨ鼀
A 鼀ᨨ� � ᆠᨨᆠ├犨�矄
I҈
졔
I

졔
I㡰
R

E

㦨




ᨨ





䰘






黨
҈



æ ҈


ö ´
Pᨨ矄矄 = I
R = ç è


÷ø

P ᨨ犨

P
㦨
VI

Þ I = V ҈� ´҈�� +



\ I 鼀� = = ҈黨


A »

A





b㡰


R ´
R
R ´҈
R +R+ + R
㦨

R
졔
R
졔

R
졔
R

㦨

R

R +
R
R + ҈
R

V
҈

矄� �

P
㦨
R
Þ


㦨
⼄
R R
㦨


҈


㦨


W
´



҈ Þµ

ᆠ㡰 
 �泐
犨�

鼀 �鼀盐鼀
犨 
矄�矄 ᆠ
犨�
 鼀� �鼀盐鼀
ᆠ㡰 ᨨ
ᨨ犨
ᆠᨨ 矄盐鼀
泐
ᆠ�ᆠ
� ᆠ矄
�
  盐�� �

ᨨ
㦨

�

␔ ␔
㦨
ᨨ
맔
鼀
�


㦨
V
맔
I

Þ

�


㦨




I
Þ I =


�

= ҈҈


»
҈
꾀




㝐� �鼀盐鼀
ᆠᆠ�泐
ᨨ
鼀�
盐矄㡰 ҈҈


ᆠ㡰

ᔴ盐 
�
ᆠ
盐
㦨


ᨨ犨 ҈

=


�꾀 㦨
ᨨ�




ᔴ盐 
 ᨨ盐�
鼀鼀
㦨



�
맔

㦨
҈
�
꾀


㦨


W 턼
㦨


W ҈



b㡰

ᔼ҈ 꾀 ᔼ


쟼⼄盐� 
矄�矄 ᆠ


R⼄
㦨


\

ᆠ㡰 ᨨ鼀
 
��
ᆠ�ᆠ盐�


矄�矄 ᆠ矄 ᔼ
㦨
҈
W

ᔼᔼ
㦨
҈⏌

W

ᔼᔼᔼ
㦨
⏌

W ᔼ
�矄
ᆠ
 

>
҈
>

 ᆠ




҈





Þ

҈


´ ҈

= 黨
奠
\
쟼 �ᆠ� ᆠ泐
= ҈


+ Pᨨ矄矄

e
 ⏌
㡰


҈҈
҈҈
´ ´҈

=҈҈´҈
� ␔
㦨
҈
҈ ҈
҈

꾀矄
ᨨ犨
㡰
=




㡰 
ᨨ犨

P 㦨
҈�
맔

�
졔
҈�
맔
҈


졔
҈�
맔
҈

졔


맔
҈


㦨
҈�
맔
҈��
졔





␔

e
e
= ⼄


ᨨ


'
㦨


㦨




␔ Q


㦨



␔㡰 㡰 ᨨ犨


㦨
ᔼ




㦨


맔
҈


맔

Þ
㦨
҈
҈
맔
҈

W ␔ 
矄盐泐
ᨨ
҈҈

�矄
ᆠᨨ

ᆠ ᨨ犨

㦨


R =
W
V =













㦨
҈

␔
�

和�鼀�泐



㦨
�


㦨

␔ b㡰 ␔ 
犨ᨨ
矄�矄 ᆠ矄

ᆠᨨ

ᆠ
�
矄�矄
⼄
㦨



犨
ᨨ犨

 ¢ =

b㡰 ෌�


ᨨ犨

P
㦨
҈
├␔
㦨
҈



␔

P ҈


= V




æ







ö ç
� + ҈

÷ è ø

´

ᔼ

ᆠᨨ ��ᨨ

⼄
㦨
⏌



Topic-4 : Heating Effect of Current

ᔴ盐 

I =







ᨨ犨
ᆠᨨ 矄盐鼀

 =

҈
졔
I


҈㡰
R 㦨


\I
㦨
꾀

 ᆠ

ᆠ盐 
 ᨨ盐�
R

㦨
I҈
졔
I

졔
I
㦨

꾀

҈













矄ᨨ
҈ =
 

= 
� ҈




ᔴ盐 
ᨨ犨
�
= ҈ +
꾀矄
=




㦨
҈
W 和
㦨


W ҈
㦨

졔
턼
㦨


W
졔


W
㦨


W

㦨

졔
和
㦨
҈W
졔


W
㦨

W ᔼ҈ ҈
㦨
ᔼ



ᔼ҈ =


ᔼ  ᔼ
= ᔼ
=
҈



ᨨ
ᔼ
=
ᔼ҈ 
ᨨ犨

㦨
ᔼ



WWW.JEEBOOKS.IN

PS-211

Current Electricity

æ
R ö ÷
㦨



䰘I
R  ø

ᔼ


ᨨ
턼

 턼 = ᔼ҈ 턼 =
´
 = ᔼ


ᨨ犨
�矄矄��ᨨ

P

㦨
I
çè







ᨨ
和

 和 = ᔼ
和 = ᔼ
×
 =
ᔼ


R

\
෌矄
鼀ᨨ盐 
ᨨ

� 
泐
和
҈


㡰 ␔
├ ᨨ犨
ᨨ犨
P =

ᔼ
�矄
��
 
 
ᨨ犨矄
ᨨ
 
盐矄

�
 
ᨨ ҈

␔
>



␔
>


␔

\

R҈



ᔴ矄
�㡰

1W 1W




㡰 \

2W

2W

1W

V V
\
R
㦨
P R

V V
R
= = R 錐 R҈
㦨
҈





꾀矄
犨
├ ᨨ犨

P
㦨


R


V҈

V
V
㦨 VR
R ҈졔R


V҈㦨 VR ҈ R ҈졔R


�矄
�矄

ᆠ矄
ᨨ
 ᆠ
␔ 矄ᨨ 
��
R҈
㦨
҈W ᔴ矄
��㡰

R

1W


�


1W


�
㡰 W㦨 V 㦨 R3 R

2W 1W

1W


㦨

�


҈

҈ ҈ ҈
= + + Þ R
= = ҈
�


҈

R

W҈
㦨


V҈


=

R҈

ᔴ矄
���㡰

1W

 
W =

1W

2W

2W

ᨨ

1W

1W

㦨


盐矄
R
㦨

W 和� ᆠ

R



R҈


R
\
P

>
P҈
>
P 꾀矄
犨
├ ᨨ犨

P
㦨
I
R 
ᔼ
ᆠᨨ 矄 
\
P
µ
R

I

R R R ᨨ犨
�矄矄��ᨨ

P҈
㦨
I
R㡰
㦨
I
R R

ᨨ


�
㡰
V


R ҈䰘


V R

´





R 


�
㡰

�
㡰
R

�
㡰

R



R҈
R  R҈ + R
㡰
 R҈ + R
㡰

�
㡰



´


�
㡰
V
V




é ҈ ҈ù
êë҈

+

úû V ´ ҈




҈

´ ҈

´

´

҈

´ ҈

´

´


҈

´ ҈

´

҈

´ ҈

´ ҈



㦨






�


҈�

 ᨨ
W҈


W



W ᆠ㡰 

I҈
맔
�
㦨
I



졔

㡰

I
I

V

V
R
= R
 R҈ + R
㡰





҈ù
é ҈ êë҈

+

úû V W


W



W҈

R

I

 R҈ + R
㡰


W

㦨



W


W



W҈

㦨







I

V
R҈

W


W



W҈
㦨


1W

㡰







2W

1W

҈�


R҈

1W

1W

1W

æ ö R÷ 㦨
҈
�
I
R
ø

\
ᔼᔼᔼ


ᔼᔼ


ᔼ


ᔼ

1W 1W

1W

҈ R ҈



1W

R⏌


ᨨ犨
�矄矄��ᨨ

P

㦨
I
çè

꾀 
V
�矄
ᆠᨨ 矄 
�

 
ᆠ矄矄

\ P µ

1W

I

R

R

V
␔
├ ᨨ犨
ᨨ犨

P = R

҈

ᆠ㡰

R I

҈ ҈ > R

R



>

R/3

R ᨨ犨
�矄矄��ᨨ

P
㦨
I
( R ⏌ )
㦨


I
R

҈ \
ᨨ

ᆠᨨ 矄 



�ᆠ盐
鼀盐
P µ R

҈

I

R

I

V
R


W


W

I

I

R

I҈

�W

WWW.JEEBOOKS.IN

EBD_8341 PS-212

Physics

I҈ \
I҈
㦨

I

Þ I
=



� 
�
�
W 矄�矄 ᆠ

矄ᆠᨨ  I҈
´ � = 
� 
�


W 矄�矄 ᆠ

矄ᆠᨨ  I҈
´



Þ

҈



�矄
犨ᨨ盐

ᨨ矄矄�
ᨨ 泐
犨  R
㦨



W

�矄
�
矄�矄
犨�
  盐
矄�矄 ᆠ
R盐
㦨


W㡰
ᆠ盐矄
�
 
ᆠ矄
ᨨ
矄�矄 ᆠ矄 犨�
矄 
⼄盐


ᨨ
҈


ᆠ

҈


҈
҈
= � Þ H�
W = =
ᆠ⏌矄 H�
W �

犨 
= r

\ 
� 
�
 

W
矄�矄ᨨ
㦨


ᆠᨨ�矄⏌矄 b㡰 
ᨨ盐ᆠ

矄ᆠᨨ 

H=




V

 

R = R
ö

ò

é
rl ù
㦨

´ =
ú

ë
p û

ᨨ
矄�矄 ᆠ
㦨
҈
졔


㦨

ê

æ
ö rl \ H = V ç pr ÷ l =

è rl ø A pr


\
 =

æ pV r r H =ç
ᨨ

H µ ÷ l l r è ø

ᨨ





㡰


´ 
⏌


ᨨ
矄�矄 ᆠ
㦨


ᔴ矄
ᆠᨨ

ᆠ
�
矄�矄 R

R

\
 =




r

E

r

æ
E ö J҈ = I
R = ç
R è
r + R ÷ø R

R

矄�矄 ᆠ
ᨨ
 

ᆠᨨ�

R 㦨


\

E

r

ᔴ盐 
�
 
ᆠ�ᆠ盐�

㦨 I 㦨


r


A




æ E ö J
= I
R = ç ´R r÷ çR+ ÷ è
ø 
E 㡰





r + R 㡰
Þ


R =


�



r + R 㡰


=

rö æ çè R + ÷ø



R




҈





\

R盐
㦨


㡰 
ᨨ鼀
ᨨ
矄�矄 ᆠ
҈
├W
�矄

ᨨ
 
矄�矄 ᆠ ᨨ



W

和ᨨ

 �
⼄盐� 
矄�矄 ᆠ
�矄

᲼Q = ҈W桌

V ҈

´ ҈

= =

W P �



҈ ҈ ҈ ҈ ҈ = + = + R Ἐ ҈
k W


W ҈








R

盐

V
ᨨ犨
P
㦨
R

R

V Ἐ = I
´ R Þ
�
㦨
�
맔
R
\R
㦨
�W

Þ


R
졔


�㡰
㦨
҈
�


졔
R㡰
\
R
㦨

W

W 

W㡰
␔
├ ᨨ犨





Topic-5 : Wheatstone Bridge and Defferent Measuring Instruments

Þ
᲼ R +

�桌
=


�᲼
+ R桌





V 䰘� = = 䰘
� A R ҈


҈


 ᨨ鼀
졠 鼀Ἐ矄
犨
矄�矄 ᆠ
ᆠᨨ矄矄
矄�矄ᨨ
R





r  R + 㡰




�

B ҈

W

෌� 㡰

E

V
҈

´ ҈

= 㦨
҈

W P ҈



�矄
ᆠ盐 
��矄
� ᨨ
犨ᨨ
矄


I҈

 
ᆠ盐 
  矄矄矄
 ᨨ盐�
 

\
V'
㦨
I҈

Þ

�
㦨
I҈
맔
҈

\ I҈
㦨


�
A

盐矄
ᆠ盐 
 ᨨ盐�
矄�矄 ᆠ
ἘRἘ
I

㦨
䰘
�



�
㦨
�꾀


I r

\
҈
㦨



�






���㡰

\ V'
㦨
´  Ἐ = ҈
´


� =

� =
� 和� ᆠ
 
ᨨ�
ᨨ
ᆠᨨ矄矄
 

�矄

�V
 ᆠ
ᨨ� ᨨ
ᆠᨨ矄矄
҈
W
矄�矄ᨨ
�矄
҈




�㡰
㦨
䰘�V


ᔴ矄
ᆠᨨ

ᆠ
�


E


´ 
⏌


ᨨ�
ᆠᨨ矄矄
 
ᆠᨨ�
ᨨ

�
�
ᨨ矄
犨�

ᨨ犨 P Ἐ
㦨



�
␔


E
r




E

҈

⏌
= ҈ = = ҈ +

҈



ᨨ鼀
⼄

�㡰
 
���㡰


㦨


�
鼀�

I

E




��㡰

ᨨ鼀
⼄

�㡰
 
��㡰


㦨


鼀�
␔ 
矄�矄 ᆠ

ᆠᨨ

ᆠ
�


盐矄

ᨨ
H㡰
�矄
ᨨ盐
�
ᨨ
 �
l㡰
 
�盐矄 r㡰

ᨨ盐
҈黨


l
rl =
꾀 p

␔ 
矄�矄 ᆠ矄

ᆠᨨ

ᆠ
�
矄�矄







æ 
ö 
 = ´



�㡰 ÷ è ø

b

㡰
ᨨ盐ᆠ
 = ç

Þ




҈ ҈ +
�

҈



= = Þ RἘ = W R Ἐ ҈



҈







矄� �
졠 鼀Ἐ矄
犨

ᆠ盐 
�
 
ᆠ�ᆠ盐�

I=

ᨨ�
= 
矄�矄 ᆠ ҈



+






WWW.JEEBOOKS.IN

PS-213

Current Electricity ᨨ �
�  ᆠ
鼀矄盐
泐
ᨨ鼀

҈�

҈




ÞV =
㦨
҈
黨�
ᨨ V = IR Ἐ = ´ ҈





䰘䰘 +








b㡰 㝐盐�鼀
矄 ᨨ犨矄
 ᆠ�ᨨ
�
ᨨ
ᆠ矄矄
�


 ᨨ
   
矄� �
 
ᨨ矄
�
 
ᆠ ᨨ矄
ᆠᨨ鼀ᨨ  
�矄
ᆠᆠ�ᨨ





㡰 ᨨ �
�� 

x =

黨


҈







b㡰

ᨨ

 ᆠ
鼀
��


 Ӹ
Þ
Ӹ
㦨
黨
�
㦨

맔
҈



黨
�㡰 = 黨
� ҈

- 黨
�㡰 ҈

� ´ 黨
� ᨨ
 = =

҈
W


�

ᨨ �
ᨨ  �

҈


=




ᨨ⏌ᆠ鼀 ҈

-
黨

矄� �
ᨨ鼀盐

�  
矄�矄 ᆠ


æ l -l ö �
-

ö r = ç ҈
÷ s = æç ÷ ´ � = ҈
�W l è

ø è
ø



ᨨ �
ᨨ
㦨
҈


 ᆠ� �
 �
ᨨ鼀

㦨
҈




黨

\x =

b㡰

҈҈


␔ 

 
Ӹ

� ᆠ  �
l҈
 
҈



l҈㡰
犨�
矄ᨨ � ᆠ  �
矄ᨨ

l

㦨



�
ᆠ鼀 㡰 
�矄
ᨨ
ᆠ  �
�
盐
ᨨ� 

�
 
ᆠ
 
  � ᨨ鼀

 ᆠ  �
�

 ᆠ
犨 矄ᨨ 
��


㡰




 
矄�矄 ᆠ
ᨨ
 
犨 ᨨ
犨� ᨨ �
�� 
ᨨ
 
ᨨ �ᨨ鼀
犨�

졠
 ᆠ� �
ᆠᨨ ��ᨨ


dV I ´ R é

´ R ù = = ê ú mv ⏌ m dl l ë l AB û æ dV ö

´ R =ç lAP = ´ ҈


mV ÷ d l ҈


è AB ø

꾀 
 ᆠ  �

Ἐ AB Ἐ = V AP

R҈ R
= R R


Þ
VAP
㦨
�


mV 꾀矄ᨨ

VAP
㦨
�
V
 ᨨ
 ᆠ
ᨨ� 

P㡰

\R = �


V AP �
´ ҈


=

� = ҈

W �
´ ҈



矄�矄 ᆠ
ᨨ
ᨨ �ᨨ鼀
犨�
㦨



҈
맔




㦨


W ᔴ盐 
�
 
犨� ᆠ㡰

�=

 ҈ V = = A RT
+

� +

�䰘 + ҈


҈ ᨨ犨
V
㦨
�R 꾀
㦨

맔



҈
맔
�
㡰
㦨



�





䰘


㡰 ᆠ㡰 


 
 �
꾀

犨 �ᆠ
� ᨨ鼀
矄 ᨨ犨矄
盐  ᆠ�ᨨ
ᆠ盐 


�=

 e æ ö e =
ᨨ

� ç ҈
 ÷ = ҈
 +  ҈ è ø


e é e é ù e ù e ҈ Þ ·҈
 ú =
Þ ê ·҈
 ú =
ᨨ
㦨


҈ êë 
҈ û ë û



졠
 ᆠ� �
ᆠᨨ ��ᨨ


㡰 ෌�

ᆠᨨᨨ盐
ᆠᨨ
ᨨ
矄�矄 ᆠ
҈
㦨
졠 �


 
ᨨ犨


\
҈㦨


맔
҈

㦨


盐矄� �
 ᆠ
犨 矄ᨨ 
��
� ᆠ�


҈




= Þ = 




\

㦨
҈


�


ᆠᨨᨨ盐
ᆠᨨ
ᨨ

ᨨ犨

盐
 
ᨨ犨

҈



R҈ R = R
R

ᆠ㡰 꾀矄
 
犨ᨨ
ᆠ矄
ᨨᨨ矄
ᆠ
ᨨ 
 ᆠ

 
 ᆠ� 鼀
�
ᆠᨨ矄
ᆠ�ᆠ盐�
�矄
҈�

҈

㦨
�

 

矄�矄 ᆠ
�矄 ҈
졔




㦨
҈


W
ᆠ盐矄
�
 
ᆠᨨ矄
ᆠ�ᆠ盐�
 
�   矄�矄 ᆠ
ᨨ
犨ᨨ
ᆠ矄

�
矄�矄
ᔴ盐 
�
 
ᆠ�ᆠ盐�
I
㦨


 ᆠ� 鼀 � A = ᨨ 矄�矄 ᆠ ҈




ᨨ �
�  ᆠ
ᆠᨨ矄矄
ᨨ鼀
犨�

矄鼀
矄
  鼀� 
ᨨ�
ᨨ
� 
ᆠ
和� ᆠ
 
ᆠ盐 
�矄
犨
ᨨ鼀
 
ᆠ
ᨨ

҈�
 \ V҈
㦨
E҈

Ir҈



㦨
҈�

҈
ᆠ㡰

� ´


= ҈
҈  ҈



ᔼ
ᆠ矄
ᨨ
 ᆠ
鼀
��

R X =
෌�

X
㦨
黨

W

l
㦨





ᆠ鼀 l ҈

- l 黨
´

Xl = =

W \
R =

҈

- l DR Dl D (҈

- l ) = + \
҈

- l R l DR

҈

҈ = + \
DR =


�W Þ








 ᨨ

R = (

±


� ) W

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EBD_8341 PS-214

҈



Physics

b㡰

҈
W



R҈ A

x R

G C

R
R


B

ᔼ
�盐矄
ᨨ
 
犨�
�矄
ᨨ盐

 
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矄�矄 ᆠ
ᨨ
AC 犨�

R ᆠ  �
 
矄ᨨ
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矄�矄 ᆠ
ᨨ
CB 犨�
ᆠ  �

盐
矄� ᆠ
҈ ᨨ矄



꾀

�

ᆠ鼀




ᆠ鼀

P R = 꾀泐� �
 ᆠ
犨 矄ᨨ 
��
ᆠᨨ ��ᨨ
Q S X ҈
= l҈ l
 l҈
㦨
�

졔
 
ᆠᨨᆠ�ᨨ
㦨
�

졔
҈
㦨
�
ᆠ鼀

R


ᨨ
ᆠ  �
矄ᨨ

҈



l

㦨



졔
 
ᆠᨨᆠ�ᨨ
㦨



졔


㦨
�

ᆠ鼀 ᨨ
҈�


X ҈
� =
\ X = = ҈


W � �
�

P R = 꾀泐� �
 ᆠ
犨 矄ᨨ 
��
ᆠᨨ ��ᨨ

Q S R
㦨

W

l

҈

㜠l





G

҈黨


S

R

෌

꾀矄

␔ 矄ᨨ 
��
 ᆠ
ᆠᨨ ��ᨨ


҈
R Ἐ

Þ
�

졔
�R
㦨
҈
R ҈� ҈
R Ἐ
Þ �= = + ҈
R Ἐ ҈
+ R Ἐ ҈
�

\ RἘ = = ҈
W �

␔ 
 
矄�矄 ᆠ矄

� ᆠ  �
 
ᨨᆠ├泐
矄 � 矄


ᆠ鼀



X = Þ



X

l X

㦨

l
졔


l +



- l æ


ö ᨨ



X
㦨
l X
졔

㡰
졔



㦨
çè ÷  X +
㡰 +

X +
ø


ᨨ
X
㦨
㦨
W




\

\






E ҈
= k �

㡰 r + ҈


G

r + ҈
�
= Þ
�

r
졔
�


㦨
�

҈
�
N
Þ r = W = W
Þ
N
㦨
҈
� ҈
Þ

S

矄� �
IgG
㦨
I

Ig㡰
S

æ G ö = ҈

´ ҈
-
æ ҈

+ ҈ö +҈ çè ÷ è S ÷ø



҈ ø

I
㦨
ᔼ�
ç ҈䰘





��㡰

��� �
�㡰
泐
��㡰
犨
� I㜠Ig

ᨨ
㡰

҈
㡰 ␔
├ ᨨ犨
 

E µ l
犨 
l
�矄
 
 ᆠ� �
 � \
쟼
㦨
k
�

㡰



�㡰 ␔ 
 
 ᆠ� �
 �
ᆠ  �矄
泐



ᆠ鼀


Ig 㦨
҈


맔
҈


A錐 G
㦨
҈


W;
S
㦨


҈
W I

P S = Q R


矄�矄 ᆠ

�矄
ᆠᨨ

ᆠ
�

犨�
矄�矄 ᆠ
和
ᨨ ҈
W




ᨨ
҈


R

R l 㦨
l X

ᨨ







l
㦨
l X
ᨨ

l 㦨
X +


Ig

B

҈
㡰

B

㡰

G



㜠l

l졔



R X = l ҈

- l

҈



Q

P


㦨

W



X

 
ᨨ� 
C
ᨨ矄
ᨨ
ᆠ  �
㡰 和� ᆠ
 
ᨨ � �
ᨨ
ᆠᨨ矄� �
  矄犨�ᆠ
�s
ᨨ矄
ᨨ
ᨨ矄
ᨨ
ᆠ  �
  � �
ᨨ
�
 ᨨ鼀
�
鼀 矄
  �
ᨨ
 
ᆠ矄矄
 
�矄
ᨨ
ᆠ盐  矄矄� �
 ᨨ盐�
S

ᔼ
�矄
 
ᆠ矄
ᨨ  ᆠ
犨 
矄ᨨ 
��

盐矄 ᨨ �

A �矄
⼄盐
ᨨ
ᨨ �

B


\ IP
㦨
IQ
 
IR
㦨
IG

␔ 
 
��
�矄
 ᆠ
x
>

W

㡰

AC
矄 ᨨ盐
矄ᨨ
ᨨ
ᆠ  �

盐
ᨨ� 

 ᨨ A CB

I
㦨
҈


맔
҈


맔
҈



҈

㦨
҈



҈
鼀꾀 ᔼ
ᆠ矄
ᨨ
鼀
��

盐
ᨨ� 

R҈ R AC AC = = = R
R
CB ҈

 CB


҈





䰘㡰
꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

矄�矄 ᆠ

矄
鼀盐 ᆠᨨ �ᆠ� 

a 㦨






ᔴ҈ 꾀 
鼀盐
�矄
� ᆠ矄
泐
҈


ᔴ
�



D
㦨
҈


ᔴ Ἐ
㦨


҈
졔
aD 㡰
㦨


҈
졔
aD 㡰 \ Ἐ
㦨
҈
W

WWW.JEEBOOKS.IN

PS-215

Current Electricity

R҈
㦨



W

ᔼ

ᔼ


R

㦨
҈


W

QV = IR 㡰

黨
㦨
I

졔

㡰

Þ

I
㦨
҈A


m

 m

和



�


 �
 �
=   ᔼ
= = ҈ +  䰘

+





W


W

�
�

- �
´
㦨

҈
黨



҈

䰘
㦨



䰘 䰘





 ᆠ
ᨨ�
ᨨ
犨
和
 

㦨



䰘
ᨨ


㡰 ␔ 
 
鼀盐
ᨨ
鼀
� ᆠ矄矄錐
�矄
矄�矄 ᆠ � ᆠ矄矄

Rt
㦨
R

҈
졔
µ
t㡰  ├ ᨨ犨
矄�矄 ᆠ㡰 和 
矄�矄 ᆠ㡰 X S

黨
 ᨨ �
�  ᆠ
ᆠᨨ矄矄
W
矄�矄 ᆠ
㦨
V ᨨ �
�  ᆠ
ᆠᨨ矄矄

W
矄�矄 ᆠ
㦨

 \ 


ᆠᨨ矄矄

ҘF
ᆠᆠ�ᨨ
㦨
V   


ᆠᨨ矄矄


ҘF
ᆠᆠ�ᨨ
㦨

V \ ᔴ �
ᨨ

ҘF
ᆠᆠ�ᨨ
Q҈
㦨

맔

㦨
黨
ҘC ᔴ �
ᨨ


ҘF
ᆠᆠ�ᨨ
Q

㦨


맔


㦨


ҘC \ ᔴ �
ᨨ犨矄
ᨨ鼀
y
ᨨ
x
㦨



黨
㦨

䰘
ҘC b㡰 꾀矄
ᨨ
犨�
ᨨ
矄鼀
鼀�
\
ᔼ꾀
㦨
ᔼᔴ \

和


㦨
҈
ᔼ҈

�

ᔼ
㦨
҈
´

졔턼
턼


R

㦨
�


W

RἘ = ҈
W 

ᔼ = ҈

졔턼҈ 턼҈






r

꾀



ᔴ

ᔼ

N

 l⏌


l⏌


r l

҈



l

㡰

J G

\V AB =

ᨨ

鼀
��

盐
ᨨ� 
ᨨ

 ᆠ
ᆠᨨ ��ᨨ

X S X l = =
ᨨ
l ҈

- l S ҈

 l ᨨ
盐
ᨨ� 
ἘἘ

矄鼀
ᨨ� 
ᨨ

盐 ᆠ  �





l҈
ᨨ
鼀�

҈

- l �

\

� ᆠ矄� �
 
矄 
矄�矄 ᆠ
ἘsἘ
矄 ᨨ盐
矄ᨨ
� ᆠ矄
ᨨ矄��ᨨ 㜠
ᨨ
盐
ᨨ� 
��矄
鼀ᨨ矄
ᆠᆠ盐
� �

盐
ᨨ�  

�矄
鼀ᨨ矄

 
ᆠ 
ᨨ
矄�矄 ᆠ
犨�
犨 矄ᨨ 
�� �矄
鼀ᨨ矄
ᆠᆠ盐
犨 

 
ᨨ盐
矄�矄 ᆠ矄
ᨨ
 
ᨨ盐
鼀矄
 ᨨ
 
矄鼀
ᨨ
ᨨ
鼀� �盐


ᆠ㡰

P X =
Q R

㡰 ᨨ



ᨨ
ᨨ矄��
 
��
鼀� 矄
ᨨ
  � ᨨ鼀

␔  
 
�矄
ᨨ
ᆠ盐 

 
ᨨ� 
ᨨ
  � ᨨ鼀
�矄

ᨨ
㡰 C  X A D Ý ҈
W ᆠ㡰Q ��
�矄
 ᆠ


R AJ


r ҈
W = =

\ X 㦨


W RJB


r X

犨 
r
�矄
 
矄�矄 ᆠ

盐 �
 �


Topic-6 : Miscellaneous (Mixed Concepts) Problems ҈


㡰

X
矄 ᨨ盐
矄ᨨ
鼀�
盐 ᆠ  �

 ᆠ
�
X
�矄 S

G

VBC



ᔼ
ᆠ矄
ᨨ
 ᆠ
犨 矄ᨨ 
��
ᆠᨨ ��ᨨ






VAB I AB R AB R = = AB = VBC I BC RBC RBC

ᆠ㡰 ␔ 
 
矄犨�ᆠ
和
�矄
ᨨ
ᨨ
ᆠ �
Q҈
졔
Q

㦨


CV҈
졔 C
V

㦨



맔


졔



맔
㡰
㦨



␔ 
 
矄犨�ᆠ
�矄
ᆠᨨ矄
  矄泐
矄
�矄
ᆠ 

 
ᆠ盐 
I ᆠᨨ鼀� �
ᨨ鼀
 
泐
�矄

㡰








PBC I RBC = = PAB I
RAB

r

l
᲼p ´
r
桌 ҈ = l
r
᲼pr
桌

l
᲼p ´
r
桌 ҈ = l
r
᲼pr
桌

PAB
㦨

PBC


I J AB p ´
r
҈ = = 錐
\
JBC
㦨

JAB I J BC

p´r éV AB ù E AB êë l ⏌
úû ҈ = = 錐 \
쟼ᔴ
㦨

EAB E BC é VBC ù
ê l⏌
ú ë û

b㡰 

ᆠ  �
�
�  
 �泐
㦨

矄盐� �


㦨
DU = Q = I2Rt 㦨
҈
맔
҈
맔
҈


맔
�
맔



㦨







㦨


├ 㡰 꾀
 泐
� 矄 
ᨨ
�鼀
t
盐� �
ᆠ �� �
ᨨᆠ矄矄

   矄� 
ᆠ盐 
�
 
ᆠ�ᆠ盐�

V
-t ⏌ RC e \ ᨨ �
�  ᆠ
ᆠᨨ矄矄
矄�矄ᨨ
R é V
-t ⏌ RC ù VR = ê e ú´R ëR û

I
㦨


= V
e - t ⏌ RC

\

Q

ᨨ �
�  ᆠ
ᆠᨨ矄矄
C Vc
㦨
V


V
et/RC

㦨

V

҈
et/RC㡰

Vc = VR




�㡰


��㡰

WWW.JEEBOOKS.IN

EBD_8341 PS-216

\

Physics

(

)

\ 
�鼀
ᆠᨨ 矄  t
㦨
RC
㦨


矄ᆠ ᨨ �
ᆠᨨ矄矄

m
ᆠᆠ�ᨨ -t ù 
 泐
�鼀
� é

V
҈ - e -t ⏌ RC = V

e - t ⏌ RC

Þ ҈
et/RC 㦨
et/RC

Þ
҈
㦨

et/RC

V = V
êë҈ - e t úû

├� �
ᨨ�
ᨨ
ᨨ
矄�矄

æ t ö ᨨ�e
҈
㦨

ᨨ�e

졔
ç è RC ÷ø t Þ

㦨


맔




ᨨ�҈




RC Þ t
㦨
᲼

맔




ᨨ�҈


桌
맔


�
맔
҈


맔


맔
҈


ᨨ

t
㦨
҈



矄 �


b㡰

é

Þ
= ҈
ëê҈ - e Þ
t
㦨


矄
Þ e

 ҈



ᔴᨨ鼀� �
犨�
y
㦨
mx 졔
C

҈ æE ö ᔼ ᆠ
㦨
ln ç
÷
 
矄ᨨ
㦨

è Rø RC





䰘




和ᨨ

犨 
x
�矄
ᆠ  �
ᨨ

x
 
矄ᨨ
� ᆠ矄矄
 
ᆠ盐  ᆠᨨ鼀矄
矄矄

 ᆠ
犨
�
�矄
Q
b㡰 ᨨ矄
ᨨ �ᆠ
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矄ᨨ
犨ᨨ├矄
ᨨ
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� ᆠ�
ᨨ
 ᆠ 犨 
矄ᨨ 
��
盐矄
ᨨ
鼀矄盐
盐 ├ ᨨ犨
矄�矄 ᆠ

ᨨ   
矄�矄 ᆠ
犨�

 
ᨨ
矄�矄 ᆠ
ᨨ
犨 ᨨ
 �
ᨨ ᨨ

ᔼ
矄 ᨨ盐

ᆠᨨ

ᆠ
犨
A
 
D
ᆠ㡰 꾀矄
 
ᆠ盐 
ᔼ
�矄
�   
ᨨ
R


�
ᨨᨨ犨矄
 
  矄�矄 ᆠ
R ҈

R


R 

R

 
R

鼀盐矄
ᨨ鼀
 
 ᆠ ␔ 矄ᨨ 
��
\
R҈
R

㦨
R
R b㡰 
R

 
矄�矄 ᆠ
ᨨ
犨�
쟼 �泐
矄
�
t
矄ᆠᨨ 

H
㦨
꾀矄ᨨ

H
㦨
mcD
㦨


҈

�㡰

æ I - Ig ç Ig è

ᨨ鼀
ç

҈


⼄
㦨

ᔴ ҈ - 

҈ 錐V µ 錐V µ r
r r

-


ᔴ


ᔴ






�҈ =

㡰


  -
ᔴ    -
ᔴ 


㡰




�
=

Ҙ


㦨 ᔴ



㦨

�KW




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Ҙ

- ᔴ҈ é
ᔴ ù
㦨

ê҈ -
 úû 㦨
ë
ᔴ

�ᨨ 

҈ V
ᨨr
P µ (Q V = ᆠᨨ 矄  ) R R

꾀矄
 
ᨨ�ᨨ
�矄
ᨨ 㜠盐 � ᨨ鼀

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
�  
ᆠᨨ矄矄㜠矄ᆠ�ᨨ

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犨�
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
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犨 
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�矄
鼀 �鼀盐鼀





ö V -R ÷S = ÷ I g ø


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㦨
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C
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m

-

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꾀 


ᨨ �
�鼀
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㡰 ␔�
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ᨨ�ᨨ
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ᆠᨨ鼀
 �


 泐
ᆠ矄� �
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
ᨨ
ᆠᨨ矄矄㜠矄ᆠ�ᨨ

 
� ᆠ矄� �
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矄�矄 ᆠ

 ᨨ
 
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犨�
ᆠᨨ 矄盐鼀 矄矄
ᨨ犨
ᨨ犨矄
 
 
ᨨ
� 
꾀矄
ᨨ犨

P =

҈



黨
҈
� -





n 
=� ´ = -
黨黨

\
n =
黨







黨




҈҈



ᔴ ҈ - 

éQ � = �  -  t ù ᨨ ëê ûú ҈ �



꾀 
҈矄
ᆠ盐 
�
 
鼀鼀
ᆠᨨ鼀矄
ᨨ
 
� �� e

N

㦨
҈

맔






\


N
㦨



ᔼ  矄�泐
ᨨ
�� 
I
µ


⼄҈ 㦨

�



꾀
�鼀

㦨


矄
ᨨ鼀
犨�
�矄泐
� �
졠


mcDT 黨V
´ t ⏌ R ҈ 黨´
=

=

ᨨ

mc DT N V t ⏌
R
N


㡰



b

ᆠ

㡰 꾀
㦨


�



矄
矄ᨨᨨ
矄
 
├泐
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矄矄

ᔴᆠ�ᨨ矄
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矄 矄 ᨨ
ᆠ�ᆠ盐�
 
ᨨ鼀
�矄泐
� �
� 꾀

㦨
µ

�



 

ᨨ �
�鼀
ᨨ
├泐
矄矄

ᔴᆠ�ᨨ
ᆠ矄
矄 ᨨ
ᆠ�ᆠ盐�
 
ᨨ
ᆠ盐 
ᨨ犨矄
 ᨨ盐�
ᨨ鼀
Q 泐 ��
矄�矄 ᆠ
ᨨ
ᨨ鼀㡰
矄ᨨ
�
�矄泐
졔�


꾀
t
㦨
҈矄ᆠ


�
㦨


��� �
⼄

�㡰
泐
��㡰

Þ t =
( l �  l ) =

�

\
D 
㦨

�

맔


졔







�㡰






��㡰

҈


 �

�

꾀矄ᨨ
HἘ
㦨
m'cDT
㦨

m㡰
CDT

黨


t = l   l �


�
├W

 NV 㡰 ´ t (Q R µ l )
R

N
V
´t

鼀ᆠD
㦨

R

=


=

�





ᨨ










t


V 㡰
æ V
ö t÷ ´ t çQ H = ç R R ÷ø è

黨V
´ t

Q 盐
Q
㦨
mcDT㡰 R

t ù
û ú

�҈


RἘ

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矄�矄 ᆠ
ᨨ
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犨�
ᨨ
ᨨ盐
 �
�
l \ ᨨ犨
 �泐
矄
�
㜠矄ᆠᨨ 矄
' 㦨


-

├� �
ᨨ�
ᨨ
ᨨ
矄�矄


ᔴ盐 
�
RC ᆠ�ᆠ盐�

I
㦨
I
et/RC

æ -t ö t + lnI
ᨨ lnI
㦨
lnI



ᨨ


lnI
㦨
ç è RC ÷ø RC æE ö æ -t ö + ln ç
÷ lnI 㦨
ç è RC ÷ø è Rø


m

҈㝐W

æ è

␔ 
 
�鼀 
鼀盐
�矄
�� 
ç l n µ ҈



҈ö ÷

�
鼀�矄
��  Tø

ᨨ
ᨨ犨
犨 �
ᨨ
�� 
 
ᨨ
⼄盐 ᆠ�矄

b
㡰
 ᆠ
ᆠ盐 
矄矄� �㜠 ᨨ盐�

� 矄盐ᨨ
  矄鼀�ᆠᨨ 盐ᆠᨨ

졠⧤
�矄
ᨨ
犨 
ᨨ �
�  ᆠ
�矄

꾀

⧤

� 矄盐ᨨ
ᨨ矄
ᨨ
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 泐
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ᨨ
ᨨ犨
 ᨨ盐�
�
꾀

⧤

矄鼀�ᆠᨨ 盐ᆠᨨ
 矄
矄

� 矄盐ᨨ


WWW.JEEBOOKS.IN

PS-217

Current Electricity

C

҈�


ᔼ
矄
�矄� �




K


矄鼀
� �
ᆠ盐 
ᨨ犨矄
 ᨨ盐�
 p㜠n 盐 ᆠ�ᨨ
�ᨨ
ᔼ
ᆠ矄
ᨨ
鼀

 
ᆠ盐 
ᨨ犨� �
犨�

泐㜠泐
��
ᆠ盐矄 
鼀
ᆠᨨ鼀矄
矄盐
ᆠᨨ 盐ᆠᨨ


⧤
b

㡰
ᔼ

ᔴ

ᆠ�ᆠ盐�

�矄ᆠ �� �
ᆠ盐  I
㦨
I
et/RC C҈

ᔼ ��
ᆠ盐 
I

㦨


҈黨







C


V
R 



꾀矄
V
 
R
ᨨ
ᨨ
ᆠᆠ�ᨨ矄

矄鼀
矄ᨨ

� ��
�矄ᆠ �� � ᆠ盐 
犨�

矄鼀
盐
ᨨ 㜠ᨨ
꾀矄ᨨ

q
㦨
q
et/RC

҈䰘


\
�҈
㦨


M

e 졔 t/RC
㦨



Þ t
㦨
RC
ᨨ�e


\
t
µ
C


\

t҈ C҈ ҈ =
㦨
=

�
t
C


V 㦨
�҈R 졔


q
q

 

㦨
q

et/RC

ᨨ

t҈ 㦨


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t


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奠
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 

C

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奠
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� ��
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®
s �鼀�ᆠ
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ᨨ
矄ᨨ�矄
�



l
㦨
l
҈
졔
µ
t㡰 
®
q 和鼀
 �� 
�矄
矄
ᨨ
 �泐
ᆠᨨ 矄�ᨨ
ᔴ
®
p, q ᔼ ᆠ 矄ᆠ 
鼀
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矄
ᨨ
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ᆠᨨ 矄�ᨨ
�



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�� 
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 
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ᨨ鼀

ᨨ
ᨨ泐

®
q, r 쟼ᆠ�ᆠ
盐矄
�矄
矄
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鼀� �
ᨨ
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盐矄
鼀�
犨 �ᆠ
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盐
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� �
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ᨨ
ᆠ盐 
෌�
Q
㦨
Q
᲼҈

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Q

㦨
鼀 �鼀盐鼀
ᆠ �
 

Q + �R C

T

B

ᨨ

a
㦨






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쟼�鼀� � �
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ᨨ鼀
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�҈R
졔
ᨨ

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㦨



Q


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�҈R
㦨
V



҈ æ
Q ö çV ÷
ᨨ R è C ø


Q C

dQ ҈ æ
Q ö = çè V ÷ dt R Cø Q t dt dQ ò
æ
Q ö = ò
R çè V ÷ C ø

dQ dt

ᨨ =
Q ö  R æ çè V ÷ Cø CV ᨨ

Q
㦨
҈

e
t⏌RC㡰
ù dQ d é CV ҈ - e-
t ⏌RC ú = ê 㡰 �҈
㦨
dt ë
dt û V -
t ⏌ RC CV
e = ´ ´ e -
t ⏌  RC

= R
 RC ᨨ

(

҈ ҈ = t c C  q

ᨨ犨
 
鼀 �鼀盐鼀
ᆠ � Q

㦨
C᲼V
桌
犨 
V

㦨
鼀
ᨨ �
�  ᆠ
ᆠᨨ矄矄
C

é

ù V ´ R
ú ë R҈ + R
û

)

ᨨ鼀
⼄盐�ᨨ
�㡰

=
C ê

V+

V -
t ⏌RC e 

\ ᔴ盐 
 ᨨ盐�
AB
R


ç

V + �҈R �
㦨
=
R

\
a =

V -
t ⏌  RC e V  - e -
t ⏌  RC �

㦨
�

�҈
㦨

R R V V -
t ⏌  RC V e Þ �

㦨

ᨨ
�

㦨

矄
t
® ¥
R
R
R

æ

ö V ÷ �矄
 
矄泐
矄
ᆠ盐 
 ᨨ盐�
R

è R҈ + R
ø é R҈R
ù 꾀 

t c = C Req
= C ê ú ë R҈ + R
û (Q R҈   R

�

ᆠᨨ鼀� �ᨨ )

҈



R

dQ dt

㡰 꾀泐� �
⧤
�
 
ᨨᨨ
MNABM V
㦨
�

�҈㡰
R 졔
�R ᨨ V 㦨

�R

�҈R



�㡰 和�鼀�泐

泐� �
⧤
�
ᨨᨨ
MNSTM


␔ 
q
㦨


҈



ᨨ
 ᨨ矄
ᨨ
 
�├

ᨨ � B A ��

泐
矄 ᨨ盐

ᆠᨨ

ᆠ R ᆠᨨ矄矄
A  
B

졠盐盐
ᆠ

├
ᆠᨨ矄矄  
鼀� 矄
A  
C ᨨ
B  
C
V 꾀ᆠᆠᨨ� �
ᨨ
⼄盐矄�ᨨ

ᆠᆠ�ᨨ
�矄
� ��泐
盐 ᆠ �
 
  矄犨�ᆠ
s
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EBD_8341 PS-218

Physics

E⼄盐� 
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PS-219

Moving Charges and Magnetism

Moving Charges and Magnetism

18

Topic-1 : Motion of Charged Particle in Magnetic Field ҈


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\ = ö ÷÷ 㦨


ᆠ鼀 ø

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EBD_8341 PS-220

黨


Physics

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�
r  ᆠ
= ⼄ ´ 㡰 ᔼ 꾀ᆠᆠᨨ� �
ᨨ
鼀� �Ἐ矄
 
 
盐


犨


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�ᆠ�ᨨ
ᨨ
鼀ᨨ�ᨨ
ᨨ ᆠ �
�矄
ᨨ犨矄
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犨�
b㡰 턼

㦨
⼄쟼
 

㦨
⼄ \
쟼
㦨
 꾀 
෌盐矄矄Ἐ矄
犨
�
쟼ᆠᨨ矄�ᆠ矄
쟼 㦨

s 㦨  Þ
s 㦨 e 
e
σ҈㦨  σ
쟼㦨

R a d

dqB mv

쟼 ҈

= = ҈
 犨 ⏌ 鼀
 ҈
҈

㡰 矄
 
ᆠᨨ
ᨨ鼀
ᨨ
B
 
v
�
 
ᨨ鼀盐 ur r ur F = qv ´ B㡰
ᨨ
�
 
� 矄  ᨨ盐矄
�ᆠ�ᨨ
ᨨ
ᨨᆠ 
x
㦨
a
 
x
㦨

a
ᔼ
 
��ᨨ
a


x



a
ᨨᆠ
鼀盐矄

�
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�ᆠ�ᨨ
�ˆ ´ ˆ �.e.

졔z㜠�ᆠ�ᨨ
矄ᨨ
�ᆠ泐
盐犨

꾀 
�
 
��ᨨ


a


x


a
�
z㜠�ᆠ�ᨨ
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b㡰 㝐� �ᆠ
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ᨨ
 
ᆠ �
�ᆠ
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�矄
矄 \ u
㦨
v


ᨨᆠ
ᆠ� �
ᨨ

X B y ᆠᨨ
犨�
  �ᆠ盐
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和ᨨ  
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ᨨᨨ犨  

矄
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 ᆠ
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矄
v
㦨
u
 

y




r ҈

b㡰 ␔ 
E
㦨


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
ᨨ
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ᆠ �
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 ᆠ
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㡰
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
� ᆠᨨᆠ

 ᨨᆠ�泐

P
�矄
�
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X㜠�ᆠ�ᨨ
�� 㡰
r 
v = k�$
꾀 


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ᨨ矄��泐
ᆠ �
�ᆠ
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�
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y
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ᨨ犨

y
�ᆠ�ᨨ
 
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
�矄
ᨨ 㜠ᆠ�ᆠ盐
ur r ur ᨨ犨

F = qv ´ B㡰 ur Þ F = q ᲼k�$ ´ ckˆ + a�$㡰桌
ᨨ
ᨨ�ᨨ
㡰 Þ=








㡰

a

q éQ qV = ҈ mv
ù ê ú
û
mV ë ҈�
ᆠ㡰 和� ᆠ
�ᆠ
�矄
鼀ᨨ� �
盐  ᆠ 和ᨨ

⼄ 쟼 = ⼄



= 䰘
� ´ ҈
졠N和㡰 꾀矄矄盐鼀� �
�ᆠ
 矄
ᨨ鼀



㡰

=

mv

㦨
qvB R

矄�
a 㦨 Bd

-


҈



mv qB

\ 矄�
a =

�

㡰

҈

´ ҈
-҈黨 ҈

´ ҈
-

㡰 ᨨ鼀
��盐

矄�
a
㦨
d⏌R

s e



⧤鼀  




���㡰

ˆ $ ´ $� 桌
㦨
kcq  - $ 㡰




㦨
q ᲼kc�$ ´ kˆ + ka�



 Ä

҈黨


ur

和� ᆠ
�
ᨨ�ᨨ
㡰

ᆠ�ᆠ
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�矄
矄ᨨ
矄 
E = a�$
  ᨨ
�
犨�
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 

ᨨᆠ
�
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졔X
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


矄盐
ᨨ
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犨ᨨ
ᨨᆠ矄
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졠 泐
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㡰
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b㡰 ␔ 
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�ᆠ
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鼀ᨨ� �
�

盐 � ᨨ鼀 鼀� �ᆠ
�
ᨨ鼀
ᨨ
�矄
鼀ᨨ�ᨨ

qvB
㦨


q x=a

mv
R X v

x=b

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PS-221

Moving Charges and Magnetism ␔�
ᨨ
 
鼀� �ᆠ
�
��ᨨ
b

a㡰


R錐
犨 
ἘRἘ
�矄 �矄
�盐矄
ᨨ
ᆠ盐盐
� 矄�
鼀� �ᆠ
�


mv b - a 㡰 qB
³ b

a㡰
Þ v鼀�
㦨
qB m


b㡰 ␔ 

ᆠ �
�ᆠ
�矄
鼀ᨨ� �
ᨨ鼀
ᨨ
  鼀� �ᆠ
�
 

ᨨᆠ
ᆠ矄
ᨨ
�
犨 �ᆠ
 矄
矄
 ᆠ �
ᨨᆠ
 
鼀ᨨ矄
 
�ᆠ
�
ᆠ�ᆠ盐
鼀ᨨ�ᨨ
\ R
㦨







㡰 㝐� �ᆠ
�

 
ᆠ 
盐
ᨨ
ᆠ盐 
�
ᆠ
AB m
I҈ q B҈
㦨
犨矄㡰
p r C

I



p㜠q q

A




mv mv




Þ r
㦨
\ F
㦨
qvB
㦨
qB r ᔼ
q
 
B

矄鼀
ᨨ
ᨨ
 

r
µ
mv

B

I҈ I

B

E

㝐� �ᆠ
�

 
ᆠ 
盐
ᨨ
ᆠ盐 
�
ᆠ
ACB

B

和� ᆠ
rA
>
rB Þ
҈







mAvA  mBvB

r

ᆠ㡰 ෌�
ᆠ�ᆠ
�
ᨨ �
졔x
�ᆠ�ᨨ

E = E
�ˆ
  r 鼀� �ᆠ
�
ᨨ �
졔z
�ᆠ�ᨨ
B = B
kˆ ᨨᆠ�泐
ᨨ
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�ᨨ �矄
�ᆠ
犨�

ᨨ �
�ᆠ�ᨨ

r 鼀� 
泐
q
E r ᨨ
ᨨᆠ�泐
v = AqE�ˆ
犨 
A
�矄

ᨨ矄��
ᆠᨨ 矄 

A

E
 
B

ᨨ矄��
ᆠᨨ 矄 矄
ᔴ �
ᨨ
�ᨨ 矄
q㡰
鼀泐

ᨨ矄��
ᨨ
�� r r r 㝐� �ᆠ
ᨨᆠ
F = q  v ´ B 㡰 r \ F = q᲼ AqE�ˆ㡰 ´  Bkˆ㡰桌
㦨
q

AEB �ˆ ´ kˆ 㡰 㦨
q

AEB- ˆ 㡰
㦨
᠀q㡰

AFB

ᨨ �
��
y㜠�ᆠ�ᨨ
꾀矄
鼀� �ᆠ
ᨨᆠ
�矄
ᨨ �
��
y㜠 �矄
 ᆠ

�ᨨ 矄
犨  
ᨨ矄��
ᨨ
��
犨�
 ᆠ
ᨨ犨矄
�� y㜠 �矄
ᆠ㡰 
 �盐
鼀ᨨ鼀 盐鼀
ᨨ
 
�ᆠ L
㦨
mvr
㦨
mr
w
犨 
w
㦨

pn
᲼Q
v
㦨
rw桌

q w wq 錐
盐 
�
㦨

㦨
q 맔
n
㦨
t
p
p wq qr
w ´ pr


Þ M
㦨
㝐� �ᆠ
鼀ᨨ鼀 

M
㦨
�A
㦨

p


\
⼄盐 ᆠ泐
n
㦨


\






Ҙ
I
(
p - q) ᨨ犨 犨矄㡰
p r \ 
鼀� �ᆠ
�

 
ᆠ  m
I҈ q m
I

p - q㡰 B
㦨

p r p
p r E E EA é lù QR =r ú = = 꾀矄ᨨ

I҈
㦨
R rl ⏌ A rr q ëê Aû B

㦨



A

M q wqr
= = L
mr
w
m

r r 㡰  � ᨨ鼀
ᆠ�ᆠ
�
E
 
鼀� �ᆠ
�
B
 
ᨨ
ᆠ
ᨨ 
FE
㦨
qE  ᨨᆠ
盐
ᨨ
ᆠ�ᆠ
�㡰 FB
㦨
evB
矄�
q
㦨
qvB
矄�


㦨

 ᨨᆠ
盐
ᨨ
鼀� �ᆠ
�㡰 E 졔q

B 和� ᆠ
ᨨᆠ
盐
ᨨ
鼀� �ᆠ
�
�矄
ᨨ

和ᨨ
ᨨᆠ
盐
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犨�
鼀├
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ᆠ �
�ᆠ
ᆠᆠ
ᨨ ᆠ

ᨨ
�矄
�
矄�� 
� 


҈

 
I

㦨


҈

E E EA = = R
rl
⏌ A rr 
p - q㡰

m
é EA q EA 
p - q㡰 ù ´ ´ ê ú=

p ë r r q r rr 
p - q㡰 r û
�
ᆠ㡰 꾀 
 � �
 
鼀� �ᆠ
�


鼀� �ᆠ
ᨨᆠ ᆠ矄
ᨨ
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ᆠ �
�ᆠ
犨 �ᆠ
鼀ᨨ矄
 
ᆠ � �ᆠ
�
ᆠ�ᆠ盐
 \ B
㦨


mv
m K mv

㦨
qvB Þ R
㦨
ᨨ

R = qB qB R 

K

q

B

⼄盐 \ R

µ
m \

m҈ R҈
= m
R





盐 �

ᔴ
�矄
ᨨ
ᆠ �矄矄
�ᆠ
盐ᨨ
꾀ᆠᆠᨨ� �
ᨨ
鼀� �Ἐ矄
 
 
盐

 
ᨨᆠ
ᨨ
ᆠᨨ 矄  ��
ᆠ �㡰
犨�

ᨨ犨矄
�� 
�.e.

ᆠ├
D
꾀矄ᨨ

泐
 
矄鼀
盐
犨
� 
 
 
ᨨᆠ
ᨨ
ᨨᨨ
  a㜠�ᆠ
�.e.

졔㡰泐
ᆠ �
�ᆠ
�矄
ᨨ犨矄
 

ᨨ犨 矄� ᆠ
 
鼀� �ᆠ
ᨨᆠ
犨�
 
矄
ᆠ �
ᨨᆠ

m \ mv = qvB ᨨ 
µ
q r ҈
ᨨ
ᨨᨨ

r
µ
= ҈ 錐


ᨨ
a㜠�ᆠ
r µ =
҈
�.e.
(r㡰a
>
r㡰p \ a㜠�ᆠ
犨�
├

B

䰘
矄 ⧤� �ᆠ
 �泐

⧤ᨨᨨ
㦨
⧤ᆠᨨ
㦨
⧤  ᆠ





mv
= qvB r

Þ

r
㦨



mK mv
Þ r
㦨
qB qB

᲼Q
mv
㦨
P
㦨

mk 桌 \ rµ m ᲼ ᨨ
ᆠᨨ 矄 
q

K
 
B桌 \ rµ m 和� ᆠ
鼀矄矄
ᨨ
ᨨᨨ
>
鼀矄矄
ᨨ
ᆠᨨ
  ᨨ
�盐矄 ᨨ
ᨨᨨ
犨�

鼀ᨨ


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EBD_8341 PS-222






Physics

矄



ᨨ

�ᆠ
�矄

ᆠ�ᆠ
犨 
�
 矄
ᨨ鼀
ᨨ  
鼀� �ᆠ
�


ᨨᆠ�泐
ᆠᨨ鼀ᨨ  
v

犨�
 矄ᨨ 矄�
�
鼀ᨨ� �
 
ᆠ �
�ᆠ
�

ᆠ�ᆠ




B v v

㦨
v矄� q

q

)



v҈㦨
v
ᆠᨨ矄
q

ᆠ㡰
ᔴ  �
�
鼀ᨨ鼀 盐鼀
㦨

p


黨



ᨨᆠ�泐
ᆠᨨ鼀ᨨ  
v҈
犨�

矄ᨨ 矄�
�
鼀ᨨ� �  
ᆠ �
�ᆠ
�
ᨨ�ᨨ 
�ᆠ�ᨨ

 ᨨ
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�ᆠ
犨�

�

�ᆠ

T盐


鼀� �ᆠ
ᨨᆠ
ᆠ� �
ᨨ

ᆠ �
�ᆠ
�矄
�
 �ᆠ�ᨨ
 �ᆠ盐
ᨨ
 
�ᆠ�ᨨ
ᨨ
ᨨᆠ�泐
   ᆠ
�
ᆠ

ᨨ
ᆠ  �
 
矄
ᨨ
 
ᆠ �
�ᆠ


æ ҈
ö 鼀 ÷ ᨨ矄
ᨨ
ᆠ  �
è
ø 꾀矄
犨ᨨ├
ᨨ 
泐
 �矄
ᨨᆠ
�矄
ᨨ
矄ᨨ
 
 �泐
ᨨ
鼀ᨨ� � ᆠ  �
�ᆠ矄
鼀� 矄
盐 ᆠ  �




b㡰 㡰
ᨨ
 
ᆠ �
졔턼
ᨨ
盐
��ᨨ
҈


 ᨨ

 
├� �ᆠ
 �泐
ç =

 ù
p é mv
= QB
ê 

�盐矄
r = R ú = QvB



Þ
ë
û R R ⏌
㡰
p \

B = QR
p

 ᨨ
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p ҈QR mv
æ
p ö = Qv ç ÷ r è ҈QR ø

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= qvB
\
r =

\
r
µ
m qB r �.e.

�矄 ᆠ
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b

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 

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㦨

㡰
盐 


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㡰
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\
r =

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R ⏌




҈ 졔턼

\
q =


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CP
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æ � R ö æ R ö = ç ÷ ÷ +ç è
ø è
ø




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ᔴ

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p


맔 맔 맔 맔 턼
㝐맔졔 맔 맔

q





맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 盐� = 盐 =
�ˆ 맔 맔 맔 맔

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㦨


맔 맔 
泐 㦨

맔 q 맔 맔  = 맔
 맔 맔 맔 맔 맔 맔 맔 맔

맔 맔 맔 맔 맔 맔 맔

ᆠ 矄
맔
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´ ҈
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êQ �盐矄 = 턼 úû
M ´
⏌ QB ë

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泐
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�ᆠ =

\

B =

�
pM Q

WWW.JEEBOOKS.IN

PS-223

Moving Charges and Magnetism 


ᆠ

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º

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矄鼀桌

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r + µ
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ᨨ
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q
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Þ
v
 
B
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矄�� 
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ᆠᨨ 矄  ᨨᆠ�泐 r r Fe + FB =


r r FE = -eE = -e- E
xˆ 㡰 = eE
xˆ r r éE ù FB = qv ´ B㡰 = -e ê
yˆ ´ B
zˆ ú ë B
û r FB = -eE
xˆ

㡰

S
㦨


E
xˆ B


FB B = B
zˆ



E = E
zˆ Ӹ

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ᨨ
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鼀ᨨ�ᨨ
犨 �ᆠ
犨�

WWW.JEEBOOKS.IN

EBD_8341 PS-224






Physics

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- v
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맔 A 맔 
M 맔 맔 맔 O 맔 맔 맔 맔 맔 맔 X=


´

맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔

맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔

맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔

맔 맔 vᨨ 맔 P 맔 맔 B = B
kˆ 맔 X 맔 맔 맔 맔 맔 맔 X=L

\
AO
㦨
AP
㦨
R
㦨
�盐矄
ᨨ
ᆠ�ᆠ盐



mv
mv

= B
qv

Þ

R
㦨




�㡰 qB
R L R = L




��㡰 ᔼ
DAPM


㦨
矄�


Þ
R
矄�


㦨
L
Þ R



p m 맔
v
ᆠᨨ矄



㦨


҈ qB

B
㦨



p mv ᆠᨨ矄

° q ´

҈

盐


҈

E mv
㦨
E

Þ

v
㦨

m










S

r 㦨


҈鼀 v҈
㦨
v ᆠᨨ矄





pm
E ´ ´ ᆠᨨ矄

° \ B
㦨
q ´

҈ m 㦨


B

v G

v

㦨
v 矄�


ᨨ


p
´ 
҈
´
mE ´ ᆠᨨ矄

°
㦨
q ´

҈ ҈

´ ҈
-҈黨 ´

҈

㦨

´ 黨
҈ ´ ҈
-҈ ´
´ ҈
 ´ ҈

´ ҈
-҈黨 ´

҈


҈
黨



맔

҈

맔
҈


㦨

䰘
䰘
맔
҈

҈
-҈黨 ᨨ
B
㦨


䰘䰘
맔
҈






꾀
鼀
ᨨ
ᨨᨨ 矄  矄

盐 � ᨨ鼀 鼀� �ᆠ
�

㦨






 �
q
㦨



犨�
 ᨨᆠ�泐

㦨


맔
҈
�
鼀⏌矄 㦨



҈
ᨨ�矄
ᨨ�ᨨ 
鼀ᨨ�ᨨ
 


ᆠ�ᆠ盐
鼀ᨨ�ᨨ

ᨨ
 
鼀
ᨨ
ᨨᨨ 矄

矄ᨨ
ᨨᨨ 矄
ᨨᨨ犨

�ᆠ



WWW.JEEBOOKS.IN

PS-225

Moving Charges and Magnetism

mv

= qv
B r mv
҈
䰘
´ ҈
-
䰘 ´
´ ҈
� ´  =
㦨



҈

鼀 r 㦨
qB ҈

´ ҈
-҈黨 ´

 ´
�ᆠ
ᨨ
�
㦨
v҈
맔
T

B

\

犨 
T
㦨



p r
p r = v
v 矄� q

qvB
㦨
ma Þ
a
㦨


㦨


҈

´ ҈


-҈黨

qvB m

q ´

´ ´

O

-
䰘

V V2= ¾ Ö


´ ´

x

x

x x

x

x x

E

x x

v x x x E


�

´

B
㦨





⏌  ´ ҈
-



´ ҈


=






 ´ ҈


-

´
´ ҈




㦨


҈
矄

Topic-2 : Magnetic Field Lines, Biot-Savart’sLaw and Ampere’s Circuital Law

´ ´

ᨨ

V ¾ E´ V 1=Ö


㦨
҈
҈�
鼀⏌矄


x x


�ᆠ�ᨨ
ᨨ
ᨨᆠ
盐
ᨨ
ᆠ�ᆠ
�
⼄쟼㡰
�矄
ᨨᨨ矄� ᨨ
鼀� �ᆠ
�
⼄
맔
㡰 
쟼
�矄
�
ᨨ矄��
�ᆠ�ᨨ

\

맔

矄 ᨨ盐

�
��
x㜠�ᆠ�ᨨ
ᨨ

矄 ᨨ盐
 �
㡰
z㜠�ᆠ�ᨨ
^
ᨨ

� 犨矄


´

b

G

´ ҈
´ ҈

҈

´ ҈


\

F ´ ´

x -

෌�


V
㦨







㦨


맔
҈


鼀⏌矄
 

㦨

鼀鼀 㦨

맔
҈


鼀

´

䰘

x x

+ x


p r v 矄� q 㦨

pr
ᆠᨨ
q
㦨


맔

҈

맔



҈

맔
ᆠᨨ



㦨






鼀 m
㦨
҈


맔
҈

䰘
├�

q 㦨
҈


맔
҈
҈黨
ᔴ B
㦨
҈


v
㦨
҈
䰘
鼀⏌矄 F
㦨
q


v B
矄�
a ᆠ� �
ᨨ犨矄
O
泐
鼀� �Ἐ矄
 
 
盐㡰 Þ F
㦨
qvB ᲼Q a
㦨
黨
桌 \ �ᆠ
ᨨ
�
㦨
v ᆠᨨ矄 q ´


�


+ x x

҈


㡰

v

´

Ð
OEF
㦨

� Q OE
ᆠ
矄

�盐矄㡰 泐
矄泐鼀鼀泐
Ð OFE 㦨

� \ Ð
EOF
㦨
黨
泐
�ᨨ鼀泐㡰

�矄
�矄
 
ᆠ �
ᆠᆠ�ᨨ

졔

҈
҈
v
= ҈
҈�


Þ
r
㦨
҈� =

҈鼀
r ҈
҈ \
쟼
㦨

r
ᆠᨨ矄

�
㦨


맔


҈ ´
㦨


҈
҈
鼀

꾀 �

q
㦨

�
ᔼ
 
鼀� �ᆠ
�
�矄
�
 
ᨨ盐犨
�ᆠ�ᨨ
 
 
�ᆠ  矄
�
 
矄鼀
犨泐

E

 
ᆠᆠᨨ� �
ᨨ
鼀� �Ἐ矄
   
盐

 
�ᆠ
犨�
盐
ᨨ犨矄
ᆠᨨᆠ├犨�矄
�ᆠ�ᨨ

 
ᆠᨨ
⏌

ᨨ

ᆠ�ᆠ
矄
矄 ᨨ犨
�
 
��盐
\ ac
㦨


졔q

 �
ᨨ
 
ᆠ�ᆠ盐


l =
pr

q qv = T
pr 㝐� �ᆠ
鼀ᨨ鼀 
M
㦨
ᔴ盐 
맔
꾀 qv
´ pr






























= � ´ pr =
pr ҈ M = q×v×r
ᔴ盐 

� =

졠


�

�盐矄
ᨨ
ᆠ�ᆠ盐

�
鼀� �ᆠ
�

r =

E

mv qB

mv mv
҈ qv ´ ÞM =
qB
B r r �ᆠ�ᨨ
ᨨ
M
�矄
ᨨᨨ矄�
ᨨ
B
  ᨨ r r - mv
B M =
B
泐
鼀盐�泐� �
ᨨ
盐鼀ᨨ
 
 ᨨ鼀� ᨨ
泐
B㡰
\M =

O 졠


�
�졠

F

  æ
p ö  é
p r ù \ �鼀
⼄盐�
㦨
T = ç ÷
㦨
´ ê

è w ø
ë v úû




㦨


䰘҈
맔
҈


矄
꾀矄
ᆠᨨ
矄矄矄
盐 �
\
ᨨᆠ
盐
ᨨ
ᆠ�ᆠ
�
㦨
ᨨᆠ
盐
ᨨ
鼀� �ᆠ
�
eE = evB \ B 㦨


E V ⏌d = v v

Vù é êëQ E = d úû





㡰 ෌�


IA
㦨


꾀

RA 㦨


ᆠ鼀

q A =
p -

p p =



p �p =   m
I q 矄� �

鼀� �ᆠ
�
B = 4pR IB
㦨

꾀

RB
㦨


ᆠ鼀

q B =
p -

WWW.JEEBOOKS.IN

EBD_8341 PS-226

Physics

B A I A q A RB = ´ = BB I B qB RA




p ´


= �p � ´ ´

´





Ҙ
�
矄� a + 矄� b㡰
p r

b㡰 B
㦨



r
㦨
�
- 

㦨


ᆠ鼀 a
㦨
b
㦨
䰘

� ´
矄� 䰘°
㦨
҈
�
맔
҈
�



ᆠ㡰 
ᔼ

 
ᆠ盐 
�
ᆠ
犨�

�ᆠ
� 犨矄㡰 㝐� �ᆠ
�

�졠
盐
ᨨ

 
턼㝐
犨�

ᨨ
�





㦨
和
졔
턼 m � m � \


鼀� �ᆠ
�

=
+

p
p m �
´ ҈
-䰘 ´ � ᨨ

҈
-
=
+
p
´ ҈
-
\

�
㦨



꾀
 
 
�ᆠ�ᨨ
ᨨ
鼀� �ᆠ
�
�矄  �ᆠ盐
� ᨨ
 
  㡰 \ B
㦨
҈


ᆠ㡰





䰘


a

ᔼ a


㝐� �ᆠ
�
盐
ᨨ
ᨨ 
矄�
ᨨ
 �ᨨ

m
I 矄� 
° + 矄� 
°㡰 B= a
p






m
I æ ҈ ҈ ö m
I çè + ÷ø =


a
ap ᨨ犨

鼀� �ᆠ
�
盐
ᨨ
ᨨ 
 �ᨨ
ᆠᨨ� m I B =
´

ap 꾀��
鼀� �ᆠ
�

 
ᆠ 
ᨨ
 �ᨨ 
矄 
ᆠᨨ� ᨨ
�

盐 矄


ᨨ

B =

m
I

҈�
m
I  ´

҈ ´ p


㦨

p


= �



m
I p


㦨

맔

p

ᨨᨨ
㦨


\ 黨


m
� m
� m
� m

� = =
ᆠᨨ�
㦨


æ ö

ç ÷ è ø  ҈

 = ᔴ 


ᆠ㡰 㝐� �ᆠ
�

 
ᆠ 
ᨨ
ᨨᨨ

҈ =




(

㡰

鼀
= ᔼ
p
m
ᔼ 
=



(

m I m I m I =
[ 矄� 黨
°  矄�
�° ] +
+

᲼矄�
�
졔
矄� 黨
桌
pR
R
pR m
ᔼ æ ҈ ö m
ᔼ m
ᔼ æ ҈ ö ҈ + + ҈+ ç ÷ ç ÷ è ø è
p

p

ø

e uuur m ᔼ æ ҈ ö e
=
ç p + ÷
p è
ø

�


®

®

®

㡰 B = B + B ҈


=

( )

µ
æ � º ˆ � º ˆ ö
ç
k + -k ÷ =
ø d
p è d

 \ ҈ = 

҈



)




=

ᔼp
=
鼀҈

)

B

㦨
B҈
졔
B

졔
B
졔
B


=

m
ᔼ


�ᨨ
鼀ᨨ鼀 
ᨨ
ᆠ�ᆠ盐
ᨨᨨ
�矄
鼀
㦨
ᔼ꾀 鼀҈
㦨
ᔼ
꾀
㦨
ᔼ
p

{
㦨
�盐矄
ᨨ
 
ᨨᨨ晤 ᔼ
鼀ᨨ鼀 
�矄
ᨨ盐
├� �
ᆠ盐 
ᆠᨨ 矄 㡰

㜠 ᆠᨨ鼀矄






ᔴᨨ�

㦨

Þ

㦨


҈
é ù êëQ a = ҈

=

҈ 鼀 úû


ap





ᨨᨨ

ÞB=

B = �
´
´

-䰘

(

m
ᔼ
m
ᔼ


=


)

b㡰 ᨨ� 
P
�矄
矄�盐

 
鼀�㜠ᨨ� 
ᨨ
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� 
ᨨ� � �  
ᆠ 矄
ᨨ
 
ᆠ�ᆠ盐
犨�矄
犨 �ᆠ

矄鼀
��
 㡰


r


鼀� �ᆠ
�矄
 B 㡰


盐
ᨨ
 
ᆠ盐 矄
�
  犨�矄

�
矄鼀
�ᆠ�ᨨ
㝐� �盐
ᨨ
鼀� �ᆠ
�

ᨨ� 

P

ì ü m
NIR

m
NI m
NIR
B
㦨

ï ïï
㦨
 ⏌

㦨
⏌
ï ⏌
� R � í æ
R
ö ý ï
ç R + ï ÷

ø þï îï è

WWW.JEEBOOKS.IN

PS-227

Moving Charges and Magnetism ҈�
҈҈


㡰 ᨨ鼀
��盐


b㡰 ᔴ矄
㡰


ᔼ
D
졠꾀ᔴ






ᆠᨨ矄



㦨


Ҙ ᔼ Ҙ ᔼ ´
p
Q

p
㦨฀l㡰 꾀
㦨

´
p =

p
p l ⏌
p Ҙ
ᔼ ´ 
p㡰
㦨

p l
� ᔴ矄
b㡰

 

꾀

ᔼ



㦨

´

҈



 


矄�
q
㦨


҈ =a
㝐� �ᆠ
�

Ἐ졠Ἐ
盐
ᨨ
鼀 
꾀 㦨



ᆠᨨ矄
q

m
I é  ҈ ù m
I ҈ - ú= ´   - ҈㡰 ê
p a ë

û
pa
\ 㝐� �ᆠ
�

 
ᆠ 

盐
ᨨ
ᆠᨨ鼀
ᨨᨨ

 ll㡰�

é FB m

I ´ I ù m
ᔼ ´ ᔼl =

êQ
p
l 矄� q úû
p 2 矄� q ë l

m I é m
I ҈ ù ´   - ҈㡰 ú ´҈
=
´
  - ҈㡰 㦨
ê
p a ë
p a
û ҈



B=

m
ᔼ



҈ =


m
ᔼ
꾀

p  
 
ᆠ 

㦨
�盐矄
ᨨ
ᨨᨨ m

ᔼ m
´  ´ ( ᔼ ⏌ ) ´ p (  ⏌  ) 
=

㦨


p (  ⏌ )


҈

= ҈
 ҈

b㡰 ෌�


�盐矄
㦨
�矄 ᆠ
=

⏌







= 黨㡰⏌
=
䰘

B=0

R £ r < R



µ0


R

é R
ù êr úJ
r ûú ëê

 B

⏌


r

R

æ aö




é

pù


N
㦨
҈

A
㦨
a

졔

p ç ÷ 㦨
a
ê҈ + ú ë
û è
ø ᨨ鼀
和ᆠ犨
犨

�ᆠ�ᨨ
ᨨ
m
�矄 ᨨ盐犨
ᨨ
�
졔
㜠�ᆠ�ᨨ
\

æ 

㡰
ö = çç҈ + ÷
÷ø è

r P

R


R r
 ᆠ
�
d㡰
ᆠᨨᆠ泐
�ᆠ矄
 B 
矄盐矄
r
�
r r ҈䰘
b㡰 㝐� �ᆠ
鼀ᨨ鼀 
ᨨ

ᆠ盐 
ᆠ泐� �
ᨨᨨ
M = NIA

҈ m ᔼ ⏌
 =

m

ᔼ ⏌


ö ÷÷ ø

R



µ i ᨨ
r
>
R






B = 0
pr ҈ �.e.

B µ r

ᨨ犨

ᨨ
ᆠᨨ�
 =

ᆠ  æ
= çç҈ +
 �矄 è

㡰 ᨨ
r < ᨨ



m
ᔼ b㡰 ᨨ
ᨨᨨ
 =
 犨 


�矄
 
�盐矄
ᨨ
ᨨᨨ


\

m
I [ 矄�

° - 矄� 
°]
p a


=

pl�

 ᨨ

ᔼ =
矄� q 盐
ᆠᨨ矄 q ҈





\
졠ᔴ
㦨

a ´



矄�
q


矄�
q

矄�
q







a

᲼
a
㦨
桌

p
B Þ
A = BB

㡰 
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ἘlἘ
 �
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꾀
⼄盐���盐鼀

ᆠᨨ矄
q
㦨
l�l






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Ҙ
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m
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pl

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⏌


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ᔼ 
㦨


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a

҈

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a a

a 㡰 㡰 
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I
�
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鼀� �ᆠ
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� ur B

x

y㡰
�
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z
㦨


 


WWW.JEEBOOKS.IN

EBD_8341 PS-228

Physics

ur

ur ҈ B= $� + k$ 㡰




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I B= y�ˆ - xˆ
pr


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ᨨ




(

(

)

)

ur m
I y�ˆ - xˆ
B=
p x
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(

m
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)

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和�鼀�泐

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BGB

BCA BEA

FB

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A D BDB

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C

E

B

BGA G

BCB

BEB






 ᆠ
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p㜠q q

A

B

I҈ I

E

WWW.JEEBOOKS.IN

PS-229

Moving Charges and Magnetism 㝐� �ᆠ
�

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ᆠ 
盐
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p - q㡰 B
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p r p
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I҈
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ê R҈ rl҈ ⏌ A rr q ë Aû  
I

㦨
\ B
㦨

�


꾀矄ᨨ
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B=

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l R
r p - q㡰
A

m
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p - q㡰 ù ´ ´ ê ú=

p ë r r q r rr 
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…
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m I ém I ù m
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+


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x㡰




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…
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R …�㡰 …
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r r

和� ᆠ
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ur ᨨᨨ
�矄
ᨨ
\ 㝐� �ᆠ
�

B
㦨



WWW.JEEBOOKS.IN

EBD_8341 PS-230

Physics ¥

Þ pR
IBq =





� Q

¥

M

L R �

P

O

S

�

Q
m
㦨
IA
 
鼀ᨨ鼀 
ᨨ
� �
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ᆠ�ᆠ盐
ᨨᨨ

I =

¥

�

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IB
q = ᨨ¥

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鼀� �盐
ᨨ
 �盐㡰 D 鼀ᨨ鼀 盐鼀
�� 

㦨
tD
㦨
�p



D
㦨
p턼

᲼Q�
D
㦨
턼桌

꾀矄ᨨ

ᨨ⼄盐

t =

҈ ù é ҈ 
+ úû
㦨
҈











ë

㦨
҈
䰘
맔
҈

맔
ê

Topic-3 : Force and Torque on Current Carrying Conductor ҈


ᆠ㡰 ᨨ⼄盐
ᨨ
 
ᨨᨨ


�


㡰

t = M ´ B = MB 矄� q = MB 矄� 黨
° 㝐� �ᆠ
�

B =

m
I
pd

꾀矄

ᨨᆠ
ᨨ
 
 �
犨�
ᔴ
�矄
ᨨ


æm I ö \ t = I҈ 
a㡰
ç

÷ 矄� 黨
° è
pd ø


m
I҈ I

m I
a
´ a
=
pd pd b㡰 㝐� �ᆠ
鼀ᨨ鼀 
ᨨ
ᨨᨨ
ABCD
M҈
㦨

ᨨ
ᨨᨨ
맔
ᆠ盐  r 

ab
㦨

ᨨ
ᆠ �㡰 M҈ = abI 㡰 ˆ 㡰 㝐� �ᆠ
鼀ᨨ鼀 
ᨨ
ᨨᨨ
DEFA
r M = abI 㡰�$㡰





=













鼀� �ᆠ
鼀ᨨ鼀 
r r r r M = M ҈ + M
Þ M = abI $� + $ 㡰

䰘


r Ҙ ᔼ Ҙ
ᔼ
Þ =
҈ + =

p
p - 㡰 Ҙ
ᔼ҈ Ҙ
ᔼ
=
p
p - 㡰 ᔼ҈

ᔼ҈
㦨
ᔼ
ᔼ = ҈ ᔼ҈ - ᔼ


犨ᨨ
ᆠ矄矄
鼀泐

ᨨ矄矄�
�
ᔼ҈
>
ᔼ

ᨨ
ᔼ

>
ᔼ҈ m
æ �� �
�҈�
ö m �� ´a =
҈
ᆠ㡰 F =
p çè a
a ÷ø
p r r r b㡰  t = m ´   ᲼m
㦨
ᔼ꾀桌 㦨ᔼ꾀
맔

矄�
黨
ᨨ ᲼꾀
㦨
p
桌

Þ t 㦨ᔼp 

æ ˆ kˆ ö
abI ç + ÷
ø è
ᆠ㡰 ᨨ⼄盐
ᨨ
ᆠ�ᆠ盐
ᨨᨨ

t
㦨
MB
矄�
q 犨 
M
㦨
鼀� �ᆠ
鼀ᨨ鼀  B
㦨
鼀� �ᆠ
� ᨨ犨

盐矄� �
t 㦨
Ia \ t 㦨
MB
矄�
q
㦨
Ia r Þ M  =




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PS-231

Moving Charges and Magnetism



㡰 㝐� �ᆠ
鼀ᨨ鼀 

m = ᔼ꾀 =

҈



⼄  p
㡰
p

⼄w
҈ p 㡰 = ⼄
w
p
ᆠ㡰 㝐� �ᆠ
鼀ᨨ鼀 
ᨨ
ᆠ盐 
ᆠ泐� �
ᆠ �盐
ᨨᨨ ᨨ

꾀
�矄
��
泐
㝐
㦨
ᔼ꾀 鼀� �ᆠ
鼀ᨨ鼀 
ᨨ
ᆠ盐 
ᆠ泐� �
ᆠᨨ�
�矄

ᆠᨨ
  �矄
�ᆠ�ᨨ
�矄
��
泐
�� 
 
 盐鼀
盐

ᨨ r ᆠ �盐
ᨨᨨ



ᆠ 
盐
ᨨ
ᆠ盐 
�
ᨨᨨ
  r 㝐

犨泐矄

ᨨ
m =

黨


ᆠ㡰 
ᨨᨨ
犨�
鼀ᨨ
ᨨ犨矄
 
犨�
AB

ᨨ
  ᆠ �盐
ᨨᨨ
犨�
�
ᆠ
ᨨ
 
ᨨ �
矄�� 
犨�
矄  
ᆠ盐 矄


 
�
 
矄鼀
�ᆠ�ᨨ
犨 矄 CD

犨�


ᆠ盐矄
ᆠ盐 矄

�
ᨨᨨ矄� ҈ �ᆠ�ᨨ

盐
矄� ᆠ
 �矄
ᨨᆠ
F µ
犨 
r
�矄
 
�矄 ᆠ r 犨
 
犨�矄

 ᨨ

 
犨�



ᆠ� ᨨᆠ
ᨨ
 
ᆠ �盐
ᨨᨨ

ᨨᆠ
ᨨ
BC �矄
⼄盐
  ᨨᨨ矄�
ᨨ
 
ᨨ
AD

矄ᨨ
 矄
ᨨᆠ矄
犨�
ᆠ ᆠ
ᆠ
ᨨ 
I

B

C

I

A 
㝐 Ä ᔼ 犨矄


㝐

D




҈



 ᆠ

ᆠ㡰
ᆠᨨ矄ᨨ 矄
ᨨ
矄
⼄盐���盐鼀
r r ᆠ㡰 ҈ =
=
ᆠ盐矄
ᨨ
ᆠ�ᨨ
 
ᆠ�ᨨ
�

҈҈


㡰 ᨨ
矄
⼄盐���盐鼀
㝐  

mv mv
㦨
qvB
Þ R = R qB ҈
ᨨ
R µ ᲼



m

q

v

 
矄鼀桌 B R B \
҈ =
R
B҈

ᨨ
盐 矄
⼄盐���盐鼀
㝐  㡰 㡰 I҈
I

㦨
ᨨ矄�� ᆠ㡰
F
㦨
�� I҈
I
㦨
�� 㡰
F
㦨
ᨨ矄��  ᆠ

ᨨ�ᨨ
㡰
�矄
 
ᆠᨨᆠ
 矄犨
㡰 I҈ 㦨


꾀 I 㦨
҈

꾀 I
㦨



꾀

é ҈ ù Ҙ
҈ ´
I ê + ú
X X X p R҈
҈û ë ҈ = ᨨ
R
Ҙ
é ҈ ù ҈ ´
I ê ú
p ë X҈ X
- X҈ û X
é ù X
- X҈ + X҈ X
= 㦨
ê ෌�
X = ú X
- X҈ - X҈ X
-
X҈ ë ҈ û

e 졠盐犨矄

҈



҈


r

r

҈䰘


r

r


ᆠ鼀

\


�
ᆠ鼀 

҈



҈�


턼

㡰



꾀矄ᨨ
�� 錐
 �
ᨨ
犨�
턼 㦨

�
ᆠ鼀
㦨



�
鼀 ᨨᆠ
ᨨ
犨�
턼
盐
ᨨ
犨�

´

´ ҈
´


� FQR
㦨
҈
䰘
´


� � 㦨



맔
҈


 ᨨ犨矄
 㡰 ᨨᆠ
ᨨ
犨�
턼
盐
ᨨ
犨�

´ 
´ ҈
FQP
㦨
҈
䰘
맔
´


�


 㦨
�

맔
҈
�

 ᨨ犨矄
�� 㡰  ᆠ

F 
㦨
FQP

FQR 㦨
�

맔
҈
�





맔
҈
�
 㦨

맔
҈



ᨨ犨矄
��  b㡰 ᔼ
盐 � ᨨ鼀
鼀� �ᆠ
�


ᨨᆠ
ᨨ

ᆠ盐 
ᆠ泐� � ᨨᨨ
�矄
ᨨ

 ᆠ
ᨨᨨ
ᆠ

ᨨ
鼀ᨨ

鼀� �Ἐ矄
 
 
盐
犨
� 
 

ᨨᆠ
�矄
ᆠ� �
�
 
�泐
ᨨ盐犨
�ᆠ�ᨨ

 ᨨ盐� ᨨ盐
 
ᆠ�ᆠ盐鼀  ᆠ
ᨨ
 
ᆠᨨ 盐ᆠ� �
ᨨᨨ
b㡰 꾀矄
犨
├ ᨨ犨

 
ᨨᆠ

盐 �
 �
⏌ ҈

犨
犨ᨨ
犨�矄
ᆠ泐� �
ᆠ盐 矄
I҈
  I

矄
泐

�矄 ᆠ
r
� �

P m I I m �

㦨

҈
=
᲼Q
I҈
㦨
I

㦨
�
 
r
㦨
b桌 l
pr
pb

b

X
X҈

R҈  = = = X R
-

-
X҈

҈






ᨨᆠ
ᨨ
 
ᨨᨨ
EDCBE
犨�

ᨨ

꾀矄ᨨ
ᨨᆠ 盐
ᨨ
矄�鼀 
FE
 
BA
犨�

ᨨ

ᨨᆠ
盐
ᨨ 矄�鼀 
EB ur F = I ᲼ L�$ ´ B $ 桌
㦨
ILBk$
�
 
ᨨ矄��
z �ᆠ�ᨨ


҈黨


T盐 꾀
ᆠ盐 
ᆠ泐� �
ᆠᨨ�
�矄

鼀� �ᆠ
�ᨨ


 鼀� �ᆠ
ᨨᆠ
ᨨ

鼀� �ᆠ
�ᨨ
ᆠ
�

盐 � ᨨ鼀 鼀� �ᆠ
�
�矄
ᨨ
�㡰
㝐� �ᆠ
�
盐
ᨨ

� � �泐
ᨨ �
ᆠ盐 
ᆠ泐� � 犨�

�矄 ᆠ

�矄
��
泐






B
㦨


m
æ
� ö ç ÷
p è r ø Y

C

� A

B

P O Z d

x

X

d

ᔼ
ᆠ矄
ᨨ
 
� �ᆠ
犨�矄

矄盐 
�
ᆠ

ᨨ ᨨ 泐
�
 
ᨨ� 
P
�矄
犨
 
犨ᨨ
犨�矄

ᨨ 犨�矄

WWW.JEEBOOKS.IN

EBD_8341 PS-232

Physics

�
B
盐
ᨨ

 
犨�矄
犨�

�
 
矄鼀
�ᆠ�ᨨ
  矄ᨨ
矄盐 
B
ᆠ

ᨨ

ᨨ

 ᆠ

�
ᨨ� 
P
�矄

 �矄 ᆠ

ᨨ鼀
 
ᆠ 
犨�
矄
矄 ᨨ犨
�
��盐

 
ur ur ur ur B P = B PA + B PB + B PC ur 犨 
B PA
㦨
鼀� �ᆠ
�

P
盐
ᨨ
A ur B PB 㦨
㝐� �ᆠ
�

P
盐
ᨨ
B ur B PC 㦨
㝐� �ᆠ
�

P 盐
ᨨ C
�ᆠ�ᨨ
ᨨ
B
�矄
��
泐
�� 
 
鼀
盐


ur m ҈ ҈ ù $ é ҈ B P =

� ê + - k 㡰

p ë d + x x d - x úû ur ᨨ B P =

҈ ҈ ҈ +
㦨
d+x x dx d 和ᨨ� �
犨
�

x
㦨
±  ��㡰 
ᨨᆠ

盐 �
 �
犨
犨ᨨ

ᆠ盐  ᆠ泐� �
犨�矄 m

�҈�
㦨
f 矄泐㡰
p r 꾀ᆠ�
�
ᆠ盐 矄

�
 
矄鼀
�ᆠ�ᨨ
d

d B

A r

X

Z

f


҈


␔ 
犨�
꾀
�矄
矄�� 泐
矄矄

ᨨᆠ
ᨨ
盐矄�ᨨ
盐 ᨨ
ᆠ盐 

ᔼ
�
ᨨᨨ矄�
ᨨ
 
犨�
ᔴ
犨�
盐矄
�
盐犨矄  
꾀
鼀泐
�

ᔼ
ᨨᆠ
ᨨ
ᆠ�ᨨ
 �矄矄

ᔴ
犨� ᆠ�ᨨ
꾀
 
��ᨨ
ᨨ
꾀
犨�
ᨨ

ᨨ矄矄�
␔ 
 
ᨨ
�矄
矄矄
泐

�矄 ᆠ
x

 
 
ᨨᆠ
ᆠ� � ᨨ
 
盐
犨�
� ᆠ矄矄
 
 矄
矄

矄ᨨ� �
ᨨᆠ
Fmag I҈
㦨


A

A' r 㦨



҈鼀

mg

C

I

㦨

A

mg m

I҈I

㦨
r
p r

㝐矄矄

盐 �

 �
㦨


zù m
� � F z é =
f ᆠᨨ矄 q =

҈
´ ê 矄 ᆠᨨ矄 q = ú r
p r l r ë û ᨨ

F m

�
z = ×

l
p d + z 㡰

᲼矄
I҈
㦨
I

 
r

㦨
d

졔
z
桌








l w

㦨


m
�
Þ
f
㦨
pl
pd

ᨨ鼀
⼄

�㡰
 
��㡰

m

I҈I
m

I҈I
= ´ w


Þ

p r
p rg Þ

g r

w
㦨



p g 㦨


r




҈ r =
p =


矄 g 黨





 
鼀� �ᆠ
�
盐
ᨨ ᆠ盐 矄
�
A
 
B

P

B҈  
B

矄ᆠ�泐
�
  �ᆠ�ᨨ 矄
 �ᆠ盐
ᨨ
꾀  

矄
矄 ᨨ犨
�
��盐

Ð BPO
㦨
ÐAPO
㦨 q ur ur m
I  B҈  =
=  B




Z
p a
+ x
\ T
㦨

p

Y B a

xq O

㜠a A

P

B
矄� q

B


q

B҈矄� q

C

q

B
ᆠᨨ矄q B҈ᆠᨨ矄q

X

B҈

졠
矄ᨨ� �
B҈
 
B

犨
�
 
 
矄� q
ᆠᨨ鼀ᨨ  矄 ᆠ ᆠ
ᨨ盐
 
 
ᆠᨨ矄 q
ᆠᨨ鼀ᨨ  矄

盐
\ B
㦨

B҈
ᆠᨨ矄 q




Ҙ
� m

� ´


Þ w
㦨

p d
pd
l

� Þ

p f
㦨
d





��㡰



x 졔


F m �.e.

= - w
z l l 㦨

l w

z

犨
�

F
㦨

mw
z

m

I҈ I

p rg


Öa

m æ
� ö F = -
ç ÷ z ᲼矄
d
>>
z
 
F
�矄
ᨨᨨ矄�
ᨨ
z桌




�㡰 l
p è d ø


ᆠᨨ鼀ᨨ  矄
ᨨ
ᨨᆠ
ᨨ �
x㜠 �矄
犨�

ᆠ ᆠ

F
µ

z

 
鼀ᨨ�ᨨ
�矄
矄�鼀
鼀ᨨ �ᆠ
ᔴᨨ鼀� �
⼄

�㡰
犨�
 
矄 
⼄盐�ᨨ
ᨨ
和

㝐


ᨨ


D

m

I҈I
mg x
㦨

r
p r 和� ᆠ

F
µ
x
 
�ᆠ
ᨨ
⼄盐���盐鼀
ᨨ矄��ᨨ


 ᨨ
 
鼀ᨨ�ᨨ
�矄
矄�鼀
鼀ᨨ �ᆠ m

I҈ I
\

㦨
鼀矄矄

盐 �
 � 㡰
w
Q
a
㦨
lw
㡰



�㡰
p r


B

ᔼ
 
犨�
B
�矄
�矄ᆠ
ᨨ �
Z㜠 �矄
泐

矄鼀
�矄 ᆠ F Z

 
矄ᨨ� �
ᨨᆠ

盐 �
 �

ᨨ
 
犨�
B
盐 l ᨨ
犨�矄
A
 
C
犨�


B'

矄ᨨ� �
ᨨᆠ⏌ �
F
㦨


꾀
⼄盐���盐鼀



Z

x

m

I҈ I
m

I҈ I
矄ᨨ� �
ᨨᆠ⏌ �
㦨

p r - x
p r m
҈ ù m
I I x é ҈ - ú
㦨

҈

I҈I
ê 㦨

p ë r - x r û
p r r - x㡰 ␔ 
x
�矄
矄鼀
�.e.

x



r
 
r

㦨
x
»
r

f

q q

B

A

m
pl

㦨



m

p


I


a +x




´

x


a +x




=

m

Ix
p a
+ x
㡰 ᨨ犨矄

Y
�ᆠ�ᨨ 㡰

WWW.JEEBOOKS.IN

PS-233

Moving Charges and Magnetism 
盐矄
ᆠᨨ 矄�

矄鼀
ᨨ�ᨨ
ᨨ
犨�
OC

P
ᨨ
 � dx

ᆠ盐矄
鼀� �ᆠ
�
ᨨ盐ᆠ

�  
ᨨ� 矄 ᨨ
졠ᔴ
犨�

�  㡰


矄鼀
鼀ᨨ盐 
ᨨ
ᨨᆠ
ᆠ� � ᨨ
 
矄鼀
ᨨ�ᨨ

黨
黨

´ ҈҈ I B ´ ҈҈ = 㦨
꾀

Þ IB
㦨


҈
��㡰 
�鼀 矄�ᨨ 矄
��
矄 ᨨ犨矄
  SA

졔
SB

㦨
AB


\ ÐASB
㦨
黨
泐
泐 �ᨨᨨ盐矄
 ᨨ鼀㡰 㝐� �ᆠ
�
盐
ᨨ
A

S m

I A m

´ 黨

× = ´ BSA
㦨
�ᆠ
S
ᨨ
B㡰
p rSA
p ҈

㝐� �ᆠ
�
盐
ᨨ
B

S m

I B m

´  × = BSB
㦨
�ᆠ
S
ᨨ A㡰
p rSB
p ҈

和� ᆠ
BSA
 
BSB

鼀盐盐泐
 �ᆠ盐 \ 
鼀� �ᆠ
�
Þ

ur ur ur dF = I dx ´ B㡰


\
dF
㦨
I dx B
矄�
黨


m
Ix Þ dF 㦨
I dx
맔

´

p  a + x
㡰

xdx m Þ dF
㦨


I


p a + x
㡰 \

ᨨ
ᨨᆠ

F
㦨


L m
xdx ´
I
ò


p a + x
㡰

L

m
é҈ ù ´
I
ê ᨨ� e a
+ x
㡰 ú 㦨


p ë
û





é m
a
+ L
ù ´
I
êᨨ�e ú
p a
ûú ê ë ur


�ᆠ�ᨨ
ᨨ
F �矄
ᨨ犨矄

Z
�ᆠ�ᨨ
矄

鼀� �矄矄  
 
盐
␔ 
�ᆠ�ᨨ
ᨨ
ᆠ盐 
�
B
�矄
矄


鼀� �ᆠ �
�矄
ᨨ �
 
ᆠ盐 

 ᆠ
ᨨᆠ
F
㦨
IlB
矄�
q
 
q
㦨
҈


�矄
ᨨ



犨ᨨ

犨�矄

 �ᆠ盐
ᨨ
 
 
ᨨ
  


犨�
꾀
ᆠ�矄
ᆠ盐 
�ᆠ
� ᨨ
 
 
ᨨ 



ᆠ盐 
�
犨�
B
矄 ᨨ盐

�
盐犨
�ᆠ�ᨨ
矄ᨨ
矄 ᨨ
ᆠ ᆠ
 
鼀� �ᆠ
�
盐
ᨨ
A

P

泐
�� 
 

盐鼀
盐㡰

m æ 黨

ö æ  ö

B 㦨
BSA + BSB =
ç ÷ +ç ÷
p è


ø è


ø 㦨
҈
䰘
맔
҈
㦨
҈

맔
҈




���㡰 ᨨᆠ

盐 �
 �
ᨨ
犨�
B

Þ F 㦨





m

I A I B
p rAB ҈
-䰘 ´
´ 黨

´  㦨
㦨




맔҈
䰘
⏌鼀


㦨


Topic-4 : Galvanometer and its Conversion into Ammeter and Voltmeter ҈


A ´

㡰 ෌ ᨨ鼀
ᨨ
矄�矄 ᆠ
෌㡰
ᆠᨨ 
� ᨨ
 ᨨ鼀
ᨨ
 �

㜠҈

R҈ ෌ �g

V = ҈ = �g G + R҈ 㡰


鼀 ҈



鼀

෌ R҈

10 m 11

S

҈

鼀




�㡰

ᨨ
� ᆠ矄
 
 �
ᨨ
ᨨ鼀

㜠




= �g  R҈ + R
+ G㡰






M BA

P

BB N

�㡰

鼀� �ᆠ
�

P
盐
ᨨ
ᆠ盐 
�
犨�
A


m

I A m

´ 黨

= ´
�ᆠ�ᨨ P
ᨨ
M㡰




�㡰 ҈
ö
p rAP
p æ çè
+ ÷ø ҈҈


鼀� �ᆠ
�

P
盐
ᨨ
ᆠ盐 
�
犨�
B BA
㦨



I m
m
´ 黨

´ =
´ B ҈
ö
p æ ҈
ö
p æ çè
+ ÷ø çè ÷ø ҈҈ ҈҈




��㡰

��� �
⼄

�㡰
泐
��㡰
G + R҈ ҈ Þ =
G + R҈ + R


B

m

I B ´ BB
㦨

p æ ҈
ö çè ÷ø ҈҈ ᨨ鼀
⼄

�㡰
 
��㡰

R






Þ G + R҈ + R
=
G +
R҈ \ R
= G + R҈ 㡰 ෌�
ᔴ盐 
矄矄� �
 ᨨ盐�
� ᨨ鼀

I
㦨


鼀꾀  ᆠ�ᨨ

q
㦨

��盐
ᨨ
鼀�
ᨨ
� ᨨ鼀 I
´҈
- = =  ´҈
- 꾀⏌�
q ᨨ 盐矄㡰

㦨
��

졔
෌㡰 Þ
�
㦨
҈




맔
҈


졔
㡰
㦨

҈黨�
맔
҈

W ᆠ㡰 
㦨
��
෌
졔
㡰
㦨


맔
҈

�

졔
�


㡰
㦨


 㡰 Cq
㦨
NB�A
矄�
黨
æ p ö = ҈䰘�B҈
- 㡰 ´ ҈
-
ᨨ ҈
-
ç è ҈

÷ø \
B
㦨
҈



=





��㡰 


�


WWW.JEEBOOKS.IN

EBD_8341 PS-234





Physics

b㡰 ␔ 
├泐
⧤҈
�矄
ᆠᨨ矄
 
├泐
⧤

�矄
ᨨ

쟼 �� = = ᔴq



+ � ␔ 
ᨨ
 
├泐矄

ᆠᨨ矄





�㡰

ᔴq
쟼 � æ ö ´ = ç � � ÷  � + �㡰 � ç


+ ÷ + � è �ø ᔴq
�쟼 Þ =



��㡰

� � + ҈҈

� 쟼 = ᔴq




�㡰


+ � ��� �
�㡰
泐
��㡰

犨
�

� � + ҈҈

Þ =� ҈҈

+ � � Þ


��


졔

� �
㦨


� �
졔
҈҈




�
㦨




�
㦨


W b㡰 ෌ ᨨ鼀
矄

�
��矄�ᨨ 矄
ᔼ�
㦨



맔
҈


맔

�
㦨
҈


꾀 ෌

�

� �� =

䰘


�


W 








黨


҈



҈ ´ ҈

´
䰘
´ ҈
-

和= = ҈҈
W æ
ö
-ç ÷ è
ø ҈҈
ᆠ㡰 ෌�


ᔴ盐 
 ᨨ盐�
 
� ᨨ鼀
�g
㦨
�
맔
҈

A ෌ ᨨ鼀
矄�矄 ᆠ

G
㦨
҈�W 
矄�矄 ᆠ
R
ᨨ

盐
�
矄�矄
犨�
 
� ᨨ鼀 ᨨ
ᆠᨨ 
�
� ᨨ

ᨨ鼀
V
㦨
�g
R
졔
G㡰 ҈

㦨
�
맔
҈

R
졔
҈�㡰 \ R
㦨






҈�
㦨
҈黨
�
㦨
҈
黨
�
맔
҈

W ҈

ᆠ㡰 ᔼ�
෌
㦨

ᔼ

ᔼ�㡰矄 \
҈

맔
҈


㦨
҈


҈
㡰
맔
和 \

和
»



҈W  ҈
㡰 꾀ᆠᆠᨨ� �
ᨨ
졠 鼀Ἐ矄
犨

ᔼ =  ᔼ� 㦨 졔 ෌ 犨 

ᔼ�㜠෌ ᨨ鼀
ᆠ盐 

෌㜠෌ᨨ ᨨ鼀
矄�矄 ᆠ ᔼ෌ ෌ ҈҈ ´ ҈
 ´


㦨
ᔼ
�













�



␔ 
矄 盐 
ᨨ
矄�矄 ᆠ
和
�矄
ᆠᨨ

ᆠ

ᨨ
  ෌和 ෌ ᨨ鼀
 
෌ 㦨 ෌졔 和  \ ᔼ㦨 ෌和 졔 ෌졔 和 ᔼ෌
෌


㦨
ᔼ�
෌
졔
㡰

�
㦨
�

졔
㡰
҈


\

㦨



W ᆠ㡰  ᆠ�ᨨ
ᆠ盐  㦨
ᔼ�鼀
㦨
├
㦨



�
맔

␔ 


㦨
盐鼀
ᨨ
��矄�ᨨ 矄
㦨


 
├
㦨




�
鼀⏌��矄�ᨨ

㦨
҈�
맔
҈


㦨


҈� 
㦨
ᔼ�᲼


졔
桌 ҈�
㦨


҈�
᲼


졔
桌 ҈


㦨



졔

㦨



W 㡰 ෌�
矄�矄 ᆠ
ᨨ
� ᨨ鼀

G
㦨
҈

W ᔴ盐 

��
㦨
҈
mA 꾀
� ᨨ鼀
ᆠ

ᆠᨨ 
� ᨨ
ᨨ鼀
泐 ᆠᨨ

ᆠ� �

�
矄�矄 ᆠ

�
矄�矄
犨�
�


ᨨ
矄�矄 ᆠ
ᨨ
 
ᆠᨨ鼀� �ᨨ
㦨
G
졔
R 꾀ᆠᆠᨨ� �
ᨨ
졠 鼀矄
犨

V
㦨
��
G
졔
R㡰 \
҈

㦨
҈
맔
҈

҈


졔
R
㡰 Þ
҈





҈


㦨
黨黨


W
㦨

Þ


㦨
黨
黨
├W b㡰 ��盐
ᨨ
鼀�
ᨨ

� ᨨ鼀
�矄
 
ᆠᨨᆠ
⼄盐� ᨨ
ᨨ盐ᆠ

 ᆠ�ᨨ
ᨨ
ᨨ 
��矄�ᨨ
�
 
� ᨨ鼀 ᔼ �



��盐
ᨨ
鼀�
= q e ҈ ᔼ=
෌ = ⧤W +෌ 黨 ҈ e 和 ҈ e和 = ´ Þ =
+ ෌和 和+ ෌
和 + ෌㡰 + ෌和 和 ෌+和 ෌

෌

҈ ᔼ
和=  + ෌㡰ᔼ









e
෌ ´

和 쟼⼄盐
ᨨ �
�  ᆠ
�矄
��
泐

ᔼἘ� ෌ 㦨 ᔼ 㜠 ᔼἘ� 㡰和 ᔼ Ἐ� ෌ + 和㡰 = ᔼ和

쟼

쟼

=

Þ ෌ + 和 + ෌和 =
和 + ෌㡰 Þ ෌ =
和 + ෌㡰 - 和 + ෌㡰 \ ෌ = 和 + ෌㡰

ᔼ⏌


҈



ᔼ�

ᔼ和
෌+和   和 Þ = ´
 + ෌ 㡰 + ෌和 ෌+和 ෌ +和 ҈ 和 Þ =
 + ෌㡰 ෌ + 和㡰 + ෌和 Þ ෌ + 和㡰 + ෌和 =
和 + ෌㡰 Þ

�
�
㡰 ҈ q
㦨




 
㦨
�

 

᲼෌�


㦨






ᔴ
㦨
҈


鼀

WWW.JEEBOOKS.IN

PS-235

Moving Charges and Magnetism 꾀
㦨


맔
҈


鼀
桌 ␔
├ ᨨ犨

ᔴq
㦨
꾀
ᔼ�

҈䰘


Cq ҈

´



㦨


҈꾀
㦨
NBA �
´
´҈

´





ᨨ
ᆠᨨ 

� ᨨ鼀
ᨨ
鼀鼀


矄 盐 矄
�矄
盐矄
�


ᨨ
 
� ᨨ鼀
ᔼ�
맔
෌
㦨
ᔼ

ᔼ�㡰
和 \ ᔼ�
㦨




҈ ´ �
�
\ 和
㦨

㦨

㦨

㦨
�
�

W I  Ig ҈ 

҈ 黨 ҈�
b

㡰 

෌
㦨
҈
W錐
ᔼ�
㦨


맔
҈


꾀


㦨
҈


鼀 
㦨


҈

ᔼ
㦨
҈
꾀 矄� �


㦨
ᔼ�
෌
졔
㡰
᲼
㦨
矄�矄 ᆠ
ᆠᨨ

ᆠ
�
矄�矄 犨�
� ᨨ鼀桌 Þ


҈
㦨


맔
҈


 \

㦨
�
맔
҈

W
 矄�矄 ᆠ
ᨨ
ᨨ鼀㡰 꾀矄ᨨ
ᔼ�
෌
㦨
ᔼ

ᔼ�㡰矄

맔
҈


맔҈

㦨
҈




맔
҈

㡰矄 \
和
㦨


맔
҈


W  矄�矄 ᆠ
ᨨ
鼀鼀㡰 I gG

b

ᆠ㡰 ᨨ
ᆠᨨ 
鼀鼀
� ᨨ

ᨨ
鼀

��
矄�矄 ᆠ �矄
ᆠᨨ

ᆠ
�
矄�矄
 
ᨨ犨
矄�矄 ᆠ
�矄
ᆠᨨ

ᆠ
�


ᨨ
ᆠᨨ 

� ᨨ鼀
� ᨨ

鼀鼀
ᨨ
V
㦨
Ig
G
졔
R㡰
㦨
�
맔
҈
�
᲼҈


졔







桌
㦨
҈


æG ö é҈

ù + ҈÷ = � ´ ҈
-� ê + ҈ú
㦨
�鼀꾀
èS ø ë ҈ û

ᨨ
I
㦨
Ig ç ҈



θ NBA ᆠ㡰 和 矄���泐
ᨨ

鼀ᨨ� �
ᆠᨨ�
� ᨨ鼀

㦨
㦨
ᔼ C θ ᔼ

� ᆠ矄矄

泐
盐矄� �
�ᨨ
ᆠᨨ

�.e.錐
矄 矄���泐
� ᆠ矄矄
ᔼ 和ᨨ 
�ᨨ
ᆠ

矄�泐
鼀� �矄
 

鼀� �


Topic-5 : Miscellaneous (Mixed Concepts) Problems ҈


ᆠ㡰 
盐矄
ᆠᨨ 矄�

鼀 
 �
dl
矄盐 � �


 �
dq

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ᆠ 
ᨨ
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ᆠ�ᆠ
�졠


FB

  鼀� �ᆠ
ᨨᆠ
ᆠ� �
ᨨ
 �矄
 �


T
ᆠᨨ矄
q




෌和 ҈
´



꾀 = = »



W ෌ + 和 ҈
+





ᔼ=

T

쟼 쟼 = �




´ ҈


+



黨



҈ �҈




 =



q


 쟼 黨


黨 = ´ = 黨


黨W ᔼ 黨



҈ 쟼 ⏌ 黨



҈ 

T矄�

q



T矄�
q






q

q









I

쟼 �




´ ҈


쟼 ´ = ´ 黨


黨 黨



҈ �҈


黨



҈

\
鼀矄盐 =

T
ᆠᨨ矄
q




ᔼ

꾀

쟼

é ᆠdl 㡰 ù êQ  � q㡰 = ú �盐矄 R û ë

꾀
⼄盐���盐鼀

T 矄� ç

\ E
㦨
黨黨黨


鼀
和� ᆠ
�A
㦨
҈
꾀

҈



I

L æ q ö q ÷ = BI p

è ø

E æ ҈


ö
´ç ÷ ´ � ´҈
黨

è �҈


ø

\ 㝐 �鼀盐鼀
� �
ᨨ
R
㦨


R

Lù é æ Lö = BI ç ÷ d q êQ
pR = L Þ R =
p ú è
p ø ë û 
T

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 矄�ᨨ
�
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犨�
ᆠ� �
ᨨ �
ᨨ
 矄
ᨨ  
鼀 
 �


ᔼ
 
ᨨ鼀
矄 ᨨ犨矄
盐
矄ᆠ
 ᆠ�ᨨ

 



҈
㦨


T

\ FB 㦨
BI dl㡰

盐犨矄㡰 㦨
BI
Rdq㡰

 ҈


W 



q


Þ
T

黨黨黨

´҈
 ҈ ´ ҈



㦨
黨黨黨


W



ᆠ㡰 
 �
ᨨ
ᨨ鼀
V
㦨
IgReq
졔
G㡰 \ 㝐 �鼀盐鼀
ᨨ�
ᆠ
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ᨨ� 
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鼀 �鼀盐鼀

�.e.
犨 

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
ᆠᨨ

ᆠ
�
矄�矄



 �
ᨨ
鼀鼀

G ö æ I
㦨
I g ç ҈ + ÷ è S eq ø \ 㝐 �鼀盐鼀
ᆠ盐 
 �
ᆠ

ᨨ� 
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⼄盐�  矄 盐 
矄�矄 ᆠ
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�.e.

犨 

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ᆠᨨ鼀ᨨ  矄 
ᆠᨨ

ᆠ
�



dq L = BI dq

p

é dq ù êëQ
泐㜠泐
矄鼀 úû

BIL
p 㡰 
鼀� �ᆠ
� 矄
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矄盐ᆠ
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x

y
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矄
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ᨨᨨ矄

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盐鼀 ᨨ
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鼀ᨨ� �
ᨨ犨 犨矄
�
x

y
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
矄鼀
�

盐鼀
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盐犨矄
ᨨ
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x  y
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 ᨨ
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x

y
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v
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B
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ur r ur

 ᨨ

鼀� �ᆠ
ᨨᆠ
F m = qv ´ B㡰 =
\
T =








WWW.JEEBOOKS.IN

EBD_8341 PS-236

Physics Z

v



Y

I

㜠I

X ␔�矄

�
X㜠Y
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B

d






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矄
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x㜠 �矄

vx
㦨


y mv
mv
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r҈
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r

㦨
qB҈ qB


d҈ + d
t҈ + t


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B



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r҈
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r


r B҈

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r


x

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p
r
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;
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Z

BR B


B҈

I҈ 졠

P

Z

I Y





æ m p
r
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I
㦨
I

ᆠᨨ矄


t㡰

N
㦨

(





X

æ m pr
ö dI df = ç
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p
r
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I
矄� 

t 㡰 M
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P

r m I ⼄盐
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æ m
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矄�
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t㡰 dt

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R Y ᆠ㡰 ᔼ
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Iù m
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ᆠ�ᆠ盐
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\

I
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r҈
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b

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ᔼ 盐ᆠ
鼀
e = 

WWW.JEEBOOKS.IN

PS-237

Moving Charges and Magnetism

҈



҈҈


҈



 ᆠ

 

盐
ᨨ
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⼄



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t N�AB = q q

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EBD_8341 PS-238

Physics ᨨ鼀
⼄

��㡰
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m
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맔
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d d QwBR
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B
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r


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� m
I [ ᨨ� e x ] rr҈
p

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PS-239

Moving Charges and Magnetism

� m
I �m ´ ҈
r æ

҈
ö ᨨ� e
=
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ur

ur

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


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y v

x

N �㡰 ᔴ盐 
I
 ᨨ盐�
 
ᆠ
MN
 
矄�� 
犨�
NM 矄盐

鼀� �ᆠ
�

 
ᆠ 
P



ᆠ �
�ᆠ
�矄
ᆠ
盐ᨨ
泐
 
鼀� �ᆠ
ᨨᆠ 盐
ᨨ
 矄
�矄


鼀� �ᆠ
ᨨᆠ
ᨨ盐ᆠ矄


� 矄  ᨨ盐矄
ᆠᆠ�ᨨ
�
 
ᆠ �
�ᆠ

� 盐
ᨨ
ᆠ
MN
҈ m
I B҈
㦨

ᨨ盐犨矄 
a

҈
m
I m
I ᨨ B҈
㦨

ᨨ盐犨矄
㦨

ᨨ盐犨矄 a
a



҈
m
I ˆ r k B҈
㦨
a �
盐
ᨨ
矄�� 
犨�
NM m
I 矄�



졔
矄�


㡰

犨 
r
㦨
a
ᆠᨨ矄


B

㦨

pr m
I ᨨ B

㦨


矄�


㡰
p a ᆠᨨ矄

° m
I ´  � 犨矄㡰 ᨨ B

㦨

pa


䰘 m
I ˆ r k ᨨ B

㦨
 a r r r \ Bnet
㦨
B҈ + B
ᨨ



҈҈ m
I ˆ r k Bnet
㦨  a r v
㦨
v
ᆠᨨ矄



�ˆ 졔
v
矄�



ˆ

v ˆ r v  v
㦨
�ˆ +

\ 㝐� �ᆠ
ᨨᆠ r

҈҈m
IQv ˆ r r    �ˆ㡰 \ Fm
㦨
Q  v ´ Bnet 㡰
㦨

a r Fm r \ 꾀ᆠᆠ�ᨨ
鼀矄矄
a
㦨
m m IQv

҈҈ r
( ˆ  �ˆ ) ᨨ a
㦨

ma ��㡰 ᨨᆠ
 
ᨨ⼄盐
ᨨ
 
ᨨᨨ
r ᔼ
盐 � ᨨ鼀
鼀� �ᆠ
�
B = B�ˆ

ᨨᆠ
ᨨ
ᆠ盐 
ᨨᨨ
�矄 ᨨ
r r

ᨨ⼄盐
ᨨ
ᨨᨨ
㦨
M ´ B r M =  IA㡰 kˆ ᨨ

pa
a

矄�
҈


ᨨ
꾀
㦨



҈
a

r r \ M
㦨



҈
Ia
㡰 kˆ B = B�ˆ r r r r \ t
㦨
M ´ B ᨨ
t
㦨



҈
Ia

B㡰  kˆ ´ �ˆ 㡰 r ᨨ t
㦨



҈Ia
B 㡰 ˆ


�㡰 㝐� �ᆠ
�
盐
ᨨ
ᆠ盐 
�
犨�
P

R m

I p m

´

� Ҙ
´ = ´
㦨
҈
㦨

p
p rPR
p � ᲼�
 
 
ᨨ

ᨨ犨 犨矄桌 ᨨ A
㦨


 ᨨ

F҈ = F
=
BIR�$ r r r r \
ᨨ
ᨨᆠ
ᨨ
 
ᨨᨨ

F = F҈ + F

Þ
F
㦨

BIR �ˆ

a v

졔Q r

a P

ᨨ



�
꾀 맔 

ᔼ
꾀 맔 턼

�  v
㦨

҈
鼀⏌矄 졔


鼀  㦨
�鼀

和�鼀�泐

 
鼀� �ᆠ
�
盐
ᨨ
ᆠ盐 
�
犨�
Q

R Ҙ
´ I Ҙ
I
᲼�
 
 
ᨨ

ᨨ犨 犨矄桌 = 

㦨

´
p

p

ᨨ
鼀� �ᆠ
�

R
᲼盐
ᨨ
P
 
Q桌 Ҙ
Ҙ
Ҙ + I =
҈ + I 㡰 B 㦨
B҈
졔
B

㦨

p
p
p ᲼�
 
 
ᨨ

ᨨ犨 犨矄桌 ᨨᆠ
 � ᆠ
泐
 
ᆠᨨ

F 㦨
qvB
矄�
q Ҙ
㦨
pvB
矄�
黨

㦨
҈


맔
҈
҈黨
맔


맔
҈
�
맔
҈
졔
I㡰
p \ 


맔
҈



㦨
҈


맔
҈
҈黨
맔


맔
҈
�
맔
҈
䰘
҈
졔
I㡰 ᨨ
I
㦨


꾀 ��㡰 ᨨ

�

R
ᨨ

ᨨ

鼀� �ᆠ
�
盐
ᨨ
 � 犨�
ᆠ泐� �

ᆠ盐 
ᨨ


�
꾀
矄 ᨨ盐
 B
㦨
҈
䰘I
졔
҈㡰
㦨
�
맔
҈
䰘



r

 
�矄 ᆠ
ᨨ
 �矄
犨�
ᨨ鼀
m

´

� ´ ᨨ
r
㦨
҈鼀 \ �
맔
҈
䰘
㦨

p r �.e.

� 
 
ᨨ
�� 
҈鼀
ᨨ鼀

�ᆠ�ᨨ
ᨨ
ᆠ盐  ᨨ盐犨矄
 
� 犨矄
矄ᆠ�泐


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EBD_8341 PS-240

Physics

Magnetism and Matter

19

Topic-1 : Magnetism, Gaus's Law, Magnetic Moment, Properties of Magnet ҈









ᆠ㡰 

q = 
°

t
㦨



҈

㜠鼀

B
㦨







ᨨ⼄盐
ᨨ


鼀� 

t = MB 矄� q


҈
= M ´



´ 矄� 
° ҈ Þ


҈
= M ´



´ Þ M =



꾀㜠鼀

ᨨ矄��ᨨ
ᨨ
矄
⼄盐���盐鼀
q
㦨

㡰 ᨨ矄��ᨨ
ᨨ
盐 矄
⼄盐���盐鼀
q
㦨
҈

㡰 㝐� �鼀盐鼀
犨ᨨ├
⼄盐�
ᨨ
ᨨ

鼀� 
ᨨ鼀
矄 ᨨ
盐 矄
⼄盐���盐鼀 DU = U f - U � = -MB ᆠᨨ矄҈

° - - MB ᆠᨨ矄
°㡰 W =
MB =
´


´



\ W = 䰘

´ ҈
-
 㡰 ෌�
㝐ᨨ鼀 
ᨨ
� �
ᨨ
ᆠ�ᆠ盐
ᆠᨨ�

I
㦨




├�
鼀
㝐� �ᆠ
鼀ᨨ鼀 
ᨨ
ᆠ�ᆠ盐
ᆠᨨ�

M
㦨



꾀鼀
ᨨ�ᨨ 
├� �ᆠ
 �泐
ᨨ
ᆠ�ᆠ盐
ᆠᨨ�
҈
⧤쟼
= I w


w
㦨
 �盐
矄
ᨨ
ᆠᨨ� ᨨ �
 �泐
ᨨ

鼀� 
㦨

MB
ᆠᨨ矄
f ᨨ鼀
 �泐
ᆠᨨ 矄�ᨨ

҈
I w = U � - U f = - MB ᆠᨨ矄

° - - MB 㡰
MB ҈
Þ = Iw



´
҈ Þ = 


㡰w



Þ ҈

= w Þ w = ҈


b㡰 B҈ =





�






䰘




黨


µ

K
p ( d ⏌
)

҈



 
B
=

꾀 �盐
鼀ᨨ鼀 盐鼀

L
㦨


Iw
㦨

mR
㡰
w

µ

M
p ( d ⏌
)  µ

M
p ( d ⏌
) 

B
=
㦨
҈ \  q = B҈ µ

M
p ( d ⏌
)  ᨨ q
㦨

�º

uur


矄盐 
�
�矄

�º
ᨨ鼀
҈


 �
犨

B uur  
v
ᨨ

矄ᨨ
ᨨᆠ
ᨨ
 
�ᆠ
�矄
ᨨ
ᆠ㡰 ␔ᨨ├
ᨨ 

␔
㦨


鼀· 㦨


맔
҈


맔
҈
ᆠᨨ矄


҈
�㡰
㦨





 b㡰 ␔
├ ᨨ犨
 

鼀� �ᆠ
�ᨨ
鼀ᨨ鼀  㝐 = �꾀 ᆠᨨ矄 θ �


㝐 µ ᆠᨨ矄θ ␔ 
犨ᨨ
鼀� �ᆠ
�矄

� ᆠ� 


 �
ᨨ
䰘�  
⼄盐���盐鼀
犨�





矄ᨨ ҈ ᆠᨨ矄 θ 㦨 ᆠᨨ矄䰘� 㜠 
㡰 㦨 ᆠᨨ矄
� 㦨
҈� 㦨 \ » ҈҈

r N�A ᆠ㡰 M 鼀�

鼀ᨨ鼀 ⏌ᨨ盐鼀㡰
㦨
Al N� �

㡰҈� 㦨

㦨

㦨





꾀鼀 꾀鼀҈ l
� ´ ҈

uur ur 㡰 ᨨ �
 �泐
U
㦨
- M
B
㦨

MB
ᆠᨨ矄
q ᔼ
ᆠ矄
ᔼ
q 㦨
҈



U
㦨
졔
MB ᔼ
ᆠ矄
ᔼᔼ
q 㦨
黨


U
㦨

ᔼ
ᆠ矄
ᔼᔼᔼ
q 㦨
ᆠ盐

U 㦨
졔

矄矄
 

졔
MB㡰 ᔼ
ᆠ矄
ᔼ
q 㦨
ᨨ盐矄

U
㦨

 \ ᔼ
>
ᔼᔼᔼ
>
ᔼᔼ
>
ᔼ
㡰 㝐� �ᆠ
� 矄
ᨨ
ᨨᆠ
ᨨ鼀
ᆠᨨ矄
ᨨᨨ矄

ᔼ 矄�
 鼀� 

 矄

�ᆠ
ᨨ鼀
矄ᨨ盐
ᨨ
ᨨ
ᨨ
  ᨨ盐矄�
 
鼀� 
ᨨ鼀
ᨨ
ᨨ
矄ᨨ盐
ᆠ㡰 
 �盐
鼀ᨨ鼀 盐鼀
ᨨ
 
�ᆠ L
㦨
mvr
㦨
mr
w
犨 
w
㦨

pn
᲼Q
v
㦨
rw桌 q wq w \
⼄盐 ᆠ泐
n
㦨
錐
盐 
�
㦨

㦨
q 맔
n
㦨
t
p
p wq qr
w ´ pr


Þ M
㦨
㝐� �ᆠ
鼀ᨨ鼀 

M
㦨
�A
㦨

p
M q wqr
\


= = L
mr
w
m qw æ wö q 㡰 ᔴ盐 

�㦨 㦨
 ⼄盐 ᆠ泐㡰ᆠ �㡰㦨
çè ÷ø 
q㡰 =
p p t 㝐� �ᆠ
鼀ᨨ鼀 
æ qw ö

w M
㦨
�㡰
A㡰
㦨
çè ÷ø pR 㡰 = qwR 㡰 p

M qw R
q R R = =
L
 mR 㡰 w
m (q,m) (q,m) ᆠ㡰 ᔼ

盐 � ᨨ鼀
鼀� �ᆠ
�

鼀� �ᆠ
ᨨᆠ
ᨨ

ᆠ盐  ᆠ泐� �
ᨨᨨ
�矄
ᨨ


鼀� �ᆠ
�


�矄
 �ᆠ盐
ᨨ  
 
ᨨ
 



盐矄
ᆠᨨ 矄�
犨ᨨ
�鼀�ᆠ泐
ᨨᨨ矄� 鼀 矄

泐
鼀� �Ἐ矄
 
 
盐

ᨨ
鼀 
AB
ᨨᆠ dF 㦨
I ( d l ) B 矄� 黨
° = I ( d l ) B

 犨矄㡰

\

҈҈


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PS-241

Magnetism and Matter X

X

X

X

dF X

X

B A

X

X

X

X

X

ᔴX

l X

X

X

l

D

X

X

X

dF

X

X

X

X

X

X

ᨨᆠ
ᨨ
AD

FAD
㦨
I
dlB҈
矄�
黨

I
dlB҈ B҈ D

d

ᔼ X

҈



҈


҈



҈�


X

X

X

졠
鼀 
CD

ᨨᆠ dF 㦨
Idl㡰B ᲼ �� 
ᨨ
 
 
ᨨ
桌

矄
犨ᨨ
ᨨᆠ矄
犨�
ᆠ ᆠ
ᨨ盐
和�鼀�泐


ᨨᆠ矄
ᆠ� �
ᨨ
 
�鼀�ᆠ泐
ᨨᨨ矄� 鼀 矄
犨�
ᆠ ᆠ
ᨨ盐
�
�

 ᆠ
 

ᨨᆠ
ᆠ� � ᨨ
 
ᨨᨨ
犨�

ᨨ
ur B 㡰 
ᨨᆠ
ᨨ
ᨨ
ᨨ
㦨
mB҈ ur


ᨨᆠ
ᨨ
矄ᨨ盐
ᨨ
㦨
mB
N 和� ᆠ
 
犨ᨨ
ᨨ矄
ᨨ
 
鼀� �ᆠ


犨�

ᆠ
盐ᨨ
泐 S �  
ᨨᆠ矄
�
�   �ᆠ�ᨨ 矄
  ᨨ

 
鼀� �ᆠ

 � ᆠ矄
 ᨨᆠ
矄
犨
矄

ᨨ⼄盐
q ne � 㦨
=
㦨
ve
㦨
҈
҈

맔
҈


맔
҈
҈黨
㦨
҈


맔
҈

꾀
t t 㝐� �ᆠ
鼀ᨨ鼀 
M
㦨
�
맔
A
㦨
�
맔
pr






㦨
҈


맔
҈

맔

҈

맔


�
맔
҈
҈

맔


�
맔
҈
҈






㦨
҈

�
맔
҈


꾀鼀
␔�
ᨨ
 �
L
�矄
 
�
 
ᨨ鼀
ᨨ

ᆠ�ᆠ
\ �鼀
ᨨ
 
ᆠ�ᆠ
㦨
 �
ᨨ
犨� L Þ
pr 㦨
L



Þ
r
㦨

p pL
L
\ 꾀
ᨨ
 
ᆠ�ᆠ
㦨
pr

㦨

=
p
p 㝐� �ᆠ
鼀ᨨ鼀 
ᨨ

ᨨᨨ
�
犨 �ᆠ
ᆠ盐 
� ᨨ犨矄 �L
ᨨ
M
㦨
�A
㦨


p 

ᆠ㡰 ᨨᆠ
ᨨ
AB

鼀� �ᆠ
�
盐
ᨨ
ᆠ盐 
I҈
�矄
ᨨ � AB
\ F
㦨
IdlB
矄�
q
AB 㦨
I d l ´ B ´ 矄�
°㡰 =
C B

I


O

O' I҈ A

D

ᨨᆠ
ᨨ
CD
和�鼀�泐
 
鼀� �ᆠ
�
盐
ᨨ
ᆠ盐 
I҈
�矄 ᨨ �
DC

ᆠ盐矄
q
㦨
҈




  ᨨ
ᨨᆠ
ᨨ
DC
�矄
ᨨ
ᔴ
㦨
IdlB
矄�
q
㦨
Idl
矄�
҈


㦨

ᨨᆠ
ᨨ
BC

ᔴᨨ 矄�

矄鼀
鼀 
dl
B҈

I


C

B

FBC
㦨
I
dlB҈
矄�
黨
Þ dF
㦨
I
dlB҈ 泐
鼀� �Ἐ矄
 
 
盐

 
�ᆠ�ᨨ
ᨨ
 �矄
ᨨᆠ
�矄  �ᆠ盐
ᨨ
 
 
ᨨ
 

�ᆠ
ᨨ盐犨矄


I
A 泐
鼀� �Ἐ矄
 
 
盐

 
�ᆠ�ᨨ
ᨨ
 �矄
ᨨᆠ
�矄  �ᆠ盐
ᨨ
 
 
ᨨ

�ᆠ
� 犨矄

和� ᆠ
  ᆠ盐 
鼀 矄

ᨨᆠ
矄泐鼀鼀�ᆠ
ᨨ
ᆠ盐  I҈

  ᨨ ᨨᆠ
ᨨ
BC
犨�
ᆠ ᆠ
ᨨ盐
 
 ᆠ
ᨨ
ᨨᆠ
ᨨ
AD
 ᆠ

ᨨᆠ
ᨨ
ᨨᨨ
ABCD

F 
㦨




ᨨᆠ
ᨨ
BC
 
AD
犨�
ᆠ

ᨨ⼄盐
ᨨ
ABCD
�
ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ
ᨨ盐
OOἘ
 �矄
矄
矄
泐
 
ᨨ矄

O
҈

ᆠ

㡰 졠 泐
� 盐ᆠ
ᆠ�ᆠ
�
 
鼀� ᨨ矄�ᆠ
�
ᨨ鼀 ᆠᨨ矄
ᨨᨨ矄
ᨨ
�
� 矄
ur ҈䰘
㡰 꾀矄
 
鼀� �ᆠ
�
B
�矄
�ᆠ


 �
ᨨ

� ᨨ
 
x㜠 �矄

�
x

y
 
ᆠᨨ鼀ᨨ  矄
ᨨ
 
鼀� �ᆠ
� Bx
㦨
B
ᆠᨨ矄

�
㦨
B ⏌

  By
㦨
B
矄�

�
㦨
B ⏌
Iᨨ

S

Y

R B

t

A

ᨨ


�

P

X

Q

和ᨨ

�
ᆠᨨ
ᨨ鼀 ur uur æ B ö $æ B ö B = �$ ç + ç M = �A㡰kˆ = I
L
k$   ÷ ÷ è
ø è
ø

 ᨨ
ᨨ⼄盐
ᆠ� �
ᨨ
 
ᨨᨨ
r uur ur æ B $ B �+ t = M ´ B = I
L
k$ ´ ç è




$ ö = I
L B ( ˆ - �ˆ) ÷ø


I

L
B
 
�矄
�ᆠ
ᨨ �
� 
QS
ᨨ鼀
Q
ᨨ
S


㡰 
 
ᨨ⼄盐
�矄
�
ˆ  �ˆ
�ᆠ�ᨨ
�.e.


ᨨ
턼和
 ᆠ
 
ᨨᨨ
犨�
ᨨ
ᨨ盐

 �矄
矄矄� �
 ᨨ盐� 턼
 
和

꾀ᆠᆠᨨ� �
ᨨ
 
 ᨨ鼀
ᨨ
 �ᆠ盐  矄

鼀ᨨ鼀 
ᨨ
� �
ᨨ
 
鼀
ᨨ盐 QS
҈ ҈ æ

ö

IQS
㦨
I z = çè ML ÷ø = ML

  꾀矄ᨨ

ᨨ⼄盐
t
㦨
Ia


t I
L
B ´   I
B = = I
M
ML

a
�矄
ᆠᨨ 矄 

  ᨨ
ᨨ鼀

\ a
㦨


҈ q = w
t + a t


w

㦨



҈
҈ æ I
B ö  Dt 㡰
\ q = a t = çè


M ÷ø

ᨨ

q=

 I
B  Dt 㡰

M

WWW.JEEBOOKS.IN

EBD_8341 PS-242

҈



Physics

�㡰
㝐� �ᆠ
鼀ᨨ鼀 
M
㦨
IA 犨 
I
ᆠ盐 
盐
ᨨ
ᨨ�
鼀ᨨ�ᨨ
ᨨ
ᆠᨨ
  A
㦨

ᨨ
ᨨᨨ
鼀
泐
ᆠᨨ
M
㦨
IA
㦨






w e㜠

R

␔

҈黨














 I ´
=
´

Þ I
= =

䰘� 꾀⏌鼀







b㡰 ␔ 
鼀� �ᆠ
�
�矄
�
ᨨ

�鼀� �ᆠ 矄盐矄 ᆠ

�
ᨨ盐ᆠ矄
鼀� �ᆠ
�
�
ᨨᨨ矄�
�ᆠ�ᨨ

矄ᨨ

鼀� �ᆠ
�
� 矄�
 
ᆠ�泐
ᨨ
矄 
犨�
 ᨨ

和ᨨ

�
� 矄�
 
鼀� �ᆠ
矄盐矄 ᆠ
├ � 矄�
 
ᆠ�泐
�矄
ᨨ
㡰 鼀  
鼀� 矄
P㡰

鼀
ᨨ
鼀�矄
犨�
�  ��泐
 
�
ᆠᨨᆠ��泐

 矄 ᨨ鼀
ᆠᨨ矄
T㡰
 鼀
ᨨ
鼀�矄
犨�
ᨨ犨
 ��泐
 
ᨨ犨
ᆠᨨᆠ��泐
Þ








�


㡰





㡰 꾀ᆠᆠᨨ� �
ᨨ
ᔴ盐�
犨
ᨨ
鼀� �ᆠ
矄盐矄 ᆠ


和

Þ

҈


㦨
�䰘


꾀㜠鼀



� ෌�


l
㦨

�
ᆠ鼀㡰

m
㦨


㦨
pm
맔
B
맔
矄�



㦨
҈



맔



�
맔





c

㦨 䰘








´ �
´҈

㦨





맔
҈





㡰 㝐� �ᆠ
矄盐矄ᆠ���泐
c=

ᔼ 

犨 

ᔼ =

Topic-2 : The Earth Magnetism, Magnetic Materials and their Properties b㡰 ෌�
ᨨ盐鼀
ᨨ
�ᨨ
ᨨ

V = ҈
-
鼀 �
鼀��泐

m r = ҈


盐鼀
ᨨ
盐 矄

盐 �
 �

n
㦨
҈
㝐� �ᆠ
鼀ᨨ鼀 
ᨨ

�ᨨ
ᆠᨨ
矄ᨨ ᨨ�


ᔴ




´ ҈




㦨
c
�


ᨨ⼄盐
盐
ᨨ
 
ᨨᆠ
㦨
F
맔


҈


 ᨨ
⼄盐���盐鼀
F
맔


҈

㦨
҈



맔



�
맔

Þ F
㦨

�
黨

 ᔼ
 
ᨨᆠ

�矄
鼀ᨨ

 
 
ᨨ⼄盐
盐
ᨨ
鼀� �ᆠ
� 犨�
鼀ᨨ
 

鼀� 

ᔼ
犨�
矄
ᨨ矄ᆠ�� �
ᨨ盐
 
鼀

r r ᨨ矄��ᨨ
犨 
 
 �
犨
pm
 
B
�矄
ᨨ


c

҈

c µ
Þ
҈
㦨


c
ᔴ҈ ᔴ

ᨨ⼄盐
盐
ᨨ
鼀� �ᆠ
�

҈


҈
´

� ´ ҈
- =
黨黨
� » �


҈
-
㡰 ᨨ
鼀� �ᆠ
鼀�

꾀ᆠᆠᨨ� �
ᨨ
ᆠ盐�矄
犨 ҈ cµ T ᨨ
犨ᨨ
鼀盐矄
T҈
 
T
c҈T҈ = c
T
I 盐
c = B I I \ ҈ T҈ =
T
B҈ B


쟼





P
㦨
m
맔

l
㦨
҈




V N Þ M = mr - ҈㡰 ´ �V l l

Þ M = 黨黨黨 ´

nh nh Þ Rw

㦨 Q
V
㦨
Rw㡰
pm
p e nh nhe eh = = \
M
㦨
´ Q n
㦨
҈
ᨨ
�ᨨ盐 
矄㡰

pm
pm
pm �ᆠ�ᨨ
ᨨ
鼀� �ᆠ
鼀ᨨ鼀 盐鼀
m
 �ᆠ盐
ᨨ
   
ᨨ
ᨨ�




B nˆ ��㡰 ␔
├ ᨨ犨
 
ᨨ⼄盐 t ᨨ 
r r r


Þ

t
㦨
MB
矄�
q t = M ´B 犨 
q
�矄
 
 �
犨
M
 
B q
㦨

he heB e ´ B 矄� 
° = \ t
㦨
MB
矄�
q
㦨

pm
pm �ᆠ�ᨨ
ᨨ
ᨨ⼄盐
ᆠ

ᨨ盐 
泐
�� 
 
 盐鼀
盐



�ᆠ�ᨨ
ᨨ
ᨨ⼄盐
�矄
 �ᆠ盐
ᨨ
ᨨ
㝐
 



Þ mRw

㦨




Þ M = mr - ҈㡰 ´ N�

eö e ew æ ´ pR

Þ
M
㦨
´ pR
çQ҈ = ÷ è sø
p T

҈
\ M
㦨
ewR
盐
ᆠᆠᨨ� �
ᨨ
ᨨ Ἐ矄
 ᨨ泐
















mvx
㦨
nh⏌
p



M = m r - ҈㡰 ´ N�A

㝐� �ᆠ 鼀ᨨ鼀 

´҈


㦨



⏌鼀
= 
ᨨ盐鼀 ҈




҈ = ´҈
 = 
 ´҈



´҈
   ö æ b㡰 ᔴᨨᆠ��泐

 = m
 
 = m
� çè = ÷ø l


ᨨ犨

c =





ᨨ


㦨


 ҈

�
㦨

맔
�


㦨





꾀⏌鼀 l




WWW.JEEBOOKS.IN

PS-243

Magnetism and Matter

黨


҈

҈҈
҈



҈
҈



҈�


b㡰 ෌�
盐鼀
ᨨ
盐 矄



㦨
҈



盐 矄⏌ᆠ鼀
㦨
҈



맔
҈


盐 矄⏌鼀 ᔴᨨᆠ��泐
ᨨ
ᨨ鼀� 


㦨
҈


꾀⏌鼀 ᔴ盐 
ᨨ
鼀� �矄
 
ᨨ鼀� 

ᔼ
㦨
䁸 矄� �

㦨
ᔼ ᨨ
100 = 10�
맔
ᔼ ҈

\ ᔼ = � = ҈ 鼀꾀 ҈
b㡰 ෌
᲼꾀桌
�矄
ᨨ
鼀�
盐矄
ᨨ
鼀├� �
鼀   鼀� 矄
 ��
ᆠᨨᆠ��泐㡰 ෌
᲼桌
�矄
ᨨ
鼀├� �
ᆠᨨ鼀� 矄
 
 矄 ᨨ鼀矄
㡰 
㦨
l
㦨




맔
�
맔
҈
�
ᆠᨨ矄q㡰
맔
�
㦨



䰘
鼀 泐
�� 
 
盐

 
ᆠ �
鼀ᨨ矄
ᨨ
 
 
ᨨ
�ᨨ
ᆠ㡰 㝐� �ᆠ
�
�
矄ᨨ ᨨ�
B
㦨
m
n� B = n�
␔ 
n
㦨
盐鼀
ᨨ
盐 矄

盐 �
 � 㡰 Þ m
B N� ҈

� = Þ
 ´ ҈
 = Þ m
L ҈
´ ҈
-
Þ �
㦨
꾀 b㡰 �鼀� �ᆠ
鼀�矄


�

   鼀� �ᆠ
�

B
矄 矄
�鼀� �ᆠ
鼀�矄
㡰 ᔼ
鼀� �ᆠ
�ᨨ ҈ ᨨᆠ
µ 
ᔼ
 
��
⼄盐矄�ᨨ
ᨨᆠ
µ



 ᆠ


㦨

b㡰 ෌�


M҈
㦨
҈



Am
 M

㦨
҈



Am

錐

r =



cm =

҈鼀


 
㦨
҈
졔


졔


Bnet =

҈



 ҈ 
졠

和 

-䰘

҈
҈

+ ҈㡰 

҈㡰

TµI \





ᨨ




T҈





҈ ⏌
鼀
㝐 ᔴ泐


㦨
ᆠ ᆠ㡰 ෌�


㝐� �ᆠ
鼀ᨨ鼀 

M
㦨


䰘
맔
҈


Am
㝐� �ᆠ
�


㦨



҈
T 㝐ᨨ鼀 
ᨨ
� �

I
㦨
䰘
�
맔
҈


⧤�鼀


矄� �

T =
p

I MB

䰘
� ´ ҈
-



p ´ ҈


s

䰘 ´ ҈
´


҈ ҈


�鼀
├
ᨨ
҈

ᆠᨨ鼀
ᨨ矄ᆠ��ᨨ 矄 t
㦨
҈
T
㦨

p
맔
҈


㦨



�


»


�
矄 =
p

-


=

Topic-4 : Miscellaneous Problems ҈






T


B

æ
ö ´ çè ÷ø = B҈ 
B҈ 黨 \ B = = 0.46


㡰 矄� �

�鼀
⏌ᨨ矄ᆠ��ᨨ
�ᨨ


ᆠ
㦨

p




X

B
ᆠᨨ矄
�° B
´
æ
ö = çè ÷ø = ҈
� B҈ ᆠᨨ矄 
°
´ B҈ ´ 

鼀



㦨

p (
㝐)

+ 

´ ҈
-� =

�
´ ҈
-
犨⏌鼀


꾀 �矄
ᨨ
ᨨ�ᨨ

Bx҈

ᔼ 㝐 ␔ 

㝐
㦨
鼀� �ᆠ
鼀ᨨ鼀 

ᔼ
鼀ᨨ鼀 
ᨨ
� �
 

㦨 鼀� �ᆠ
�

和

MR
 = MR




Bx



㦨

p

㡰 ෌�
㝐
㦨


맔
҈




꾀鼀

㦨

㦨




맔
҈

鼀 m

㝐 쟼 矄
鼀� �ᆠ
�


㦨


p 
p ´ ҈
-䰘
´
´ ҈


= ´ @


෌盐矄矄
p 


´ ҈

㡰

I
= MR
+

T



=






㡰 
I҈
 
I


 
鼀ᨨ鼀 
ᨨ
� �
�
�矄
 
矄ᆠᨨ  ᆠ矄
矄ᆠ�泐
I҈ =
MR


I҈ = I





Topic-3 : Magnetic Equipment ҈


T҈


I mgd


MR

=   MR

I ␔


T
㦨

p MB x

T҈ = T


ᨨ 盐矄㡰 \

和

r

r

m  M҈ + M
㡰 =
+ BH
p r

�鼀
�ᨨ

T =
p

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B㡰

 
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TC
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ᔼ
B

>
B҈

 
ᨨ



 
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
犨 �ᆠ
  矄�矄 ᆠ
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
矄矄
b㡰 ෌�


㦨



TC
㦨
҈


⧤  


㦨
䰘
�T

TC

㦨
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⧤ ␔�
� ᆠ矄
�

TC
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꾀

㦨
�T

TC

矄 ᨨ盐

犨
䰘�
⧤
 
҈


⧤


WWW.JEEBOOKS.IN

EBD_8341 PS-244

Physics

Electromagnetic Induction

20

Topic-1 : Magnetic Flux, Faraday's and Lenz's Law ҈






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㡰


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I
t҈

t㡰 \ I
㦨
ᔼ


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t
f
=
B.A f
㦨
m
nI㡰
맔
p
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ᔼ and A = pR
㡰 f VR = dt VR = m
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 I

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t 㡰

A B w æ ҈ö A
B
w
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wt e
>
㦨
< >
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çè ÷ø R
R R

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