Advanced Problems In Physics For JEE Main & Advanced 9789384934033

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BalaJj ■

Advanced Problems in

by:

Er. Anurag Mishra B.Tech (Meeh. Engg.)

HBTI Kanpur

•

1

SHRI BALAJI PUBLICATIONS (EDUCATIONAL PUBLISHERS & DISTRIBUTORS)

AN ISO 9001-2008 CERTIFIED ORGANIZATION Muzaffarnagar (U.P.) - 251001

I

I

I

B Published by:

SHRI BALAJI PUBLICATIONS (EDUCATIONAL PUBLISHERS & DISTRIBUTORS)

6, Gulshan Vihar, Opp. Mahalaxmi Enclave, Jansath Road, Muzaffarnagar (U.P.) Phone : 0131-2660440 (O), 2600503 (R) website: www.shribalajibooks.com email: [email protected]

a

I i

Edition

: 2018

B ©All rights reserved with author

B Price:

B

? 600.00

All the rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the author and publisher. Any violation/breach shall be taken into legal action.

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PREFACE

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It is a matter of great pleasure for me to present the edition of “Advanced Problems in i Physics” for JEE aspirants. This book brings out the experience gained during many years of

I

teaching to the JEE aspirants. The objective of this book is to provide proper guidance and relevant material, which is really needed for the preparation of JEE. In the book, each chapter

consists of problems to cover the wide subject of physics in a nut shell.

I

The salient features of each chapter are given below.

□ Challenging problems based on twists and wide applications of facts. Single answer type problems, More than one type answer problems. □ Problems based on Comprehension, matching the column, Assertion-Reason and Integer type.

The problems are completely supported by answers. In the last, hints and solution have also been provided wherever necessary, to save precious time of students.

I sincerely wish that this book will fulfill the needs of all the aspirations of the readers. I

Although no stone has been left unturned to make the book free from errors but some errors inadvertently may creep in. Author and Publisher shall be highly obliged if suggestions, regarding

i improvements of the book are pointed out by readers.

In the last, I also pay my sincere thanks to all the esteemed members of ; Shri Balaji Publications in bringing out this book in the present form.

.1I ii:

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Anurag Mishra i i

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CONTENTS 1. UNIT, DIMENSIONS, ERROR AND VECTORS

r

L

O Only one alternative is correct Answers O More than one alternative is/are correct; Answers O Comprehension based problems; Matching type problems; Integer type problems Answers 2. KINEMATICS O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems Assertion and Reason type problems; Integer type problems Answers 3. NEWTON LAW'S OF MOTION ZZZZZZZ.ZZZ. O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems Assertion and Reason type problems Integer type problems Answers

4. CIRCULAR MOTION

i

1-7

'

.ZZZZZZ

O Only one alternative is correct Answers • O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems; Assertion and Reason type problems; Integer type problems Answers 5. Work, Power and Energy J O Only one alternative is correct Answers O More than one alternative is/are correct; Answers O Comprehension based problems Matching type problems Assertion and Reason type problems; Integer type problems Answers

1 4 5 6 7 8-20 8 11 12 13 14 16

18 20 21-36 21 25 26

29 30 31 33 34 36 37-44 | 37 39 40 41

42 43 44 45-53 j

45 47 48 49 50 52 53

6. Momentum and Centre of Mass O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems; Assertion and Reason type problems Integer type problems Answers 7. Rotational Motion O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems; Matching type problems Assertion and Reason type problems; Integer type problems Answers 8. Gravitation

O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems; Matching type problems Assertion and Reason type problems Integer type problems; Answers 9. Solids and Fluids □ Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems Assertion and Reason type problems; Integer type problems Answers 10. Temperature, Heat and Equation of State, Heat Transfer O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems Assertion and Reason type problems; Integer type problems Answers 11. Thermodynamics O Only one alternative is correct Answers O More than one alternative is/are correct

Answers

54-66 54 57 58 60 61 64 65 66 67-81 67 70 71 75 76 78 81 82-89 82 85 85 86 87 88 89 90-100 90 94 95 96 96 97 98 100 101-123 101 110 111 112 113 117 120 123 124-136 124 127 128 129

12.

13.

L_14.

L 15.

O Comprehension based problems Matching type problems Assertion and Reason type problems; Integer type problems Answers Sound, Wave and String . O Only one alternative is correct Answers O More than one alternative is/are correct; Answers O Comprehension based problems Matching type problems Integer type problems Answers Simple Harmonic Motion O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems; Matching type problems Assertion and Reason type problems; Integer type problems Answers Electrostatics O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems Assertion and Reason type problems Integer type problems Answers Electric Current ' ■

O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems Assertion and Reason type problems Integer type problems Answers [ 16. Capacitors . .

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O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems Assertion and Reason type problems; Integer type problems Answers

130 133 135 136 137-144 137 139 140 141 142 143 144 145-154 145 148 149 150 151 152 154 155-185 155 165 166 171 172 176 180 182 185 186-207 186 194 195 198 199 200 204 205 207 208-229 208 215 216 220 221 226 228 229

Ill

17. The Magnetic field O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems Assertion and Reason type problems; Integer type problems Answers 18. Electromagnetic Induction and A.C. Circuit

L.

r 19.

O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems Integer type problems Answers Geometrical Optics

O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems Assertion and Reason type problems Integer type problems Answers r [ 20. Wave Optics

L

O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems Assertion and Reason type problems; Integer type problems Answers 21. Modern Physics O Only one alternative is correct Answers O More than one alternative is/are correct Answers O Comprehension based problems Matching type problems Assertion and Reason type problems; Integer type problems Answers Hints and Solutions (Chapterwise)

230-255“ 23C 242 2438 24£ 24F 251 253 255 256-282 256 269 270 272 273 279 280 282 ^ 283-306 283 293 294 296 297 300 303 304 306 307-319 307 310 311 312 313 315 316 319 320-332

320 323 324 325 326 328 329 332 335-568

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1

V" '-""OWm

1

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—-.-J O50

UNIT, DIMENSION, ERROR & VECTORS I Onli/ one Alternative is correct | 1. A particle experiences a force F = Ar 2r, where r is the unit vector along position vector r. The dimensional formula of A is : (a) [MLT-2] (b) [ML’3T’2] (c) [ML“2T-1] (d) [ML-1T"2]

2. A calorie is a unit of heat energy and its value is 4.18 J where 1 J = 1 kg-m2/s2. Suppose we use a new system of units in which unit of mass equals a kg, the unit of length equals p m and the unit of the time is y sec. Then the value of a calorie in the new system of units is : y2 (b) 4.18^ (a) 4.18^-

ap2

y2

(c) 4.18— a

Y R2

(d) 4.18-P— ay

3. Which of the following functions of A and B is possibly correct if A and B possess different dimensions?

(a) —— e(A/B)

(b)------ ---log(A/B)

(c) -

(d) (d)-

~eA+B

B

4. The distance travelled by an object is given by bt2 , . . , , x = at +------- where t is time and a, b, c are

(c + a)

constants. The dimensions of b and c respectively are : (a) [LT“2],[LT’1] (b) [L2T-3],[LT-‘]

(c) [LT-1 ], [L2T-1]

.

■

3 K* .. iiiiiilik' -

°"Rm

M

(d) [LT’1],[LT'2]

5. Which of the following is not one of ±e seven fundamental SI units? (a) henry (b) ampere (c) candela (d) mole

6. In a metre bridge experiment, we try to obtain the null point at the middle. This : (a) reduces systematic error as well as random error. (b) reduces systematic error but not the random error. (c) reduces random error but not the systematic error (d) reduces neither systematic error nor the random error. 7. An approximate value of number of seconds in an year is nx 107. Determine the % error in this value. (a) 0.5 % (b) 8 % (c) 4 % . (d) 15 % 8. On the basis of detail given about two measuring instruments, select the correct statement. (i) Vernier callipers having main scale division = 0.05 2 45 cm and vernier scale division = —— cm. 50 (ii) Screw gauge having pitch 0.5 mm and its circular scale division measures 0.01 mm. (a) Both the instruments have same least count. (b) Least count of screw gauge is lesser than that of vernier callipers. (c) Both the instruments have same least count but screw gauge is more precise. (d) Both the instruments have different least count and screw gauge is more precise.

2

Advanced Problems in Physics

9. We can compensate backlash error by: (a) Measuring the objects in ascending order of their sizes. (b) Measuring the objects in descending order of their sizes. (c) Either (a) or (b) (d) It is not possible as it is random error.

10. A projectile is thrown with velocity U = 20m/s ± 5% at an angle 60°. If the projectile falls back on the ground at the same level then which of following can not be a possible answer for range? (a) 39.0 m (b) 37.5 m (c) 34.6 m (d) 32.0 m

11. When the gap is closed without placing any object in the screw gauge whose least count is 0.005 mm, the 5th division on its circular scale coincides with the reference line on main scale, and when a small sphere is placed reading on main scale advances by 4 divisions, whereas circular scale reading advances by five times to the corresponding reading when no object was placed. There are 200 divisions on the circular scale. The radius of the sphere is: (a) 4.10 mm (b) 4.05 mm (c) 2.10 mm (d) 2.05 mm

I

r20

c=

Object 0

-KM"50

ex

Now if we measure the same with help of vernier callipers [lMSD = lmm, 10 divisions of vernier coinciding with 9 divisions of main scale] having a negative zero error of 0.5 mm, then find which of the following figures correctly represents the reading: 2 (cm)

1

0 111111111

(a)

11 ii 11

5I

0

o

1

1111111

(b)

10

I in'

o

Main scale

Vernier scale

2 (cm) Main m111111 scale

5 I 10

Vernier scale

12. If a = 2i + 3j + 6k and b = 3i + 4j, then 0

projection of a on b —>

(a) 7/5

(c) 4/9

Aw

(c)

—>

projection of b on a

0

(b) 5/7 (d) None of these

13. A student makes 10 one-second measurements of the disintegration of a sample of a long-lived radioactive isotope and obtained the following values 3, 0, 2,1, 2, 4, 0, 1, 2, 5. How long should the student want to establish the rate to an uncertainity of 1 per cent? (Given standard deviation of count rate of a radioactive decay = ^/mean decay rate) (a) 80 sec (b) 160 sec (c) 2000 sec (d) 5000 sec

14. When a piece of wire is held diametrically in a screw gauge [pitch = 1 mm, number of division on circular scale = 100]. The readings obtained are as shown:

5

11 111111112 (cm) 10

0 1 1111111111111111

(d)

|iiii|lni|

0 5f 10

Main scale

Vernier scale

2 (cm)

J

Mam scale

Vernier scale

15. A student obtained following observations in an experiment of metre bridge to find unknown resistance of given wire: S.No. R I 100-I S = 100 -1 R I

1.

2Q

43

57

2.65

2.

3Q

52

48

2.77

3.

4Q

58

42

2.89

4.

5Q

69

31

2.25

Unit, Dimension, Error & Vectors

The most accurate value of unknown resistance will be: (a) 2.65Q (b) 2.77Q (c) 2.89Q (d) 2.25Q

16. In which of the following instruments used in the lab there exists an error of random category known as back lash error? (ii) Spherometer (i) Screw gauge (iii) Searle’s apparatus (iv) Vernier callipers (a) (i) and (ii) only (b) (i), (ii) and (iii) only (c) (i) only (d) all four 17. In Searle’s apparatus, when experimental wire is loaded and unloaded, the air bubble in spirit level gets shifted: (a) towards reference wire while loading and towards experimental wire while unloading (b) towards experimental wire while loading and towards reference wire while unloading (c) towards experimental wire, both the times, during loading and unloading (d) towards reference wire, both the times during loading and unloading 18. A student performing experiment using optical bench observes that when concave mirror is at zero and tip of object needle is at 10 cm from the pole of the mirror, the fixture of object needle mark reads x cm, and it’s image appears to cover the whole aperture of the mirror. Now, when the object needle is moved to y cm, it’s inverted image coincides with it and parallax is removed, then the bench correction (index correction) can be expressed as (Given bench error is positive for object needle): (i) x-10 (ii)y-20 (iii) x + 10 (iv)y+ 20

Choose the correct option: (a) (i) and (ii) (b) (ii) and (iii) (c) (i) and (iv) (d) (iii) and (iv)

19. A student is experimenting with resonance tube apparatus in Physics lab to find the speed of sound at room temperature. He got resonating lengths of air column as 17 cm and 51cm, using tuning fork of frequency 512 Hz. Find speed of sound at room temperature and specify, whether the side water reservoir was moved upward or downward to obtain the second resonance (51 cm)? (a) 348 m/s, downwards (b) 348 m/s, upwards (c) 332 m/s, downwards (d) 332 m/s, upwards

3

20. Accuracy and precision are and these are (i) respectively linked with (ii) and (iii) Fill the blanks above in correct order: (a) (i) same, (ii) systematic error, (iii) random error (b) (i) different, (ii) systematic error, (iii) random error (c) (i) same, (ii) random error, (iii) systematic error (d) (i) different, (ii) random error, (iii) systematic error 21. A plane travelling north at 200 m/s turns and then travels south at 200 m/s. Its change in velocity is: (a) Zero (b) 200 m/s north (c) 400 m/s south (d) 400 m/s north 22. You have ring balanced at the center of the table by three forces

+y

F1,F2 and F3. The forces Fj and F2 have components. = 10N

Fly =-50N

= -40 N

F2y =100N

The force F3 is required to make the ring stationary at the center of the table. Which one of the 4 vectors in

the diagram could represent F3? (a) A

(b) B

(c) C

(d) D

23. If block A shown is moving with speed V as observed by a person on ground, then the magnitude of velocity of A with respect to the block B is : (the wedge is fixed)

(a) V

(c) VV3

(d) Zero

24. Figure shows two vectors a

z

(in y-z plane) and b (in x-y

a7 X370 \37° b\

plane) such that |a|=|b|= 5 units. What is angle between X‘

a and b? (a) 9 = cos

(c) 9 = cos

12 25 9 25

(b) 9 = cos

(d) 9 = 90°

16 25

*y

4

Advanced Problems in Physics

25. A particle is moving westward with a velocity

(a) ' ' (b) (c) • (d)

where

|| = 5 m/s. Its velocity changed to v2 where |V2I = 5 m/s and is towards north. The change in velocity

5 m/s towards north-west zero 5V2 m/s towards north-west 5V2 m/s towards north-east

vector (Av = v*2-) is:

AN9WER9 (d)

2.

(a)

3.

(0

4.

(b)

5.

(a)

6.

(a)

7.

(a)

8.

(a)

9.

(d)

10.

(a)

11.

(d)

12.

(a)

13.

(d)

14.

(a)

15.

(b)

16.

(b)

17.

(a)

18.

(a)

19.

(a)

20.

(b)

21.

(c)

22.

(b)

23.

(0

24.

(b)

25.

(d)

Unit, Dimension, Error & Vectors

5

(a)

More than One Alternative is/are Correct^

a+ b

(b) a-b

75

(c) a 1. A student took the following readings ’in an experiment to determine the linear expansively of brass. Which reading are necessary? (a) Original length of brass rod (b) Final length of brass rod (c) Initial and final temperatures of rod (d) Density of the brass rod

(d) 6

6. Mark the correct statements: (a) Two equal vector have same magnitude same direction always. —> —>

—>

(b) Vector equation (e.g., A+ B =C) is dimensionally homogeneous. (c) Component of a vector is effective value of a vector in a prescribed direction. (d) n non-zero vectors are added so that the resultant vector is equal to one of them, then n has to be an integer > 3.

2. Choose the correct statements. Assume ABCDEF to be a regular hexagon.

7. Mark the incorrect options. A

(a) if d - e = f and f = d + e then d and e are opposite in direction.

B

(a) ED+DB+BE = 0

(b) FE = BC

(c) AD = 2FE

(d) DC = - AF

(b) if d+e = f and f = J2d; d = e then d and e are perpendicular.

(c) if d-e = f and f = d + e then d and e are in the same direction.

3. A boy is pulling a block by a force of 10 -10N N at an angle of 37° to the horizontal. vbr (b)vr =vbr (c)vrt

AN9WER9 (b)

2.

(a)

3.

(b)

4.

(d)

5.

(c)

6.

(b)

7.

(c)

8.

(b)

9.

(b)

10.

(c)

11.

(a)

12.

(d)

13.

(d)

14.

(a)

15.

(a)

16.

(0

17.

(a)

18.

(a)

19.

(c)

20.

(b)

21.

(d)

22.

(b)

23.

(c)

24.

(b)

25.

(a)

26.

(b)

27.

(0

12

Advanced Problems in Physics V

V

More than One Alternative is/are Correct^ (d) for III

(c) for II 1. The position versus time graph of a particle moving in a straight line along x-axis is shown in figure, from the graph it can be said that:

t

t

■>

4. Figure shows position versus time graph of two Rabbits running opposite to each other between two trees. Which of the following statements are true ? x

O h t2 (a) From t = 0 to t =tj particle is moving in positive x direction 0?) From t = 0 to t =t2 particle is continuously moving towards positive x direction (c) At t=t1 particle has changed its direction of motion (d) At t =t2 particle stops.

2. The position of a particle varies according to the expression x=t(t - l)(t -2), then : (a) Velocity will be zero at t2 =1--^= second and t2 = 1 + -7= sec

V3

(b) Acceleration changes its direction between =0 and t2 =2 (c) Acceleration remains constant in direction between tj =0 and t2 =2 (d) None of the above 3. Figure shows the dependence of the coordinates of the body moving in a straight line on time for three cases: I, II, III. Enter the law of motion and construct the graphs of velocity and acceleration versus time in eveiy case. Curve I is a parabola.

(a)

(b) (c) (d)

t Rabbit A has greater magnitude of average velocity. Rabbit B has greater magnitude of average velocity. Both the Rabbits have same displacement. Both the Rabbits have constant speed.

5. Which of the following statements are true about a ground to ground projectile motion ? (a) Average velocity for time of flight is u cos 9. (b) Change in velocity from the time of projection to the time it reaches maximum height has magnitude u sin 0. (c) Average acceleration during entire time of flight is zero. (d) Horizontal component of velocity remains constant. 6. Trajectories of two stones projected from level ground is shown. Let T1? T2 be their time of flights and ult u2 their speeds of projection then :

X.m

120100-

'III I

80-■

60-40--

i II

2°> 0^i

i i i i i l i i I

2 4 6 8 10

t.c

v

‘ a

(b) for I

(a) for I

t

t

(a)T2>T1 (b)u2>u1 (c)T2=Ti (d)u2=u1 7. Two boats A and B having same speed in still water are moving in a river; A moves normal to river current as observed from ground. B moves normal to river current as observed from the river frame. Then : (a) For a ground observer, B is moving faster than A (b) For a ground observer A and B are moving with. same speed (c) For an observer in river frame, B is moving faster than A (d) For an observer in river frame A and B are moving with same speed

13

Kinematics

8. The velocity time graph of a particle moving in a straight line is given in the figure. Then starting from t =0, the particle:

instant of projection cart is moving with velocity vc. Which of the following remains same in both ground and cart reference frame? Assume that particle does not collide with celling and any wall.

v

1 (a) (b) (c) (d)

(a) (b) (c) (d)

continuously speeds up first slows down and then speeds up moves with constant acceleration moves with acceleration of constant magnitude which changes direction at t =t0.

Time of flight Horizontal component of velocity Horizontal range Maximum height attained

9. A particle is projected with a velocity v relative to cart

moving with an acceleration a towards right. At the

ANSWERS 1.

(a,b)

9.

(a, d)

2.

(a,b)

3.

(a,b,d)

4.

(b,d)

5.

(a, b, d)

6.

(b, c)

7.

(a, d)

8.

(b, c)

Advanced Problems in Physics

14

Comprehension Based Problems|

| Pass age: 1 | A car travels south for 8 minutes with a speed of v = 60 kmph, then turns and begins to move in a direction 53° north of west and travels at the same speed for 10 minutes. 1. Find the total displacement of the car (both magnitude and direction). (b) 12 km north-east (a) 8 km south-west (c) 6 km west (d) 10 km east

2. If the average force exerted by wind on the car is 1000 N in north-east direction what is the work done by that force on the car? (b) 3V2 x 106 J (a) 3x106J (c) -3x106J (d) -3-72 x 106 J 3. Determine the distance travelled by the car? (a) 10 km (b) 12 km (c) 15 km (d) 18 km

I Passage:2 I Refer to the diagram below when answering the next two questions. This diagram represents a multiflash photograph of an object O moving along a horizontal surface. The successive positions as indicated in the diagram are separated by equal time intervals of 1 sec. The first flash occurred just as the object started to move and the last just as it came to rest. ---------- ► 000 0 0 0 0 0 00 [llllllllllllllllllllllllllllllllllllllllllllllllll 1

2 3

4

5

6

7

8

9

10 (in m)

1. Which of the following graphs best represents the objects velocity as a function of time? V

V

(a) v*

(c)

4++4++\f->

(b)

f IH11 [ UI »

V*

(d) •/

I

2. The average speed of the object during the motion is:

(a) — m/s 16 (c) 1 m/s

(b) — m/s 15 (d) 0.96 m/s

| Pass age: 3 | A problem that has fascinated philosophers and scientists is how the universe began, if it had a beginning, and how long ago it happened. Much has been learned in the last few­ decades about the conditions of the early universe. Current thinking assumes that the universe began as a big-bang, at which time everything was condensed into a very small space, where there were both extremely high densities and exceptionally high temperatures (actually, it was beginning of space and time as we know them). Since that time the universe has been in continuous expansion, so that the average density and temperature have been decreasing continuously. The rate of expansion of the universe (that is, the rate at which galaxies are receding from each other) is given by Hubble’s Law, discovered in 1929 by the astronomer Edwin Hubble, it states that the rate of separation of any two galaxies in the universe is directiy proportional to their separation. Thus, if we have two galaxies a distance R apart, their receding relative speed is given by v = HR where, H is a proportionality factor called as Hubble’s parameter. The currently accepted value for H is: 22kms-1 MLyr’1 = 2.32x 10’18 s 1 where one MLyr (mega light - year) is 9.46 x 1021 m.

1. If we define a time tH, which corresponds to the time when two galaxies have reached a separation of 2R from R, using Hubble’s Law, then tH is: f(a). /•tH = 1 rw ln(2-> (b) tt H = — ri

(c)tH=£

ri

(d)tH=£ln(2)

2. Some recent observations by Hubble Space Telescope has fascinated astronomers and baffled the scientists that even with the presence of gravitational attraction to retard the separation between the galaxies, it seems that they are accelerating which according to some scientists proves the existence of dark energy which is supplying the increasing kinetic energy to accelerate the expansion. To measure the amount of dark energy supply we need to calculate the rate at which separation velocity is increasing at different levels of separation, find the ratio of relative acceleration at separation of R and 2R respectively: (a) 2 : 1 (b) 1 : 1 (c) 1 : 2 (d) 1 : 4 3. There is a distance called as the horizon of the universe, which is equal to the farthest distance that can be observed from earth, it corresponds to the time taken by light to reach earth almost very close to the age of universe. The oldest and farthest objected observed from earth is near the constellation of ursa major. It is a star like object called a quasar and is

15

Kinematics

estimated to be at a distance of 1.3 x 1026 m, therefore we see it as it was about how much years ago: (a) 1.36x 1010 years (b) 4.3 x 1017 years (d) 9.46 x 1021 years (c) 2.32 x 1018 years

1. What is the total distance travelled by the particle from 0 to 20 sec. ? (a) 0 m (b) 50 m (c) 100 m (d) 200 m 2. Displacement vs time (s-t) graph: s

| Passage:4 j Once upon a time in the Lush Green romantic village of the Punjab, there were two lovers: Soni and Mahiwal. They were in deep love but the society was against them as they belonged to different communities. So they had to meet secretly. Soni and Mahiwal are living on the same side of river bank 3 km apart. The river flows with a velocity of 2.5 km/hr and is 3 km wide. Both of them have a boat each which can travel with the velocity of 5 km/hr in still water. On the first day, both of them decide to meet on the same bank as they live. They start at the same time. Soni travels upstream and Mahiwal travels downstream to meet each other. On the second day they decide to cross the river and meet on the other side of the bank. Mahiwal rows the boat at an angle of 90° to river flow. 2.5 km/sec -

si

(a)

(b)

t

t si

Si

(d)

(C) t

t

3. Which of the following statements about the particle is correct? (a) Slows down for entire 20 sec (b) Slows down for first 10 sec and then speeds up (c) Speeds up for first 10 sec and then slows down (d) Speeds up for entire 20 sec

fPassage:6 j

3 km

Hl

Soni

Mahiwal !♦

3 km

1. What is the time for which they row the boat till they meet on the first day ? (a) 18 min (b) 30 min (c) 22.5 min » (d) 36 min 2. On the second day, what is the angle at which Soni should row the boat (w.r.t. river flow) to reach the same point as Mahiwal on the other bank ? (a) 135° (b) 127°

Katrina and Kareena were crossing a river which is flowing towards right at constant speed 5m/min. travelling in two different boats having same speed in still water. They have taken photographs of the river banks at t=0, t=T,t=2T times. B0,C0 are two trees on other bank of river. Distance shown in diagram are in m. (T=1 min.) Photograph taken by Katrina Bo| t=0 ^Co •

'

Bo| t=T £C0 Bo| t=2T £C0

I

L / 15J3 £1,'

(d) 120°

(c) 143°

Photograph taken by Kareena

I Passage:5~| A particle is moving along a straight line. Ram observes the motion and draw its v-t graph which is given below. v

15J3:

10 m/s

10 sec 0

Bo^ t = 0 ;^C’,0 : o’/1

20 sec t

But he has to do a lot of home work in mathematics so he call his brother Shyam and let him draw the remaining graphs. Shyam was confused and he made certain conclusions now you have to help Ram to choose appropriate conclusions.

/

Bo| t=T £C0 BoI t=2T £C0

\ 10J3 \

V /

5J3

\ eV'

1'

D'

•/

1. Mark correct option. (a) Katrina’s boat has crossed the river in minimum time and Kareena’s boat has crossed the river along the shortest path. (b) Kareena’s boat has crossed the river in minimum time and Katrina’s boat has crossed the river along the shortest path. (c) Kareena’s boat as well as Katrina’s boat has crossed in the minimum time. (d) None of the above

Advanced Problems in Physics

16 2. What is the distance between B and C? (a) 10 m (b) 30/3 m (c) 30 m (d) None of these 3. Mark the correct option: (a) Velocity of boat in still water will be 10 m/min. (b) Velocity of boat in still water will be 10^3 m/min. (c) Velocity of boat in still water will be 10/V3 m/min. (d) Velocity of boat in still water will be 5^3 m/min.

1 Passage:? | A particle starts moving at t = 0 in x-y plane such that its coordinates (mm) with time (in sec) as x = 2t and y=5sin(2t). 1. If magnitude of its acceleration a, then at all the times. (a) a cc x (b) a a yx2 +y 2

(a) A particle has its position as a (p) Angular momentum of the system about function of time given by the origin is r(t) = r0 + Vo t where r0 and v0 conserved. are non-zero constants. (b) A particle has its position vector (q) The particle may be charged and moving as function of time given by in a uniform r(t) = votk + (Oq cos cot )i magnetic field no other force acts on it. +(00 sin cot )j where ao and v0 are non-zero constant, (angle between v0 and a’no is 9O° t4

(d) Speeding up

Column I

Parabola

Parabola

(c)

\

:

‘3 U t5

(a) Slowing down

3. Column-I shows certain situations with certain conditions and column-II shows the parameters in which situations of Column-I match. Which can be possible combination? (a) U1 ~ U2> 61—02

Parabola Straight Straight Line /I ; • \ Line »i

A particle is moving in a plane (r) Projection of particle on x or y axis is with its position vector given by simple harmonic r (t) = (Oq sin cot )i + (b0 cos cot )j motion. where a^, b0(oQ * b0) and co are positive constant.

(d) A particle is moving in a plane (s) Kinetic energy of the particle is constant. with position vector r(t) = (oq cos cot )i + (oq sin cot )j where 0$ and co are positive constant.

xa

(a)

Column-ll

Column-I

(C)

(d) a = 0

(c) a x y

2. Column-I shows position vector as function of time for a particle. Match the characteristic of the motion with their description in column-II.

(s)

(t)

f5

f6

Two projectiles are project- ed from a height such that they strike ground at the same time.

Kinematics (b)

17

Uj > U2J 0| > 02

Ul

(q)

A/

Zm

(b)

a

r—/•

b 4—/•

(q) t2 > t3

(i)

(ii)

(iii) 45° 45°" vo Ri\

*—r2—>

14. A man walking from town A to another town B at the rate of 4 km/hour starts one hour before a coach (also travelling from A to B). The coach is travelling at the rate of 12 km/hr and on the way he is picked up by the coach. On arriving at B, he finds that his coach journey lasted 2 hours. Find the distance (in km) between A and B. 15. An aircraft is flying horizontally at 80 m/s. It accidentally drops a nut from its surface. How much time passes before the nut’s speed becomes 170 m/s? 80 m/s

80

V \l70 16. A ball is projected inside a room of height 7 m with a speed of 20 m/s and at an angle of 37° with the horizontal. If its collision with ceiling results in reversal of direction only of y-component of velocity, What is minimum value of total time of flight (in sec) till it strikes floor ?

17. Figure shows as enemy tank moving with a speed of 9 m/s in direction shown in figure. A gun can fire shells in y-z plane only with a speed 100 m/s at an angle of 53°. Tank is initially at a perpendicular distance of 360 m from the plane of firing. If tank started at t =0, then time interval (in sec) after which shell is to be fired to hit the tank? z

E

o

13. Two balls are thrown from the top of a cliff of unknown height with equal initial speed v0 =10 m/s one is projected at an angle 0=45° above horizontal and second is projected at the same angle below

Enemy tank

20

Advanced Problems in Physics

ANSWERS

^Comprehension Based Problemsj Passage-1

(0

2.

(d)

3.

(d)

Passage-2

(b)

2.

(b)

(b)

2.

(c)

3.

(a)

Passage-4 1.

(a)

2.

(c)

1.

(c)

2.

(b)

3.

(b)

Passage-6

(d)

2.

(c)

1.

(0

2. __

(0

1.

Passage-3 Passage-5 Passage-7

3.

(a)

(d)

| Matching Type Problems 1. [a-> r; b-> p,s,t; c-> q, r; d-> p, s]

2.

3. [a-> p,q,r; b-> q,r,s; c-> q,r,s]

4.

5. [a -> q; b -> p; c -> q]

| Assertion & Reason Type Problems^ 1.

2.

(a)

(a)

3.

(d)

4.

(d)

5.

(a)

| Integer Ti/pe Problems 1. 6. 1116-

2 25 1 2

2. 7. 12. 17.

400 0 750 34

3. 1 8. 16 13. 10

4. 100 9. 5 14. 30

5. 1440 10. 200 15. 15

3 .*-

J

I

NEWTON LAW’S OF MOTION | Onli/ one Alternative is correct j 1. A horizontal force of 10.0 N is acting on a 10 kg box that is sliding to the right along the floor with velocity v (as depicted in the adjacent figure). The coefficient of kinetic friction between the box and the floor is 0.20. The box is having :

(a) 3 m/s2 (b) 9 m/s2 (c) 18 m/s2 (d) 36 m/s2 4. Two blocks are connected by a string and pulled by a force F = 50 N. If the coefficient of friction between the blocks and the ground is 0.2, friction force between the block B and the ground is: B 20 kg

A 30 kg

>F

7777/77777/777777777/77777/7777777777777

10 kg

10.0

(a) acceleration towards left (b) acceleration towards right (c) constant speed and constant velocity (d) constant speed but not constant velocity /////////// 2. A10 kg monkey is climbing a massless rope attached to a 15 kg mass and is passing over a smooth pulley. The mass is lying on the ground. In order to raise the mass from the ground he 10 kg must climb with : (a) uniform acceleration greater than 15 kg 5 m/sec2 — (b) uniform acceleration greater than 2.5 m/sec2 (c) high speed (d) uniform acceleration greater than 10 m/sec2 3. An astraunaut in a strange planet observer that he can jump a maximum horizontal distance of 2m, if his initial speed is 6 m/s. What is the acceleration due to gravity of the planet?

(a) 20 N (b) 30 N (c) 40 N (d) zero 5. A small body is released from v (m/s) rest on a smooth inclined plane A ' at t = 0. Velocity versus time graph of the body is as shown in the figure. The angle of 10 inclination of the plane with 5 the horizontal is: (a) 30° 0 1 (b) 45° (c) tan

2

-----► t(sec)

n

2J

(d) tan"1 (5) 6. A physicist hanged a cylinder-shaped container of base area 100 cm2 to a spring. He slowly poured water into the container and found that the surface of water remained at the same level. Find the spring constant k of the spring. Take density of water as 1000 kg/m3. (a) 50 N/m (b) 100 N/m (c) 1000 N/m (d) 500 N/m

Advanced Problems in Physics

22

7. A block of mass m is pushed up against a spring, compressing it a distance x, and the block is then released. The spring projects the block along a frictionless horizontal surface, giving the block a speed v. The same spring projects a second block of mass 4m, giving it a speed 3t». What distance was the spring compressed in the second case? (a) 6x (b) x/6 (c) 36x (d) None of these 8. A block is projected on a rough horizontal surface with ps =0.5 and = 0.4. Its initial velocity is 8 m/s. The total distance travelled by it till t = 3 sec is: (a) 6m (b) 8m (c) 1.5m (d) 9m 9. You are pushing a wooden crate across the floor at a constant speed. You decide to change the orientation of crate such that area of contact with floor become half. In the new orientation, to push the same crate across the same floor with the same constant speed, the force that you apply must be about as the force required before you changed the crate’s orientation, (a) 4 times as great (b) 2 times as great (c) equally great (d) 1/2 as great 10. The moon does not fall to Earth because: (a) The net force on it is zero (b) It is beyond the pull of Earth’s gravity (c) It is being pulled by the Sun and planets as well as by Earth (d) The statement is wrong: moon is in a state of free fall 11. A block slides down on the track r\ shown below. Comment on its \ speed and acceleration in the direction of motion? (friction is \ absent everywhere) \

Speed Acceleration (a) decreases decreases (b) decreases increases (c) increases decreases (d) increases increases 12. A ball is thrown vertically upward under the influence of gravity. Suppose observer A films this motion and play the tape backwards (so the tape begins with the ball at its highest point and ends with it reaching the point from which it was released), and another observer B observes the motion of the ball from a frame of reference moving at constant velocity which is equal to the initial velocity of the ball. The ball has a downward acceleration according to observer. (a) (A)and(B) (b) only (A) (c) only (B) (d) neither

13. An astronaut floating weightlessly in space shakes a large iron block rapidly back and forth. She compares that to her moving it on earth on a smooth horizontal surface. She find that: (a) the shaking costs her no effort because there is no weight of block. (b) the shaking costs her some effort but considerably less than on Earth. (c) although weightless, the effort required is same (d) the shaking costs her more effort than on earth 14. A 44 kg chandelier is suspended 1.5m below a ceiling by three wires, each of which has the same tension and the same length of 2.0 m (see the drawing). Find the tension in each wire.

2.0 m

(a)

(b)

1760n 9

(c)—N (d) 220 N 3 15. The person in the drawing is standing on crutches.Assume that the force exerted on each crutch by the ground is directed along the crutch. If the coefficient of static friction between a crutch and the ground is 0.90, determine the largest angle 0max that the crutch can have just before it begins to slip on the floor.

(a) tan (0.9) (b) cot-1 (0.9) (c) sin"1 (0.9) (d) cos’1 (0.9) 16. Two persons of equal weight are hanging by their hands from the ends of a light rope hung over a frictionless pulley. They begin to climb the rope. One person can climb with twice the speed of the other (with respect to the rope). Who will get to the top first? (a) The faster climber (b) The slower climber (c) Both will get to the top simultaneously (d) Nothing can be said as it is indeterminate

Newton Law’s of Motion

23

17. A horizontal force F=2N is applied on the system shown in the figure. All surface are smooth. All pulleys and string are ideal. Mass of A and B each is 1kg. What is the acceleration of B with respect to A ?

ZZZ/ZZZZZZZZZZZZZZZZZZZZZ?ZZZ/ii

p

(a) 8p mg

(b) 9p mg (d) 5p mg

(c) 7p mg

F

A

a ////////////////////// (a) 1 m/s2 (b) 4 m/s 2 (c) 2 m/s2 (d) Zero 18. If two ends of an ideal rope are pulled with forces of equal magnitude of 100N in opposite direction, the tension at the centre of the rope must be: (a) 200 N (b) 100 N (c) 50 N (d) 0 N 19. The system shown in figure is released from ///////////////// rest with spring at natural length. The spring starts extending as soon as it is released. (Neglect friction and masses of pulley, string m and spring) (a) if M > m (b) if M > 2m (c) if M > m/2 (d) for any value of M 20. A train of mass M stops after travelling a distance of 10m on applying brakes. If more passengers are boarded in the train so that its mass becomes 2M, at what distance will the train now stop with the same initial speed on applying brakes? (a) 10 m (b) 5 m (c) 2.5 m (d) 7.5 m 21. The system is released from rest 2 second after the release the bigger block is held for an instant and again released. Find ///////////// the time elapsed after that before the string again becomes tight.

(a) - sec 3

1

(c) - sec 3

(b) - sec 3

5

(d) - sec 3

[m] 4m

22. On a table, three blocks (including the first block) are placed as shown in the figure. Mass of each block is m and coefficient of friction for each blocks is p. A force F is applied on the first block so as to move the system. The minimum value of F should be :

!

23. Two spaceships are travelling in the same direction. An astronaut in first ship drops a coin and finds that the coin stays at rest. An astronaut in second ship drops a coin and finds that the coin starts moving in forward direction with an acceleration of 2m/s2. Assume that the coin is uncharged and the whole system is in gravity free space: (a) the first ship can be used as an inertial frame but second cannot be (b) the second ship can be used as an inertial frame but first cannot be (c) neither the first ship nor the second can be used as an inertial frame (d) both the first ship and the second can be used as an inertial frame 24. Consider a 2 kg block B connected to block A through a string passing over a pulley as shown in figure. It is found that when the block A is 1 kg or more, the block B slides on table when the car is stationary. Now car starts moving with an acceleration on horizontal road. It is seen that now the block B moves on the table when block A is 1.2 kg or more. The acceleration of the car is: B

20

□a O

iftunnffn

(a) 3 m/s22 (c) 2 m/s2

-Cl—' _

HimiUHiiiiHiuiUHiiiHHiniHiiHUillHi.UiiiliHfliifUimut

.

(b)lm/s2 (d)4m/s2

25. Figure shows a block kept on a rough inclined plane. The maximum external force down the incline for which the block remains at rest is 2N while the maximum external force up the incline for which the block is at rest is 10N. The coefficient of static friction p is: (b)-^

T

(c) V3

V6 (d)-L V3

26. The coefficient of friction between the block and the horizontal surface is p as shown in figure. The block moves towards right under action of horizontal force

Advanced Problems in Physics

24

Fffigure -a). Sometime later another force P is applied to the block making an angle 9 (such that tan9 = p) with vertical as shown in (figure-b). After application of force P, the acceleration of block shall:

is friction between block and plank, coefficient of friction is p. Block rr^ has initial velocity v01 and plank has initial velocity v02 with (v01 >v02). Which of following graphs is correct ? VQ1

M|

m Fig. (a)

m2 Fig. (b)

v02

a

(a) increase (b) decrease (c) remain same (d) information insufficient for drawing inference 27. A non-zero net force acts on an object. Which of the following quantities could be constant? (b) velocity (a) kinetic energy (c) (a) and (b) both (d) None of these 28. A block of mass 5 kg is pulled with a constant force of 25N with the help of a string. The coefficient of friction between the block and the ground is 0.2. Pulley is massless and string is massless and inextensible. When the string makes an angle of 37° with the horizontal, the acceleration of the block is :

m1 +Mkm29

-Pk9

(b) The tension in the rope is 10,000 N. (c) The Nucleus are pulling with a force of more than 5,000 N if they are winning. (d) The Nucleus are pulling with a force of 5,000 N. 30. A block is sliding up a rough slope as shown. The angle X. /\X between the force of kinetic / friction acting on the block and the force of gravity acting on the block is : (a) less than 90° (b) equal to 90° (c) greater than 90'.0 (d) not enough information to answer 31. Block is projected on a long plank of mass m2Plank is placed on a smooth horizontal surface. There

(2)

v VO1

(b) v02

•t V

vOi

(1)

(C)

(2)

v02

5kg /ZZ/ZZ/X/ZZ/ZZ/ZZ/////^

(a) 2 m/s2 (b)4m/s2 (c) 5 m/s2 (d) 2.6 m/s2 29. Two football teams, the Sterling and the Nucleus, are engaged in a tug-of-war with a massless rope. The Sterling are pulling with a force of 5,000 N. Which of the following is an accurate statement? (a) The tension in the rope depends on whether or not the teams are in equilibrium.'

•t

(a)

F=25 N

'^TTTTTTTTn

(1)

t a (2)

+^m;9

t

(d) ~Mk9

(1)

32. A long piece of rubber is wider than it is thick. When it is stretched in length by some amount: (a) Its thickness decreases but its width increases (b) Its thickness decreases but its width remains constant (c) Its thickness increases but its width decreases (d) both its thickness and width decrease 33. A spring block system is y, fixed in a train. Suddenly ■X -►a0 train starts moving with ■0

//////Zz/Z/ZZ//ZZ//////////////////////,.

velocity along with the blocks C and A. The value of m is: (b) 3 kg (a) 2 kg (d) 5 kg (c) 4 kg

| Passage:3 I The drawing shows box 1 resting on a table, with box 2 resting on top of box 1. A massless rope passes over a massless, frictionless pulley. One end of the rope is connected to box 2 and the other end is connected to box 3. The weights of the three boxes are Wj =55 N, W2 =35 N, and W3=28N.

\

a4 aA 3aa a a

[m]

'z

1. Acceleration of m is: (a) 1.4 ms"2 (b) 5.6 ms"2 (c) 2.8 ms'2 (d) 8.4 ms"2 2. The magnitude of the velocity of M at t = 5 sec is: (a) 7 ms’1 (b) 14 ms"1 (c) 21 ms"1 (d) 5 ms"1 3. Distance covered by M in 5 sec is: (a) 12.5 m (b) 17.5 m (c) 25.5 m (d) 30 m

| Passage:? | Two blocks A and B of masses 3kg and 1kg respectively are connected by light inextensible string passing over a smooth pulley as shown.

B

/\0 = 37°

The coefficient of kinetic friction the blocks and the incline is 0.5. After the system is released. 1. The acceleration of the block B will be: (a) 3 m/s2 (b)4m/s2 (c) 5 m/s2 (d) 6.5 m/s2 2. The magnitude of acceleration of A w.r.t. B will be: (a) 4^3 m/s2 (b) 4^5 m/s2 (c) 5^3 m/s 2 (d) 4>/2 m/s2 3. After some time a block C of mass m is placed on the block B such that the block B moves with constant

JL

3

1. Determine the magnitude of the normal force that the table exerts on box 1. (a) 55 N (b) 62 N (c) 48 N (d) 90 N 2. If the pulley is pulled upward with an acceleration that increases with time, a=t/4 where t is the time in seconds, what is the time when the block 2 is lifted off? (a) 2.5 sec (b) 5 sec (c) 1.25 sec (d) 3.75 sec 3. Instead of pulling the pulley upwards, we start lowering the pulley down with a constant velocity of 4m/s. If the block 3 is initially 2 m above the ground, at what time does it strike the ground? (a) 0.25 sec (b) 0.5 sec (c) 0.75 sec (d) 1 sec

fPassage:4 ] In the figure, a horizontal force of 100N is to be applied to a 10 kg slab that is initially stationary on a frictionless surface. A 10 kg block lies on the top of the slab, there is no information about friction and coefficient of friction between the block and the slab. Block

Slab

H►100N

Z/7/Z7777777777777777Z

1. What can be a possible value of the acceleration of the slab? (a) 7 m/s2 (b) 10 m/s2 (c) 2 m/s2 (d) 9 m/s2

31

Newton Law’s of Motion

2. What can be a possible value of the acceleration of the block? (a) 4 m/s 2 (b) 7 m/s 2 (c) 10 m/s2 (d)2m/s2 3. If the ground and the top surface of the slab both are rough, which of the following can be a possible free body diagram? f is friction between block and slab, N is normal between block and slab, is friction between slab and ground, NI is normal between slab and ground. N

N

(a) I mg block

| mg block

f

(C)

n

100N N

Mg slab

Column I

| Passage:5 | Consider the system shown in figure of a two block system connected with two identical springs of spring constant Zc=100 N/m. Friction coefficient between the blocks is p =0.5 and floor is smooth. The system was such that both the springs were in their natural length (70). Then both the blocks are shifted to right by x0 =1 cm such that one spring is compressed and other is elongated. Now the system is released from rest at t = 0. (4)+ *0) $ *k [—iaA %/ i 'mi (/0-x0) ; / Rough(Wr~4z[fiO^ ' Smooth B m2 k / \

Column II

(b) M2>Mx

(q) M2 accelerates up

Mx=M2

(d) Mi » M2

Mg slab

\

(P) M2 accelerates down

WON

(d)^

\

M2 30°

(a) Mj >M2

(c)

Ni

Ni

1. For the system shown in the figure, the incline is frictionless and the string is massless and inextensible pulley is light and frictionless. As the system is released from rest, the possible situation are: \

(b)

f

| Matching Ti/pe Problems

(r) MltM2 in equilibrium (s)

Tension in string equals the weight of either block

(t)

accelerates up the in­ cline.

2. In the column-II some arrangements with light string and frictionless and light pulley are shown. In string AB. Tension may be written as T = qmg. Some values of q is given in column-I, match the values with arrangements of column-II. All the surface shown are smooth. Column II

Column I

(a)

1 n=2

(P)

m 77777777777777777/

_ B m]

(b)

2' n=3

(q)

/////////// —

77777777777777777777777777777

1. If the mass of each block is m4 = m2 =m. By the time the two springs come in their natural length (0, which of the following statement is correct ? (a) Work done by friction on block A is zero. (b) Work done by friction on block B is zero. (c) Work done by friction on system A + B is zero. (d) Work done by friction on (A + B) is negative. 2. Suppose mass of block A is =0.5 kg. Then what should be the mass of block B (m2) so that the friction force acting between the blocks can reach its limiting value ? (b)|kg (a) i kg o

(c) I kg !

(d) none of these

A[m] | [mJ

B[m]

(c)

4 n=3

(r)

A ///////////

□

m/2 [m] (s)

A

7R □

B

m/2 [m]

32

Advanced Problems in Physics

3. A block placed on a rough inclined . plane. Angle of inclination 0 of the plane as shown is varied starting from zero. The coefficient of static friction and kinetic friction between — the block and the plane is and respectively >pk). Column-II shows the graphs which necessarily contains 0 taken on x-axis. Column-I represents the quantities taken on /-axis of column-I. Match the quantities of column-I with graphs of column-II.

Column II

Column I

(a) Friction force between the (P) block and plane

(c) V A = V B = 4j,

(r)

Direction of acceleration of A is +ve /-direction.

aB = 8i- 6k

(d) V a = V b = 0, aB = 8i + 6k (s) (t)

a/i ~ aB

Friction force on A is neither in x-direction nor /-direction but somewhere in between.

5. There are five mechanical setups as shown in columr along with respective free body diagrams of the block In each free body diagram force represented have magnitude proportional to length of arrows. In each case block is at rest initially. Match the column-I with column-II.

Column II

Column I (b) Normal force between the (q) block and the plane

(a) Fi+ Fa = 0

Fl

(P)

F3

F4

//////////////

F2

(c) Total contact force (r) between the block and the plane

(b) Fi + F2+ F3+ F4 = 0

(d) Acceleration of the block

(s)

Boy pulling a block on a rough surface. | -> F1

(q)

f3

—> •F4

F2 A spring block system placed on a rough surface.

(t) (c) Fi+ F2+ F3+ F4 = zna (r) with magnitude of

acceleration | a| # 0

z 4. A block ‘A’ of mass 10 kg is placed on rough horizontal plank ‘B’ (x-y plane) where gravity act vertically downward. ps=1.2,pfc=l. (All units are in SI). Forces acting on block A is due to plank B and earth only. For x mg the situation in column-I match the appropriate description in column-II.

/ 9a/*

\Column I (a) v\ = 3i, Vb = 3i+4j,

\

\

y

aB = 8i + 6j

(d) F3 is static friction.

(s)

F

'////////////

Column II

(P) Direction of friction on A is +ve /-direction.

aB = 3i + 4j

(b) Vx=VB=3i + 4i

Block placed on a rough incline and pushed downward.

(q) Direction of friction on A is +ve x-direction.

Block being pushed on a rough horizontal surface.

Newton Law’s of Motion

33

(t)

Column-I gives various values of u, v and p while column-II gives the subsequent motion.

F4

F1

5 u

IIf2, Block is being pulled on a rough horizontal surface.

1* 6. A box is being pulled on a rough horizontal floor but the box does not move in any case. Magnitude of F is constant and 0 is always acute angle < 90°. Column I

1= 10m I

r^ ZZZZZZZZZZ2ZZZZZZZZ

(a)

Block will fall off the left end of belt

u = 1 m/s; v = 4m/s; p = 0.2

(q)

Block will fall off the right end of belt

(q) Increases when 0 is decreased.

(c)

u = 5m/s; v - 2m/s;

(r)

Slipping stops before block falls off the belt

(d)

(s)

Block will fall off the belt at time t = 1 sec

(t)

Block will fall off the belt at time t = 25/4 sec

changed.

(c) Weight of box

(r)

Decreases when 0 is increased.

(d) Total contact force

(s)

Increases when 0 is increased.

7. A block is placed on a rough horizontal surface. A constant force F is acting on the block as shown in the figure. F 0 < 6 < 90 m=2/Tkg

^p=1

iiiiniiiillliiniiiiih

Column-I gives the magnitude of force F and column-II gives information about friction acting on the block. Match the entries in column-I to all possible entries in column-II. Column-I

(P)

(b)

(P) Remains constant as 0 is

(b) Normal force

, . 21 , u = lm/s; u = —m/s;

p = 0.1

Column II

(a) Friction force

Column-ll

Column-I

Column-ll

(a) 15 N

(P) Static friction

(b) 20 N

(q) Kinetic friction

(c) 25 N

(r) Zero friction

(d) 30 N

(s) Limiting friction

(t) Magnitude of friction is equal to magnitude of normal 8. A belt is moving with constant speed u (clockwise, as shown in figure) on two rotating fixed wheels. The length of the belt between the wheels is I = 10 m, as shown. At time t = 0, a small block starts moving from the upper right end of the belt with leftward speed v (w.r.t. ground), p is friction coefficient between belt and block. The moving part of the belt between the wheels is always horizontal.

p =0.4 u = 2m/s; u = lm/s;p =0

^ Assertion & Reason Type Problems^ Direction: In the questions that follow two statements are given. Statement-2 is purported to be the explanation for statement 1. Study both the statements carefully and then select your answers, according to the codes given below: (a) Statement-1 is true, Statement-2 is true; Siatement-2 is the 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-1 is false; Statement-2 is true. 1. Statement-1 : In an at wood machine, when the masses are in the motion, for a heavy rough pulley, tension on both side of pulley are different.

S Statement-2 : Acceleration of both masses are different.

Advanced Problems in Physics

34

2. Figure shows a smooth can on a smooth surface (with small wheels) with an at wood machine on it. An external force P acts on cart. Following statements are based on given situation. The system starts from rest.

u

3.

4.

5.

6.

I — a

m2

mi

y

>P

o

Statement-1: The magnitudes of velocities of blocks mj and m2 with respect to pulley are same. Statement-2: If the string connecting the two blocks is to remain taut, relative to the centre of the pulley, velocities of blocks must be equal and opposite. Statement-1 : A man is standing on a bathroom scale. Suddenly he squats (sits down) with acceleration a. The scale reading will increase. Statement-2 : Man pushes down on the scale while squatting. Consider a small block placed with zero velocity on a rough conveyor belt moving with constant velocity. The block eventually comes to rest with respect to belt. Statement-1: Net work done by kinetic friction on belt is zero. Statement-2: The kinetic energy of the belt does not change. Statement-1: We can walk on ground due to frictional force exerted by ground on our feet. Statement-2: Friction force acts opposite to our motion. Statement-1: For a body resting on level ground normal reaction and weight form Newton’s 3rd law action-reaction pair. Statement-2: Action-reaction pair of Newton’s 3rd law are equal and opposite.

2. A driver has a personal rule never to drive faster than his safe speed, which he defines as the speed at which his car can just brake to a halt in 30 m. On a certain dry road, his safe speed is 20 m/s, but suppose that it starts to snow and the coefficient of friction drops to half of its dry value. What now is his safe speed (in m/s)? (Neglect reaction time) round off to nearest ' integer. 3. Two blocks with masses =10 kg and m2=20 kg connected by a light inextensible cord, are kept on a horizontal surface. The cord can withstand a tension force, T=100N before it breaks. The coefficient of friction between each block and the surface is equal to p = 0.1. With what maximum force F (in N) can we pull the first load (mj) parallel to the cord so that the cord does not break? Initially the cord is not taut. k-\

F

rn 100N

4. A crate of mass m=20 kg rests on a boat which has a mass M =100 kg and initially at rest. The coefficient of kinetic friction between crate and boat is p=0.2. The rope is pulled by another ship suddenly such that boat gets an initial velocity v0 =24 m/s. Determine the time (in sec) the crate slides on the boat before coming to rest relative to boat. Neglect frictional resistance of water. » (Take: g =10 m/s2) y v0.

Smooth

g Integer Tqpe Problems [1 to 4 Digit] | 1. One end of a string is connected to top of the fixed vertical ring as shown in the figure. The other end is connected to a bead of mass 3 kg free to slide smoothly on the ring. If the string is taut in the shown configuration (take 9=37°) tension in it (in newton) is equal to.

M

F

5. Figure shows a block placed on a bracket. Bracket is placed on a smooth floor, it is pulled by a force F=6 N horizontally. Block is projected with velocity v0 relative to bracket as shown in figure. Find time in second after which it stops relative to bracket. Horizontal surface of bracket is smooth while vertical surface is rough (Given: m=l kg, M = 5 kg, v0 =5 m/s, p=0.5)

Newton Law’s of Motion

35 m M

\

Rough

/

* vo

7777777777777777777777777777777777

6. A 2.5 kg wooden block initially at rest on a fix horizontal rough table of height lm. The block is initially 2m away from edge. It is pushed with a constant force of 50 N for a distance of lm and then let go. The block falls off the edge and lands 2m from the bottom of the table. Find the coefficient kinetic friction (p) between the block and the table. Fill lOp in OMR sheet. 7. In the figure shown a plank of length I = 64 m, mass m2 = 5 kg rests on a smooth surface. Upper surface of block is rough with coefficient of kinetic friction pfc =0.5 and ps = 0.6. A small block of mass m} = 2 kg is placed over it. A force F of magnitude 30 N is applied on block . What is displacement of plank (in m) till the small block falls over from plank ? >----- Rough ITI2

/—Smooth

p, q, r, s, t; b-> p, s, t; c-> p, t; d-> q] 2. [a -> p ; b -> q ; c -> r, s] 3. [a -> t; b -> r; c -> s; d -> q] 4. [a -> p, r ; b -> s, t; c -> q ; d -> q, s]

5. [a -> p, q ; b -> s , t; c -> p, q , r ; d -> p, q , r, s , t or s , t] 6. [a -> q, r ; b -> q, r ; c -> p ; d -> q, r]

7.

[a -> p; b -> p, s, t; c -> p, q, s, t; d -> q, r, t]

8. [a-> p,s;b-> q,r,t;c-> q,s;d-> p]

| Assertion & Reason Tqpe Problems^ (b)

2.

(b)

3.

(d)

4.

(d)

5.

(c)

6.

(d)

I Integer Tqpe Problems I. 48 6. 5 II. 7

2. 14 7. 16 12. 1

3. 150 8. 5

4. 9.

10 10

5. 10 10. 3000

H Only one Alternative is correct | 1. The Earth’s gravity provides the centripetal force on a orbiting satellite to keep it moving in circle at constant speed. Which statements best explains why the speed of the satellite does not change even through there is a net force exerted on it? (a) The satellite is in equilibrium with respect to ground (b) The acceleration of the satellite is zero with respect to ground (c) The centripetal force has magnitude mv2/r and it is balanced by a centrifugal force as seen from the earth (d) The centripetal force is always perpendicular to the velocity 2. An unbanked circular highway curve on level ground makes a turn of 90°. The highway carries traffic at 108 km-hr-1, and the centripetal force on a vehicle is not to exceed 1/10 of its weight. What is the approximate minimum length of the curve, in km? (a) 1.4 km (b) 1 km (c) 0.6 km (d) None of these 3. In a circular motion of a particle, the tangential acceleration of the particle is given by at = 9 m/s2. The radius of the circle is 4m. The particle was initially at rest. Time after which total acceleration of the particle makes an angle of 45'o with the radial acceleration is: (a) 1/3 sec (b) 2/3 sec (d) 4/3 sec (c) 1 sec

4. Consider the setup of a Ferris wheel in an amusement park. The wheel is turning in a counter clockwise manner. Contrary to the illustration, not all seats are aligned horizontally, i.e., parallel to the x-axis. Determine the orientation of the normal to the seat as it passes the point A. (a) parallel to the x-axis (b) in the first/third quadrants (c) parallel to the y-axis (d) in the second/fourth quadrants 5. Two water slides at a pool are shaped differently but start at the same height. Two riders Sita and Gita start from rest at the same time on different slides. Neglecting friction, and assume same path length for both. Mark the correct statement.

(a) Gita reaches ground earlier than Sita (b) Sita reaches ground earlier than Gita (c) Sita and Gita arrive on horizontal ground level simultaneously. (d) Information is insufficient

38

Advanced Problems in Physics

6. Figure shows path followed by a particle and position of a particle at any instant. Four different students have represented the velocity vectors and acceleration vectors at the give instant. Which vector diagram can not be true in any situation? (In each figure velocity is tangential to the trajectory).

(b) 798 m/s

(a) 7 m/s

(d) — m/s 2 12. A car is travelling on a banked road with a speed of 10 m/s on a banked road of inclination 37° in a circle of radius 30 m. The car does not skid. The correct Free body diagram of car is : (c) 3.5 m/s

v trajectory of particle

P.

v

v V

A particle at a • \ given instant

0>90' sita

7.

8.

9.

10.

11.

e>9Qj,

Ro­ a

Gita

kRam

(b)

(a)

Shyam

a

(b) Gita (a) Sita (c) Ram (d) Shyam Two pendulums A and B having equal lengths and equal bob masses are attached to a common point O and were initially in the position, as shown in the figure. The bob of o A h pendulum A is released from rest and simultaneously that to B was * given a horizontal velocity y]2gh as h shown in the figure, when the two bobs collide, what will be the ratio 2gh of their kinetic energy just before B collision ? (a) 1 : V2 (b) J2 : 1 (c) 1 :1 (d) 1: 2 Airplane of mass ‘ m’ travels in a vertical circle of radius ‘r’ at constant speed ‘v’. At what angle 0, measured from the 0 lowest point in the circle, is the net force on the airplane horizontal? (a) 0° (b) 30° (c) 60° (d) 90° A particle is moving on a circle of radius lm and its speed is changing as v=2t. The magnitude of the acceleration of particle at t =1 sec is : (a) 4 m/s2 (b) 2 m/s2 (c) 2^3 m/s2 (d) 2^5 m/s2 A particle is projected at an angle 53° to the horizontal at speed of 10 m/s. Find its tangential acceleration at t=1.4 sec. (a) 5 m/s2 (b) 5^2 m/s2 2 (c) 10V2 m/s2 (d) 10 m/s2 A mass m is suspended by a (massless) string forming a simple pendulum of 4.9 m length. The pendulum is initially at an angle of 60° with the vertical when the mass is released. What is the maximum speed of the mass?

N

f

(c)

(d)f

(W few 37° mg

13. A pendulum bob is released from rest from horizontal position as 'Ju shown in figure. Which of the following cannot represent direction of instantaneous acceleration during its motion upto lowest position along circular path for the first time. (a) Vertically up (b) Vertically down (c) Horizontal towards left (d) Horizontal towards right 14. A jet travelling at a constant speed of 1.20 x 102 m/s executes a vertical loop with a radius of 5.00xl02 m. (See ■ Fig.) Find the magnitude of « the force of seat on a 70.0 kg pilot at the top of the loop. v '*• (Take g = 10 m/s ) (a) 1316 N (b) 2700 N (c) 700 N (d) 2000 N 15. In figure particle is shown travelling counterclockwise in circle of radius 10 m. The acceleration vector is indicated at a specific time. Find the value of ‘v’ at this time.

Circular Motion

39

conical pendulum with the string making 60° with the vertical. Then : (a) its period of revolution is — sec.

V

!37° r = 10ml ------- * -?*( a= 50 m/s2 a:

(b) the tension in the string is double the weight of the particle (c) the velocity of the particle = 2.8^3 m/s (d) the centripetal acceleration of the particle is 9.8^3 m/s2.

(a) 10 m/s (b) 15 m/s (c) 20 m/s (d) 7 m/s 16. A heavy particle is tied to the end A of a string of length 1.6 m. Its other end 0 is fixed. It revolves as a

ANSWERS 1.

(d)

2.

(a)

3.

(b)

11.

(a)

12.

(b)

13.

(d)

14.

(d)

5.

(b)

6.

(d)

(a)

15.

(c)

16.

(b)

7.

(c)

8.

(d)

9.

(d)

10.

(b)

40

Advanced Problems in Physics oa

| More than One Alternative is/are Correct^ 1. Ram and Shyam see a trolley moving on a straight line. They are both stationary and located as shown.

(c) time

0 is the angle that Jai’s position vector makes with positive x-axis.

Ram

vx

Shyam

—► time

(d)

(a) The angular displacement measured by both in same time interval is same. (b) The displacement measured by both in the same time interval is the same. (c) The angular velocity of the line joining both the observers to the trolley any instant is the same. (d) The velocity of the trolley as observed by both the observers at any instant is the same. 2. Little Jai is sitting on a seat of merry-go-round moving with constant angular velocity. At t = 0, Jai is at position A shown in figure. YA A v att=O

vx is the x component of Jai’s velocity. 3. A point object of mass m is slipping down on a smooth hemispherical body of mass M and radius R. The point object is tied to a wall with an ideal string as shown. At a certain instant, speed of the hemisphere is v and its acceleration is a. Then speed vp and acceleration ap of a particle has value (assume all the surfaces in contact are frictionless): (a) vp =vsin60° (b) vp =v

(c) ap = a A

(d) ap =

v

-

+ a2

x

4. A particle is projected from ground at an angle 0. At a Top view of merry-go-round

Which of the graphs shown in figure are correct? All graphs are sinusoidal.

certain instant the velocity vector v of particle makes an angle a with horizontal.

(a)dTg

FyA

(b) modulus of

(a) time

Fy is the y-component of the force keeping Jai moving in a circle

XA (b)

= g sin a

(c) Tangential acceleration has magnitude of g sin a (d) Normal/radial acceleration = g cos a 5. On a train moving along east with a constant speed v, a boy revolves a bob with string of length I on smooth surface of a train, with equal constant speed v relative to train. Mark the correct option(s).

time

(a) Maximum speed of bob is 2v in ground frame. x is the x component of Jai’s position.

(b) Tension in string connecting bob is —-— at an instant. 2

(c) Tension in string is —~ at all the moments. (d) Minimum speed of bob is zero in ground frame.

41

Circular Motion

(a) According to A centripetal acceleration is provided by friction force (b) According to A friction is kinetic and according to B friction is static. (c) According to B centrifugal force is balanced by friction force. (d) According to A centrifugal force is balanced by friction force.

6. A block is resting on a rotating rough table as shown. co

B

ANSWERS (b, d)

2.

(a, b, c, d) 3.

(b, d)

4.

(a, b, c, d) 5.

(a, c, d)

6.

(a, c)

42

Advanced Problems in Physics

| Comprehension Based Problems^

| Passage:1 | Two indentical particles are attached by an inextensible and massless string which passes over a smooth fixed cylinder of radius R as shown in figure. Initially particles are lie on same horizontal line and they are slightly displaced.

1. The velocity of particles as a function of 0 is given by : (a) ylgR sin 0 (b) 7gH(l-cosO) (c) %/gRCl-sin9)

(d) 7gK(9~sin 0) 2. The angle 0 at which the particle slipping on the cylinder leave the contact with the cylinder satisfy the relation. (a) cos 0=2/3 (b) sin 0=0 (c) sin 0=0/2 (d)cos0 = l/3 3. The tangential acceleration of particle in contact with cylinder is : (a) a=g sin0 (b) a=g sin(G/2) Cc) a=g sin2 0 (d) a=g sin 2 (0/2)

| PASSftGE:2 I A pendulum is formed by attaching a small body of mass, M =0.2 kg to the end of a string 0.5 m long. The other end of the string is attached to a nail so that the pendulum can swing freely in a circle in the vertical plane. The pendulum is set in motion so that its speed at the bottom of the swing is 2 m/s. Nail

0.5 m

0.2 kg

1. The maximum height of the mass M above its lowei position is CLOSEST to : (a) 0.1 m (b) 0.2 m (c) 0.4 m (d) 0.5 m , 2. If the mass M of the body and the length of the strinj are each doubled, but the speed of the mass at th bottom is still 2 m/s, which of the following quantitie would change? (a) The maximum height of the mass above its lowes position. (b) Average torque about nail during one complet oscillation. (c) The work done by the gravitational force on th mass during one complete oscillation. Cd) The tension in the string at the bottom of it swing.

| Passage:3 | A small toy train is moving with a constant speed of 10 m/i 16 in a circle of radius — m. The circle lies in horizontal plane

At time t = 0 train is at point P and moving in anticlockwise direction. At this instant a stone is projected from the trait with velocity (—6i + 8k) m/s relative to train. y

•rigin

P

*x

at t=0

1. What is angular displacement of train when the ston is at highest point of its trajectory? (a)

(b) 2 4 (c) n (d) 2n 2. What is velocity vector of stone relative to train whe the stone is at highest point of its trajectory? (a) -4i + 10j (b) 4i-10j (c) 4i + lOj (d) —4i — lOj

3. What is displacement of the stone till it finally lands o the ground? (a) -9.2i-8j .(b) 9.2i + 8j (c) 9.6i-16j (d) -9.6i + 16j

Circular Motion

43

| Matching Ti/pe Problems

I op

1. A bob of mass 1 kg is hanging by an inextensible of length 1 m as shown. The bob is given a velocity v0 at lowest point as shown. 4 kg wwxww

2kg

Column I

Column II

(a) v0 = 2 m/s

(P) Tension becomes zero at some point in motion.

(b) v0 = 4 m/s (c) v0 = 6 m/s

(q) The path is always circular. (r) The string may be horizontal at some point in its motion.

(d) v0 = 8 m/s

(s) The

bob completes vertical circle.

| Assertion & Reason Type Problems! Direction: In the questions that follow two statements are given. Statement-2 is purported to be the explanation for statement 1. Study both the statements carefully and then select your answers, according to the codes given below: (a) Statement-1 is true, Statement-2 is true; Statement-2 is the 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-1 is false; Statement-2 is true. 1. Statement-1: Angular velocity of body A with respect to another body B is always given by

Statement-1: As point ‘P’ moves from right side to left side of pulley, the magnitude of it’s acceleration remains same throughout. Statement-2: The tension in massless thread remains constant in magnitude.

| Integer Ti/pe Problems [1 to 4 Digit] j 1. An elastic cord having an unstretched length I, force constant k and mass per unit length mQ is stretched around the drum of radius r(2irr > I). Determine the speed of the cord, due to rotation of the drum, which will allow the cord to loosen its contact with the drum. v2 (in m2/s2) (for the given data Express value of 22 m0 = 40g/cm, k = lOON/m,n = —,

I = 40 cm, r=70 cm).

2. What should be the minimum initial velocity of ball u (in m/s) with which it must be thrown such that it just reaches the ceiling of room after impact from the floor? ///////////////////////////

coA -coB =^ab- Where cox and are angular velocities of the two rotating bodies. Statement-2: Velocity of body A with respect to

1.2m

S K

S

another body B is VA - VB.

2. Point ‘P’ is on a massless thread in an ideal pulley arrangement as shown.

K Sb

u

s*

\

s s \b

s. V

,_______5

777777777777777777777777777^ e=0.5

5m

Advanced Problems in Physics

44

3. A circular platform rotates around a vertical axis with angular velocity (0 (o=10rad/sec. On the platform is a ball of mass 1 kg, attached to the long axis of rhe platform by a thin rod of length ] 10 cm (a = 30°), Find normal force t exerted by the ball on the platform (in newton). Friction is absent. 4. Three aircrafts make a turn in the horizontal plane at uniform speed, moving along concentric circular trajectories that are shown in figure. The aircrafts move such that they are at constant distance of 600 m from each other at any time. The aircraft closest to the centre moves in a circle of radius R=600 m. The aircraft 2 is moving at a speed of v2 =720 km/h. Find the acceleration of third aircraft (in m/s2).

2

3

1

5. A rod is arranged at an angle of 30° from th: horizontal. Attached to the rod with two strings is th: mass m, as shown. The rod is rotated, maintaining ic direction in space, so that m travels in a circular pad The strings are of equal length, and make angles of 60 with the rod as shown. Take the length of the strings ai 2.4m. Calculate the minimum value of the tangentid speed (in m/s) of the mass such that the string witl tension T2 does not becomes slack when the mass i directly above the rod. o. t

**

>£60*

•IT1

q; c -> p, r; d -> q, r, s]

Assertion & Reason Type Problems| 1.

2.

(d)

j(d) |

|

|

|

|

|

I Integer Type Problems 1.

1

2.

18

3.

5

4.

50

5.

6

,4r

5

4'

I

WORK, POWER AND ENERGY h2 > h3 (c) hy = h3 < h2

B

C

(b) hi 0. Which of the following is an unstable equilibrium point? (a) 0 (b) 7^/2b (c) -yla/2b (d) Ja/b 8. When force F is applied to the 7 combination of two springs (shown in at the figure), the elongation in upper spring will be (the whole system is I inside a lift which is moving upwards

(c) Pof 0 (d) Can not be predicted, data insufficient 10. A graph of v(x)* potential energy 'F Ar............... T.............. V(x) versus x is shown in figure. A particle of Eo energy Eo is B executing ►X motion in it. D C Which of the following graph denotes the variation of kinetic energy with x? KE*I Ar

Eq B

KE*I

C.

»X

D

C

D

*

B : Eo

(b)

1

A

KE*I

F

c

*-X

D

*

I

I

F

(a)

IL

with an acceleration a). The upper spring is ideal while the lower spring has mass M. (M(g +q)) (a) K] +K2 (F + M(g+ayKKi + K2) (b)

/

I

B

;e0

(C)

±

A

F

►X

*1*2

(C)

(d)

KE*

(F + MQg+q)) +K2 (F + M(g +a))

Ki

Ppto2 (t+t0)2’ Where Po and t0 are constants. The machine starts at t = Oand runs forever. What is maximum work that the machine can perform: (a) Infinite (b) zero

C

T

D

B

:e0

(d)

9. A machine delivers power given by P =

A

±

F

►x

11. A mass is constrained so that it can only move in one dimension along the x-axis. The potential energy o: the mass as a function of x is shown on the diagrair There is a force in the negative x-direction at:

47

Work, Power and Energy

u

K

K

(d)

(C)

■x

X

-10 (b)E (a) B (c) C (d)D 12. When the cart maximally compresses the spring at the bottom of the track, the cart’s: y.

x

h

(b)

(a)

-►x

-10

"1

14. The potential energy of an object is given by UM = 3x2 -2x3, where U is in joules and x is in

metres. (a) x=0 is stable and x=l is unstable (b) x=0 is unstable and x=l is stable (c) x=0 is stable and x=l is stable (d) x=0 is unstable and x=l is unstable 15. In figure, A sphere suspended from the ceiling starts from rest at position (1) and ends its upward swing at position (2). Ignoring friction and air resistance, the correct relation among the following is :

0

(a) velocity and acceleration are zero (b) velocity is non-zero but its acceleration is zero (c) acceleration is non-zero, but its velocity is zero (d) velocity and acceleration are both non-zero 13. A particle initially at rest is F(N) displaced from x=-10 m h+10N to x=+10 m. Under the influence of force F as 10 shown in the figure. Now x(m) -10 the kinetic energy vs position graph of the -10N particle is :

3

X

-10

-10

1.

(d)

2.

(a)

3.

(a)

4.

(a)

5.

11.

(b)

12.

(0

13.

(0

14.

(a)

15.

. . LsinO-Icosd) (a) cosy =------ —-----

... . L cos 0 -1 sin a P • (c) --r r 2. Find the ionization energy Eo of the particle B.

The elevator cable can withstand a maximum tension of 28000N. It is being driven by a motor fixed at the top. 1. How much power must the motor deliver to lift the elevator car and its passengers at a constant speed of 3 m/s? (a)6xlO4W (b)4.8xlO4W (c)1.2x104W (d)3.6xlO4W 2. The elevator is moving upwards. What can not be a possible acceleration of the elevator? (a) 5m/s2T (b) 10 m/s2'!' (c) 14 m/s2 7 (d) 8 m/s2^

3. Consider a time when elevator is moving up and retarding: (a) The power supplied by the motor must be negative (b) The power supplied by the motor must be positive (c) The power supplied by the motor must be zero (d) The power supplied by the motor may be positive

[Passage:2 i The potential energy at a point, relative to the reference point is defined as the negative of work done by the conservative force as the object moves from the reference point to the point considered. The value of potential energy at the reference point itself can be set equal to zero because we are always conservative only with differences of potential energy between two points and the associated change of kinetic energy. A particle A is fixed at origin of a fixed coordinate system.

«£

•»¥

B2 B2 (d) 4a cl 3. If particle B is transferred slowly from point

(c)

Pi(42r0,y[2r0') to point P2 -7=>-t= in the Ay-plane an 2 y2 J

external agent, calculate work required to be done by it in the process.

(a)|g 64 a

(b)li

B2 (0 T~ 4a

(d) None of these

16a

[ Passage:3 j A small block of mass m, can move without friction on the outside of a fixed vertical circular track of radius R. The block is attached to a spring of natural length R/2 and spring constant k. The other end of spring is connected to a point at height R/2 directly above the centre of track. A

R/2 0

/

1. If the block is released from rest when the spring is in horizontal state (see figure) then at that moment: (a) tangential acceleration is g — - — (75 -1) 2 4m p-

(b) radial acceleration is — + 2 4m

50

Advanced Problems in Physics ( x . . . . . g kR\f~3 (c) tangential acceleration is —--------- (V3-1) 2 4m _/"o

l-D

(d) radial acceleration is g----------- (V3 -1) 2 4m 2. Consider block to be at rest at topmost point A of track. If the block is slowly pushed from rest at the highest point A. When the spring reaches in horizontal state, then : 3kK2>| (a) Spring potential energy is (2-V3) k 4 f ZcR kR 2 1 r~ (b) Spring potential energy is ----- (V3 — 1) 2 8 (c) Gravitational potential •otential energy (taking U=Q U=0 at

9=0°) is M I 2 ) (d) Gravitational potential energy (taking U=0 at 3mgR ) 0 = 0°) is

8 J

3. If the complete setup is in a gravity free space, then the minimum speed (v0) required at the highest point A to just reach the lowest point is :

(a) 2rJ£ \m

3R 2 Vm

(0 r J^ Vm (d) Motion not possible in gravity free space

I Passage:4 | Consider two frames of reference S and S', the first one being fixed to the ground and the second one fixed to a moving train moving with 5.00 m/s with respect to the ground (figure). A block of mass 4.00 kg, initially at rest with respect to S', is acted upon by a 14.0 N force for 3.00 s in the positive x direction. Neglect friction. y +/

A it

5m/s

lmr*

If •

(c) The change in kinetic energy 220.5 J. (d) The work done by the force on the block is 220.5 J 3. Mark the correct option(s) : (a) Work energy theorem cannot be applied in frame S\ (b) Work energy theorem is derived from Newton’s second law it is valid in all inertial reference frames. (c) Work done by force is same in both the frames. (d) Change in kinetic energy of both the blocks is independent of reference frames S and S'.

| Passage:5 I A particle of mass 2 kg is moving along x-axis under the

influence of force F=(4-x)t Answer the following questions, assuming positive x-axis towards right.

1. If the particle is displaced slightly towards left from x=-2 and released from rest, choose the correct option. (a) The particle moves with increasing retardation. (b) The particle moves with decreasing speed. (c) The particle moves with constant acceleration. (d) The particle moves with increasing acceleration. 2. Which of the following options is INCORRECT ? (a) The particle is at stable equilibrium position at x=2 (b) The particle is at stable equilibrium position at x=-2 (c) The particle is at unstable equilibrium position at x=-2 (d) Force acting on the particle is conservative. 3. If the particle starts at rest from origin then the maximum x-coordinate of the particle satisfies, which of the following relations ? (a) *max 2a/3 (c) •^max 5

| Matching Type Problems 1. Match the column:

\

Column I

Column II

7////////// x 77777777777777777?

1. According to an observer in S : (a) the initial kinetic energy of the block is 50 J. (b) final kinetic energy 480.5 J. (c) the change in kinetic energy 430.5 J. (d) the work done by the force on the block is 430.5 J. 2. According to an observer in S', what are the corresponding quantities ? (a) The initial kinetic energy of the block is zero (b) Final kinetic energy 220.5 J.

(a) Displacement of particle

(p) Path dependent

(b) Work done by conserva­ tive force

(q) Path independent

(c)

Work done by non-conservative force

(d) Angular displacement

(r) Frame dependent (s)

Frame independent

(t)

Dependent on location of observer in a given frame

Work, Power and Energy

51

2. Column-I represents potential energy graph for certain system. Column-II gives statements related to graphs. Column I

\

\

\

3. Column-I shows constrained motion of a particle in vertical plane. The constraint force (tension and

Column II

normal reaction) are collectively denoted by F. Match the characteristic shown in column-II with that in column-I. Consider the motion by the time the particle returns to starting point or comes to permanently at rest.

U(0?

(P) If total energy is E3, it is not possible for the body to 2.5 mg/ = E3 have any turning point in its 2 mg/= E2 motion. "7! mg/ = E1

(a)

-H

Q

+7t

Column I

>

U(x)

E3 e2 Ei x

*-v0=V4gR

(q) For a small displacement about point 0 potential energy function is quadratic in variable plotted on x-axis.

0 A particle moving along x-axis with potential energy function as

l/(x) = [l-e

2

E3 -a

+a

A particle is projected inside a smooth tube, it fits slightly loosely inside it.

(b) IllllUJtlll (q)

o

].

U(x)

(c)

Constraint force may be (P) directed radialy outward at an instant during subsequent motion.

(a)

U vs. 0 graph for a bob hanging vertically from a string with its lowest position as reference level and 0 is angle of string from vertical line.

(b)

Column II

(r) For a small displacement about position 0 motion is simple harmonic.

instant during motion. ac is centripetal acceleration. v0=V2gR

A bob constrained on a string. (c) iiiiiiitiiu

e2

(r) Tangential acceleration decreases speed of particle for entire motion.

*x Ei

0 Potential energy function of a particle in an arbit­ rary force field.

(d)

U(r)

a Ro b r

...e3 ^-e2 — E1

F+ mg = mac at a particular

o

v0=\'3gR

A bob constrained on a rod. (s) If total energy is E[otal < E2 particle executes periodic and oscillatory motion for all energy values greater than energy atO.

O Graph represents poten­ tial energy for a particle.

(d) (s)

The particle completes circular path.

•v0=v8gR

(t)

Point 0 is position of stable equilibrium.

A small block is projected inside a spherical ball along the inner surface.

(t) By the time the particle returns to starting point or comes to permanently at rest, the impulse of constraint force is zero.

Advanced Problems in Physics

52

Statement-2: Less work has to be done in case (ii) to lift the block from rest to rest by a distance h.

Assertion & Reason Ti/pe Problems | Direction: In the questions that follow two statements are given. Statement-2 is purported to be the explanation for statement 1. Study both the statements carefully and then select your answers, according to the codes given below: (a) Statement-1 is true, Statement-2 is true; Statement-2 is the 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. Cd) Statement-1 is false; Statement-2 is true. 1. Consider a one dimensional head on collision of two balls. Statement-1 : The loss in kinetic energy of the system during the collision does not depend on the velocity of the observer. Statement-2 : Kinetic energy of a body is independent of velocity of observer. 2. Statement-1: A particle moving at constant speed and constant magnitude of radial acceleration not be undergoing uniform circular motion. Statement-2 : In uniform circular motion speed cannot change as their is no tangential acceleration. 3. Statement-1: Workdone by conservative force along closed path is zero. Statement-2: When an object is moved along closed path beginning and ending at same point its displacement is zero. 4. Consider a pendulum bob of length I hanging on a thin rod. Rod is given just sufficient velocity' such that it negotiates vertical circle. Statement-1: Minimum velocity at topmost point of circle is nearly zero. Statement-2: In order to complete circle tension in rod is always non-zero during motion from lowest to topmost point. 5. Statement-1: Force F1 required to just lift block A of mass m in case (i) is more than that in case (ii).

K

Integer Ti/pe Problems [1 to4 Digit]j

1. A body with a mass of 3kg falls from a certain height with an initial velocity of 2 m/s, directed vertically downward. Find the work done (in J) to overcome the forces of resistance during 10 s if it is known that the body acquired a velocity of 50 m/s at the end of the 10s interval. Assume that the force of resistance is constant. 2. Guide angles have been attached to a conveyor belt at equal distances d=200 mm. Four packages, each having a mass of 4 kg, are placed as shown on the belt, which is at rest. If a constant force of magnitude 840 N is applied to the belt, determine the velocity of package 2 as it falls off the belt at point A. Assume that the mass of the belt and pulleys is small compared with the mass of the packages. Assume that the radius of pulley is negligible in comparison with d.

a]

840 N

3. Potential energy (sinusoidal curve) is shown graphically for a particle. The potential energy does not depend on y and z co-ordinates. For range 0 < x < 2 maximum value of conservative force (in magnitude) is (Pk). Find the value of 0. [Here this force is corresponding to above potential energy and all units are in Sil U(Joule)

6

3

0

F2

Fi

h

m A

A

(*)

(■■)

m

GJ El BI E? ■ B

BE 1/2

1

3/2

2

x(metre)

Work, Power and Energy

53

4. A 2 kg block is gently pushed from rest at A and it slides down along the fixed smooth circular surface as shown in figure. If the attached spring has a force constant k = 20N/m. What is unstretched length of spring (in m) so that it does not allow the block to leave the surface until angle with the. vertical is 0 = 60°.

p

^^0 = 37° Muiriiit.i7rinain77iJ7TnTTnTnfTTWT7f77nn/nnn7

6. In what proportion should the power developed by the engine of a steamer be increased in order to double the maximum speed, if the water resistance to the motion grows as the square of the velocity ? 7. A constant force F is applied on block, the block is placed on a plank as shown in figure. Block and Plank are connected with a spring. There is no friction between block and Plank but friction exist between Plank and ground. Find minimum constant force in newton by external agent so that plank mass just begins to move. (Take : M =1 kg, k=l N/m, p =0.2)

R=1.5m 5. A block of mass m is being pulled up the rough incline, inclined at an angle 37° with horizontal by an agent delivering constant power P. The coefficient of friction between the block and the incline is p. Find the maximum speed (in m/s) of the block during the course of ascent. [Take: P = 60W, m = 1 kg, p = 0.5]

----- F _____ Smooth

-WW1- M M

X

■Rough

(b)£

(c)I

Student-B: Equation holds for case II only. (a) Student-A is incorrect, Student-B is correct. (b) Student-A is correct, Student-B is incorrect. (c) Both are correct (d) Both are incorrect 3. A bomb of mass 3m is kept inside a — closed box of mass 3m and length 4L at its centre. It explodes in two parts of mass m and 2m. The two parts < •4Lmove in opposite direction and stick to the opposite side of the walls of box. Box is kept oi a smooth horizontal surface. What is the distance moved by the box during this tinu interval?

(d) zero

2. The following set of figures show two cases of collision

(a) 0

between two balls. Let Uj and u2 be the velocities

before collision and collision.

”i> v2

be the velocities after

The coefficient of restitution, c ^1^2-^11 |^1-u2|

Student-A: Equation holds for both cases as e is property of material of the colliding bodies.

(b) 4o (d) None of these

4. A mass ‘m’ moves with a velocity ‘v’ and collide inelastically with another identical mass at rest. Afte collision the 1st mass moves with velocity -^= in ; V3 direction perpendicular to the initial direction o motion. Find the speed of the 2nd mass after collision. ( 2v (a) -=

V3

(b) -^

V3

(c) v (d) the situation of the problem is not possibh without external impulse

Momentum and Centre of Mass

55

5. Eight solid uniform cubes of edge I are stacked together to form a single cube with centre O. One cube is removed from this system. Distance of the centre of mass of remaining 7 cubes from O is: 7>[3l (b) — (a) 16 16 , ' ^3/ (d) zero c — 14 6. A and B initially at rest on frictionless surface, move towards each other under a mutual force of attraction. At the instant when the speed of A is v and the speed of B is 2v, the speed of the centre of mass of the system is: (a) 0 (b) v (c) 1.5 v (d) 3v 7. A ball A suffers an oblique elastic collision with a ball B that is at rest initially. If their masses are the same, then after the collision : (a) They will move in opposite directions. (b) A continuous to move in the original direction while B remains at rest. (c) They will move in mutually perpendicular directions. (d) A comes to rest and B moves in the direction of ±e original motion of A. 8. Sachin (55 kg) and Kapil (65 kg) are sitting at the two ends of a boat at rest in still water. The boat weighs 100 kg and is 3.0 m long. Sachin walks down to Kapil and shakes hand. The boat gets displaced by : (a) zero m (b) 0.75 m (c) 3.0 m (d) 2.3 m 9. A ball A moving with certain velocity in positive x-axis collides with a stationary ball B. After collision their directions of motion make angles a and p with the x-axis. The possible values of a and p are :

(a) a = 30° and p=-45° (b) a=90° and p=-120° (c) a = 0° and p=-30° (d) a=45° and P=0° 10. A metal sheet 14 cm x 2 cm of uniform thickness is cut into two pieces of width 1 cm each. The two pieces are joined and laid a long xy plane as shown. The centre of mass has the coordinates. yf

14 cm

0

x 14 cm

(a) (1,D

(b)

19 19 7 ’ 7

8 8 (d) None of these 7’7 11. An initially stationary box on a frictionless A floor explodes into two pieces; piece A with mass mA and piece 8 with mass mB. —t These pieces then move across the floor B along x-axis. Graph of position versus time for the two pieces is given in figure: (a) the graph is not possible (b) mA > mB (c) mA < mB (d) mA = mB 12. A 4.0 kg particle-like object is located at x=0, y=2.0m; a 3.0 kg particle-like object is located at x=3.0m, y=1.0m. At what (a) x and (b) y-coordinates must a 2.0 kg particle-like object be placed for the center of mass of the three-particle system to be located at the origin ? (a) (-5.5m,-4.5m) (b) (-3.5m,-4.5m) (c)(-4.5m,-5.5m) (d) (-4.5m-3.5m) 13. A 10.0 kg object, with an initial velocity of 8.0 m/s to the east, collides with a stationary 4.0 kg cookie tin. Just after the collision, the object has a velocity of 4.0 m/s at an angle of 37° north of east. Just then, what is the velocity of the cookie tin ? (a) 12i-6j (b) 12i-7j (c) 8i-13j (d) 5i-9j

(c)

14. A ball collides elastically with a V 2v massive wall moving towards it with a velocity v. The collision h • occurs at a height of h above ground level and the velocity of the ball just before collision is 2v in horizontal direction as shown in figure. Then the distance from the foot of the wall and the point on the ground where the ball lands (at the instant the ball lands) is: 2h < ™


g 2h

g 15. Figure shows a block A of mass 5 kg kept at rest on a horizontal smooth surface. A spring (K=200N/m) which is compressed by 10 cm and tied with the help of a string to maintain the compression is attached to block A as shown in nB-^m/s figure. Block B also of mass 5 kg moving with 2 m/s collides with A, as shown. During the collision the string breaks and after the collision the spring is in its natural state. Assume

56

Advanced Problems in Physics

the bodies to be elastic and let the velocities of A and B be V] and v2 respectively assuming positive direction towards right, after collision. Then: (a) Vj + v2 > 2

(b) Initial kinetic energy of system = final kinetic energy of system (C) v2 + =4.4{m/s)2 -v2 =2 (d) 16. Consider a one-dimensional collision where a body of mass originally moving in the positive x direction with speed v0 collides with a second body of mass m2 originally at rest. The collision could be completely inelastic, with the two bodies sticking together, completely elastic, or somewhere in between. After the collision,

17.

18. .

19. v

moves with velocity v p while m2 moves

with velocity v2. If mj > m 2, then: (b) v2

at right angles to each other experience same force F for time T simultaneously. Consequently the particle m moves with velocity 4v in its original direction. Find the new magnitude of the velocity v' (in m/s) of the particle 4m. Given v=100m/s.

y

m

m

o o v

4v

4m

xO v Before

after

5. A 10 g bullet is fired into a 9.99 kg wood block that is at rest on a wood table. The block, with the bullet embedded, slides 5.0 cm across the table. What was the speed (in m/s) of the bullet? Take p=0.3& 6. Two blocks of mass 1 kg and 2 kg are travelling in the opposite direction with velocity of 3 m/s and 6 m/s respectively. What is their total kinetic energy (in J) in centre of mass frame of reference? 7. The carts in figure are sliding 300g to the right at 1.0 m/s on a smooth 777777777777777777777777/777777777 level ground. The ---------- ► 1.0 m/s spring between them has a spring constant of 120 N/m and is compressed 40 cm. The carts slide past a flame that bums through the string holding them together. The carts are not attached to the spring. Afterward, what is the speed (in m/s) of 300 g cart after it loses contact with the spring? 8. A square plank of mass y mj =100 kg and edge length L = 20>/2 m is placed on a smooth surface. Two person each of mass m2=m3=50 kg *«x are at comer of a plank as shown in figure. Two person begin walk on the plank along two different paths as shown in figure and reach nearest comers. What magnitude of displacement of plank (in m) in the process ?

66

Advanced Problems in Physics

ANSWERS

Comprehension Based Problems Passage-1 Passage-3 Passage-5

1.

(d)

2.

(d)

3.

(b)

(d)

2.

(a)

3.

(c)

(b)

2.

(b)

3.

(a)

4.

Passage-2 Passage-4 Passage-6

(c)

(d)

2.

(b)

3.

(d)

1.

(a)

2.

(b)

3.

(b)

1.

(b)

2.

(c)

3.

(c)

| Matching Tijpe Problems [a->q; b -> s; c -> p]

2.

| Assertion & Reason Type Problemsj (d)

2.

(d)

3.

4.

(a)

(d)

| Integer Ti/pe Problems 1. 5 6. 27

2. 7.

2 5

3. 8.

20 10

4.

125

5.

600

ROTATIONAL MOTION V >

Onlq one Alternative is correct |

1. A rigid equilateral triangular plate ABC of side 2m, is in motion in the x-y plane. At the instant shown in the figure, the point B has velocity vB = (3i + 8j) m/s and the plate has angular velocity

z2k

that the disc continues to roll with a constant velocity. Initial position of the point C is shown in the figure. The correct plot of variation of couple M with time t will be: (Take clockwise torque as positive)

o

—

777777777777777777

i----------- 2m------------ ►

co = 2k r/s. Find the speed of point A. (a) 5 m/s (b) 4 m/s (c) 3 m/s (d) None of these 2. The wheels on the old-time bicycle shown in figure have radii of 60.0 cm and 10.0 cm. If the larger wheel is rotating at 12.0 rad/s, what is the angular speed of the smaller wheel?

vo

0

mgr

(a)

m|

m9r.......

y

t

-mgr.........

(b) m|

X

t

-mgr

mgr

(c)

mJ

♦t

(d) m|

t

-mgr

4. A uniform rod AB of length 4 m and mass 12 kg is thrown such that just after the projection the centre of mass of the rod moves vertically upwards with a velocity 10 m/s and at the same time it is rotating with 71

an angular velocity - rad/sec about a horizontal axis (a) 12.0 rad/s (b) 60.0 rad/s (c) 72.0 rad/s (d) 2.0 rad/s 3. A disc of mass m and radius R is under pure rolling with a constant velocity v0 on a smooth surface. The centre of mass of the disc C is offset from centre O by a distance r. A time varying couple M is applied such

passing through its mid point. Just after the rod is thrown it is horizontal and as shown in the figure. The acceleration (in m/s2) of the point A when the centre of mass is at the highest point is: (Take: it2 = 10)

68

Advanced Problems in Physics 10 m/s

y

2

L A

B

n/2 rad/sec

(b) -5j (d)5j-10j

(a) 5j (c) -lOj

5. A rod of mass M and length I stands along z-axis. Its lower end is hinged at the centre of a disc of same mass M and radius R. The whole arrangement is rotating freely about z-axis with an angular velocity w0. The rod falls on disc and rotates with disc. Find the angular speed of the arrangement. 3w0R 2 (b) (a) 2l2 + 3R2 I2 + R2 6w0R 6cooR2 (d) (c) 1+6R l2 + 6R2 6. Starting from rest on her swing at initial height h0 above the ground, Saina swings forward. At ho the lowest point of her motion, she grabs her bag that lies on the ground. Saina continues swinging forward to reach maximum height She then swings backward and when reaching the lowest point of motion again, she simple lets go off the bag, which falls freely. Saina’s backward swing then reaches maximum height h2. Neglecting air resistance, how are the three heights related? (a) h0 > h} > h2 (b) h0=h1- h2 (c) h0 > hj = h2 (d) h0=h2> hx 7. An uniform ring of radius R, is fitted with a massless rod AB along its diameter. An ideal horizontal string (whose one end is attached with the rod at a height r) passes over a smooth pulley and other end of the string is attached with a block of mass double the mass of ring as shown. The coefficient of friction between the ring and the surface is p. When the system is released from rest, the ring moves such that rod AB remains vertical. The value of r is:

ZZZZZ?/z7zzZZZZZZZZZZZZZZZZZzXl) '^77777777777777777777,

B

(a) R 1-

3p

2(1+ g), / 3m ' (c) R 22(1 + n),

Fl

2m

(b)R 1(d)R 1-

2(1+ g), 3g ' Cl + m)>

8. A uniform circular ring of radius R is fixed in plane. A particle is placed on the axis of the ring at a distance much greater than R and allowed to fall towards the ring under the influence of the ring’s gravity. The particle achieves a maximum speed v. The ring is replaced with one of the same (linear) mass density but radius 2R, and the experiment is repeated. What is the new maximumspeed of the particle ?

(a)

(b) -}=v 42

2

(c) v (d) 42 v 9. In the figureshown, the mass of the disc as well as that of the trolley is M. The spring is ideal and has stiffness k. The trolley can L-g—— move horizontally on smooth floor and the disc can roll on the trolley surface without slipping. The spring is compressed and the system released so that oscillations begin. The: (a) acceleration of centre of disc = twice of that of trolley (b) acceleration of centre of disc = thrice of that of trolley (c) acceleration of centre of disc = half of that of trolley (d) acceleration of centre of disc = that of trolley 10. Find force F required to keep the system in equilibrium. The dimensions of the system are d = 0.3m and a = 0.2m. Assume the rods to be massless:

I

he

d

A

c

y*

100N

X

F B

(a) 150C0 (b) 150(-k) (c) 150(-i) (d) It cannot be in equilibrium 11. A small solid sphere A rolls without slipping inside a large fixed hemispherical bowl of radius R as shown in figure. If the sphere starts from rest at the top point of the hemisphere, find the normal force exerted by the small sphere on the hemisphere when it is at the bottom B of the hemisphere:

69

Rotational Motion

17 (a) -y mg

2 (b) - mg

5 (c) ^mg

7 (d)^mg

12. A disc is given an angular speed coo Released from rest and released from a certain height 0)“° (as shown in figure). Motion of disc is observed after collision with the rough surface. Velocity of centre of iiiiiiiiiii^ininiiii Rough surface mass of ball and direction of co is shown in figure after the collision. Mark possible path, disc CAN follow after the collision: v jfV

(a)

p,t; b -» p,s; c-> q,t; d -> r]

1.

Assertion & Reason Tqpe Problems^ 1.

(a)

2.

(d)

3.

(a)

4.

(c)

5.

(c)

6.

(c)

£ Integer Tqpe Problems I. 100

2. 108

6. 7

7. 6

II. 32

3. 400 8. 4

4. 5 9. 6

5. 6 10.120

10

/'■

1001?)

140 '60'' Y; ’-X.

6C ♦0 ■W.'

TEMPERATURE, HEAT & EQUATION OF STATE, HEAT TRANSFER H OnLij one Alternative is correct 1. The bar shown in the figure is made of a single piece of material. It is fixed at one end. It consists of two segments of equal length Lo but different 2 cross-sectional area A and 2A. What is the change in length of the entire system under the action of an axial force F? Consider the shape of joint to remain circular (Given: y is young's modulus). L/2 L/2

2A

A

A=area of cross-section

(a) 3H 4Ay

(b)^L

(c)^L

(d) None of these

2Ay

8Ay

2. A solid body of linear expansion coefficient 2x 10"5 (°C)-1 floats in a liquid with 20% of its volume out of the liquid at a temperature 20°C. The volumetric expansion coefficient of liquid is 3 times that of the solid body's volumetric expansion coefficient. If the temperature is increased to 70°C, the new fraction of the body's volume out of the liquid will be approximately: (b) 20.1% (a) 19.7% (d) 20.7% (c) 19.5%

3. A sample of gas is heated by three different methods from same initial state as shown. In each methods heat supplied is the same. In method I piston moves up by some amount. In method II piston moves down and in method III piston does not move. Specific heat of the gas is calculate in each of the methods to beCpCjj and Cuim

t

m

Tn

I

\iuiiiuuumnmiinnuuium\ (I)

Gas

Heater

(ID

Gas wwtw Heater

|7/////////W////»//M

(III)

fixed

Gas

WJWtJV

Heater

(a) C [ > C [[ > C [jj (b) C [] > C | > C m (c) C [jj > C jj > C [ (d) C [ > C in > C [j 4. One mole of an ideal gas undergoes a process whose molar heat capacity is 4R and in which work done by gas for small change in temperature is given by the

relation dW = 2RdT, then the ratio — is: Cy (a) 7/5 (b) 5/3 (c) 3/2 (d) 2 5. A gas undergoes an adiabatic process in which / g8 \ g 3 pressure becomes —-= times and volume become 4 \ 3v 3 J 4 of initial volume. If initial absolute temperature was T, the final temperature is: 2T , , 32T (a) — 9^3 Jyf (c) T32 (d) —— 2

102

Advanced Problems in Physics

6. Strips of steel and brass are fused together as shown in the figure (1) at room temperature. Steel

'Brass Room temperature

0)

(2)

Which of the changes cause this fused object to attain the shape shown in the figure (2)? Physical properties of steel and brass are given in the table.

Physical Property

Steel

Brass

Density (kg/m3)

7.9 xlO3

8.9 xlO3

Coefficient of expansion (/°C)

llxlO’6

19xl0’6

Bulk modulus (Pa)

16xlO10

6.1 xlO10

Tensile strength (N/m2)

SOxlO8

2.0 xlO8

Compressive strength (N/ m2)

SOxlO8

SOxlO8

(a) Increasing the temperature of the fused strip (b) Decreasing the temperature of the fused strip (c) By providing support for the fused strip only on the left-hand-side overnight (d) By placing it in a high pressure (P = 100 atm) chamber at room temperature 7. Two resistances of equal magnitude R and having temperature coefficient a: and a2 respectively are connected in parallel. The temperature coefficient of the parallel combination is, approximately: (b) aiCX2 (a) 2(04 +a2) a! + a2 (*i + a2 -a2 (d) 2 2 8. The figure shows a rectangular brass plate at 0°C in which there is cut a rectangular hole of dimensions indicated. If the temperature of the plate is raised to 150°C.

(d) the changes in x and y depend on the dimension z 9. A copper rod and a steel rod are heated. At 0°C, the copper rod has a length Lc and the steel one has a length Ls. When the rods are being heated or cooled, a difference of 5.00 cm is maintained between their lengths. Determine the values of Lc-aCu = 5x 10-5/°C as =2x 10’5/°C. (a) 3.33 cm (b) 5 cm (c) 8.33 cm (d) 15 cm 10. Solid A, with mass M, is at its melting point TA. It is placed in thermal contact with solid B, with heat capacity CB and initially at temperature TB(TB >TA) The combination is thermally isolated. A has latent heat of fusion L and when it has melted has heat capacity C A. If A completely melts the final temperature of both A and B is : (a) (CATA +CbTb -ML)/(£a +Cb) (b) (CaTa -CbTb + ML)/(Ca + Cb) (c) (CATA-CBTB-ML)/(CA+CB) (d) (CaTa + CbTb + ML)/(Ca -Cb) 11. The phase diagram for water is shown in figure. If the temperature of a certain amount of ice is increased by following the path represented by the dashed line from A to B in the phase diagram, which of the graphs of temperature as a function of heat added is correct Treat water vapour as an ideal gas. Liquid

2?

Solid

3w

A

64. A ball of surface temperature T is in therm; equilibrium with its environment. Which of the cun gives the energy E radiated by the sphere as a functia of time ‘t’?

(a) 1 (b) 2 (c) 3 (d) 4 65. A silver ball, painted black is kept inside a box which maintained at a temperature of 27°C. The ball maintained initially at a constant temperature < 127°C by making the radiation to fall on it through small hole in the box. Later on due to some chemic reaction between silver and paint, the paint uniform evaporates from the surface of ball exposing the silv If same amount of radiation continues to fall on bi then temperature of ball as a function of time is show as : (Assume emissivity of silver is zero, paint to I black body and radiation to be the only mode of he transfer):

109

Temperature, Heat and Equation of State, Heat Transfer 27°C ^27°c)

1

T 127°C

2

4

3

(a) 35°C

27°C t

T 127°C'

(b) 27°C t

30°C

0°C

20°C

-15°C

(a) 3, 4, 2,1 (b) 2, 1, 3, 4 (c) 3,4,1,2 (d) 4, 3, 2,1 68. Figure shows three different arrangements of materials 1, 2 and 3 to form a wall. Thermal conductivities are kY >k2 >k3. The left side of the wall is 20°C higher than the right side. Temperature difference AT across the material 1 has following relation in three cases:

T

1

127°C

2

3

1

3

2

3

1

2

(C)

t T 127’C

(d) 27°C t

66. Three products are being considered as possible thermal insulators. The thicknesses and conductivities of the three products are as follows: Thickness Conductivity (arbitrary units) (arbitrary units) 4 12 Product I 6 Product II 6 2 Product III 4 For a given cross-sectional area, which product would ! make the best thermal insulator? (a) Product I i (b) Product II (c) Product III I (d) They would all give the same insulation ' 67. The diagram shows four slabs of different materials with equal thickness, placed side by side. Heat flows 3 from left to right and the steady-state temperatures of the interfaces are given. Rank the materials according to their thermal conductivities, smallest to largest.

(a)

(b)

(c)

(a) ATa > ATb > ATC (b) ATQ = ATb = ATC (c) ATa = ATb >ATC (d) ATa = ATb T2>T1 72. Due to thermal expansion, with rise in temperature : (a) metallic scale reading becomes lesser than true value (a of the metal is greater than a of the object) (b) pendulum clock becomes slower (c) a floating body sinks a little more (assuming temperature of liquid remains unchanged) (d) the apparent weight of a body in a liquid may decrease (assuming temperature of liquid remains unchanged)

(c) A rocket moves forward by pushing the surrounding air backwards (d) According to Newton’s law of cooling of a body inside an enclosure of constant temperature is proportional to the temperature difference between the body and the enclosure. (Assume the temperature difference between the body and the enclosure to be small) 71. According to Wien’s displacement law: 41

•X

X3

AN9WER9 1.

(a)

2.

(c)

3.

(d)

4.

(c)

5.

(b)

6.

(a)

7.

(d)

8.

(c)

9.

(a)

10.

11.

(b)

12.

(b)

13.

(b)

14.

(c)

15.

(b)

16.

GO

17.

(c)

18.

(c)

19.

(b)

20.

(a) ' (d)~|

21.

(d)

22.

(a)

23.

(a)

24.

(c)

25.

(c)

26.

(b)

27.

(a)

28.

(c)

29.

(b)

30.

(d) •

31.

(b)

32.

(b)

33.

(c)

34.

(b)

35.

(d)

36.

(a)

37.

(b)

38.

(b)

39.

(b)

40.

41.

(a)

42.

(b)

43.

(0

44.

(a)

45.

(c)

46.

(b)

47.

(a)

48.

(a)

49.

(a)

50.

51.

(0

52.

(a)

53.

(b)

54.

(d)

55.

(a)

56.

(c)

57.

(b)

58.

(a)

59.

(c)

60.

to to J to i

61.

(b)

62.

(d)

63.

(b)

64.

(a)

65.

(c)

66.

(b)

67.

(a)

68.

(b)

69.

(b)

70.

(d)

71.

(d)

72.

(b)

Temperature, Heat and Equation of State, Heat Transfer

| More than One Alternative is/are Correct^ 1. Two metal rods X and Y having equal cross-sectional areas are joined end to end to form a composite bar, one end of which is heated. After some time has elapsed, the temperature gradient along each rod is found to be uniform, but greater in X than that in Y. Which of the following can be inferred? (a) Both the rods are well lagged (b) The heat current is the same in both the rods (c) Both the rods are of equal lengths (d) X is better conductor of heat than Y 2. A rectangular narrow U-tube has equal arm lengths and base length, each I equal to 1. The vertical arms are filled with 1/2 I mercury up to 1/2 and then one end is sealed. By heating the enclosed gas all the mercury is expelled. If atmospheric pressure is Po, the density of mercury is p and cross-sectional area is S, then [Neglect thermal expansion of glass and mercury] (a) Work done by the gas against the atmospheric . 51 n pressure is — P0S

(b) by....... the gas against the gravity is - Spgl 2 •"Work ............done -

(c) Work done by the gas against the atmospheric pressure is PQSl (d) Word done by the gas against the gravity is Spgl 2

1

-i

I !

I

»

I

I 2

3. The total energy of a blackbody radiation source is collected for one minute and used to heat water. The temperature of the water increases from 20°C to 20.5°C. If the absolute temperature of the blackbody is doubled and the experiment repeated, which of the following statements would be most nearly CORRECT? (a) The temperature of the water would increases from 20°C to a final temperature of 28°C (b) The temperature of the water would increases from 20°C to a final temperature of 36°C (c) Rate of heat emission by the body will increases 8 ' times (d) Rate of heat emission by the body will increases 16 times

111

4. A sample A of liquid water and a sample B of ice of identical mass are T ' kept in two neighboring chambers in ' an otherwise insulated container. The ‘ ‘’ chambers can exchange heat with I ‘ bach other. The graph of temperatures of the two chambers is plotted with time, c water c Sice "

2

’

(a) Finally the contents in sample A is water. (b) Equilibrium temperature is freezing point of water. (c) Ice melts partly. (d) Finally the contents in sample B is ice only. 5. A metal cylinder of mass 0.5 kg is heated electrically by a 12 W heater in a room at 15°C. The cylinder temperature rises uniformly to 25°C in 5 min and finally becomes constant at 45°C. Assuming that the rate of heat loss is proportional to the excess temperature over the surroundings: (a) the rate of loss of heat of the cylinder to surrounding at 20°C is 2W (b) the rate of loss of heat of the cylinder to surrounding at 45°C is 12W (c) the rate of loss of heat of the cylinder to surrounding at 20°C is 5W (d) the rate of loss of heat of the cylinder to surrounding at 45°C is 30W 6. Consider the shown case of a -e°c freezing lake due to negative environmental temperature (-0°C). Thickness (x) of ice ■; water layer is small in comparison to depth of lake. Rate of increase in x will be greater : (a) if environmental temperature increases (b) for larger thickness of ice layer (c) if environmental temperature decreases (d) for smaller thickness of ice layer 7. The ends of aa long I 0 + C 3t3 + L -C3t3 -L > 0 + C3t3 — L > 0

3. Water equivalent of calorimeter is:

(a) mC, (c)

mC2

(b)

C2

(d) none of these

Ci

| Pass age: 8 | Consider a gas at temperature T occupying a volume V consisting of a mixture of two gases having Na & Nb atoms of masses ma&.mb respectively.

1. Give an expression for the total pressure exerted by the gas : (Na+Nb)kT 1.5(Nfl + Nb)kT taj -----------------(b) V V (Na+Nb)RT (d) none of these (c) V 2. Suppose now that Na =Nb and that the different atoms combine at constant volume to form molecules of mass ma + mb. Once the temperature returns to its original value, what would be the ratio of the pressure after combination to the pressure before ? (b) 1/3 (a) 1 (d) none of these (c) 1/2

[/PASSAGE^ i All bodies, no matter how hot or cold, continuously radiate photons. At a given temperature, the intensities of the electromagnetic waves emitted by an object vary from wavelength to wavelength throughout the visible, infrared, and other regions of the spectrum. Figure illustrates how the intensity per unit wavelength depends on wavelength for a perfect black body emitter. Although this figure can strictly be applied only to a black body, yet this will approximately describe the behaviour of many of the self radiatings systems. For example, Sun has an approximate temperature of 6000K. It is not a black body; it has an emissivity of nearly 0.6. But its peak almost occurs at that predicted by the Wein’s law. Suppose we have a bulb of power 100 W. It emits only about 5W as visible light. Rest is emitted as infrared radiation. Assume that the bulb filament has a surface area of 10 mm2, (he = 120 eV-nm).

116

Advanced Problems in Physics

(c) The unit is not in steady state but is in thermal . equilibrium > ■ (d) The unit neither in steady state nor in thermal equilibrium 2. The thermal conductivity of air is: (a) A W/m K (b)lw/mK

Visible cn w c c a>

c S

6000K

4000I

ro J; (X Q.

0

:-4 + 500 1000 1500 Wavelength(nm)

1

1. What is the approximate temperature of the filament ? (a) 500 K (b) 350 K (c) 2500 K (d) 10000 K 2. Assume that the light emitted by the bulb in the visible region is entirely of wavelength 500 nm. What is the number of photons emitted per second in the visible region ? (a)1.25xl019 (b)5x!019 (d) 4xl019

(c) 2.5xl019

3. If we want of increase the number of photons emitted by the bulb in the visible region without changing the wattage, which method would be most appropriate ? (a) increasing emissivity by a factor of 2 (b) increasing the radius of the filament by a factor of 2 and the length by a factor of 4 (c) decreasing the radius of the filament by a factor of 2 and the length by a factor of 4 (d) doubling the voltage and decrease the length of the filament by a factor of 2

9

(c) — W/m K (d) — W/m K 14 130 3. In the above process which modes of heat transfer other than conduction is involved? (a) Convection and radiation (b) Radiation (c) Convection (d) Neither convection nor radiation.

| Pass age :11| Heat generation may occur in a variety of radial geometries. Consider a long, solid cylinder as shown in the figure, which could represent a current-carrying wire or a fuel element in a nuclear reactor. For steady state conditions, the rate at which heat is generated within the cylinder must equal the rate at which heat is convected from the surface of the cylinder to a moving fluid. (Ts)

y

cold fluid

1 Passage :10i The figure shows a cross-section of a double glass unit of a window on a vertical wall. A graph of the temperatures at different points within the unit is shown next to it. The temperature difference across the unit is 13 K. It has a cross-sectional area of 1.3m2 and the rate of heat flow through it is 65 W. Glass has a thermal conductivity of 1 W/m K. Jemp. (in K)

This condition allows the surface temperature to be maintained at a fixed value of Ts. To determine the temperature distribution in the cylinder, we begin with energy conservation principle. Consider a cylindrical section of radius r. The energy is generated within the volume and is conducted radially outwards. qnr2l = -k2nrl(1

I dr J

T+13

air

air

Where q is the energy generated per unit time per unit dT volume, k is the thermal conductivity and — is the dr temperature gradient at radius r. If q is constant

air

T

glass

glass

0

4

22

26

distance (in mm)

1. Select the correct statement: (a) The unit is in steady state and in thermal equilibrium (b) The unit is in steady state but not in thermal equilibrium

T(r) = -—r2+C

4k

Atr = r0,T(r0)=Ts.

(

2A

Therefore, T(r) = — r02 4k 0 I

ro)

The rate of heat convected to the surrounding fluid (at temperature Tf) by the surface at temperature Ts is

Temperature, Heat and Equation of State, Heat Transfer proportional to the temperature difference (Ts -Tf) and the surface area in contact with the fluid. Thus, rate of heat convection = /i(2nr0Z) (Ts -Tf) where h is a constant called heat convection coefficient. By overall energy balance, q(nr02/) = /i(2nroO (Ts -Tj)

=>TS = 7} s f

2h

1. The dimension of heat convection coefficient is: (a) [ML2T-10-1] (b) [ML°T"30-1] (c) [ML°T"20-1] (d) [ML4T“2e-1] 2. In the given passage, the difference in temperature at the axis and surface of the cylinder is: „_2

(a)

4k

(b)

117

| Matching Type Problems

1. A sample ‘A’ of liquid water and a sample B of ice of equal mass are kept in 2 nearby containers so that they can exchange heat with each other but are thermally insulated from the surroundings. The graphs in column-II show the sketch of temperature T of samples versus time t. Match with appropriate description in column-I. Column-I

Column-li

(a) Equilibrium temperature (p) is above melting point of ice.

T

k

(d(d)) 2k k 3. In the above passage, the ratio of temperature gradient at r = r0/2 and r = r0 is: (a) 1 (b) 1/4 (d) 1/8 (c) 1/2 M

(b) At least some of water (q) freezes.

I Passage:1! 2 A metal block is placed in a room which is at 10°C for long time. Now it is heated by an electric heater of power 500 W till its temperature becomes 50°C. Its initial rate of rise of temperature is 2.5°C/sec. The heater is switched off and now a heater of 100 W is required to maintain the temperature of the block at 50°C. (Assume Newton’s law of cooling to be valid)

1. What is the heat capacity of the block? (a) 50J/°C (b) 100J/°C (c) 150J/°C (d) 200J/°C 2. What is the rate of cooling of block at 50°C if the 100W heater is also switched off ? (a) 5°C/s (b) 0.5°C/s (c) l°C/s (d) O.l°C/s 3. What is the heat radiated per second when the block was at 30°C? (b) 80 W (a) 100 W (d) 30 W (c) 50 W

i

(c) At least some of ice (r) melts.

(d) Equilibrium temperature (s) is below freezing point of water.

(t)

T

118

Advanced Problems in Physics

2. The surface of a house hold radiator has an emissivity of 0.55 and an area of 1.5 m2. Its equilibrium

flr

temperature is 50°C and the surroundings are at 22°C. (o = 5.67 x 10 8W/m2K4) Column I

k0

3R

tm

354

:—*-x

__________ Column II

(a)

If 72 >71

(P)

no effect on submergence

(b)

72 = 71

(q)

fraction of the volume of metal sub­ merged in mercury

(c)

(d)

If 72 q;

4. 6.

[a-»q,p;

7.

| Assertion & Reason Ti/pe Problems (a)

2.

(d)

3.

(d)

4.

(b)

5.

(a)

Integer Ti]pe Problems 1. 6. 11. 16. 21. 26.

375 8 3 357 48 1

2. 7. 12. 17. 22. 27.

500 1 150 580 27 1

3. 8. 13. 18. 23.

4 7 9 4 1

4. 9. 14. 19. 24.

200 20 4 217 5

Onli/ one Alternative is correct 1. A spherical soap bubble (surface tension = S) encloses n moles a monoatomic ideal gas. The gas is heated slowly so that the surface area of the bubble increases 3riR by per unit increment in temperature. The

specific heat for this process is : (a) — (b) — 2 2 , A 7R f9R (c) — (d) — 2 2 2. If all the heat produced were used, how many litres of natural gas at NTP (heat of combustion of the gas = 37.3 x 106MJ / m3) is needed to heat 4.54 kg of water in a 0.45 kg copper cup from 298 to 373K ? (Sp. heat of copper = 389 J / kg K) (a) 95 (b) 31 (c) 39 (d) 45 3. A process 1 -> 2 using diatomic PA gas is shown on the P-V diagram ,2 below. _ 1 :3 P2 =2P1 =106N/ m2, V2 =4Vi =0.4m3. The molar •V heat capacity'of the gas in this process will be : (b) 25R/13 (a) 35R/12 (d) 22R/7 (c) 35R/11 I

4. A cylinder of ideal gas is closed by an 8 Piston I \fjjfffrniiin kg movable piston (area 60 cm2) as shown in Fig. Atmospheric pressure is Gas 100 kPa. When the gas is heated from 30°C, to 100°C the piston rises by 20 cm. The piston is then fixed in its place and the ga. is cooled back to 30°C . Let AQ} be the heat added k the gas in the heating process and | AQ2| the heat lo." during cooling. Then the value of [AQ2 -| AQ2|] w£ be: (a) zero (b) 136 J (c) -136 J (d) -68 J 5. A vessel of volume 20 litres contains a mixture o: hydrogen and helium at temperature of 27°C an: pressure 2.0 atm. The mass of the mixture is 5t Assuming the gases to be ideal, the ratio of the mas. of hydrogen to that of helium in the given mixture wii be : (a) 1 : 2 (b) 2 : 3 (c) 2 : 1 (d) 2 : 5 6. The P-V relation for a monatomic ideal gas undergoing an adiabatic change is : (a) PV53 = Constant (b) PV2 = Constant (c) PV2'3 = Constant (d) PV7/3 = Constant 7. A liquid of density 0.85 g / cm 3 flows through : calorimeter at the rate of 8.0 cm3/s. Heat is added by means of a 250 W electric heating coil and temperature difference of 15°C is established if steady-state conditions between the inflow and th;

Thermodynamics

outflow points of the liquid. The specific heat for the liquid will be : (a) 0.6 kcal/kgK (b) 0.3 kcal/kgK (c) 0.5 kcal/kgK (d) 0.4 kcal/kgK 8. If are the molecular weights of different gases, RX,R2,R3 are the gases constants respectively then which of the following is valid ?

125

initial pressures as shown and same temperatures. Now the pistons are released. Then the final equilibrium length of part /I after long time will be: L A

B

C

2P

P

X

s

P

s s S'

. . M i + Af t +■ M □ +..............

9.

10.

11.

12.

(a) —!----- £------ ------------ = 1 Ki + R2 + R3 +............ (b) = ^2 M3 = Constant R2 R3 (c) M}R} =M2R2 = M3R3 = = Constant (d) none of these The Cp/Cy ratio for a gas mixture consisting of 4 gms helium and 32 gms of oxygen is : (a) 1.45 (b) 1.6 (c) 1.5 (d) 1.66 One mole of an ideal gas is kept enclosed 1 under a light piston (area = 10-2m2) | connected by a compressed spring (spring i constant 100N/m). The volume of gas is ____ 0.83m3 and its temperature is 100K. The gas is heated so that it compresses the spring further by 0.1 m. The work done by the gas in the process is (Take R = 8.3 J/mole and suppose there is no atmosphere) : (a) 3J (b) 6J (c) 9J (d) 1.5J One mole of an ideal gas requires 207 J heat to raise the temperature by 10 K when heated at constant pressure. If the same gas is heated at constant volume to raise the temperature by 5K, then the heat required may be: (a) 62.1 J (b) 124J (c) 12.4J (d) 6.2J A long glass tube of length L, sealed at both ends, contains a small column of mercury (density = p) of length ‘ a’ (a « L) at its middle and air at pressure P on both sides. The tube is fixed horizontally. If the mercury column gets a small displacement, the time period of its oscillations would be (assuming that the air on the sides undergoes isothermal expansion or compression): (a) n[p La/ P]1/2 (b) 2it[p La/P]12 (c) n[2pLa/P]V2 (d) Tt[pla/2P]L'2

13. Two pistons having low thermal conductivity divide an adiabatic container in three equal parts as shown. An ideal gas is present in the three parts A, B & C having

L/3

L/3

(a) L/8 (b) L/4 (c) L/6 (d) L/5 14. One mole of an ideal gas at pressure Po and temperature To volume Vo is expanded isothermally to twice its volume and then compressed at constant pressure to (Vo/2) and the gas is brought to original state by a process in which P^2 •*i

-Ji

10O

20

40

60 Q(J)

80

Select the correct statements. (a) Curve 3 corresponds to isothermal process (b) Curve 1 corresponds to a polyatomic gas (c) Curve 2 corresponds to a monatomic gas (d) Process 1 and 2 are isobaric process. 2. An ideal gas has molar heat capacity at contant pressure Cp - 5R/2. The gas is kept in a cylindrical vessel fitted with a piston which is free to move. Mass of the frictionless piston is 9 kg. Initial volume of the gas is 0.0027m3 and cross-section area of the piston is 0.09m2. The initial temperature of the gas is 300 K. Atmospheric pressure Po =1.05x 10s N/m2. An amount of 2.5 x 104 J of heat energy is supplied to the gas, then :

4. A closed vessel contains a mixture of two diatomic gases A and B. Molar mass of A is 16 times and that o B and mass of gas A, contained in the vessel is 2 time that of B. (a) Average kinetic energy per molecule of gas A i equal to that of gas B (b) Root mean square value of translational velocity o gas B is four times that of A (c) Pressure exerted by gas B is eight times of tha exerted by gas A (d) Number of molecules of gas B in the cylinder i; eight times that of gas A 5. A partition divides a container having insulated wall into two compartments I and II. The same gas fills the two compartments whose initial parameters are giver. The partition is a conducting wall which can movt freely without friction. Which of the following statements is/are correct, with reference to the fina equilibrium position ? P.V.T 2P, 2V,T I II

(a) The pressure in the two compartments are equal (b) Volume of compartment I is 3V/5 (c) Volume of compartment II is 12 V/5 (d) Final pressure in compartment I is 5P/3 6. The molar heat capacity for an ideal gas can be: (a) negative (b) equal to either Cv or Cp (c) lie in the rangeCv

x=L

___

z

Fixed end

® t=V

s«L L

(p) 1

■►Free end

4

.... 2L (n) t = — (q)j-----v 3L (iii)t= — v (a) (i) (b) (ii) (c) (iii) (d) None of these 13. Two strings, A and B, of lengths 4L and L respectively and same mass M each, are tied together to form a knot ‘O’ and stretched under th same tension. A transverse wave pulse is sent alon the composite string from the side A, as shown to th right. Which of the following diagrams correctly show the reflected and transmitted wave pulses near th knot ‘O’ ?

w|----

(a) (c) y = +0.015 sin 8n t + — 20

(d) y = -0.015 sin 8k t + — 20 (b)

Sound, Wave and String

I I

139 15. String I and II have identical lengths and linear mass densities, but string I is under greater tension than string II. The accompanying figure shows four different situations, (a) to (d), in which standing wave patterns exist on the two strings. In which situation it is possible that strings I and II are oscillating at the ? • same resonant frequency? String I String II

(c)

(d)

(a)

14. Two slits separated by a distance of 1 mm are illuminated with red light of wavelength 6.5 x 10"7 m.

(b) k>

The interference fringes are observed on a screen placed Im from the slits. The distance between the third dark fringe and the fifth bright fringe is equal to: (a) 0.65 mm (b) 1.625 mm (c) 3.25 mm (d) 0.975 mm

(c)

(d) |C?

ANSWERS 1.

Cb)

2.

(c)

3.

(a)

4.

(c)

5.

(c)

11.

(a)

12.

(b)

13.

(a)

14.

(b)

15.

(c)

6.

(b)

7.

(d)

8.

(a)

9.

(c)

10.

(d)

140

Advanced Problems in Physics

More than One Al ternatii/e is/are Correct^ 1. A source is moving across a circle given by the equation x2+y2=fi2, with constant speed 330 tc 6x/3 m/s, in anti-clockwise sense. A detector is at rest at point (2R, 0) w.r.t. the centre of the circle. If the frequency emitted by the source is f and the speed of sound, c = 330 m/s .Then: (a) the position of the source when the detector ( V3 R' records the maximum frequency is + —R, —


(b) the co-ordinate of the source when the detector records minimum frequency is (0, R) (c) the minimum frequency recorded by the detector . 6^3 . '

1S--77^ f 7t+ 6v3 (d) the maximum frequency recorded by the detector 6V3 - 7t ... 2. A narrow steel rod of length ‘L’ is rigidly clamped at its mid-point and transverse standing waves of frequency */’ are set up in it. The speed of transverse waves in the rod is ‘c’. Then: (a) The free ends of the rod must be antinodes (b) The fundamental frequency * f' of the rod is c/(L) (c) The second overtone frequency of the rod is 5c/(2L) (d) can be any integral multiple of the fundamental frequency

1.

(a, b, c, d) 2.

3. A pulse is started at a time t = 0 along the +x directi: on a long, taut string. The shape of the pulse att = C given by function /(x) with:

fW

here f and x are in centimeters. The linear ma density of the string is 50 g/m and it is under a tensic of 5N: (a) The shape of the string is drawn at t = 0 then tl area of the pulse enclosed by the string and th x-axis is 2.5 cm2 (b) The shape of the string is drawn at t = 0 then tl area of the pulse enclosed by the string and tl x-axis is 5 cm2 (c) The transverse velocity of the particle at x = 13 a and t = 0.015s will be -250cm/s (d) The transverse velocity of the particle at x = 13a and t = 0.015 s will be 250 cm/s 4. String of length L whose one end is x = 0 vibrate according to the relations given by different equation Choose the correct statements:

(a) y = A sin — sin cot has 1 antinode, 2 nodes It TtX

(b) y = A cos — sin cot has 2 antinodes, 1 node It

2tlx (c) y = A sin---- sin cot has 3 nodes, 2 antinodes II

. (d) y = A cos

answers (a, c)

3.

(a, c)

4.

(a, b,c,d)

— + 1 for - 4 < x < 0 4 -x +1 for 0 < x < 1 0 otherwise

sin cot has 3 antinodes, 2 nodes

141

Sound, Wave and String

| Comprehension Based Problems^ | Passage:1 In an organ pipe (may be closed or open) of length Im standing wave is setup, whose equation for longitudinal 2k displacement is given by = (0.1 mm)cos— (y) cos (400)t 0.8 where y is measured from the top of the tube in meters and t in second.

y

.... ,(a)10 (b) 2 (c) 0 (d) greater than 10 2. If one of the trains remains stationary while the other ' , train recedes from the observer with a speed which is 1% of the velocity of sound in air, the number of beats heard by the observer per second will be (in Hz): ■' " ' (a) 0 (b) greater than 10 (c) 3.5 (d) 9.9 3. If both the trains remain stationary, but the observer runs towards one train parallel to the track with such a speed that he hears exactly 10 beats per second, his speed (given the speed of sound is 350 m/s) is: (a) 6.3 km/h (b) 4.5 km/h (c) 12.6 km/h (d) 3.6 km/h

| Passage:3 1. The upper end and the lower ends of the tube are respectively: (a) open - closed (b) closed - open (c) open - open (d) closed - closed 2. The air column is vibrating in: (a) First overtone (b) Second overtone (c) Third harmonic (d) Fundamental mode 3. Equation of the standing wave in terms of excess pressure is (Bulk modulus of air B = 5x 105 N/m2): (a) Pex =(125nN/m2)sin —(y)cos(400t) 0.8

(b) Pex = (125kN/m2)cos— (y)sin(400t) 0.8

(c) pex = (225tiN/m2)sin —(y)cos(200t) 0.8

(d) Pex = (225kN/m2)cos— (y)sin(200t) 0.8

| Passage:2 | A and B are two identical point sources of acoustic waves, each producing sound of 1 kHz. They are located in two trains, one in each of the engines of the trains which runs on two close parallel tracks. An observer is standing in between the tracks and listening to the sound received from each of the sources.

J

1. Firstly, the two trains are approaching the stationary observer from opposite sides with the same speed which is 1% of the velocity of sound in air. The number of beats that he will hear between the frequencies of A and B as heard by him will be (per second):

A metallic rod of length 1 m has one end free and other end rigidly clamped. Longitudinal stationary waves are set-up in the rod in such a way that there are total six antinodes present along the rod. The amplitude of an antinode is 4xl0“6m. Young’s modulus and density of the rod are 6.4x 1010 N/m2 and 4x 103 kg/m3 respectively. Consider the free end to be at origin and at t = 0 particles at free end are at positive extreme. 1. The equation describing displacements of particles about their mean positions is: . (a) s = 4xl0’6cos| 1. Shin chan and his mother have a tin whistle each. The pipe length of Shin chan's tin whistle is 52 cm long while the pipe length of mother's tin whistle is 50 cm long. They both play at the same time, sounding the whistles at their fundamental resonant frequencies. They note that they are not in tune with each other. The velocity of sound in air is 325 m/s. Assume the whistle is a pipe with one end open find the beat frequency (in Hz) that is heard when both whistles are playing simultaneously. (Round off to nearest integer) 2. Two coherent monochromatic sources P and Q emit light of wavelength 563.3 nm. The distance between P and Q is 5pm. If the detector is moved along a line QR perpendicular to PQ and starting from Q, what is the total number of maxima observed ? (Don't count the maxima at infinity). p; Q1-—......... -R

3. If we run towards a resonance column apparatus vibrating in the fundamental mode with a speed of 5m/s, we hear a frequency of 165 Hz. When we run away from the apparatus with a speed of 5m/s, we hear a frequency of 160 Hz. What is the length of the resonance column (in cm) that we see ? Neglect end correction.

144

Advanced Problems in Physics

measured by the observer in Hz. [Take speed of soul in air as 340 m/s]

4. If the figure given below shows the displacement-time curve of common medium particle for two sound waves A and B propagating in the same medium, the ratio of their intensities/A//B = X. Find the value ofX.

A«-

LPU

j,Displacement (mm)

2x10"2

A

10~2 -10'2

B«-

1 100

2 10
q,t; b -> q,r; c-> s,t;

3. 4.

[a -> q, t or q, s, t; b -> p, s or p; c -> p or p, r; d —> s, t]

->q]

[a -> p; b -> r,s; c-> q,t; d

[a->p,q; b-> r, s; c-> t; d-> p, s, t]

| Integer Type Problems 1. 6 6. 375

2. 8

3. 50,

4. 1

5. 5

| OnLij one Alternqtii/e is correct | 1. A physical pendulum consists of two stick each Im long and having same mass. Sticks are joined together as shown in figure. What is the pendulum’s period of oscillation about a pin inserted through point A ? (a) -^Lsec (b) V2nsec y[2

:t^7

H

A

(c) —sec (d) sec 4V2 V3 2. A particle is executing a simple harmonic motion of period 2 s. When it is at its extreme displacement from its mean position, it receives an additional energy . equal to what it had in its mean position. Due to this in its subsequent motion: (a) its amplitude will change and become equal to V2 times its previous amplitude (b) its periodic time will become doubled (i.e.,) 4 s (c) its potential energy will be decreased (d) it will continue to execute simple harmonic motion of the same amplitude and period as before receiving the additional energy 3. An insect of negligible mass is sitting on a block of mass M, tied with a spring of force constant k. The block performs simple harmonic motion with amplitude A infront of a plane mirror placed as shown in figure. The maximum speed of insect relative to its image will be:

(c) A^

™AJ?

4. A particle executes SHM with amplitude of oscillations A and time period T. Find the magnitude of average acceleration for the period of time in which it moves A from mean position by a distance —. (a)^ Y* l

(b)

12M Y’ 2

127tA(2-j3) (d) None of these T2 5. A particle moves according to the law x=acosrt. The distance covered by it in 2.5 second is : (a) 3a (b) 5a (c) 2a (d) 9a 6. A body A of mass m moving with velocity v while passing through its mean position collides elastically with a body B of same mass which is connected to a vertical wall through a spring whose spring constant is k. After collision it sticks to B and executes S.H.M. Find the amplitude of resulting motion : (c)

146

Advanced Problems in Physics ■

(b)

(a)

N 2k m rm r rjy m r y/v (c) — 7v (d) — y/v k 2k 7. At t = 0 the displacement of the block in a linear oscillator as shown is -0.08 m. At the same moment t = 0, its velocity is -1.6 m/s and acceleration is 32m/s2. Choose the incorrect statement: ,, xv

K .

i ..

—Xm

(a) 2

(b) 2nJ—

(c) 2.

(d) 2n

1

11. A particle of mass m is subjected to a force

-----' 0 0 OOOOWOWO’ .

10. A block of mass m is placed on an ideal spring and is in equilibrium. The spring is , . compressed by x0 in equilibrium. If the spring , . is further compressed by 2x0 and released find the time period of oscillations:

..

m

■

■ 1 1—

X=0

+X,m T/X

(a) Angular frequency of the motion is 20 rad/s * ‘(b) Amplitude of the motion is 11.3 cm (c) Phase constant of the motion if the equation of 7t motion is expressed as x = A sin(cot + is 4 (d) Phase constant of the motion if the equation of 5n motion is expressed as x = A sin (cot + ) is — 4 8. Two particles P and Q are executing S.H.M. across same straight line whose Q O p equation are given as yp =A sin (cot + ^ ) and y q =A sin (cot + i) is equal to : (a) h/12 (b) 7n/12 (c) 5n/12 (d) none of the above 9. A block of mass m kg moving with kinetic energy 3 J on a smooth m horizontal surface hits the free end of a 77777777777T7TrnrrrnTrrrrn7t spring, whose other g"° end is attached to a rigid wall as shown in the figure. If restoring force applied by the spring is given by F(x) =-2x-x3(N) then the maximum compression in the spring is approximately : (b) 1.51 m (a) 2.20 m (d) 1.67 m (c) 1.41 m X.

F=F0[cos(t)i +sin(t)j]. If initially (r =0) the particle was at rest, the kinetic energy of the particle as a < function of time is given by : p2

p2

(a) —[l-cos(2t)] m

(b) — (1-cost) m

(c) -2-sin(r)

W—t m m 12. A heavy container containing an M ideal gas is kept on a horizontal surface. A smooth piston of L k mass M is at rest as shown in the figure. The natural length of the spring is L. Now the piston is 777777777t7777777tTnT given a small downward push. Assuming the temperature of the gas to be constant, and there is vacuum over the piston, the time period for small oscillations is :

(a) 2nJI (c) 2k

ML Mg + kL

(d) none of these

13. The acceleration of a particle moving rectilinearly varies with displacement as a = -4x. At x = 4m and t = 0, particle is at rest. Select the incorrect alternative: (a) The maximum speed of the particle is 8 m/s (b) The distance travelled by the particle in first second is 20 m (c) The velocity-acceleration graph of the particle is an ellipse (d) The kinetic energy-displacement graph of the particle is a parabola 14. A particle is executing a simple harmonic motion When it is at its extreme displacement from its mean position, it receives an additional kinetic energy equal to what it had in its mean position. Due to this in its subsequent motion, the amplitude will change to fc times its previous amplitude. The value of k is :

Simple Harmonic Motion

147

(a) 72 (b) 2 (c) 4 (d) none of these 15. A simple pendulum is making oscillations with its bob immersed in a liquid of density n times less than the density of the bob. What is its period ? I (a) 2rc i— (b) 2n \nS k nJ I (d) 2n (n-l)g

[1--^

16. A block of mass m is oscillating on smooth surface between two light springs of spring constant k separated by a distance I colliding elastically with the springs. If the velocity of the block is increased by an external impulse when it is not touching either of the spring then time period. M—(—M

m nnnununnnxn

(a) Increases (b) Decreases (c) Remains same (d) Time period is independent of I 17. Two mass and m2 joined by a massless inextensible rope and two massless springs of force constant k: and k2 lie in a vertical plane. The pulley is smooth. When the system is released with the springs at their natural lengths, the maximum elongation of the spring (kj) is: (a) 2mim^ (b) kjfmj+m2) /qfmj + m2) (c) —1-— k^m-L + m2)

length is L and the acceleration due to gravity is g. What is distance between masses as function of time? m

k •m

(a) Lo + (L-£0)cos

(b) Lo cos (c) Lo sin

[2k Nm

[2k Nm

[2k

Cd) L0+(L-I.0)sin.& Vm 20. Two simple harmonic motions are represented by equations : =4sin (lOt + ) y2 =5coslOt What is the phase difference between their velocities ? (a) (b) .

(d)U-5

(0

21. A block of mass m is hanging from a massless spring of spring constant k. It is in equilibrium under the influence of gravitational force. Another particle of same mass m moving upwards with velocity v0 hits the block and sticks to it. For the subsequent motion choose the correct statement(s):

/it/////////

‘2

(d) none of these

18. From what minimum height h must the system be released when spring a is unstretched so that after perfectly inelastic collision (e = 0) with ground, B may be lifted off the ground? (Spring constant =k) 2m B (a) mg /(4k) (b) 4mg / k h (c) mg/(2k) (d) none of the above 19. A mass m is hung on an ideal massless spring. Another equal mass is connected to the other end of the spring. The whole system is at rest. At t = 0, m is released and the system falls freely under gravity. Assume that natural length of the spring is £0, its initial stretched

H

1

WMNMMMTOlWWAMaM

k

0 (a) Velocity of combined mass must be zero at natural length of the spring. (b) Velocity of combined mass must be maximum at the new equilibrium position (c) Momentum of combined mass must be maximum just after particle hits the block (d) Velocity of combined mass may be maximum at a point lying between original equilibrium position and natural length 22. A particle of mass m moving along the x-axis as a potential energy U(x) - a + bx2 where a and b are positive constants. It will execute simple harmonic motion with a frequency determined by the value of: (a) b alone (b) b and a alone (c) b and m alone (d) b, a and m alone

148

Advanced Problems in Physics

23. A block of mass m is pushed Lp/2 against a spring whose spring —''tHKOWJW 1— m constant is k fixed at one end with liiiiuiiiiiiiiliiiiiiiiiiii a wall. The block can slide on a frictionless table as shown in figure. If the natural length of spring is Lo and it is compressed to half its length when the block is released, find the velocity of the block, when the spring has natural length. [nT Lq (b) (a) Vm 2 2

(c)

Kl

figure. At t =0, and m2 are given velocities ant .v2. Just after t=0, what is the value of x so that the point P remains at rest ? . .

mi

V2

... 1

(d) - sec 2 25. In the question no. 24, the velocity of the rear 2 kg block after it separates from the spring will be : (a) 0 m/s (b) 5 m/s (c) 10 m/s (d) 7.5 m/s 26. Two blocks of masses m} and m2 are kept on a smooth horizontal surface. A spring of mass m and natural length L connects the two blocks as shown in the

P

777777

-*v2

77///////////////// '--------- L-----------

Lv1 vx +v2 Lm1 (c) m1 +m2

(a)

Vm 24. A 2 kg block moving with ■10 m/s 10 m/s strikes a spring of 2kg 2kg constant n2 N/m attached to 2 kg block at rest kept on a smooth floor. The time for. which rear moving block remain in contact with spring will be : 1 sec (a) 72 sec (b). —

. |

x

V1^-J

(d)

(c) 1 sec

1^

(b)

Cd)

Lv2

Vi + v2

Lm2 ml +m2

27. Two particles move parallel to x-axis about the origin with same amplitude ‘a’ and frequency co. At a certain instant they are found at a distance a / 3 from the origin on opposite sides but their velocities are in the same direction. What is the phase difference between the two ? 7 5 (a) cos (b) cos 9 9 4 1 (c) cos (d) cos 9 9

ANSWERS 1.

(d)

2.

(a)

3.

(c)

4.

(0

5.

(b)

6.

(b)

7.

(d)

8.

(b)

9.

(c)

10.

(c)

11.

(b)

12.

(c)

13.

(b)

14.

(a)

15.

(b)

16.

(b)

17.

(c)

18.

(b)

19.

(a)

20.

(d)

21.

(b)

22.

(c)

23.

(b)

24.

(c)

25.

(a)

26.

(a)

27.

(a)

-

149

Simple Harmonic Motion

More than One Alternative is/are Correct^ 1. A spring mass system is hanging from the ceiling of an elevator in equilibrium k as shown in figure. The elevator suddenly starts accelerating upwards with acceleration a, consider all the m options in the reference frame of elevator. 1 l~k (a) the frequency of oscillation is — — 2k V m ___ _ ma (b) the amplitude of the resulting SHM is — k m(g + u) (c) amplitude of resulting SHM is k (d) maximum speed of block during oscillation is

5. A simple pendulum of length I has a bob of mass m, with a charge q on it. A non-conducting thin vertical - • sheet of charge, with charge a per unit area, passes through the point of suspension of the pendulum. At equilibrium, the string makes an angle 0 with the vertical. Its time period of oscillation is T. In this ' ' * position: (a) tan 0 =

(b) tan 0 =

2e0mg

zomg

(d)T>2nJI

WT —minimum tension in string is mg k 1 l~k~ (c) Frequency of oscillation of system is — J—, for 2k V 3m all non-zero values of x0 (d) The motion will remain simple harmonic for < 3mg 0 k 3. Two simple pendula of length 36 cm and 64 cm are initially at opposite extremes of their swings. They will be in phase after approximately. (Take g = k2) (a) 7.2 s (b) 3.6 s (c) 5.4 s (d) 2.4 s 4. The system shown is hanging in equilibrium well above the ground. When 60kg the string is cut, the time taken by the spring to reach its natural length can be |l800N/m (Take k2 =10)

(a) is

(b)ls

(c) is

(d)is

6

[30kg]

downward in case-2, — upward in case-3 and Q q downward in case-4. The speed at mean position is same in all cases. Select the CORRECT altemative(s). (a) Time periods of oscillation are equal in case-1 and case-3 (b) Amplitudes of displacement are same in case-2 and case-3 (c) The maximum elongation (increment in length from natural length) is maximum in case-4. (d) Time periods of oscillation are equal in case-2 and case-4 7. The springs are identical and initially relaxed. The block is pushed to left and released. k

m

a

(a) The time period of oscillation T = (b) The time period of oscillation T = (c) If one of the springs is removed, time period increases (d) If one of the springs is removed, time period decreases 8. A ball is dropped from some height on an elastic floor undergoes: (a) periodic motion (b) oscillatory motion (c) simple harmonic motion (d) straight line motion

150

Advanced Problems in Physics

9. A particle of mass 2.5 kg force F=-10x + 20, it is at rest at t = 0 and x=4, its equation of motion is/are : (a) x = 2+2cos2t (b) x=2cos2t (c) x=2sin^2t-~^ (d) x=2+2sir 2t + -

(b) when it is 4 cm from the mean position, i« velocity is 6n/10cm/s (c) the kinetic energy of the particle when it displacement is 5 cm from the mean position i

10. A particle of mass 1 kg, executing simple harmonic motion of period 20 s, crosses the mean position at t = 0 with velocity n cm/s: (a) the maximum acceleration of the particle is 10 cm/s2

8xl06 (d) its velocity at displacement 6 cm from the mea position is 8n/10cm/s

2J

ANSWERS 9.

(a, d)

(a, b, d)

2.

(a, d)

10. (c,d)

3.

(a, d)

4.

(a, b)

5.

(a,c)

6.

(a, b, c, d) 7.

(a, c)

8.

(a, b, d)

Simple Harmonic Motion

151

| Comprehension Based Problems^

| Passage:! | A particle of mass 1.5 kg moves along x-axis in a conservative force field. Its potential energy is given by I7(x) = 2x3 - 9x2 + 12x, where all quantities are written in SI units. The plot of this potential energy is given below.

(d) *

(0

. ,,

8 8 2. At what time does the acceleration of B first point • directly towards A? T T (b)_

,?a);

• -:(o?

Cd) r

|| Matching Type Problems X

It is seen that the particle can be in stable equilibrium at a point on x-axis, x0. When it is displaced slightly from this equilibrium position, it executes SHM with time period T.

1. What is the range of total mechanical energy of the particle for which its motion can be oscillatory about a point? (a)E