Physics Formulas 1586636634, 9781586636630

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PHYSICAL CONSTANTS Acceleration due to gravity

Avogadro’s number

ELECTROMAGNETIC CONSTANTS WAVELENGTHS OF LIGHT IN A VACUUM (m)

g

9.8 m/s

NA

6.022 × 10

2

23

k

9 × 109 N·m2 /C2

Gravitational constant

G

6.67 × 10−11 N·m2 /kg 2

Green

Planck’s constant

h

6.63 × 10

Blue

Ideal gas constant

R

Permittivity of free space

ε0

8.8541 × 10−12 C/(V·m)

Permeability of free space

µ0

4π × 10−7 Wb/(A·m)

J·s

331 m/s

3.00 × 108 m/s

Electron charge

e

1.60 × 10

Electron volt

eV

1.6022 × 10

Atomic mass unit

u

1.6606 × 10 kg = 931.5 MeV/c2

Rest mass of electron

me

9.11 × 10−31 kg = 0.000549 u = 0.511 MeV/c2

mp

1.6726 × 10−27 kg = 1.00728 u = 938.3 MeV/c2

...of proton

−19

J

−27

Mass of Earth

5.976 × 1024 kg

Radius of Earth

6.378 × 10 m

4.9 – 5.7 × 10−7

1011

1012

microwaves

1 10-1 10-2 = wavelength (in m)

4.2 – 4.9 × 10−7

10-3

1013

1014

1015

10-4

1016

1017

1018

ultraviolet

infrared 10-5

10-6

10-7

10-8

10-9

1020

1019

gamma rays

X rays 10-10

R O Y G B I

10-11

10-12

V

= 780 nm visible light

4.0 – 4.2 × 10−7

Violet

360 nm

INDICES OF REFRACTION FOR COMMON SUBSTANCES ( l = 5.9 X 10 –7 m) Air

1.00

Alcohol

1.36

Corn oil

1.47 1.47

Diamond

2.42 1.33

Glycerol

Water

incident ray

θinciden t = θreflected c n= (v is the speed of light in the medium) v

Law of Reflection

Index of refraction

angle of incidence

01 0'

angle of reflection

n1 sin θ1 = n2 sin θ2 � � θc = sin −1 nn21

Snell’s Law

Critical angle

02

normal angle of refraction

refracted ray reflected ray

LENSES AND CURVED MIRRORS

1.6750 × 10−27 kg = 1.008665 u = 939.6 MeV/c2

…of neutron

1010

REFLECTION AND REFRACTION

C

−19

109

radio waves

OPTICS

c

Speed of light in a vacuum

108

5.7 – 5.9 × 10−7

Yellow

8.314 J/(mol·K) = 0.082 atm ·L/(mol·K)

Speed of sound at STP

ƒ = frequency (in Hz)

Orange 5.9 – 6.5 × 10−7

Coulomb’s constant

−34

6.5 – 7.0 × 10−7

Red

molecules /mol

q image size =− p object size

1 1 1 + = f q p

Optical instrument Lens: Concave Convex

Focal distance f

Image distance q

Type of image

negative positive

negative (same side) negative (same side) positive (opposite side)

virtual, erect 1 virtual, erect 2 real, inverted 3

negative (opposite side)

virtual, erect

negative (opposite side) positive (same side)

virtual, erect 5 real, inverted 6

pf

p

h

Convex

negative

Concave

positive

pf

DYNAMICS

V

F

Mirror:

6

4

q

6

NEWTON’S LAWS 1. First Law: An object remains in its state of rest or motion with constant velocity unless acted upon by a net external force. dp F = 2. Second Law: Fnet = ma dt 3. Third Law: For every action there is an equal and opposite reaction. Weight

Fw = mg

Normal force

FN = mg cos θ (θ is the angle to the horizontal)

h

h

F p

1

V

p

q

Kinetic friction fk = µk FN

µs is the coefficient of static friction. µk is the coefficient of kinetic friction. For a pair of materials, µk < µs .

W = F · s = F s cos θ � W = F · ds

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mv 2 Centripetal force Fc = r

ˆ a = axˆi + ayˆi + az k

Magnitude

a = |a| =

Dot product

a · b = ax bx + ay by + az yz = ab cos θ

�

a2x + a2y + a2z

Cross product

a × b = (ay bz � � ax � = �� ax � ˆi

axb

Gravitational potential energy

Ug = mgh

Total mechanical energy

E = KE + U

Average power

Pavg =

Instantaneous power

MOMENTUM AND IMPULSE Linear momentum

p = mv

Impulse

J = �Ft = ∆p J= F dt = ∆p

a

COLLISIONS b

ˆ − az by ) ˆi + (az bx − ax bz) ˆj + (ax by − ay bx ) k � ay az �� ay bz �� ˆj ˆ � k

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∆W ∆t

P =F·v

a b

p2 1 mv 2 = 2m 2

∆U = −W

Potential energy

Notation

|a × b| = ab sin θ a × b points in the direction given by the right-hand rule:

KE =

Kinetic energy

All collisions

m1 v1 + m2 v2 = m1 v1� + m2 v2�

Elastic collisions

1 1 1 1 2 2 m1 v12 + m2 v22 = m1 (v1� ) + m2 (v2� ) 2 2 2 2

v1 − v2 =

− (v1�

−

v2� )

q

V

h

F

V

F

p

h

F

V F

q

4

3

WORK, ENERGY, POWER Work

F p

(for conservative forces)

VECTOR FORMULAS

(θ is the angle between a and b)

F

Work-Energy Theorem W = ∆KE

UNIFORM CIRCULAR MOTION v2 Centripetal acceleration ac = r

q

2

FRICTION Static friction fs, max = µs FN

h

V

p

q

5

KINEMATICS Average velocity

vavg =

∆s ∆t

DISTANCE s (m)

Instantaneous ds v= velocity dt

Displacement ∆s =

Average acceleration

�

aavg =

v dt

∆v ∆t

Instantaneous dv a= acceleration dt

Change in velocity

vf = v0 + at 1 vavg = (v0 + vf ) 2

s = s0 + v0 t +

1 at 2

= s0 − vf t +

1 at 2

= s0 + vavg t

=

v02

VELOCITY v (m/s)

+

t (s)

∆v =

�

CONSTANT ACCELERATION

vf2

t (s)

a dt

– ACCELERATION a (m/s2)

+

t (s)

–

+ 2a(sf − s0 ) CONTINUED ON OTHER SIDE

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WAVES

ELECTRICITY

T

WAVE ON STRING Tension in string FT

Mass density µ =

Length L

mass length

F =k

Electric field

E=

Potential difference

W ∆V = q

Fon q q

F = Eq

CIRCUITS ∆Q ∆t

Current

I=

Resistance

R=ρ

Ohm’s Law

I=

SOUND WAVES

Power dissipated by resistor

P = V I = I 2R

Beat frequency

Heat energy dissipated by resistor

W = P t = I 2 Rt

Speed of standing wave

v=

Wavelength of standing wave

λn =

FT µ

2L n

fbeat = |f1 − f2 |

DOPPLER EFFECT Motion of source Stationary

Motion of observer

Stationary

v λ

f

veff = v + vo λeff = λ � � o feff = f v+v v

Towards source at vo

Toward observer at vs

Away from observer at vs

veff = v � � s λeff = λ v−v � v � v feff = f v−v s

veff = v � � s λeff = λ v+v � v � v feff = f v+v s

veff = v ± vo � � s λeff = λ v±v v � � o feff = f v±v v±vs

Away from source at vo veff = v − vo

λeff = λ � � o feff = f v−v v

ROTATIONAL MOTION Angular position

Angular velocity

ωavg =

∆θ ∆t

ω=

v r dθ dt at r dω dt

ω=

α=

Angular acceleration

αavg =

s r

∆ω ∆t

α=

a

CONSTANT ωf = ω0 + αt

T = 2π

�

v=0 U = max KE = 0

MASS-SPRING SYSTEM

R

R

sphere

R

MR 2 ring

disk

Elastic potential energy

Period 2 MR 2 5

L

rod

TORQUE AND ANGULAR MOMENTUM Torque

τ =

dL dt

F = −k(∆)x ∆x is the distance the spring is stretched or compressed from the equilibrium position, and k is the spring constant.

1 ML2 12

R

R2 R3

Magnetic force on moving charge

F = qvB sin θ

F = q (v × B)

Magnetic force on current-carrying wire

F = BI� sin θ

F = I (� × B)

MAGNETIC FIELD PRODUCED BY… Magnetic field due to a moving charge

B=

µ0 qv × ˆr 4π r2

Magnetic field produced by a current-carrying wire

B=

µ0 I 2π r

Magnetic field produced by a solenoid

B = µ0 nI

Biort-Savart Law

dB =

Lenz’s Law and Faraday’s Law

ε=−

MAXWELL’S EQUATIONS � Gauss’s Law

�s

Gauss’s Law for magnetic fields

�s

Restoring force

MOMENTS OF INERTIA (I ) � I= r 2 dm Moment of inertia 1 MR 2 2

mg cos 0

v=0 U = max KE = 0

equilibrium position

= θ0 + ωavg t

MR 2

v = max U = min KE = max

τ = F r sin θ τ =r×F τ = Iα

Angular momentum

L = pr sin θ

L=r×p

L = Iω

Rotational KE rot = 12 Iω 2 kinetic energy

GAS LAWS Universal Gas Law

P V = nRT

Combined Gas Law

P2 V2 P1 V1 = T2 T1

2π T

=

�

1 k(∆x)2 2 � m T = 2π k Ue =

x = A sin(ωt)

Equation of motion

where ω =

k m

is the angular frequency

and A = (∆x)max is the amplitude.

THERMODYNAMICS 1. First Law ∆ (Internal Energy) = ∆Q + ∆W 2. Second Law: All systems tend spontaneously toward maximum entropy. ∆Qout Alternatively, the efficiency e = 1 − ∆Qin of any heat engine always satisfies 0 ≤ e < 1. Boyle’s Law

P1 V1 = P2 V2

Charles’s Law

P2 P1 = T2 T1

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R3

R1

mg sin 0

1 = (ω0 + ωf ) 2

ωf2 = ω02 + 2α(θf − θ0 )

R2

Parallel circuits Ieq = I1 + I2 + I3 + · · · Veq = V1 = V2 = V3 = . . . 1 1 1 1 + ··· + + = R2 R2 R1 Req

T

� g

1 αt 2

particle

0

Period

mg

ωavg

θ = θ0 + ω 0 t +

2g� (1 − cos θmax )

R1

MAGNETISM

Velocity at equilibrium position

�

Series circuits Ieq = I1 = I2 = I3 = . . . Veq = V1 + V2 + V3 + · · · Req = R1 + R2 + R3 + · · ·

Loop rule: The sum of all the (signed) potential differences around any closed loop is zero. Node rule: The total current entering a juncture must equal the total current leaving the juncture.

PENDULUM

v=

V R

KIRCHHOFF’S RULES

SIMPLE HARMONIC MOTION

θ=

L A

Faraday’s Law

c

�

Ampere’s Law

�c

Ampere-Maxwell Law

c

E · dA =

r) µ0 I (d� × ˆ r2 4π

dΦB dt

Qenclosed ε0

B · dA = 0 E · ds = −

4

�

SPARKCHARTS

�� t

1 q1 q2 q1 q2 = 4πε0 r 2 r2

Coulomb’s Law

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Wave speed v = f λ Wave equation � � y(x, t) = A sin(kx − ωt) = A sin 2π λx −

ELECTROSTATICS

$3.95

2π T

∂ ∂ΦB =− ∂t ∂t

�

s

B · dA

B · ds = µ0 Ienclosed B · ds = µ0 Ienclosed + µ0 ε0

∂ ∂t

�

s

E · dA

GRAVITY m1 m2 r2

Newton’s Law of Universal Gravitation

F =G

Acceleration due to gravity

a=

Gravitational potential

U (r) = −

Escape velocity

vescap e

20593 36340

ω = 2πf =

7

1 2π = f ω

Angular frequency ω

TM

Period T

Contributors: Bernell K. Downer, Anna Medvedovsky Design: Dan O. Williams Illustration: Dan O. Williams, Matt Daniels Series Editors: Sarah Friedberg, Justin Kestler

T =

Wavelength λ

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Frequency f

Amplitude A

GM Earth 2 rEarth

GM m r � GM = r

KEPLER’S LAWS OF PLANETARY MOTION 1. Planets revolve around the Sun in an elliptical path with the Sun at one focus. 2. The imaginary segment connecting the planet to the Sun sweeps out equal areas in equal time. 3. The square of the period of revolution is directly proportional to the cube of the length of the semimajor axis of revolution: T 2 is constant. a3

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