Physics formulas

PhysicsForm 8.0 4/15/03 12:16 PM Page 1

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

pp>f

p

h

Convex

negative

Concave

positive

pp>f

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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PhysicsForm 8.0 4/15/03 12:16 PM Page 2

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

$5.95 CAN

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

Report errors at
www.sparknotes.com/errors

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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