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Frequently Used Equations

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Reference

Mechanics

velocity
v = ∆s
∆t
v = ds
dt
acceleration
a = ∆v
∆t
a = dv
dt
equations of motion
v = v0 + at
s = s0 + v0t + ½at2
v2 = v02 + 2a(s − s0)
v = ½(v + v0)
newton's 2nd law
∑ F = ma
∑ F = dp
dt
weight
W = mg
dry friction
fs ≤ μsN
fk = μkN
centripetal accel.
ac = v2
r
ac = − ω2r
momentum
p = mv
impulse
J = F∆t
J = ⌠
⌡		F dt
impulse-momentum
F∆t = m∆v
⌠
⌡		F dt = ∆p
work
W = F∆s cos θ
W = ⌠
⌡		F ⋅ ds
work-energy
F∆s cos θ = ∆E
⌠
⌡		F ⋅ ds = ∆E
kinetic energy
K = ½mv2
K = p2
2m
general p.e.
∆U = − ⌠
⌡		F ⋅ ds
F = − ∇U
gravitational p.e.
∆Ug = mg∆h
efficiency
η = Wout
Ein
power
P = ∆W
∆t
P = dW
dt
power-velocity
P = Fv cos θ
P = F ⋅ v
angular velocity
ω = ∆θ
∆t
ω = dθ
dt
v = ω × r
angular acceleration
α = ∆ω
∆t
α = dω
dt
a = α × r − ω2 r
equations of rotation
ω = ω0 + αt
θ = θ0 + ω0t + ½αt2
ω2 = ω02 + 2α(θ − θ0)
ω = ½(ω + ω0)
torque
τ = rF sin θ
τ = r × F
2nd law for rotation
∑ τ = Iα
∑ τ = dL
dt
moment of inertia
I = ∑ mr2
I = ⌠
⌡		r2 dm
rotational work
W = τ∆θ
W = ⌠
⌡		 τ ⋅ dθ
rotational power
P = τω cos θ
P = τ ⋅ ω
rotational k.e.
K = ½Iω2
angular momentum
L = mrv sin θ
L = r × p
L = Iω
angular impulse
H = τ∆t
H = ⌠
⌡		τ dt
angular i.m.
τ∆t = m∆ω
⌠
⌡		τ dt = ∆L
universal gravitation
Fg = − Gm1m2 r̂
r2
gravitational field
g = − Gm r̂
r2
gravitational p.e.
Ug = − Gm1m2
r
gravitational potential
Vg = − Gm
r
orbital speed
v = √Gm
r
escape speed
v = √2Gm
r
hooke's law
F = − k∆x
spring p.e.
Us = ½k∆x2
s.h.o.
T = 2π√m
k
simple pendulum
T = 2π√ℓ
g
frequency
f = 1
T
angular frequency
ω = 2πf
density
ρ = m
V
pressure
P = F
A
pressure in a fluid
P = P0 + ρgh
buoyancy
B = ρgVdisplaced
mass flow rate
qm = ∆m
∆t
qm = dm
dt
volume flow rate
qV = ∆V
∆t
qV = dV
dt
mass continuity
ρ1A1v1 = ρ2A2v2
volume continuity
A1v1 = A2v2
bernoulli's equation
P1 + ρgy1 + ½ρv12 = P2 + ρgy2 + ½ρv22
dynamic viscosity
F = η ∆vx
A∆y
F = η dvx
Ady
kinematic viscosity
ν = η
ρ
drag
R = ½ρCAv2
mach number
Ma = v
c
reynolds number
Re = ρvD
η
froude number
Fr = v
√gℓ
young's modulus
F = E ∆ℓ
Aℓ0
σ = Eε
shear modulus
F = G ∆x
Ay
τ = Gγ
bulk modulus
F = K ∆V
AV0
P = Κθ
surface tension
γ = F
ℓ

Thermal Physics

solid expansion
∆ℓ = αℓ0∆T
∆A = 2αA0∆T
∆V = 3αV0∆T
liquid expansion
∆V = βV0∆T
sensible heat
Q = mc∆T
latent heat
Q = mL
ideal gas law
PV = nRT
molecular constants
nR =Nk
maxwell-boltzmann
p(v) = 4v2⎛
⎜
⎝		m⎞
⎟
⎠		32e
− mv2
2kT
√π2kT  
molecular k.e.
⟨K⟩ = 32kT
molecular speeds
vp = √2kT
m
⟨v⟩ = √8kT
πm
vrms = √3kT
m
heat flow rate
P = ∆Q
∆t
P = dQ
dt
thermal conduction
P = kA∆T
ℓ
stefan-boltzmann law
P = εσA(T4 − T04)
wien's law
λmax = b
T
fmax = b′T
internal energy
∆U = 32nR∆T
∆U = 32Nk∆T
thermodynamic work
W = −⌠
⌡		P dV
1st law of thermo.
∆U = Q + W
entropy
∆S = ∆Q
T
S = k log w
efficiency
ηreal = 1 − QC
QH
ηideal = 1 − TC
TH
c.o.p.
COPreal = QC
QH − QC
COPideal = TC
TH − TC

Waves & Optics

periodic waves
v = fλ
f(x,t) = A sin(2π(x/λ − ft) + φ)
frequency
f = 1
T
beat frequency
fbeat = fhigh − flow
intensity
I = ⟨P⟩
A
intensity level
LI = 10 log⎛
⎜
⎝		I⎞
⎟
⎠
I0
pressure level
LP = 20 log⎛
⎜
⎝		∆P⎞
⎟
⎠
∆P0
doppler effect
fo = λs = c ± vo
fsλoc ∓ vs
∆f ≈ ∆λ ≈ ∆v
fλc
mach angle
sin μ = c
v
cerenkov angle
cos θ = c
nvp
interference fringes
nλ = d sin θ
nλ ≈ x
dL
index of refraction
n = c
v
snell's law
n1 sin θ1 = n2 sin θ2
critical angle
sin θc = n2
n1
image location
1 = 1 + 1
fdodi
image size
M = hi = di
hodo
spherical mirrors
f ≈ r
2

Electricity & Magnetism

coulomb's law
F = k q1q2
r2
F = 1 q1q2 r̂
4πε0r2
electric field, def.
E = FE
q
electric potential, def.
∆V = ∆UE
q
field & potential
E = ∆V
d
E = −∇V
−⌠
⌡		E ⋅ dr = ∆V
electric field
E = k ∑ q r̂
r2
E = k ⌠
⌡		dq r̂
r2
electric potential
V = k ∑ q
r
V = k ⌠
⌡		dq
r
capacitance
C = Q
V
plate capacitor
C = κε0A
d
cylindrical capacitor
C = 2πκε0ℓ
ln(r2/r1)
spherical capacitor
C = 4πκε0
(1/r1) − (1/r2)
capacitive p.e.
Uc = ½QV = ½CV2 = ½ Q2
C
electric current
I = ∆q
∆t
I = dq
dt
charge density
ρ = Q
V
current density
J = I
A
J = ρ v
ohm's law
V = IR
E = ρ J
J = σ E
resitivity-conductivity
σ = 1
ρ
electric resistance
R = ρℓ
A
electric power
P = VI = I2R = V2
R
resistors in series
Rs = ∑ Ri
resistors in parallel
1 = ∑ 1
RpRi
capacitors in series
1 = ∑ 1
CsCi
capacitors in parallel
Cp = ∑ Ci
magnetic force, charge 
FB = qvB sin θ
FB = qv × B
magnetic force, current
FB = IℓB sin θ
dFB = I dℓ × B
biot-savart law
B = μ0I⌠
⎮
⌡		ds × r̂
4πr2
solenoid
B = μ0nI
straight wire
B = μ0I
2πr
parallel wires
FB = μ0 I1I2
ℓ2πr
electric flux
ΦE = EA cos θ
ΦE = ⌠
⌡		E ⋅ dA
magnetic flux
ΦB = BA cos θ
ΦB = ⌠
⌡		B ⋅ dA
motional emf
ℰ = Bℓv
induced emf
ℰ = − ∆ΦB
∆t
ℰ = − dΦB
dt
inductance
ℰ = − L dI
dt
capacitive reactance
XC = 1
2πfC
inductive reactance
XL = 2πfL
impedance
Z = ±√[R2 + (XL − XC)2]
Z = R + j(XL − XC)
gauss's law
∯E ⋅ dA = Q
ε0
∇ ⋅ E = ρ
ε0
no one's law
∯B ⋅ dA = 0 
 
∇ ⋅ B = 0 
 
faraday's law
∮E ⋅ ds = − ∂ΦB
∂t
∇ × E = − ∂B
∂t
ampere's law
∮B ⋅ ds = μ0ε0 ∂ΦE + μ0I
∂t
∇ × B = μ0ε0 ∂E + μ0 J
∂t
electromagnetic plane wave
E(x,t) = E0 sin [2π(ft − x + φ)] ĵ
λ
B(x,t) = B0 sin [2π(ft − x + φ)] k̂
λ
em energy density
η = ε0E2
η = 1 B2
μ0
poynting vector
S = 1 E × B
μ0
em pressure
P = ½η

Modern Physics

lorentz factor
γ = 1
√(1 − v2/c2)
time dilation
t = t0
√(1 − v2/c2)
t = γt0
length contraction
ℓ = ℓ0√(1 − v2/c2)
ℓ = ℓ0
γ
relative velocity
u′ = u + v
1 + uv/c2
relativistic energy
E = mc2
√(1 − v2/c2)
E = γmc2
relativistic momentum
p = mv
√(1 − v2/c2)
p = γmv
energy-momentum
E2 = p2c2 + m2c4
mass-energy
E = mc2
relativistic k.e.
K = ⎛
⎜
⎝		1 − 1⎞
⎟
⎠		mc2
√(1 − v2/c2)
K = (γ − 1)mc2
relativistic doppler eff.
λ = f0 = √⎛
⎜
⎝		1 + v/c⎞
⎟
⎠
λ0f1 − v/c
photon energy
E = hf
E = pc
photon momentum
p = h
λ
p = E
c
photoelectric effect
Kmax = E − φ
Kmax = h(f − f0)
schroedinger's equation
iℏ ∂ Ψ(r,t) = − ℏ2 ∇2Ψ(r,t) + U(r)Ψ(r,t)
∂t2m
Eψ(r) = − ℏ2 ∇2ψ(r) + U(r)ψ(r)
2m
uncertainty principle
∆px∆x ≥ ℏ 
2
∆E∆t ≥ ℏ 
2
rydberg equation
1 = −R∞ ⎛
⎜
⎝		1 − 1⎞
⎟
⎠
λn2n02
activity
A = n
t
half life
N = N02−t/T½
absorbed dose
D = E
m
equivalent dose
H = wRD
effective dose
E = wTH