Frequently Used Equations

Mechanics

velocity
 = Δs
Δt
v = ds
dt
acceleration
 = Δv
Δt
a = dv
dt
equations of motion
v = v0 + at
x = x0 + v0t + ½at2
v2 = v02 + 2a(x − x0)
 = ½(v + v0)
newton's 2nd law
∑ F = m a
∑ F = dp
dt
weight
W = m g
dry friction
ƒ = μN
centrip. accel.
ac = v2
r
ac = − ω2 r
momentum
p = m v
impulse
J =  Δt
J = 
F dt
impulse-momentum
 Δt = m Δv

F dt = Δp
work
W = Δs cos θ
W = 
F · ds
work-energy
Δs cos θ = ΔE

F · ds = ΔE
kinetic energy
K = ½mv2
general p.e.
ΔU = − 
F · ds
F = − ∇U
gravitational p.e.
ΔUg = mgΔh
efficiency
ℰ = Wout
Ein
power
 = ΔW
Δt
 = F̅v cos θ
P = dW
dt
P = F · v
angular velocity
ω̅ = Δθ
Δt
ω = dθ
dt
v = ω × r
angular acceleration
α̅ = Δω
Δt
α = dω
dt
a = α × r − ω2 r
equations of rotation
ω = ω0 + αt
θ = θ0 + ω0t + ½αt2
ω2 = ω02 + 2α(θ − θ0)
ω̅ = ½(ω + ω0)
2nd law for rotation
∑ τ = I α
∑ τ =  dL
dt
torque
τ = rF sin θ
τ = r × F
moment of inertia
I = ∑ mr2
I = 
 r2 dm
rotational work
W = τ̅Δθ
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 = I ω
universal gravitation
Fg = − Gm1m2 
r2
gravitational field
g = − Gm 
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 Δx
elastic p.e.
Us = ½kΔx2
s.h.o.
T = 2π √ m
k
simple pendulum
T = 2π √ 
g
frequency
ƒ = 1
T
angular frequency
ω = 2πƒ
density
ρ = m
V
pressure
P = F
A
pressure in a fluid
P = P0 + ρgh
buoyancy
B = ρgVdisplaced
mass flow rate
I = m
t
volume flow rate
φ = V
t
mass continuity
ρ1A1v1 = ρ2A2v2
volume continuity
A1v1 = A2v2
bernoulli's equation
P1 + ρgy1 + ½ρv12 = P2 + ρgy2 + ½ρv22
dynamic viscosity
η = /A
Δvxz
η = F/A
dvx/dz
kinematic viscosity
ν = η
ρ
aerodynamic drag
R = ½ρCAv2
mach number
Ma = v
c
reynolds number
Re = ρvD
η
froude number
Fr = v
g
young's modulus
F = E Δℓ
A0
shear modulus
F = G Δx
Ay
bulk modulus
F = K ΔV
AV0
surface tension
γ = F

Thermal Physics

solid expansion
Δℓ = αℓ0ΔT
ΔA = 2αA0ΔT
ΔV = 3αV0ΔT
liquid expansion
ΔV = βV0ΔT
sensible heat
Q = mcΔT
latent heat
Q = mL
ideal gas law
PV = nRT
molecular constants
nR =Nk
maxwell-boltzmann
− mv2
p(v) = 4v2
m3/2
e2kT
√π2kT
molecular k.e.
K⟩ = 3 kT
2
molecular  speeds
vp = √  2kT
m
v⟩ = √ 8kT
πm
vrms = √ 3kT
m
heat flow rate
Φ̅ = ΔQ
Δt
Φ = dQ
dt
thermal conduction
Φ = kAΔT
stefan-boltzmann law
Φ = εσA(T4 − T04)
wien displacement law
λmax = b
T
internal energy
ΔU = 32nRΔT
thermodynamic work
W = −
P dV
1st law of thermo.
ΔU = Q + W
entropy
ΔS = ΔQ
T
S = k log w
efficiency
real = 1 − QC
QH
ideal = 1 − TC
TH
c.o.p.
COPreal = QC
QH − QC
COPideal = TC
TH − TC

Waves & Optics

periodic waves
v = ƒλ
frequency
ƒ = 1
T
beat frequency
fbeat = fhigh − flow
intensity
I = P
A
intensity level
LI = 10 log
I
I0
pressure level
LP = 20 log
Pmax
P0
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
ƒdodi
image size
M = hi = di
hodo
spherical mirrors
ƒ ≈ r
2

Electricity & Magnetism

coulomb's law
F = k q1q2
r2
electric field, def.
E = FE
q
electric field, around charges
E = k ∑ q 
r2
E = k 
dq  
r2
field and potential
 = −  V
d
E = − ∇V
electric potential, def.
ΔV = ΔUE
q
electric potential, around charges
V = k ∑ q
r
V = k 
dq
r
capacitance
C = Q
V
plate capacitor
C = κε0A
d
cylindrical capacitor
C = 2πκε0
ln (b/a)
spherical capacitor
C = 4πκε0
(1/a) − (1/b)
capacitive p.e.
U = 1 CV2 = 1 Q2 = 1 QV
22C2
electric current
 = Δq
Δt
I = dq
dt
ohm's law
V = IR
E = ρ J
J = σE
resitivity-conductivity
ρ = 1
σ
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 = q v × B
magnetic force, current
FB = IB sin θ
dFB = I d × B
biot-savart law
B = μ0I
ds × 
r2
solenoid
B = µ0nI
straight wire
B = μ0I
r
parallel wires
FB = μ0 I1I2
r
electric flux
ΦE = EA cos θ
ΦE = 
E · dA
magnetic flux
ΦB = BA cos θ
ΦB = 
B · dA
motional emf
V = Bv
induced emf
ℰ = − ΔΦB
Δt
ℰ = − dΦB
dt
gauss's law
 E · dA = Q
ε0
∇ · E = ρ
ε0
no one's law
∯ B · dA = 0
 
∇ · B = 0
 
faraday's law
E · ds = − dΦB
dt
∇ × E = − B
t
ampere's law
B · ds = μ0ε0 dΦE + μ0I
dt
∇ × B = μ0ε0 E + μ0 J
t

Modern Physics

time dilation
t' = t
√(1 − v2/c2)
length contraction
ℓ' = ℓ √(1 − v2/c2)
relativistic mass
m' = m
√(1 − v2/c2)
relative velocity
u' =  u + v
1 + uv/c2
relativistic energy
E = mc2
√(1 − v2/c2)
relativistic momentum
p = mv
√(1 − v2/c2)
energy-momentum
E2 = p2c2 + m02c4
mass-energy
E = mc2
relativistic doppler effect
λ  =  ƒ0  = √ 
1 + v/c
λ0 ƒ 1 − v/c
photon energy
E = hf
photoelectric effect
Kmax = E − ϕ = h(ƒ − ƒ0)
photon momentum
p = h
λ
schroedinger's equation
iℏ  Ψ(r,t) = − 2 ∇2Ψ(r,t) + V(r)Ψ(r,t)
∂t2m
Eψ(r) = − 2 ∇2ψ(r) + V(r)ψ(r)
2m
uncertainty principle
Δpx Δx ≥ ℏ 
2
ΔE Δt ≥ ℏ 
2
rydberg equation
1 = −R 
1 − 1
λn2n02
half life
N = N02t