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
a c = − ω2 r
 
momentum
p = m v
 
 
 
efficiency
ℰ =  Wout
Ein
 
 
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
gravitational p.e.
ΔUg = mgΔh
 
potential energy
ΔU = − 
F · ds
F = − ∇U
 
power
 =  ΔW
Δt
P =  dW
dt
 = F̅v cos θ 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
 
frequency
ƒ =  1
T
ω = 2πƒ
 
hooke's law
F = − k Δx
elastic p.e.
Us = ½kΔx2
 
universal gravitation
F g = −  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
 
s.h.o.
T = 2π √  m
k
 
simple pendulum
T = 2π √ 
g
 
density
ρ =  m
V
 
pressure
P =  F
A
 
pressure in a fluid
P = P0 + ρgh
 
 
buoyancy
B = ρgVdisplaced
 
 
volume flow rate
φ =  V
t
 
mass flow rate
I =  m
t
 
volume continuity
A1v1 = A2v2
 
 
mass continuity
ρ1A1v1 = ρ2A2v2
 
 
bernoulli's equation
P1 + ρgy1 + ½ρv12 = P2 + ρgy2 + ½ρv22
   
aerodynamic drag
R = ½ρCAv2
 
 
kinematic viscosity
ν =  η
ρ
 
dynamic viscosity
η =  /A
Δvxz
η =  F/A
dvx/dz
 
reynolds number
Re =  ρvD
η
 
 
 
mach number
Ma =  v
c
 
froude number
Fr =  v
g
 
young's modulus
F  = E  Δℓ
A 0
shear modulus
F  = G  Δx
A y
bulk modulus
F  = K  ΔV
A V0
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
m 3/2
e 2kT
√π 2kT
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
λ max =  b
T
 
internal energy
ΔU = 32nRΔT
 
thermodynamic work
W = −
P dV
 
first law
ΔU = Q + W
 
entropy
ΔS =  ΔQ
T
S = k log w
 
efficiency
real = 1 −  QC
QH
ideal = 1 −  TC
TH
coefficient of performance
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
power level field level
β = 10 log  I
I0
β = 20 log  P
P0
   
interference fringes
nλ = d sin θ
nλ  ≈  x
d L
index of refraction
n =  c
v
 
 
snell's law
n1 sin θ1 = n2 sin θ2
 
 
critical angle
sin θ c =  n2
n1
 
 
mirrors, lenses
1  =  1  +  1
ƒ do di
M =  hi  =  di
ho do
spherical mirrors
ƒ =  r
2
 
 

Electricity & Magnetism

coulomb's law
F = k  q1q2
r2
 
electric field, def.
E =  F e
q
 
electric field, around charge.
E = k ∑  q  
r2
E = k 
dq  
r2
field and potential
 = −  V
d
E = − ∇V
 
electric potential, def.
ΔV =  ΔUE
q
 
electric potential, around charge
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 potential energy
U =  1  CV2 =  1   Q2  =  1  QV
2 2 C 2
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
Rp Ri
capacitors in series
1  = ∑  1
Cs Ci
capacitors in parallel
Cp = ∑ Ci
 
 
magnetic force, charges
FB = qvB sin θ FB = q v × B
   
magnetic force, currents
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
motional emf
V = Bv
 
 
 
electric flux
Φ E = EA cos θ
Φ E = 
E · dA
magnetic flux
Φ B = BA cos θ
Φ B = 
B · dA
induced emf
ℰ = −  ΔΦ B
Δt
ℰ = −  dΦ B
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
λ0  =  ƒ  = √ 
1 − v/c
λ ƒ0 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)
∂t 2m
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
λ n2 n02
half life
N = N02t