Frequently Used Equations

Mechanics

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

F dt = Δp
W = 
F · ds

F · ds = ΔE
       
kinetic energy potential energy power  
K = ½mv2
ΔU = − 
F · ds
 = ΔW
Δt
P = dW
dt
gravitational p.e.
ΔUg = mgΔh F = − ∇U  = F̅v cos θ P = F · v
       
angular velocity angular acceleration equations of rotation 2nd law for rotation
ω̅ = Δθ
Δt
α̅ = Δω
Δt
ω = ω0 + αt ∑ τ = I α
θ = θ0 + ω0t + ½αt2
∑ τ = dL
dt
ω = dθ
dt
α = dω
dt
ω2 = ω02 + 2α(θ − θ0)
ω̅ = ½(ω + ω0)  
v = ω × r a = α × r − ω2 r    
       
torque moment of inertia rotational work rotational power
τ = rF sin θ I = ∑ mr2 W = τ̅Δθ P = τω cos θ
τ = r × F
I = 
 r2 dm
W = 
 τ · dθ
P = τ · ω
   
       
rotational k.e. angular momentum frequency hooke's law
K = ½Iω2 L = mrv sin θ
ƒ = 1
T
F = − k Δx
  L = r × p elastic p.e.
  L = I ω ω = 2πƒ Us = ½kΔx2
       
universal gravitation gravitational field gravitational p.e. gravitational potential
Fg = − Gm1m2 
r2
g = − Gm 
r2
Ug = − Gm1m2
r
Vg = − Gm
r
       
orbital speed escape speed s.h.o. simple pendulum
v = √ Gm
r
v = √ 2Gm
r
T = 2π √ m
k
T = 2π √ 
g
       
density pressure pressure in a fluid buoyancy
ρ = m
V
P = F
A
P = P0 + ρgh B = ρgVdisplaced
   
       
volume flow rate mass flow rate volume continuity mass continuity
φ = V
t
I = m
t
A1v1 = A2v2 ρ1A1v1 = ρ2A2v2
   
       
bernoulli's equation   aerodynamic drag kinematic viscosity
P1 + ρgy1 + ½ρv12 = P2 + ρgy2 + ½ρv12 R = ½ρCAv2
ν = η
ρ
     
       
dynamic viscosity reynolds number mach number froude number
η = /A
Δvxz
Re = ρvD
η
Ma = v
c
Fr = v
g
η = F/A
dvx/dz
     
     
       
young's modulus shear modulus bulk modulus surface tension
F = E Δℓ
A0
F = G Δx
Ay
F = K ΔV
AV0
γ = F
       

Thermal Physics

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

Waves & Optics

periodic waves frequency beat frequency intensity
v = ƒλ
ƒ =  1
T
fbeat = fhigh − flow
I =  P
A
   
       
power level   interference fringes  
β = 10 log  I  = 20 log  P
I0 P0
nλ = d sin θ  
nλ  ≈  x
d L
 
     
       
index of refraction snell's law mirrors and lenses spherical mirrors
n =  c
v
n1 sin θ1 = n2 sin θ2
1  =  1  +  1
ƒ do di
ƒ =  r
2
critical angle
 
sin θc =  n2
n1
M =  hi  =  di
ho do
 
   
       

Electricity & Magnetism

coulomb's law electric field    
F = k q1q2
r2
E = Fe
q
E = k ∑ q 
r2
E = k 
dq 
r2
       
field and potential electric potential    
 = − V
d
ΔV = ΔUE
q
V = k ∑ q
r
V = k 
dq
r
E = − ∇V      
       
capacitance plate capacitor cylindrical capacitor spherical capacitor
C = Q
V
C = κε0A
d
C = 2πκε0
ln (b/a)
C = 4πκε0
(1/a) − (1/b)
       
capacitive p.e.   electric current  
U = 1 CV2 = 1 Q2 = 1 QV
22C2
 = Δq
Δt
I = dq
dt
 
       
ohm's law resitivity–conductivity electric resistance electric power
V = IR
ρ = 1
σ
R = ρℓ
A
P = VI = I2R = V2
R
E = ρ J
J = σE
       
resistors in series resistors in parallel capacitors in series capacitors in parallel
Rs = ∑ Ri
1 = ∑ 1
RpRi
1 = ∑ 1
CsCi
Cp = ∑ Ci
   
       
magnetic force      
FB = qvB sin θ FB = q v × B FB = IB sin θ dFB = I d × B
       
biot-savart law solenoid straight wire parallel wires
B = μ0I
ds × 
r2
B = µ0nI
B = μ0I
r
FB = μ0 I1I2
r
 
       
motional emf electric flux magnetic flux induced emf
V = Bv ΦE = EA cos θ ΦB = BA cos θ
ℰ = − ΔΦB
Δt
ℰ = − dΦB
dt
 
ΦE = 
E · dA
ΦB = 
B · dA
   
       
gauss's law no one's law faraday's law ampere's law
 E · dA = Q
ε0
∯ B · dA = 0
 
E · ds = − dΦB
dt
B · ds = μ0ε0 dΦE + μ0I
dt
∇ · E = ρ
ε0
∇ · B = 0
 
∇ × E = − B
t
∇ × B = μ0ε0 E + μ0 J
t
       

Modern Physics

time dilation length contraction relativistic mass relative velocity
t' = t
√(1 − v2/c2)
ℓ' = ℓ √(1 − v2/c2)
m' = m
√(1 − v2/c2)
u' = u + v
1 + uv/c2
 
       
relativistic energy relativistic momentum energy-momentum mass-energy
E = mc2
√(1 − v2/c2)
p = mv
√(1 − v2/c2)
E2 = p2c2 + (mc2)2 E = mc2
   
       
relativistic doppler effect photon energy photoelectric effect photon momentum
λ0 = ƒ = √ 
1 − v/c
λƒ01 + v/c
E = hf Kmax = E − ϕ = h(ƒ − ƒ0)
p = h
λ
   
       
schroedinger's equation   uncertainty principle rydberg equation
iℏ  Ψ(rt) = − 2 ∇2Ψ(rt) + V(r)Ψ(rt)
∂t2m
Δpx Δx ≥ ℏ 
2
1 = −R 
1 − 1
λn2n02
Eψ(r) = − 2 ∇2ψ(r) + V(r)ψ(r)
2m
ΔE Δt ≥ ℏ 
2
half life
N = N02t