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# Frequently Used Equations

## 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 = Ft
 J = ⌠⌡ F dt
impulse-momentum
Ft = mv
 ⌠⌡ F dt = ∆p
work
W = Fs cos θ
 W = ⌠⌡ F · ds
work-energy
Fs 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 = mgh
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 = τt
 H = ⌠⌡ τ dt
angular i.m.
τt = mω
 ⌠⌡ 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 = − kx
spring p.e.
Us = ½kx2
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 A dy
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 A ℓ0
σ = Eε
shear modulus
 F = G ∆x A y
τ = Gγ
bulk modulus
 F = K ∆V A V0
P = Κθ
surface tension
 γ = F ℓ

### Thermal Physics

solid expansion
∆ℓ = αℓ0T
A = 2αA0T
V = 3αV0T
liquid expansion
V = βV0T
sensible heat
Q = mcT
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 = bT
internal energy
U = 32nRT
U = 32NkT
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(ft − x/λ) + 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 fs λo c ∓ vs
 ∆f ≈ ∆λ ≈ ∆v f λ c
mach angle
 sin μ = c v
cerenkov angle
 cos θ = c nv
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 θc = n2 n1
image location
 1 = 1 + 1 f do di
image size
 M = hi = di ho do
spherical mirrors
 f ≈ r 2

### Electricity & Magnetism

coulomb's law
 F = k q1q2 r2
 F = 1 q1q2 r̂ 4πε0 r2
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 = 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, charge
FB = qvB sin θ
FB = qv × B
magnetic force, current
FB = IℓB sin θ
dFB = I d × B
biot-savart law
 B = μ0I ⌠⎮⌡ ds × r̂ 4π r2
solenoid
B = μ0nI
straight wire
 B = μ0I 2πr
parallel wires
 FB = μ0 I1I2 ℓ 2π r
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
 ∮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 ⎞⎟⎠ λ0 f 1 − 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) ∂t 2m
 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 ⎞⎟⎠ λ n2 n02
activity
 A = n t
half life
N = N02t/T½
absorbed dose
 D = E m
equivalent dose
H = wRD
effective dose
E = wTH