physics all equation

Mechanics

velocity

v̅ =  Δs Δt

v =  ds dt

acceleration

a̅ =  Δv Δt

a =  dv dt

equations of motion

v = v0 + at x = x0 + v0t + ½at2 v2 = v02 + 2a(x − x0) v̅ = 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 = F̅ Δt

J =  ⌠ ⌡ F dt

impulse-momentum

F̅ Δt = m Δv

⌠ ⌡ F dt = Δp

work

W = F̅Δs cos θ

W =  ⌠ ⌡ F · ds

work-energy

F̅Δs 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
P̅ =  ΔW Δt	P̅ = 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  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 Δx

elastic p.e.

Us = ½kΔx2

s.h.o.

T = 2π √  m k

simple pendulum

T = 2π √  ℓ 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 =  F̅/A Δvx/Δz

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 √gℓ

young's modulus

F  = E  Δℓ A ℓ0

shear modulus

F  = G  Δx A y

bulk modulus

F  = K  ΔV A V0

surface tension

γ =  F ℓ

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⟩ = √8kTπm vrms = √  3kT m

heat flow rate

Φ̅ =  ΔQ Δt

Φ =  dQ dt

thermal conduction

Φ =  kAΔT ℓ

stefan-boltzmann law

Φ = εσA(T4 − T04)

wien displacement law

λmax =  b T

internal energy

ΔU = 3⁄2nRΔT

thermodynamic work

W = − ⌠ ⌡ P dV

1st law of thermo.

ΔU = Q + W

entropy

ΔS =  ΔQ T

S = k log w

efficiency

ℰreal = 1 −  QC QH

ℰideal = 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 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 ƒ do di

image size

M =  hi  =  di ho do

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  r̂ r2	E = k  ⌠ ⌡ dq  r̂ r2

field and potential

E̅ = −  ∆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 =  2πκε0ℓ ln (b/a)

spherical capacitor

C =  4πκε0 (1/a) − (1/b)

capacitive p.e.

U =  1  CV2 =  1 Q2  =  1  QV 2 2 C 2

electric current

I̅ =  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, charge

FB = qvB sin θ

FB = q v × B

magnetic force, current

FB = IℓB sin θ

dFB = I dℓ × 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

ℰ̅ = −  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) = −  ℏ2  ∇2Ψ(r,t) + V(r)Ψ(r,t) ∂t 2m
Eψ(r) = −  ℏ2  ∇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 = N02−t/τ