PHYSICS IMPORTANT FORMULA

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

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