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 = ma

∑F = dp

dt

• Weight W = mg dry friction ƒ ≤ μN centripetal accel.

ac = v2

r

ac = − ω2r momentum p = mv impulse J = F̅Δt

J = ⌠

⌡ F dt

• Impulse-momentum F̅Δt = mΔv

⌠

⌡ F dt = Δp

• Work

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

η = Wout

Ein

• Power

P̅ = ΔW

Δt

P = dW

dt

• Power-velocity P̅ = 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 = Iα

∑ 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 = Iω 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

• Escape speed

v = √ 2Gm

• Hooke's law F = − kΔx elastic p.e. Us = ½kΔx2 s.h.o.

T = 2π √ m

k

• Simple pendulum

T = 2π √ ℓ

• kinematic viscosity

ν = η
ρ
drag R = ½ρCAv2 mach number
Ma = v
c

1. reynolds number

Re = ρvD

η

1. froude number

Fr = v

√gℓ

1. young's modulus

F = E Δℓ

A ℓ0

1. shear modulus

F = G Δx

A y

1. bulk modulus

F = K ΔV

A V0

1. surface tension

γ = F

ℓ