## Formulas 3

#### Center of mass

xCM = ∑mixi/M,  yCM = ∑miyi/M,  zCM = ∑mizi/M,  M = ∑mi.

#### Rotational kinematics

ω = dθ/dt,   ωavg = (θf - θi)/(tf - ti) = Δθ/Δt.
α = dω /dt,  αavg = Δω/Δt.
Rotation about an axis with constant α:
ωf = ωi + α(tf - ti),  θ= θi + ωi(tf - ti) + ½α(tf - ti)2.

v = ds/dt = rdθ/dt = rω,
at = dv/dt = rdω/dt = rα,  ar = v2/r = rω2,
a = (at2 + ar2)½ = (r2α2 + r2ω4)½ = r(α2 + ω4)½.

Moment of inertia:  I = ∑miri2.
Kinetic energy:  K = ∑Ki = ∑½mivi2 = ∑½mri2ω2 = ½ω2 ∑mri2 = ½Iω2.
Rolling:  KEtot = ½mv2 + ½Iω2,  v = rω.

#### Rotational dynamics

Angular momentum:  L = Iω.
Torque:  τ = r × F = Iα = dL/dt.
Work and power:  W = τdθ,  P = dW/dt = τdθ/dt = τω.

T2 = (4π2/Gm1)R3

#### Oscillations:

Harmonic motion:  F = -kx.
x(t) = Acos(ωt + φ),  v(t) = -ωAsin(ωt + φ),   a(t) = -ω2Acos(ωt + φ) = -ω2x.
ω = sqrt(k/m) = 2πf =  2π/T.
Energy:  K = (1/2)mv2,  U = (1/2)kx2,  E = K + U = (1/2)kA2.

Pendulum:  θ(t) = θmaxcos(ωt+φ),  ω2 = g/L,  T = 2π(L/g)1/2.

#### Waves:

Traveling waves:  y(x,t) = A sin(kx ± ωt + φ)
Waves on a string:  v = (F/μ)1/2
Sound level:  β = 10 log10(I/I0)
Beat frequency:  |f1 - f2|

Standing sound waves:
tube of length L with two open ends:  L = nλ/2, n = 1, 2, 3, ...
tube of length L with one open end and one closed:  L = nλ/4, n = odd integer

Doppler effect:
f = f0(v - vobs)/(v - vs)  (velocity components in the direction of v are positive)