Work and energy

Work:  W = F·d
Scalar product:  A·B = ABcosθ = A_xB_x + A_yB_y + A_zB_z

Hooke's law:  F = -kr,  F_x= -kx
Elastic potential energy:  W = (1/2)kx²
Conservative systems:  E = K + U,  F_x = -dU/dx

Work-kinetic energy theorem:  W_net = ∆K = (1/2)m(v_f² - v_i²)
Power:  P = F·v  or  P = ∆W/∆t
coefficient of restitution:  (outgoing speed)/(incoming speed)

Gravitational force:  F₁₂ = (-G m₁m₂/r₁₂²) (r₁₂/r₁₂)
Gravitational potential energy:  U_f - U_i = -G m₁m₂(1/r_12f - 1/r_12i)

Momentum and impulse

p = mv,   F = ∆p /∆t,  I= ∆p = F∆t.
Center of mass:  x_CM = ∑m_ix_i/M,  y_CM = ∑m_iy_i/M,  z_CM = ∑m_iz_i/M,  M = ∑m_i.

Center of mass

x_CM = ∑m_ix_i/M,  y_CM = ∑m_iy_i/M,  z_CM = ∑m_iz_i/M,  M = ∑m_i.

Rotational kinematics

ω = dθ/dt,   ω_avg = (θ_f - θ_i)/(t_f - t_i) = Δθ/Δt.
α = dω /dt,  α_avg = Δω/Δt.
Rotation about an axis with constant α: 
ω_f = ω_i + α(t_f - t_i),  θ_f = θ_i + ω_i(t_f - t_i) + ½α(t_f - t_i)².

v = ds/dt = rdθ/dt = rω,
a_t = dv/dt = rdω/dt = rα,  a_r = v²/r = rω²,
a = (a_t² + a_r²)^½ = (r²α² + r²ω⁴)^½ = r(α² + ω⁴)^½.

Moment of inertia:  I = ∑m_ir_i².
Kinetic energy:  K = ∑K_i = ∑½m_iv_i² = ∑½mr_i²ω² = ½ω² ∑mr_i² = ½Iω².
Rolling:  KE_tot = 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.

Kepler's laws

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) = θ_maxcos(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 - v_obs)/(v - v_s)  (velocity components in the direction of v are positive)