Mechanics Equations (Rotational & Oscillations)

Angular velocity (definition)

ω = dθ/dt

Angular acceleration (definition)

α = dω/dt

Angular velocity with constant angular acceleration

ω = ω_0 + αt

Angular position with constant angular acceleration

θ = θ_0 + ω_0 t + (1/2)αt²

Angular velocity-squared (no time)

ω² = ω_0² + 2α(θ - θ_0)

Tangential velocity

v = rω

Tangential acceleration

a_T = rα

Torque (vector)

τ = r × F

Total rotational inertia

I_tot = ΣI_i = Σm_i r_i²

Rotational inertia (continuous)

I = ∫r² dm

Parallel axis theorem

I' = I_cm + Md²

Angular acceleration of a system

α_sys = Στ/I_sys = τ_net/I_sys

Rotational kinetic energy

K_rot = (1/2)Iω²

Rotational work

W = ∫τ·dθ

Angular momentum (vector)

L = r × p = Iω

Change in angular momentum

ΔL = ∫τ dt

Arc length from angular displacement

Δx_cm = rΔθ

Period (rotational)

T = 2π/ω = 1/f

Period of a spring (SHM)

T_s = 2π√(m/k)

Period of a simple pendulum

T_p = 2π√(ℓ/g)

Period of a physical pendulum

T_phys = 2π√(I/(mgd))

Position in SHM

x = x_max cos(ωt + φ)