Simple Harmonic Motion

Simple Harmonic Motion (SHM): periodic motion where restoring force is proportional to displacement and opposite in direction. F = −kx.

Equation of motion: m·d²x/dt² = −kx ⇒ d²x/dt² = −ω²·x with ω = √(k/m).

General solution: x(t) = A·cos(ωt + φ). A = amplitude, ω = angular frequency, φ = initial phase.

Period T = 2π/ω = 2π·√(m/k). Frequency f = 1/T.

SHM is the simplest periodic motion — many oscillators (pendulum, spring, sound, LC circuit) reduce to SHM at small amplitudes.

Two SHMs combine to give beats, interference, Lissajous figures.

Phase angle φ: shifts the wave in time; ωt + φ = 0 at maximum displacement.

Mathematical basis: any small oscillation around a stable equilibrium is SHM to leading order.