Simple Harmonic Motion Calculator — Spring & Pendulum

Simple Harmonic Motion Guide

Mass-Spring System

For a mass m on a spring with spring constant k: T = 2π√(m/k). f = 1/T = (1/2π)√(k/m). ω = 2πf = √(k/m). The period is independent of amplitude — this is a key property of SHM. Example: m=0.5kg, k=20 N/m. T = 2π√(0.5/20) = 2π×0.158 = 0.994 s. f = 1/0.994 = 1.006 Hz. ω = 2π/0.994 = 6.32 rad/s. With amplitude A=0.1m: v_max = Aω = 0.1×6.32 = 0.632 m/s. a_max = Aω² = 0.1×39.9 = 3.99 m/s².

Simple Pendulum

For a pendulum of length L: T = 2π√(L/g). Valid for small angles (< 15°) — at larger angles, the period increases slightly. Pendulum length of 1.0m on Earth: T = 2π√(1.0/9.81) = 2π×0.319 = 2.006 s ≈ 2 seconds. The seconds pendulum (T=2s exactly) has length exactly 0.994m. Note: the period of a pendulum is independent of its mass and of the amplitude (for small angles). This was used historically in precision clocks.

SHM Equations of Motion

Displacement: x = A cos(ωt + φ). Velocity: v = −Aω sin(ωt + φ). Acceleration: a = −Aω² cos(ωt + φ) = −ω²x. The key relationship: a = −ω²x. Acceleration is proportional to displacement and directed toward equilibrium — the defining equation of SHM. Energy: total mechanical energy E = ½kA². At equilibrium (x=0): all KE = ½mv². At amplitude (x=A): all PE = ½kA². At any point: ½mv² + ½kx² = ½kA². Velocity at displacement x: v = ω√(A² − x²).

Damping and Resonance

Real oscillators lose energy to damping (friction, air resistance, internal heating). Light damping: amplitude decreases slowly. Heavy damping: returns to equilibrium without oscillating. Critical damping: returns to equilibrium fastest without oscillating — used in door closers, car suspension. Resonance: when a driving frequency equals the natural frequency f₀ = (1/2π)√(k/m), amplitude increases dramatically. In lightly damped systems, this can cause structural failure. Examples: Tacoma Narrow