Definition

Simple harmonic motion

a = - ω^{2} x, x (t) = A cos (ω t + ϕ)

Statement

An acceleration proportional to and opposing displacement; the resulting motion is sinusoidal with angular frequency ω.

Symbols and units

$a$	acceleration	m/s²
$ω$	angular frequency	rad/s
$A, ϕ$	amplitude, phase constant	m, rad
$x$	displacement from equilibrium	m
$t$	time	s

When it applies

Any restoring force/acceleration linear in displacement — springs, pendulums (small angle), LC circuits alike.

Don't use when

Does not describe motion with a restoring force that isn't linear in displacement — a large-angle pendulum or a non-Hookean spring falls outside SHM.

Derivation

A restoring acceleration proportional to displacement defines simple harmonic motion:

a = - ω^{2} x

This is a linear second-order differential equation whose general solution is a sinusoid of angular frequency ω:

x (t) = A cos (ω t + ϕ)

Watch out for

ω is the angular frequency (rad/s), not the ordinary frequency f (Hz); they are related by ω = 2πf.