Simple harmonic motion occurs when the restoring force is proportional to displacement and directed toward equilibrium: $F = -kx$. Combining this with Newton’s second law gives $m\ddot{x} + kx = 0$.

The motion is sinusoidal: $x(t) = A\cos(\omega t + \phi)$, where $\omega = \sqrt{k/m}$. The period is $T = 2\pi\sqrt{m/k}$ and the frequency is $f = 1/T$.

In ideal SHM, total mechanical energy is constant and oscillates between kinetic and potential forms. At equilibrium, speed is maximum; at the turning points, speed is zero and spring potential energy is maximum.

For small angles, a simple pendulum behaves approximately like SHM with period $T = 2\pi\sqrt{L/g}$. This approximation relies on the small-angle condition where $\sin\theta \approx \theta$.