Simple Harmonic Motion

An object moves with $\text{simple harmonic motion}$ if its acceleration is directly proportional to its displacement from a fixed equilibrium point and is always directed towards that point.

The minus sign in $a=-{\omega }^{2}x$ captures "towards equilibrium": when $x$ is positive, $a$ is negative, and vice versa.

$\omega$ is the $\text{angular frequency}$, in $\mathrm{rad}{\mathrm{s}}^{-1}$, related to frequency and period by $\omega =2\pi f=2\pi /T$.

Released from rest at maximum displacement $A$ (the $\text{amplitude}$), the displacement varies as $x=A\cos \omega t$.

Velocity is the gradient of displacement: $v=-A\omega \sin \omega t$. Its maximum magnitude, at equilibrium, is $v_{\max }=A\omega$. At any displacement, $v=\pm \omega \sqrt{A^2-x^2}$.

Acceleration is the gradient of velocity: $a=-A{\omega }^2\cos \omega t=-{\omega }^{2}x$. It is greatest at the extremes and zero at equilibrium.

Velocity is a quarter-cycle ($\pi /2$) out of phase with displacement; acceleration is in antiphase ($\pi$) with displacement.

For a mass $m$ on a spring of stiffness $k$, the restoring force is $F=-kx$. Newton’s second law gives $a=-\frac{k}{m}x$ — the SHM condition, with ${\omega }^2=k/m$.

So the period is $T=2\pi \sqrt{m/k}$. A heavier mass oscillates more slowly; a stiffer spring more quickly.

The period is independent of amplitude and of $g$ — a spring oscillator would have the same period on the Moon.

A simple pendulum is a small mass on a light string of length $l$. For small displacements, the restoring force component is $mg\sin \theta \approx mgx/l$, so $a=-\frac{g}{l}x$.

The period is $T=2\pi \sqrt{l/g}$, independent of mass and — for small angles under about $10^{\circ }$ — of amplitude.

To find $g$ experimentally, time many oscillations for several lengths, and plot $T^2$ against $l$. The gradient is $4{\pi }^2/g$. Timing from the equilibrium position, where the bob moves fastest, reduces reaction-time uncertainty.