Simple Harmonic Motion

An object moves with $\textbf{simple harmonic motion}$ 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 $\textbf{angular frequency}$, in $\mathrm{rad\,s^{-1}}$, related to frequency and period by $\omega = 2\pi f = 2\pi/T$.

Released from rest at maximum displacement $A$ (the $\textbf{amplitude}$), 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 $a = -\dfrac{k}{m}x$ — 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 $mg\sin\theta \approx mg\,x/l$, so $a = -\dfrac{g}{l}x$.

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.