C.1 Simple harmonic motion

Why do pendulum clocks keep such precise time? Why does a guitar string vibrate the way it does? These phenomena share a common thread: oscillatory motion. At the heart of this motion lies simple harmonic motion (SHM) , a special, highly regular type of oscillation that underpins everything from mechanical systems to wave behaviour.

Simple Harmonic Motion (SHM): A type of oscillatory motion in which the restoring force acting on an object is directly proportional to its displacement from the equilibrium position, and always directed toward that equilibrium position.

The defining equation of SHM links acceleration and displacement:

$a=-{\omega }^{2}x$

where $\omega$ is the angular frequency of the oscillation and $x$ is the displacement from equilibrium. The negative sign is critical , it tells us the acceleration always opposes the displacement, pulling the system back toward equilibrium.