In simple harmonic motion, displacement follows a specific periodic pattern with time: sinusoidal. The restoring force, proportional to displacement, produces inherently periodic motion described by sine or cosine functions.

The displacement from equilibrium is given by

Here $A$ is the amplitude (maximum displacement from equilibrium) and $\omega$ is the angular frequency. Use cosine when the object starts at maximum/minimum displacement. Use sine when it starts at equilibrium.

The angular frequency relates to period $T$ and frequency $f$ through

This ensures that when $t = T$, the argument increases by $2\pi$, completing one full cycle.

Creating a Displacement Function for SHM

To write a displacement function to model a simple harmonic oscillator, first compute the angular frequency, then substitute into the appropriate form based on initial conditions. For an object starting at maximum positive displacement with amplitude $A$ and period $T$:

Find the position at any time by direct substitution of $t$.