Simple harmonic motion: the maths of anything that oscillates

Simple harmonic motion is the physics of repetitive back-and-forth: a pendulum, a mass on a spring, a guitar string. They look different but share one defining condition, and once you spot it, the same equations solve all of them.

The defining condition

A body moves with SHM when its acceleration is proportional to its displacement from a central point, and always points back towards that point:

The minus sign is the whole idea: the further the object is pushed from the middle, the harder it's pulled back. This restoring "always towards the centre" force is what makes the motion repeat. Here $\omega$ is the angular frequency, linked to the period by $\omega = \dfrac{2\pi}{T}$.

The key equations

$A$ is the amplitude, the maximum displacement. A defining feature of SHM: the period $T$ does not depend on the amplitude. A pendulum takes the same time to swing whether the swing is wide or narrow — the property that made it the basis of clocks.

Where the energy is

Energy sloshes between two stores, and the total stays constant:

• At the centre ($x = 0$): moving fastest → all kinetic energy.
• At the extremes ($x = \pm A$): momentarily still → all potential energy.

So velocity is greatest in the middle and zero at the ends, while displacement is the opposite. That swap is the source of most graph confusion.

Reading the graphs

If displacement follows a cosine curve, then velocity (its rate of change) is a negative sine curve, and acceleration is a negative cosine — back in step with displacement but flipped. Each graph is the gradient of the one before: velocity is the slope of displacement, acceleration the slope of velocity.

Velocity is maximum where displacement is zero — not where displacement is maximum. At the extremes the object has stopped to turn around. Lining up "fast in the middle, still at the ends" with the graphs is what this topic is really testing.

Last revised 2 June 2026.