What Conditions Cause Simple Harmonic Motion (SHM)?

Simple harmonic motion (SHM) occurs when a system has a stable equilibrium position and its acceleration is directly proportional to, but opposite in direction to, its displacement from equilibrium. Mathematically, the defining condition is $a=-\omega^2x$.

The Conditions for SHM

This is an SL and HL concept in C.1 Simple harmonic motion. Two features must be present:

1. A displacement from a stable equilibrium position produces a restoring acceleration directed back toward equilibrium.
2. The magnitude of that acceleration is directly proportional to the displacement: $|a|\propto |x|$.

Combining these gives

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

where $x$ is displacement from equilibrium and $\omega$ is angular frequency. The minus sign shows that acceleration and displacement have opposite directions.

This condition must hold throughout the displacement range being considered. A restoring force alone is not sufficient: if its magnitude varies non-linearly with displacement, the motion may remain periodic but is not SHM. In an ideal SHM model, damping and external driving forces are absent or negligible.

For a mass attached to an ideal spring, Hooke's law gives $F=-kx$. Using Newton's second law,

$$ma=-kx \quad \Rightarrow \quad a=-\frac{k}{m}x.$$

Therefore, the motion is SHM with $\omega^2=\frac{k}{m}$.

A common misconception is that any oscillation is SHM. Oscillation only requires repeated motion about equilibrium; SHM additionally requires the specific linear relationship $a=-\omega^2x$. At equilibrium, acceleration is zero and speed is maximum. At maximum displacement, acceleration has maximum magnitude and speed is zero.

Exam Technique

If asked to show that a system undergoes SHM, derive an expression in the form $a=-Cx$, where $C$ is a positive constant, and identify $C=\omega^2$. Examiners expect both proportionality and the restoring direction; stating only that the motion oscillates is insufficient.