Explain why a simple pendulum swinging with a small amplitude is described as undergoing simple harmonic motion.

AQA A-Level Physics (7408) — 3.6.1 Periodic Motion · Explain · 5 marks · View as Markdown

Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).

A simple pendulum consists of a small bob suspended on a light string. When displaced slightly from its rest position and released, the bob oscillates back and forth.

Model answer (5 marks)

A simple pendulum has a restoring force that is the component of gravity along the arc,
F=−mg sinθ. For small angles sinθ≈θ≈x/L, so F≈−(mg/L)x. Thus the force is directed towards the equilibrium position and its magnitude is proportional to the displacement x. The acceleration a=F/m=−(g/L)x is therefore proportional to the displacement and always points opposite to it. This satisfies the definition of simple harmonic motion.

Examiner tips

Show the small‑angle approximation sinθ≈θ to link force to displacement
Explicitly state that acceleration is proportional to displacement and opposite in direction
Use the correct sign convention (negative sign) to indicate direction

Common mistakes

Using F=−mgθ without the L factor, giving wrong proportionality
Forgetting to mention that the force is directed towards equilibrium
Confusing displacement with angle and not showing the small‑angle approximation

Mark scheme (5 marks)

There is a restoring force acting on the bob
The restoring force is directed towards the equilibrium/rest position
The magnitude of the restoring force is proportional to the displacement from the equilibrium position
The acceleration of the bob is proportional to its displacement from the equilibrium position
The acceleration (and restoring force) is always directed towards the equilibrium position, i.e. opposite to the displacement

Key terms in this question

simple harmonic motion · amplitude