# Explain why the motion of a simple pendulum released from a small angle approximates simple harmonic motion, and identify one condition under which this approximation breaks down.

> IB DP Physics Higher Level (2023 syllabus) — C.1 Simple harmonic motion · Explain · 4 marks

> A simple pendulum consists of a small bob of mass m suspended by a light, inextensible string of length L. The pendulum is displaced through a small angle θ from its equilibrium position and released from rest.

## Mark scheme (4 marks)

1. The restoring force (or the component of gravity along the arc) is proportional to the displacement from equilibrium for small angles.
2. For small angles, sin θ ≈ θ (in radians), so the restoring force becomes approximately mgθ, which is linearly proportional to the angular displacement.
3. A linear restoring force directed towards the equilibrium position satisfies the defining condition of SHM: acceleration (or force) is proportional to displacement and directed towards the equilibrium position.
4. The approximation breaks down when the angle of displacement is large (typically θ greater than about 10–15°), because sin θ is no longer approximately equal to θ and the restoring force becomes non-linear, so the motion is no longer SHM.

## Key terms

- [simple harmonic motion](https://www.gradenine.co.uk/glossary/simple-harmonic-motion)

## Related

- [Revision notes for IB DP Physics Higher Level (2023 syllabus)](https://www.gradenine.co.uk/learn)
- [How to answer "Explain" questions](https://www.gradenine.co.uk/tools/command-word-cheatsheet)
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