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.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
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.
Model answer (4 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 (in radians), so F≈−mgθ, which is proportional to the angular displacement θ and directed towards equilibrium. A linear restoring force proportional to displacement is the defining condition of simple harmonic motion, so the pendulum behaves as SHM for small angles.
The approximation fails when the displacement is large (≈10–15° or more); then sinθ is no longer ≈θ, the restoring force becomes non‑linear and the motion is no longer simple harmonic.
The approximation fails when the displacement is large (≈10–15° or more); then sinθ is no longer ≈θ, the restoring force becomes non‑linear and the motion is no longer simple harmonic.
Examiner tips
- Show the force component and the small‑angle approximation explicitly; link proportionality to SHM definition.
- Mention the angle range where the approximation breaks down to earn the extra point.
Common mistakes
- Using sinθ≈θ for all angles; not recognising the linearity condition; forgetting to state the failure condition.
Mark scheme (4 marks)
- The restoring force (or the component of gravity along the arc) is proportional to the displacement from equilibrium for small angles.
- For small angles, sin θ ≈ θ (in radians), so the restoring force becomes approximately mgθ, which is linearly proportional to the angular displacement.
- 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.
- 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 in this question
Related
- All IB DP Physics Higher Level (2023 syllabus) revision notes →
- How to answer a "Explain" question →
- Decode the mark scheme abbreviations →
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