Explain why the motion of a simple pendulum released from a small angle can be modelled as simple harmonic motion, and identify one condition under which this model would no longer be valid.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
A simple pendulum consists of a small bob suspended from a fixed point by a light, inextensible string. When displaced from its equilibrium position and released, the bob oscillates back and forth.
Model answer (4 marks)
The restoring force is the component of gravity along the arc,
F=−mg sinθ, which is directed towards the equilibrium position.
For small angles sinθ≈θ (in radians), so F≈−mgθ. The displacement from equilibrium is s=Lθ, giving F≈−(mg/L)s. Thus the restoring force (or acceleration) is proportional to the displacement.
These two facts – force directed towards equilibrium and proportional to displacement – satisfy the definition of simple harmonic motion.
The model breaks down when the displacement angle is large (≈10°–15° or more) because the small‑angle approximation sinθ≈θ no longer holds, so the restoring force is no longer proportional to displacement.
F=−mg sinθ, which is directed towards the equilibrium position.
For small angles sinθ≈θ (in radians), so F≈−mgθ. The displacement from equilibrium is s=Lθ, giving F≈−(mg/L)s. Thus the restoring force (or acceleration) is proportional to the displacement.
These two facts – force directed towards equilibrium and proportional to displacement – satisfy the definition of simple harmonic motion.
The model breaks down when the displacement angle is large (≈10°–15° or more) because the small‑angle approximation sinθ≈θ no longer holds, so the restoring force is no longer proportional to displacement.
Examiner tips
- Show the force component and the small‑angle approximation explicitly
- Explain why proportionality gives SHM
- Mention the angle limit where the approximation fails
Common mistakes
- Using degrees instead of radians in the sinθ≈θ approximation
- Failing to state that the force must be directed towards equilibrium
- Not specifying the angle range where the model fails
Mark scheme (4 marks)
- The restoring force (or acceleration) is directed towards the equilibrium position
- For small angles, the restoring force (or acceleration) is proportional to the displacement from equilibrium
- These two conditions (restoring force directed towards equilibrium and proportional to displacement) satisfy the definition of simple harmonic motion
- The model is no longer valid when the angle of displacement is large (greater than approximately 10°–15°), because the small-angle approximation breaks down and the restoring force is no longer proportional to displacement
Key terms in this question
Related
- All IB DP Physics Standard Level (2023 syllabus) revision notes →
- How to answer a "Explain" question →
- Decode the mark scheme abbreviations →
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