Explain how the restoring force acting on a mass–spring system gives rise to simple harmonic motion, and state the condition on this force that must be satisfied for the motion to be classified as simple harmonic.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
A block of mass m is attached to a horizontal spring of spring constant k and rests on a frictionless surface. The block is displaced from its equilibrium position and released.
Model answer (4 marks)
When the block is displaced a distance x from equilibrium the spring exerts a restoring force F = −kx, directed towards the equilibrium position. The negative sign shows the force is opposite to the displacement. Because F = ma, the acceleration a = −(k/m)x is also proportional to the displacement and directed towards equilibrium. This means the block is always pulled back, overshoots, and then pulled back again, producing an oscillatory motion about the equilibrium point. The condition for the motion to be simple harmonic is that the restoring force (or acceleration) must be directly proportional to the displacement and directed towards the equilibrium position (i.e. F = −kx).
Examiner tips
- Use the equation F = −kx to show proportionality and direction; mention a = −(k/m)x; explain overshoot and oscillation; state the proportionality condition explicitly.
Mark scheme (4 marks)
- When displaced, the spring exerts a restoring force directed towards the equilibrium position (opposing the displacement).
- The magnitude of the restoring force is proportional to the displacement from equilibrium (F = −kx), so the acceleration is also proportional to displacement.
- Because the net force (and hence acceleration) is always directed towards equilibrium, the block overshoots, creating an oscillatory motion about the equilibrium position.
- The condition for SHM is that the acceleration (or restoring force) must be directly proportional to displacement and directed towards the equilibrium position / opposite to the displacement.
Key terms in this question
restoring force · simple harmonic motion
Related
- All IB DP Physics Higher Level (2023 syllabus) revision notes →
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