The big idea: Pull something away from its resting place and let go — a guitar string, a swing, a mass on a spring.
What pulls it back: A restoring force pulls it back toward the middle. It overshoots, comes back, and swings to and fro.
When it counts as SHM: When the acceleration is always proportional to the displacement and points back toward the middle, the motion is simple harmonic motion (SHM).
New words — equilibrium & restoring force: Equilibrium is the resting position in the middle, where the object would sit still (displacement x = 0).
A restoring force is any force that always pushes or pulls the object back toward that middle position.
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The two conditions for SHM: 1. The acceleration is proportional to the displacement (twice as far out → twice the acceleration).
2. The acceleration is always directed back toward equilibrium (opposite to the displacement).
Where the acceleration comes from
Spotting SHM in a question: Force against displacement: a straight line through the origin with a negative gradient (−k) means SHM; so does a straight line of a against x with gradient −ω².
A spring obeying Hooke's law gives SHM, because its force is proportional to its extension. A floating cork pushed down by x gets extra upthrust ρgAx, proportional to x and upwards, back to equilibrium: SHM too.
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Those two conditions are captured by one equation, given in the data booklet. It says the acceleration a is a fixed multiple of the displacement x, with a minus sign for 'back toward the middle'.
- acceleration of the object (m s⁻²)
- angular frequency — how fast it oscillates (rad s⁻¹)
- displacement from equilibrium (m)
How to read the equation: a ∝ x — bigger displacement gives bigger acceleration (that's the proportional bit).
The minus sign is the back-toward-the-middle bit. The constant ω² is always positive, so it never changes that direction.
Plot acceleration against displacement and you get the signature of SHM: a straight line through the origin (proportional) that slopes down (the minus sign). The steepness of that line is ω².
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A trolley on a spring has its acceleration measured at several displacements. The data fit a straight line through the origin: a = −9.0x (with a in m s⁻² and x in m). State whether the trolley moves with SHM, and find its angular frequency ω.
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How this is tested — 3.1.1 is mostly an understanding question, not a calculation:
Paper 1A
- A quick identify question — e.g. spotting the correct description of a lightly damped oscillation.
Paper 2
- Outline why a displaced object (a floating cork, a mass on a spring) oscillates with SHM — name the restoring force and link it to a = −ω²x.
The classic trap: Forgetting the direction. SHM needs the acceleration back toward equilibrium, not just proportional to x.
How to OUTLINE why something does SHM: Two sentences earn the marks:
1. When displaced, there is a restoring force (so an acceleration) directed back toward equilibrium.
2. That force/acceleration is proportional to the displacement — which is exactly a = −ω²x, the condition for SHM.
The words to know
| Word | What it means |
|---|---|
| Equilibrium position | Where the resultant force is zero (x = 0) |
| Displacement x | Distance and direction from equilibrium |
| Restoring force | A force that always points back towards equilibrium |
| Simple harmonic motion | Motion with a = −ω²x: acceleration proportional to displacement and opposite to it |
A small cork floats at rest on the surface of a pond. It is pushed down a little and released, then bobs up and down. Outline why the cork undergoes simple harmonic motion.
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