The big idea: Pull something away from its resting place and let go — a guitar string, a swing, a mass on a spring.
A restoring force pulls it back toward the middle. It overshoots, comes back, and swings to and fro.
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.
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).
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.
IB-style question — is this motion SHM?
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.
Solution
- Compare the data with the given defining condition:
- The data a = −9.0x has the same form (a ∝ x, with a minus sign), so it is SHM. Match the constants:
- Take the square root — keep the unit:
Final answer
Yes, it is SHM (a is proportional to −x), and ω = 3.0 rad s⁻¹.
See how examiners mark answers
Access past paper questions with model answers. Learn exactly what earns marks and what doesn't.
How this is tested: 3.1.1 is mostly an understanding question, not a calculation.
- 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. - Paper 1A: a quick identify question — e.g. spotting the correct description of a lightly damped oscillation.
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.
IB-style question — outline why the cork does SHM
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.
Solution (a two-mark 'outline')
- Restoring force. Pushed below the surface, the extra upthrust is bigger than the cork's weight, so there is a net force pushing it back up toward its resting position (equilibrium).
- Proportional to displacement. The further down it is pushed, the more extra water it displaces, so the net upward force — and the acceleration — is proportional to the displacement and always directed back toward equilibrium.
- Conclude. Acceleration proportional to displacement and directed back to equilibrium is the condition a = −ω²x, so the cork moves with SHM.
Final answer
It does SHM because there is a restoring force (acceleration) proportional to the displacement and always directed back toward equilibrium — exactly a = −ω²x.