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Advanced Higher Physics · Simple Harmonic Motion
Answer everything, then mark it in one go.
Have your jotter and a pen beside you. Do every calculation and every derivation in your jotter, showing each line exactly as you would in the exam — this page asks you only for the relationship you used and your final answer with its unit, and the two derivations are marked against the scheme, line by line.
Nothing is timed and nothing is sent anywhere — your answers stay on this device, so you can close the tab and come back.
| Quantity | Relationship |
|---|---|
| Angular frequency, and period | ω = 2πf T = 1 / f T = 2π / ω |
| The condition for SHM | a = d2y/dt2 = −ω2y |
| Displacement | y = A cos ωt or y = A sin ωt |
| Velocity at displacement y | v = ±ω√(A2 − y2) |
| Kinetic and potential energy | Ek = ½mω2(A2 − y2) Ep = ½mω2y2 |
The mass–spring result ω = √(k/m) is not on the relationships sheet. The exam gives it to you when it needs it — or expects you to reach it from F = −ky and F = ma, which is exactly the derivation in this test.
| Quantity | Symbol | Value |
|---|---|---|
| Gravitational acceleration on Earth | g | 9.8 m s⁻² |
✏️ Working goes in your jotter. On this page you give the relationship and the final answer only — the mark scheme at the end shows the substitution you should have written down.
Everyone should aim for full marks here. One step, no rearranging.
A pupil makes the following statements about a body moving with simple harmonic motion.
A body oscillates with simple harmonic motion. The period of its motion is 0.24 s. The angular frequency of the motion is
A system is displaced from its rest position and released. Which graph shows the displacement of a critically damped system varying with time?
A body oscillates with simple harmonic motion between displacements +A and −A. A pupil makes the following statements about the motion.
A mass hanging on a spring oscillates with angular frequency ω. The mass is now replaced by one of four times the original mass, on the same spring. The angular frequency of the new oscillation is
The demand of the real test — one rearrangement, one conversion or one intermediate step.
A hanging basket is suspended from a spring bracket outside a shop. The basket has a mass of 2.4 kg. It is pulled down 0.055 m from its equilibrium position and released, and then oscillates vertically. The basket completes 20 oscillations in 17.0 s. Frictional forces are negligible.
(a) The vertical displacement y of the basket can be described by the expression y = A cos ωt, where the symbols have their usual meaning. Using calculus methods, show that this expression is a solution to the equation d2y/dt2 + ω2y = 0.3
Write this out in your jotter, one line at a time. When you have finished, reveal the mark scheme and award yourself the marks line by line.
(b) Calculate the angular frequency of the oscillation of the basket.2
(c) Calculate the maximum speed of the basket during its oscillation.2
A tall building is protected from swaying by a tuned mass damper — a heavy block hung from springs near the top of the building, free to oscillate. A working model of one is built in a laboratory. The block of the model has a mass of 4.8 kg, its oscillation has an angular frequency of 2.6 rad s⁻¹, and it is released from an amplitude of 0.14 m.
(a) Calculate the maximum potential energy stored in the springs of the model.3
(b) The model is displaced and released, and its motion is recorded. State the type of damping shown in the graph.1
(c) The model should return to its rest position in the shortest possible time, without oscillating. State the type of damping needed, and suggest one change to the model that would achieve it.2
Two or more steps, an unfamiliar context, and a judgement to justify.
A body oscillates with simple harmonic motion of amplitude A and angular frequency ω. Its displacement at time t is y = A sin ωt.
(a) Using calculus methods, derive the relationship v = ±ω√(A2 − y2).3
Write this out in your jotter, one line at a time. When you have finished, reveal the mark scheme and award yourself the marks line by line.
(b) State the displacement at which the speed of the body is greatest, and the speed it reaches there.1
As people walk across it, the deck of a footbridge oscillates vertically. The motion of the middle of the deck can be modelled as simple harmonic motion. Its vertical displacement, in metres, at time t seconds is given by y = 0.018 sin 9.4t.
(a) Determine the speed of the middle of the deck at the instant its displacement from the rest position is 0.011 m.3
Bands are indicative only — they are a guide to where you are, not a grade.
Sources & credits: questions are adapted from Advanced Higher Physics past papers, reproduced and adapted for educational use — past papers and marking instructions © Qualifications Scotland (SQA). Numbers, apparatus and contexts have been changed, and two questions are original to this site. Oscillation diagrams and graphs drawn for this site © R Stewart, 2026.