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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 simple harmonic motion.
A body oscillates with simple harmonic motion. The angular frequency of the motion is 12 rad s⁻¹. The period of the motion is
A body oscillates with simple harmonic motion of amplitude A. Which graph shows how the kinetic energy of the body varies with its displacement y?
A mass on a spring is pulled down to its maximum displacement and released from rest at time t = 0. A pupil models its displacement as y = A sin ωt. A second pupil makes the following statements.
A block of mass m rests on a spring of spring constant k, compressing it by a distance Δy. The block is replaced by one of three times the mass, on the same spring. The new compression is
The demand of the real test — one rearrangement, one conversion or one intermediate step.
A machine that fills bags of flour is mounted on springs so that its vibration is not passed into the floor. The machine has a mass of 18 kg. When it is running it oscillates vertically with simple harmonic motion of period 0.65 s and amplitude 0.035 m. For a mass on springs, ω = √(k / m), where k is the combined spring constant of the mounting.
(a) The vertical displacement y of the machine can be described by the expression y = A sin ωt, where the symbols have their usual meaning. Show that this expression is a solution to the relationship F = −mω2y.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) Determine the combined spring constant of the mounting.2
(c) Calculate the maximum speed of the machine as it oscillates.2
A child bounces on a pogo stick. Once the bouncing is steady the motion of the child can be modelled as simple harmonic motion. The child has a mass of 32 kg, the angular frequency of the motion is 6.2 rad s⁻¹, and the amplitude of the motion is 0.18 m.
(a) Calculate the kinetic energy of the child at the instant the displacement is half the amplitude.3
(b) Calculate the maximum acceleration of the child during the motion.2
(c) The child is at the top of the bounce, at maximum displacement, at time t = 0. Which graph shows how the acceleration of the child varies with time over the first full oscillation?1
Two or more steps, an unfamiliar context, and a judgement to justify.
A mass m hangs on a spring of spring constant k and is set oscillating with simple harmonic motion. Hooke's law gives the restoring force on the mass as F = −ky.
(a) Using F = −ky and F = ma, derive the relationship ω = √(k / m).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) The same mass and spring are now set oscillating with a larger amplitude. State the effect this has on the angular frequency of the oscillation.1
A diver stands on the free end of a springboard and bounces on the spot. The vertical displacement of the free end, in metres, at time t seconds is given by y = 0.19 sin 7.8t.
(a) Determine the time taken for the free end of the board to travel from its rest position to its maximum displacement.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.