Simple Harmonic Motion — Quick Summary

Mr Toogood's Physics · Periodic motion

AQA 3.6.1.2
a = −ω²x
Defining equation
x = A cos ωt
Displacement
v=±ω√(A²−x²)
Speed
v_max=ωAa_max=ω²A
Maxima

Conditions for SHM

An oscillator moves with SHM only if all three hold:

  • Acceleration is directed towards a fixed equilibrium point.
  • Acceleration is in the opposite direction to displacement.
  • Acceleration is directly proportional to displacement from that point.
Graph of acceleration against displacement for SHM, a straight line with negative gradient through the origin

a–x graph: straight line through the origin, gradient = −ω².

Whenever displacement is positive, acceleration is negative (and vice versa) — always directly proportional, never a curve.

x, v, a — how they relate

Linked displacement-time, velocity-time and acceleration-time graphs for SHM

v–t is the gradient of x–t; a–t is the gradient of v–t.

  • At maximum displacement (x=±A): v=0, a is maximum (opposite direction).
  • At equilibrium (x=0): v is maximum (v_max=ωA), a=0.

Link to circular motion

Diagram showing SHM as the projection of circular motion, linking radius to amplitude

SHM = the projection of circular motion onto a diameter.

Picture a particle spinning at radius r; its shadow on a diameter oscillates with SHM. This gives x = A cos θ = A cos ωt, where:

Circular motionSHM
Radius, rAmplitude, A
Angular displacement, θPhase, ωt
Angular velocity, ωAngular frequency, ω

Key definitions

  • Amplitude, A — maximum displacement from equilibrium (m).
  • Period, T — time for one complete cycle (s).
  • Frequency, f — complete oscillations per second (Hz).
  • Phase — position within a cycle; phase difference = 2πΔt/T = ωΔt (rad).
In phase / antiphase: a phase difference of a multiple of 2π rad means back in phase; π rad means exactly out of phase (antiphase).

Exam essentials

Reading the graphs

  • x–t: cosine shape (starts at max displacement).
  • v–t: sine shape — a quarter-cycle (π/2 rad) out of phase with x–t.
  • a–t: mirror image of x–t (a = −ω²x throughout).

Maximum values

  • v_max=ωA at x=0 (equilibrium).
  • a_max=ω²A at x=±A (extremes).

Common slips

  • Don't drop the minus sign in a=−ω²x — direction matters.
  • The ± in the speed equation reflects direction, not measurement error — state which way if asked.
  • Keep phase differences in radians unless a question explicitly asks for degrees.