What is the defining condition?

This single condition produces sinusoidal motion. The constant ω relates to the period by ω = (2π)/(T). For a mass on a spring T = 2((m)/(k)), and for a simple pendulum T = 2((L)/(g)) (for small angles).

An object in SHM has ω = 10 rad per second and amplitude 0.030 m. Find its maximum speed. [2 marks]

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What defines simple harmonic motion and how does it lose or gain energy?

The defining condition for simple harmonic motion, displacement, velocity and acceleration in SHM, energy interchange in an oscillator, and free, damped and forced oscillations with resonance.

A focused answer to the Edexcel 9PH0 SHM content, covering the defining condition, displacement, velocity and acceleration, energy interchange in an oscillator, and free, damped and forced oscillations with resonance.

Generated by Claude Opus 4.812 min answerUpdated 2026-06-02

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What this dot point is asking

Edexcel wants you to state the defining condition for simple harmonic motion (SHM), describe how displacement, velocity and acceleration vary, analyse the interchange between kinetic and potential energy in an oscillator, and distinguish free, damped and forced oscillations, including resonance.

The answer

The defining condition

This single condition produces sinusoidal motion. The constant $\omega$ relates to the period by $\omega = \frac{2\pi}{T}$ . For a mass on a spring $T = 2\pi\sqrt{\frac{m}{k}}$ , and for a simple pendulum $T = 2\pi\sqrt{\frac{L}{g}}$ (for small angles).

Displacement, velocity and acceleration

The speed is greatest at the centre (where displacement is zero) and zero at the extremes; the acceleration is greatest at the extremes and zero at the centre. Velocity leads displacement by a quarter cycle and acceleration is in antiphase with displacement.

Energy interchange

A graph of energy against displacement shows potential energy as a parabola rising to the amplitude, kinetic energy as the inverted parabola, and their sum as a constant horizontal line.

Free, damped and forced oscillations

A free oscillation happens at the system's natural frequency with no driving and no friction. Real oscillators are damped: resistive forces remove energy so the amplitude decays. Light damping decays slowly; critical damping returns to equilibrium fastest without overshoot; heavy (over) damping returns slowly. A forced oscillation is driven by a periodic external force; when the driving frequency matches the natural frequency, resonance occurs and the amplitude becomes very large (limited only by damping).

Mass-spring period and energy

A $0.20$ kg mass on a spring of stiffness $80$ N per metre oscillates with amplitude $0.050$ m. Find the period and the total energy.

step 1: Period

$T = 2\pi\sqrt{\frac{m}{k}} = 2\pi\sqrt{\frac{0.20}{80}} = 2\pi\sqrt{2.5 \times 10^{-3}} = 2\pi \times 0.050 = 0.31$ s.

step 2: Total energy (maximum elastic PE at the amplitude)

$E = \frac{1}{2}kA^2 = \frac{1}{2} \times 80 \times (0.050)^2 = \frac{1}{2} \times 80 \times 2.5 \times 10^{-3} = 0.10$ J.

This total energy is constant; at the equilibrium it is all kinetic, at the extremes all potential.

Examples in context

A pendulum clock keeps time because its period depends only on length and $g$ , not amplitude (for small swings). Car suspensions are deliberately damped (near critical) so the car settles quickly after a bump without bouncing. Resonance can be destructive: soldiers break step on bridges, and engineers design structures to avoid resonating with wind or earthquakes (the Tacoma Narrows bridge is the classic failure). Resonance is also useful, tuning a radio to a station's frequency and producing the rich tone of musical instruments.

Try this

Q1. State the defining condition for simple harmonic motion. [1 mark]

Cue. The acceleration is proportional to the displacement from equilibrium and directed towards it, $a = -\omega^2 x$ .

Q2. An object in SHM has $\omega = 10$ rad per second and amplitude $0.030$ m. Find its maximum speed. [2 marks]

Cue. $v_{\max} = \omega A = 10 \times 0.030 = 0.30$ m per second.

Q3. Explain what is meant by resonance. [2 marks]

Cue. When a system is driven at its natural frequency, energy is transferred most efficiently and the amplitude of oscillation becomes very large.

Exam-style practice questions

Practice questions written in the style of Pearson Edexcel exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Edexcel 20194 marksA mass on a spring oscillates with simple harmonic motion of amplitude

0.040

m and period

0.50

s. Calculate the maximum speed and maximum acceleration of the mass.

Show worked answer →

Angular frequency: $\omega = \frac{2\pi}{T} = \frac{2\pi}{0.50} = 12.6$ rad per second.

Maximum speed: $v_{\max} = \omega A = 12.6 \times 0.040 = 0.50$ m per second.

Maximum acceleration: $a_{\max} = \omega^2 A = (12.6)^2 \times 0.040 = 158 \times 0.040 = 6.3$ m per second squared.

Markers reward $\omega = \frac{2\pi}{T}$ , $v_{\max} = \omega A$ , $a_{\max} = \omega^2 A$ , and the values $0.50$ m per second and $6.3$ m per second squared.

Edexcel 20225 marksA simple pendulum has a period of

2.0

s. Determine its length and explain, using the idea of resonance, why a child can be pushed to a large amplitude on a swing with small pushes. Take

g = 9.81

m per second squared.

Show worked answer →

Pendulum period: $T = 2\pi\sqrt{\frac{L}{g}}$ , so $L = \frac{gT^2}{4\pi^2} = \frac{9.81 \times 2.0^2}{4\pi^2} = \frac{39.2}{39.5} = 0.99$ m.

Resonance: a swing has a natural frequency. If the pushes are applied at that natural frequency, energy is transferred efficiently each cycle and the amplitude builds up, so small periodic pushes produce a large amplitude.

Markers reward the rearrangement to $L = \frac{gT^2}{4\pi^2}$ , the value about $1.0$ m, and a correct resonance explanation (driving at the natural frequency).

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Sources & how we know this

Pearson Edexcel A-Level Physics (9PH0) specification — Pearson Edexcel (2015)