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A Level Physics 9702 · A Level · Topic 17 · ~5 min read

Oscillations

Describing oscillations, simple harmonic motion defined by a = −ω²x, displacement, velocity and acceleration equations and graphs, energy in SHM, and damping, forced oscillations and resonance.

Amplitude, period, frequency, phaseSHM definition a = −ω²xx = x₀ sin ωtv = ±ω√(x₀² − x²)Energy in SHMLight, critical & heavy dampingForced oscillationsResonance
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Paper tag = where it is examinedDATA & FORMULAE given in the examC1A1B1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • SHM: acceleration is proportional to displacement from a fixed point and always directed towards it: a = −ω²x.
  • Solutions: x = x₀ sin ωt (starting at the centre) or x₀ cos ωt (starting at an extreme). ω = 2πf = 2π/T.
  • Speed: v = ±ω√(x₀² − x²), so v₀ = ωx₀ at the centre and zero at the extremes. Maximum acceleration ω²x₀ is at the extremes.
  • Energy: total E = ½mω²x₀², constantly swapping between kinetic and potential. Damping removes energy; resonance happens when the driving frequency equals the natural frequency.

By the end you can

  • Understand amplitude, period, frequency, angular frequency and phase difference.
  • Define SHM by a = −ω²x and use x = x₀ sin ωt, v = v₀ cos ωt and v = ±ω√(x₀² − x²).
  • Describe the interchange of kinetic and potential energy, and use E = ½mω²x₀².
  • Describe light, critical and heavy damping, forced oscillations and resonance.

1Simple harmonic motion

The defining equation

Simple harmonic motion — motion in which the acceleration is proportional to the displacement from a fixed point and is directed towards that point.

a = −ω²x ; x = x₀ sin ωt ; v = v₀ cos ωt ; v = ±ω√(x₀² − x²) DATA & FORMULAE
ω = 2πf, v₀ = ωx₀
txtvtax = x₀ sin ωtv leads xby π/2a = −ω²xopposite to xx max: v = 0, a max
x, v and a are each a quarter-cycle apart: v leads x by π/2, and a is in antiphase with x.
  • Phase difference: the fraction of a cycle between two oscillations, as an angle (2π rad = one cycle).
  • The period of SHM doesn't depend on the amplitude (isochronous) — the basis of pendulum clocks.
Worked example

A mass on a spring oscillates in SHM at 2.5 Hz with amplitude 0.040 m. Find (a) the maximum speed, (b) the maximum acceleration, (c) the speed at x = 0.020 m.

  1. ω = 2π × 2.5 = 15.7 rad s⁻¹ C1
  2. (a) v₀ = ωx₀ = 0.63 m s⁻¹ A1
  3. (b) amax = ω²x₀ = 9.9 m s⁻² A1
  4. (c) v = 15.7√(0.040² − 0.020²) = 0.54 m s⁻¹ A1
Common mistakeCalculator in degree mode for x = x₀ sin ωt. ωt is in radians.

2Energy and damping

Energy in SHM

xEETEpEk−x₀x₀Ep = ½mω²x², Ek = ½mω²(x₀² − x²): they add to ET = ½mω²x₀²
Kinetic energy is greatest at the centre; potential energy greatest at the extremes; the total is constant.
Etotal = ½mω²x₀² ; EK = ½mω²(x₀² − x²)
undamped
Worked example

The mass above is 0.30 kg. Find the total energy of the oscillation.

  1. E = ½ × 0.30 × 15.7² × 0.040² C1
  2. E = 0.059 J A1

Damping and resonance

txlight: decaying oscillationcritical: fastest returnheavy: slow return
Light damping: amplitude decays gradually; critical damping: returns to equilibrium fastest without oscillating.
DampingWhat happens
Lightoscillates with gradually decreasing amplitude; the period is almost unchanged
Criticalreturns to equilibrium in the shortest time without oscillating (e.g. car suspension)
Heavyreturns to equilibrium slowly, without oscillating

Resonance — the maximum amplitude of a forced oscillation, which occurs when the driving frequency equals the natural frequency of the system.

famplitudef₀red: light dampingorange: more dampingblue: heavy dampingMore damping: lower, broader peak, shifted slightly below f₀.
More damping lowers and broadens the resonance peak and shifts it slightly lower.
  • Useful resonance: microwave heating of water, tuning a radio, MRI. Problem resonance: bridges, car panels vibrating; reduced by damping.
Common mistakeDefining SHM as "a ∝ x" without the direction (towards the fixed point), or describing resonance without "driving frequency = natural frequency". Both lost marks in 2024.
Linking questionSHM is the projection of circular motion; wave motion is in Waves; and energy ideas are from Work, energy and power.

Test yourself — cover the answers first

Quick check

  1. State the defining equation of SHM.
  2. Where is the speed greatest in SHM?
  3. What is the phase difference between x and a in SHM?
  4. Which damping returns a system to rest fastest without oscillating?
  5. What happens to the energy of an SHM oscillator if its amplitude doubles?

Exam-style

  1. A pendulum has period 2.0 s and amplitude 0.10 m. Find its displacement 0.25 s after passing through the centre.
  2. For a body in SHM with ω = 4.0 rad s⁻¹, find the acceleration when x = −0.050 m.
  3. Explain why heavy damping makes the resonance peak lower and broader.
Show answers
  1. a = −ω²x
  2. at the equilibrium position
  3. π rad (antiphase)
  4. critical damping
  5. it quadruples
  6. ω = π; x = 0.10 sin(π × 0.25) = 0.071 m
  7. a = −16 × (−0.050) = +0.80 m s⁻²
  8. more energy is removed each cycle, so the maximum amplitude is smaller and less sharply dependent on frequency
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by Cambridge University Press & Assessment. Syllabus codes 9709 and 9702 refer to the Cambridge International AS & A Level syllabuses.