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A2 · Practice questions · Damping and resonance

Lose energy, or gain it.

Six original Cambridge-style questions on light, critical and heavy damping, forced oscillations, and resonance when the driving frequency matches the natural frequency.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
Keep these straight

Damping drains energy; resonance pours it in.

01
Analysis
[3 marks]

The suspension of a car is designed so that, after the wheel passes over a bump, the body of the car is critically damped. State what is meant by critical damping, and explain why light damping and heavy damping would each be unsuitable.

  • Critical damping returns the system to its equilibrium position in the shortest possible time without it oscillating. ✓
  • With light damping the car body would continue to bounce with a slowly decaying amplitude after every bump, which is uncomfortable and lets the tyres lose contact with the road. ✓
  • With heavy damping the body would creep back to equilibrium so slowly that the suspension would still be compressed when the next bump arrived, leaving little travel to absorb it. ✓
02
Calculation
[3 marks]

A lightly damped pendulum is released with an amplitude of 60 mm. After five complete oscillations the amplitude is 48 mm. Calculate the fraction of the oscillator's energy that has been dissipated in these five oscillations. Assuming the same fraction is lost in every five oscillations, determine the amplitude after ten oscillations, and state the effect of the damping on the period.

  • The energy of an oscillation is proportional to the amplitude squared, so the fraction remaining is (48 / 60)² = 0.64 and the fraction dissipated is 1 − 0.64 = 0.36, or 36%. ✓
  • If each five oscillations keep 64% of the energy, they keep 80% of the amplitude: after ten oscillations the amplitude is 0.80 × 48 = 38 mm. ✓
  • Light damping leaves the period (and so the frequency) almost unchanged; only the amplitude decays. ✓
03
Analysis
[3 marks]

A singer can shatter a thin wine glass by holding one particular note loudly. Explain, in terms of forced oscillations and resonance, why only that note works and why it must be sung loudly.

  • The glass has a natural frequency at which it vibrates freely when tapped; the sound wave applies a periodic driving force to it, so the glass performs forced oscillations at the frequency of the note. ✓
  • When the driving frequency equals the natural frequency the glass resonates: energy is transferred to it most efficiently and its amplitude of vibration is a maximum, so only the matching note has this effect. ✓
  • The glass is damped, so the amplitude at resonance depends on the amplitude of the driving force: the note must be loud enough for the vibration amplitude to exceed the strain at which the glass cracks. ✓
04
Calculation
[3 marks]

A washing machine stands on rubber feet which give it a natural frequency of vertical oscillation of 12 Hz. During the spin cycle an unbalanced load makes the drum turn at 720 revolutions per minute. Explain why the machine vibrates violently at this speed, and calculate the maximum acceleration of the machine if the amplitude of its vertical oscillation reaches 2.5 mm.

  • The unbalanced drum applies a periodic driving force at 720 / 60 = 12 Hz, equal to the natural frequency, so the machine resonates and its amplitude is a maximum. ✓
  • ω = 2πf = 2π × 12 = 75.4 rad s⁻¹. ✓
  • a₀ = ω²x₀ = 75.4² × 2.5 × 10⁻³ = 14 m s⁻², more than g, so the machine's feet momentarily leave the floor and it "walks". ✓
05
Analysis
[3 marks]

A graph of amplitude against driving frequency is drawn for a driven oscillator, first with light damping and then with heavier damping. Describe three ways in which the curve for the heavier damping differs from the curve for the light damping, and state one feature that both curves share.

  • The peak amplitude is lower, because more energy is dissipated per cycle so a smaller amplitude is reached before the energy supplied by the driver balances the energy lost. ✓
  • The peak is broader, spanning a wider range of driving frequencies, and the maximum occurs at a slightly lower frequency than the natural frequency. ✓
  • Both curves share the same amplitude at very low driving frequencies, where the oscillator simply follows the driver, and both fall towards zero at very high driving frequencies. ✓
06
Calculation
[4 marks]

A footbridge has a natural frequency of vertical oscillation of 1.8 Hz. When a column of people crosses it, walking at 108 steps per minute, the bridge begins to oscillate with an amplitude that grows to 12 mm.

(a) Explain why this walking pace causes a large amplitude. (b) Calculate the maximum vertical acceleration of the bridge deck at this amplitude. (c) Engineers fit dampers to the bridge. State two effects this has on the graph of amplitude against walking frequency.

  • (a) 108 steps per minute is 108 / 60 = 1.8 Hz, so the periodic force of the footsteps matches the natural frequency of the bridge and it resonates. ✓
  • (b) ω = 2πf = 2π × 1.8 = 11.3 rad s⁻¹. ✓
  • a₀ = ω²x₀ = 11.3² × 0.012 = 1.5 m s⁻², about 15% of g, easily felt by the people on the deck. ✓
  • (c) The resonance peak becomes much lower, and it becomes broader and moves to a slightly lower frequency. ✓

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