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9702 · 17.3

Damped and forced oscillations, resonance — practice questions

Practice and worked examples for 9702 Damped and forced oscillations, resonance. Short previews only — attempt the full question in MarkScheme against the official scheme.

Worked example 1

An oscillating system experiences resonance. Describe how its amplitude-driving frequency graph would change if the damping in the system was increased significantly.

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  1. Reduced Peak Amplitude: The most noticeable change would be a substantial decrease in the maximum amplitude achieved at the resonant frequency. The peak of the curve would be much lower.
  2. Broader Resonance Curve: The curve would become wider and less sharp. This means the system would still oscillate with a relatively large amplitude over a wider range of driving frequencies around the natural frequency, not just a very narrow band.
  3. Slight Frequency Shift (Optional but accurate): In cases of heavy damping, the frequency at which the maximum amplitude occurs might shift slightly to a lower value compared to the original natural frequency, though this effect is often less prominent than amplitude reduction and broadening.

Worked example 2

A mechanical oscillator of mass 0.50 kg is attached to a spring, giving it a natural frequency of 2.0 Hz. The system is lightly damped and is driven by a periodic force at its resonant frequency. The system reaches a steady-state amplitude of 5.0 cm. The damping force is given by F_d = -0.8v, where v is the velocity in m/s. Calculate the average power that must be supplied by the driving force to maintain this constant amplitude.

Show solution outline
  1. Identify knowns and the goal:
    • Natural frequency (f₀) = 2.0 Hz.
    • Amplitude at resonance (A) = 5.0 cm = 0.050 m.
    • Damping coefficient (b) = 0.8 Ns/m (from the relation F_d = -0.8v).
    • Goal: Calculate the average power supplied (<P_supplied>).
  2. State the principle: For the amplitude to remain constant, the average power supplied by the driver must equal the average power dissipated by damping.
    • <P_supplied> = <P_dissipated>
  3. Calculate the angular frequency (ω): Resonance occurs when the driving frequency equals the natural frequency.
    • ω = 2πf₀
    • ω = 2π × 2.0 Hz = 4.0π rad/s ≈ 12.57 rad/s.
  4. Use the formula for average power dissipated: The average power dissipated by a damping force F_d = -bv in simple harmonic motion is given by:
    • <P_dissipated> = ½ * b * (Aω)²
  5. Substitute values and calculate:
    • <P_dissipated> = ½ × (0.8 Ns/m) × (0.050 m × 12.57 rad/s)²
    • <P_dissipated> = 0.4 × (0.6285)²
    • <P_dissipated> = 0.4 × 0.3950
    • <P_dissipated> ≈ 0.158 W
  6. Final Answer: The average power that must be supplied by the driving force is 0.16 W (to 2 significant figures).
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