Explain forced oscillations and resonance
Connect a periodic driving force to the natural frequency and explain why resonance produces the maximum amplitude.
A Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 17. These original explanations and calculations are study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use displacement, velocity, acceleration, force, energy and trigonometric functions with SI units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
When does resonance occur in a forced oscillator?
What you will learn
- Understand resonance in a forced oscillator
- Relate maximum amplitude to driving at the natural frequency
- Explain why damping limits real oscillation amplitude
A periodic force drives an oscillator
A forced oscillation occurs when a system is made to oscillate by an external periodic force. The system has a natural frequency at which it oscillates when disturbed and then allowed to move freely, subject to its conditions.
The driving frequency is the frequency of the applied periodic force. If it differs from the natural frequency, energy transfer is generally less effective than at resonance.
At resonance the amplitude is greatest
Resonance occurs when the system is forced to oscillate at its natural frequency; the oscillation amplitude is then a maximum. Energy is transferred efficiently from the driving force to the oscillator on each cycle.
In real systems, resistive forces dissipate energy and limit the amplitude. Do not describe resonance as making amplitude infinite or as requiring the driving force to stop; the continuing force supplies energy while damping removes it.
Pause and explain
What happens to oscillation amplitude at resonance?
Worked example
A child pushes a swing periodically at the swing's natural frequency. Name the phenomenon and state the amplitude response.
Show the worked solution
- A continuing periodic force is driving the swing.
- Its frequency matches the natural frequency.
- Apply the definition of resonance.
Answer Resonance occurs, and the swing reaches its maximum amplitude for the driven response.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A system has natural frequency 3.0 Hz. At what driving frequency does resonance occur, and what is the amplitude condition?
Give me a hint
Match the driving and natural frequencies.
Compare my reasoning
- Resonance occurs when fdrive = fnatural.
- Set fdrive = 3.0 Hz.
- At this condition the amplitude is a maximum.
Drive at 3.0 Hz; the oscillation amplitude is maximum at resonance.
Look for these in your work
- Frequency condition stated
- Frequency applied
- Amplitude condition stated
A lightly damped oscillator is driven at frequencies below, near and above its natural frequency. Where is the response amplitude greatest?
Give me a hint
Identify the frequency that satisfies the resonance condition.
Compare my reasoning
- The system is driven periodically at each trial frequency.
- The largest response occurs when the driving frequency matches the natural frequency.
- That condition is resonance.
The response amplitude is greatest near the natural frequency, at resonance.
Look for these in your work
- Driving frequency compared
- Natural frequency identified
- Maximum response placed at resonance
Explain why a real oscillator's amplitude does not grow without limit at resonance even though energy is supplied each cycle.
Give me a hint
Account for both energy input and energy dissipation.
Compare my reasoning
- The periodic driving force transfers energy to the oscillator.
- Resistive forces dissipate some mechanical energy to the surroundings each cycle.
- A steady response is reached when energy input balances energy loss, limiting amplitude.
Damping dissipates energy; in a steady state, input balances loss so the resonant amplitude is finite.
Look for these in your work
- Energy input noted
- Dissipation explained
- Finite steady response concluded
Common mistakes
- Defining resonance as any forced oscillation
- Confusing driving frequency with amplitude
- Claiming a real resonant amplitude is infinite
- Omitting that resonance is associated with the natural frequency
What can you explain now?
When does resonance occur in a forced oscillator?
Compare with the explanation
When the driving frequency equals the natural frequency
That is the frequency condition for resonance.
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Make it stick
What must match for resonance, and what happens to the driven amplitude?
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