A student sets up a simple pendulum and observes that it oscillates with a fixed period. The student then increases the amplitude of the pendulum's swing while keeping the length of the pendulum the same. Explain why the period of the pendulum does not change when the amplitude is increased, and describe what would happen to the period if the student instead doubled the length of the pendulum.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
Model answer (5 marks)
For a simple pendulum the period is given by T=2π√(L/g). The expression contains only the length L and the acceleration due to gravity g, so it does not depend on the amplitude of the swing.
When the amplitude is increased the restoring torque (and hence the restoring force) is larger, which gives the bob a larger acceleration. This makes the bob move faster, but the extra speed exactly compensates for the longer path, leaving the time for one complete oscillation unchanged.
If the length of the pendulum is doubled, the period becomes T'=2π√(2L/g)=√2·T. Thus the period increases by a factor of √2 (about 1.41 times the original period).
When the amplitude is increased the restoring torque (and hence the restoring force) is larger, which gives the bob a larger acceleration. This makes the bob move faster, but the extra speed exactly compensates for the longer path, leaving the time for one complete oscillation unchanged.
If the length of the pendulum is doubled, the period becomes T'=2π√(2L/g)=√2·T. Thus the period increases by a factor of √2 (about 1.41 times the original period).
Examiner tips
- State the formula T=2π√(L/g) and note it contains no amplitude term. Explain the speed–distance compensation when amplitude changes. Mention the √2 factor when length is doubled. Use correct UK spelling and units (s).
Common mistakes
- Claiming the period changes with amplitude. Forgetting to include the √2 factor when length is doubled. Using the wrong formula for a pendulum (e.g. T=2π√(L/2g)).
Mark scheme (5 marks)
- For simple harmonic motion / a simple pendulum, the period is independent of amplitude
- A larger amplitude means a greater restoring force acts on the pendulum (bob)
- The greater restoring force causes a greater acceleration / the bob moves faster, exactly compensating for the longer distance travelled
- Period increases when the length is doubled (the period does not stay the same / the period gets longer)
- Period is proportional to the square root of the length, so doubling the length increases the period by a factor of √2 (approximately 1.41 times longer)
Key terms in this question
Related
- All AQA A-Level Physics (7408) revision notes →
- How to answer a "Explain" question →
- Decode the mark scheme abbreviations →
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