Q.Four pendulums A, B, C and D hang side by side in a row from a common horizontal elastic (flexible) support. Their lengths are related as follows: A and C have exactly the same length; B is shorter than A; and D is longer than A. Pendulum A is now given a transverse displacement and set oscillating. Because the support is elastic, A's oscillation drives the other three through the support. Which pendulum will eventually vibrate with the maximum amplitude?
You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.
Start your 14-day free trial to unlock the full solution →This is a case of forced oscillation and resonance. Pendulum A, once displaced, oscillates at its own natural frequency and forces the elastic support to wobble at that frequency. Any other pendulum whose natural frequency equals A's will be driven at resonance and build up a large amplitude. A pendulum's natural frequency depends only on its length, so the pendulum of the same length as A responds most strongly — that is C.
Concept: coupled pendulums and resonance
The four pendulums are coupled through the common elastic support. When A swings, it periodically shakes the support, acting as a driver at its own natural frequency. Each of the other pendulums is a driven oscillator.
Why length decides it
The natural frequency of a simple pendulum is
which depends only on its length (and ). Resonance — the condition for maximum energy transfer and hence maximum amplitude — occurs when the driven pendulum's natural frequency matches the driver's frequency, i.e. when their lengths are equal.
Applying it …
Unlock everything free for 14 days
- Full step-by-step solutions
- Concept-first explanations
- Methods, shortcuts & mistakes
- PYQ mapping + timed mock tests
Full access for 14 days. No credit card required.