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Q87 (NDA-I/2024) Science & Technology › Basic Science (Physics, Chemistry, Biology) Official Key

Which one of the following statements regarding simple pendulum is correct ? Simple pendulum has a time period independent of amplitude :

Result
Your answer: —  ·  Correct: D
Explanation

A simple pendulum performs Simple Harmonic Motion (SHM) only when the restoring force acting on the bob is directly proportional to its displacement from the equilibrium position. The restoring force is given by F = -mg sin(θ), where θ is the angular displacement.

  • For small amplitudes, the approximation sin(θ) ≈ θ (in radians) is valid. Consequently, the restoring force becomes F ≈ -mg(x/L), where x is the arc displacement and L is the length.
  • Since the force is now proportional to the displacement (F ∝ x), the motion is SHM, and the time period T = 2π√(L/g) is independent of the amplitude.
  • At larger amplitudes, the linear relationship fails, and the time period becomes dependent on the amplitude.

Thus, Option D is correct as it identifies that proportionality of force to displacement is the necessary condition for amplitude independence, which only holds for small angles.

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