UPSC NDA I 2024 General Ability Paper

Concept:The time period of a simple pendulum is independent of amplitude only for small angular displacements, where the motion approximates simple harmonic motion (SHM).Explanation:For a simple pendulum, the time period is given by $T = 2\pi \sqrt{\frac{l}{g}}$, where $l$ is the length and $g$ is the acceleration due to gravity.This formula is derived under the assumption of small amplitude (typically $\theta < 10^\circ$).For small amplitudes, the restoring force is approximately proportional to the displacement ($F \propto -x$), which is the condition for SHM.Because of this proportionality, the time period remains constant regardless of the exact amplitude within this range.For larger amplitudes, the restoring force depends on $\sin\theta$ instead of being linear, so the motion is no longer strictly SHM.Hence, the time period becomes slightly amplitude-dependent for large swings.Thus, the correct statement is that the time period is independent of amplitude only for small amplitudes, because then the net force on the bob is proportional to its displacement.Answer:Option D: "only for small amplitudes because then the net force on its bob is proportional to its displacement."