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NDA 2026
Exam
NDA
Year
2026
Subject
General Ability
Chapter
General Science

NDA 2026 · General Ability · General Science

A pendulum of length L oscillates with an angular amplitude of θ =...

A pendulum of length LL oscillates with an angular amplitude of \theta = 60^{}^{\circ} and time period TT. Let T0=2πLgT_0 = 2\pi\sqrt{\dfrac{L}{g}} be the time period for small angle of oscillations, where gg is the acceleration due to gravity. If air resistance is negligibly small and the string remains straight, then which one of the following is correct?

Options

  1. ATT will be slightly greater than T0T_0.
  2. BTT will be slightly smaller than T0T_0.
  3. CTT will be exactly equal to T0T_0.
  4. DTT will depend upon the mass of the bob.

Correct answer

A. TT will be slightly greater than T0T_0.

Step-by-step solution

The restoring torque acting on the pendulum is τ=−mgLsin⁡θ\tau = -mgL \sin\theta.

The angular acceleration is α=−gLsin⁡θ\alpha = -\dfrac{g}{L} \sin\theta.

For small angular amplitudes, sin⁡θ≈θ\sin\theta \approx \theta, which gives the time period T0=2πLgT_0 = 2\pi\sqrt{\dfrac{L}{g}}.

For a large angular amplitude (\theta = 60^{}^{\circ}),sin⁡θ<θ), \sin\theta < \theta. This implies that the restoring torque is smaller in magnitude compared to the linear approximation τ=−mgLθ\tau = -mgL\theta.

Due to the smaller restoring torque, the angular acceleration is reduced, which increases the time taken to complete an oscillation.

The exact time period for large amplitudes is given by the series expansion T = T_0 \left(1 + \dfrac{1}{4}\sin^2 \left(\dfrac{\theta_0}{2}\right) + \dots \right).<br><br>Since \theta_0 = 60^{}^{\circ}, the terms in the bracket are positive, yielding T>T0T > T_0.

Therefore, TT will be slightly greater than T0T_0.

Answer: TT will be slightly greater than T0T_0.
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