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 oscillates with an angular amplitude of \theta = 60^{}^{\circ} and time period . Let be the time period for small angle of oscillations, where 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
- A will be slightly greater than .
- B will be slightly smaller than .
- C will be exactly equal to .
- D will depend upon the mass of the bob.
Correct answer
A. will be slightly greater than .
Step-by-step solution
The restoring torque acting on the pendulum is .
The angular acceleration is .
For small angular amplitudes, , which gives the time period .
For a large angular amplitude (\theta = 60^{}^{\circ}. This implies that the restoring torque is smaller in magnitude compared to the linear approximation .
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 .
Therefore, will be slightly greater than .
Answer: will be slightly greater than .
The angular acceleration is .
For small angular amplitudes, , which gives the time period .
For a large angular amplitude (\theta = 60^{}^{\circ}. This implies that the restoring torque is smaller in magnitude compared to the linear approximation .
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 .
Therefore, will be slightly greater than .
Answer: will be slightly greater than .