Question

Two simple pendulums having lengths $l_{1}$ and $l_{2}$ with negligible string mass undergo angular displacements $\theta_{1}$ and $\theta_{2}$, from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct?

• A. $\theta_{1} l_{2}^{2}=\theta_{2} l_{1}^{2}$
• B. $\theta_{1} l_{1}=\theta_{2} l_{2}$
• C. $\theta_{1} l_{1}^{2}=\theta_{2} l_{2}^{2}$
• D. $\theta_{1} l_{2}=\theta_{2} l_{1}$

Hint:

The angular acceleration of a pendulum depends on its length and angular displacement.

Answer 1

The Correct Option is D

1. Angular acceleration: \[ \alpha = -\omega^2 \theta \] \[ \omega = \sqrt{\frac{g}{l}} \]
2. Equating angular accelerations: \[ \frac{g}{l_1} \theta_1 = \frac{g}{l_2} \theta_2 \] \[ \theta_1 l_2 = \theta_2 l_1 \] Therefore, the correct answer is (4) $\theta_{1} l_{2}=\theta_{2} l_{1}$.