JEE MAIN - Physics (2025 - 4th April Morning Shift - No. 3)
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?
$\theta_1 l_2=\theta_2 l_1$
$\theta_1 l_1=\theta_2 l_2$
$\theta_1 l_2^2=\theta_2 l_1^2$
$\theta_1 l_1^2=\theta_2 l_2^2$
Explanation
$ \omega = \sqrt{\frac{g}{\ell}} $
where $g$ is the acceleration due to gravity and $\ell$ is the pendulum length.
Angular Acceleration: The angular acceleration ($\alpha$) can be expressed in terms of angular displacement ($\theta$) and angular frequency:
$ \alpha = -\omega^2 \theta $
Equating Angular Accelerations: Since the angular accelerations of the two pendulums are equal, we equate them:
$ \frac{g}{\ell_1} \theta_1 = \frac{g}{\ell_2} \theta_2 $
Simplifying the Expression: By canceling out $g$ on both sides, we derive:
$ \theta_1 \ell_2 = \theta_2 \ell_1 $
Thus, the correct expression that relates the displacements and lengths of the pendulums is $\theta_1 \ell_2 = \theta_2 \ell_1$.
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