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?

<p<strong>Angular Frequency</strong>: The angular frequency ($\omega$) of a simple pendulum is given by:</p> <p>$ \omega = \sqrt{\frac{g}{\ell}} $</p> <p>where $g$ is the acceleration due to gravity and $\ell$ is the pendulum length.</p></p> <p><p><strong>Angular Acceleration</strong>: The angular acceleration ($\alpha$) can be expressed in terms of angular displacement ($\theta$) and angular frequency:</p> <p>$ \alpha = -\omega^2 \theta $</p></p> <p><p><strong>Equating Angular Accelerations</strong>: Since the angular accelerations of the two pendulums are equal, we equate them:</p> <p>$ \frac{g}{\ell_1} \theta_1 = \frac{g}{\ell_2} \theta_2 $</p></p> <p><p><strong>Simplifying the Expression</strong>: By canceling out $g$ on both sides, we derive:</p> <p>$ \theta_1 \ell_2 = \theta_2 \ell_1 $</p></p> <p>Thus, the correct expression that relates the displacements and lengths of the pendulums is $\theta_1 \ell_2 = \theta_2 \ell_1$.</p>