### Related Formula Angular acceleration definition for a simple pendulum swinging at small angle theta$\theta$: alpha = -omega^2 theta$\alpha = -\omega^2 \theta$ where: omega = sqrtfracgl implies omega^2 = fracgl $\omega = \sqrt{\frac{g}{l}} implies \omega^2 = \frac{g}{l} $ Hence, the magnitude of angular acceleration is: alpha = fracgltheta$ alpha = \frac{g}{l}\theta$ ### Core Logic Given that angular accelerations are exactly identical in magnitude (alpha_1 = alpha_2$\alpha_1 = \alpha_2$): fracgl_1theta_1 = fracgl_2theta_2$\frac{g}{l_1}\theta_1 = \frac{g}{l_2}\theta_2$ ### Step 1: Simplify Expression Cancelling out the constant gravitational field factor g$g$: fractheta_1l_1 = fractheta_2l_2 implies theta_1 l_2 = theta_2 l_1$\frac{\theta_1}{l_1} = \frac{\theta_2}{l_2} implies \theta_1 l_2 = \theta_2 l_1$ ### Pattern Recognition Angular acceleration scales inversely with length for a fixed angle. To preserve equal acceleration values, the product theta cdot l^-1$\theta \cdot l^{-1}$ must be constant, resulting in cross-multiplication balance. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Oscillations