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  4. A simple pendulum consisting of a mass M attached to a string of length L is released from rest at an angle α. A pin is located at a distance l below the pivot point. When the pendulum swings down, the string hits the pin as shown in the figure. The maximum angle θ which the string makes with the vertical after hitting the pin is

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Q. A simple pendulum consisting of a mass $M$ attached to a string of length $L$ is released from rest at an angle $\alpha$. A pin is located at a distance $l$ below the pivot point. When the pendulum swings down, the string hits the pin as shown in the figure. The maximum angle $\theta$ which the string makes with the vertical after hitting the pin isPhysics Question Image

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Solution:

Since the pendulum started with no kinetic energy, conservation of energy implies that the potential energy at $Q_{\max }$ must be equal to the original potential energy, i.e., the vertical position will be same. Therefore,
$L \cos \alpha=l+(L-l) \cos \theta$
$\Rightarrow \cos \theta=\frac{L \cos \alpha-l}{L-l}$
$\Rightarrow \theta=\cos ^{-1}\left[\frac{L \cos \alpha-l}{L-l}\right] .$