Question

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 is

• A. ${\cos }^{-1}\left[\frac{L\cos \alpha +l}{L+l}\right]$
• B. ${\cos }^{-1}\left[\frac{L\cos \alpha +l}{L-l}\right]$
• C. ${\cos }^{-1}\left[\frac{L\cos \alpha -l}{L-l}\right]$
• D. ${\cos }^{-1}\left[\frac{L\cos \alpha -l}{L+l}\right]$

Answer 1

Answer: (C)

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,
$L\cos \alpha =l+(L-l)\cos \theta$
$\Rightarrow \cos \theta =\frac{L\cos \alpha -l}{L-l}$
$\Rightarrow \theta ={\cos }^{-1}\left[\frac{L\cos \alpha -l}{L-l}\right].$