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 figure. The maximum angle $\theta$ which the string makes with the vertical after hitting the pin is:

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

Answer 1

The correct option is C $co{s}^{-1}\left[\frac{Lcos\alpha -l}{L-l}\right]$

System: Pendulum + Earth

As the only external force on the system will be the normal force by the peg, so mechanical energy will be conserved.

By conservation of energy

$k_i+u_i=k_f+u_f$

$K_i=K_f=0$

$\Rightarrow -mgL\cos \alpha =-mg[l+(L-l\left)\cos \theta \right]$

$\Rightarrow \cos \theta =\frac{L\cos \alpha -l}{L-l}$

$\Rightarrow \theta ={\cos }^{-1}\left(\frac{L\cos \alpha -l}{L-l}\right)$