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

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$\cos^{- 1} [\frac{Lcos α - l}{L - l}]$

$\cos^{- 1} [\frac{Lcos α + l}{L + l}]$

$\cos^{- 1} [\frac{Lcos α + l}{L - l}]$

$\cos^{- 1} [\frac{Lcos α - l}{L + l}]$

answer is C.

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Detailed 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,

$\begin{array}{l} Lcos α = l + (L - l) \cos θ \\ \Rightarrow \cos θ = \frac{Lcos α - l}{L - l} \\ \Rightarrow θ = \cos^{- 1} [\frac{Lcos α - l}{L - l}] \end{array}$