# A simple pendulum , made of a string of length l and a bob of mass m , is released from a small angle $\theta_{0}$ . It strikes a block of mass M , kept on a horizontal surface at its lowest point of oscillations, elastically. It bounces back and goes upto angle $\theta_{1}$. Then M is given by : Option 1)$\frac{m}{2}\left ( \frac{\theta_{0} + \theta_{1}}{\theta_{0} - \theta_{1}} \right )$Option 2)$m\left ( \frac{\theta_{0} + \theta_{1}}{\theta_{0} - \theta_{1}} \right )$Option 3)$\frac{m}{2}\left ( \frac{\theta_{0} - \theta_{1}}{\theta_{0} + \theta_{1}} \right )$Option 4)$m\left ( \frac{\theta_{0} - \theta_{1}}{\theta_{0} + \theta_{1}} \right )$

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Coffecient of Restitution (e) -

$e= \frac{v_{2}-v_{1}}{u_{2}-u_{1}}$

- wherein

Ratio of relative velocity after collision to relative velocity before collision.

Perfectly Elastic Collision -

$e=1$

- wherein

$e\: :\: coefficient \: of\: restitution$

before Collision                                           After collision

apply momentum conservation

$mv=-mv_{1}+mv_{2}\, \, \, \, \, -\left ( 1 \right )$

from energy balance

we get

$v=\sqrt{29l\left ( 1-cos\Theta _{0} \right )}$

$v_{1}=\sqrt{29l\left ( 1-cos\Theta _{1} \right )}$

Put value of  $v$ & $v_{1}$  in $eq^{n}\, \, \, \left ( 1 \right )$

$m\times \sqrt{29l\left ( 1-cos\Theta _{0} \right )}=-m\sqrt{29l\left ( 1-cos\Theta _{1} \right )}+mv_{2}\, \, \, \, \left ( 2 \right )$

For elastic collision

$e=1=\frac{V_{2}-\left ( -V_{1} \right )}{V}=\frac{ \sqrt{29l\left ( 1-cos\Theta _{1} \right )}+V_{2}}{ \sqrt{29l\left ( 1-cos\Theta _{0} \right )}}$

$\Rightarrow V_{2}=\sqrt{29l}\left ( \sqrt{\left ( 1-cos\Theta _{0} \right )}-\sqrt{\left ( 1-cos\Theta _{1} \right )} \right )\rightarrow$ put in equation(2)

We get $\frac{m}{m}=$$\frac{\sqrt{1-cos\Theta _{0}}+\sqrt{1-cos\Theta _{1}}}{\sqrt{1-cos\Theta _{0}}-\sqrt{1-cos\Theta _{1}}}$

By componendo & dividendo

$\frac{m-m}{m+m}=\frac{\sqrt{1-cos\Theta _{1}}}{\sqrt{1-cos\Theta _{0}}}=\frac{Sin\left ( \frac{\Theta _{1}}{2} \right )}{Sin\left ( \frac{\Theta _{0}}{2} \right )}$

$\frac{m}{m}=\frac{\Theta _{0}-\Theta _{1}}{\Theta _{0}+\Theta _{1}}$

$m=m=\left ( \frac{\Theta _{0}-\Theta _{1}}{\Theta _{0}+\Theta _{1}} \right )$

Option 1)

$\frac{m}{2}\left ( \frac{\theta_{0} + \theta_{1}}{\theta_{0} - \theta_{1}} \right )$

Option 2)

$m\left ( \frac{\theta_{0} + \theta_{1}}{\theta_{0} - \theta_{1}} \right )$

Option 3)

$\frac{m}{2}\left ( \frac{\theta_{0} - \theta_{1}}{\theta_{0} + \theta_{1}} \right )$

Option 4)

$m\left ( \frac{\theta_{0} - \theta_{1}}{\theta_{0} + \theta_{1}} \right )$

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