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$A$ uniform rod of mass $M$ is hinged at its upper end. $A$ particle of mass $m$ moving horizontally strikes the rod at its mid point elastically. If the particle comes to rest after collision find the value of $M/m =?$
$3/4$
$4/3$
$2/3$
none
Solution
Given, mass of the particle $=$ $m$
Mass of the rod $=M$
Let the length of the rod be $L$
Applying conservation of angular momentum about the hinge, we get
$m v \frac{L}{2}=I \omega$
$m v \frac{{L}}{2}=\frac{M L^{2}}{3} \omega$
$\Rightarrow \omega=\frac{3 m v}{2 M L}$
Applying principle of conservation of energy, we have
$\frac{1}{2} m v^{2}=\frac{1}{2} I \omega^{2}$
$\frac{1}{2} m v^{2}=\frac{1}{2} \times \frac{M L^{2}}{3} \times\left(\frac{3 m v}{2 M L}\right)^{2}$
$\frac{1}{2} m v^{2}=\frac{1}{2} \times \frac{M L^{2}}{3} \times \frac{9 m^{2} v^{2}}{4 M^{2} L^{2}}$
$m v^{2}=\frac{3 m^{2} v^{2}}{4 M}$
$\frac{M}{m}=\frac{3}{4}$
