Question

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, the value of $\frac{M}{m}$ is

• A. $\frac{3}{4}$
• B. $\frac{4}{3}$
• C. $\frac{2}{3}$
• D. None

Answer 1

The correct option is A $\frac{3}{4}$



After collision rod rotates about O, let angular velocity of rotation is $\omega$.
Initial velocity of particle $u_1=V$
Final velocity of particle $V_1=0$
Initial velocity of point of rod at collision $u_2=0$
Final velocity of point of rod at collision $V_2=\frac{\omega L}{2}$


Coefficient of restitution is
$[e=-\frac{V_1-V_2}{u_1-u_2}]$


As collision is elastic $e=1$
$1=\frac{\frac{\omega L-0}{2}}{V}\Rightarrow \frac{\omega L}{2}=V$

Applying the conservation of angular momentum about the point O
$L_{initial}=L_{Final}$
$\Rightarrow \frac{mVL}{2}=\frac{M{L}^2}{3}.\omega$
$\Rightarrow \frac{mVL}{2}=\frac{M{L}^2}{3}\times \frac{2V}{L}\Rightarrow \frac{M}{m}=\frac{3}{4}$