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 midpoint elastically. If the particle comes to rest after collision find the value of M/m?

Answer 1

Given: A uniform rod of mass M is hinged at its upper end. A particle of mass m moving horizontally strikes the rod at its midpoint elastically. If the particle comes to rest after collision .

To find the value of $\frac{M}{m}$

Solution:

Applying the conservation of angular momentum about the point O

$L_{initial}=L_{final}⟹mv\times \frac{l}{2}+v=0+I\omega$

where $\omega$ is the angular velocity and $I$ is the impulse

$⟹\frac{mvl}{2}=\frac{M{l}^2}{3}\times \omega ⟹\frac{v}{\omega l}=\frac{2M}{3m}..............\left(i\right)$

Applying the conservation of energy

$\frac{1}{2}m{v}^2=\frac{1}{2}\times I\times {\omega }^2⟹m{v}^2=\frac{M{l}^2}{3}\times {\omega }^2⟹\frac{M}{m}=\frac{3v^2}{l^2{\omega }^2}=3{\left(\frac{v}{\omega l}\right)}^2$

From eqn(i),

$\frac{M}{m}=3\times {\left(\frac{2M}{3m}\right)}^2⟹\frac{M}{m}=\frac{4M^2}{3m^2}⟹\frac{M}{m}=\frac{3}{4}$

is the required value.