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 find the value of $\frac{M}{m}=?$

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

Answer 1

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

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

$mv\frac{L}{2}=I\omega$

$mv\frac{L}{2}=\frac{M{L}^2}{3}\omega$

$\Rightarrow \omega =\frac{3mv}{2ML}$

Applying principle of conservation of energy, we have

$\frac{1}{2}m{v}^2=\frac{1}{2}I{\omega }^2$

$\frac{1}{2}m{v}^2=\frac{1}{2}\times \frac{M{L}^2}{3}\times {\left(\frac{3mv}{2ML}\right)}^2$

$\frac{1}{2}m{v}^2=\frac{1}{2}\times \frac{M{L}^2}{3}\times \frac{9m^2{v}^2}{4M^2{L}^2}$

$m{v}^2=\frac{3m^2{v}^2}{4M}$

$\frac{M}{m}=\frac{3}{4}$

Hence, the correct answer is OPTION A.