Question

A particle of mass $m$ is moving horizontally at speed $v$ perpendicular to a uniform rod of length $d$ and mass $M=6m$. The rod is hinged at centre $O$ and can freely rotate in horizontal plane about a fixed vertical axis passing through its centre $O$. The hinge is frictionless. The particle strikes and sticks to the end of the rod. The angular speed of the system just after the collision is :

• A. $2v/3d$
• B. $3v/2d$
• C. $v/3d$
• D. $2v/d$

Answer 1

The correct option is A $2v/3d$

Moment of inertia of system after the particle is attached

$=\frac{1}{2}\left(6m\right)d^2+m{\left(\frac{d}{2}\right)}^2=\frac{m{d}^2}{2}+\frac{m{d}^2}{4}=\frac{3m{d}^2}{4}\overrightarrow{L_f}=mv\frac{d}{2}$

$\overrightarrow{L_f}=I\omega \Rightarrow \frac{mvd}{2}=\frac{3mv{d}^2}{4}\omega ($conservation of angular momentm)

$\Rightarrow \omega =\frac{2v}{3d}$