Question

A thin uniform rod of mass m and length l is kept on a smooth horizontal surface such that it can move freely. At what distance from centre of rod should a particle of mass m strike on the rod such that the point P at a distance l/3 from the end of the rod is instantaneously at rest just after the elastic collision?

• A. $l/2$
• B. $l/3$
• C. $l/6$
• D. $l/4$

Answer 1

Answer: (A)

$l/2$

Let the impulse due to the particle be J,
$J=m{v}_{CM}$
$\Rightarrow v_{CM}=\frac{J}{m}$
Angular impulse about centre mass is
$Jx=\frac{m{l}^2}{12}\omega \Rightarrow \omega =\frac{12Jx}{m{l}^2}$
For point A to be at rest instantaneously,
${\overrightarrow{v}}_{trans}+{\overrightarrow{v}}_{rot}=0$
or $v_{cm}=\frac{\omega l}{6}$
Thus $\frac{J}{m}=\frac{l}{6}\times \frac{12Jx}{m{l}^2}$
Which gives $x=\frac{l}{2}$