Question

A rod of mass $m$ and length $l$ is lying along the y-axis such that its end is at the origin. Suddenly an impulse is given to the rod such that immediately after the impulse, the end on the origin has a velocity $v_0\hat{i}$ and the other end has a velocity $2v_o\hat{i}$. The magnitude of angular momentum of the rod about the origin at this instant is $\frac{5}{n}m{v}_{0}l$. Find n.

Answer 1

Angular velocity of rod $\omega =\frac{2v_0-v_0}{l}=\frac{v_o}{l}$

Velocity of CoM of the rod
$v_{cm}=\frac{2v_0+v_0}{2}=\frac{3}{2}{v}_0$

Therefore angular momentum of rod about the origin
$L=m{v}_{cm}\left(\frac{l}{2}\right)+I_{end}\omega$
$=m\left(\frac{l}{2}\right)\times \frac{3}{2}{v}_0+\frac{m{l}^2}{12}\times \frac{v_0}{l}$
$=m{v}_{0}l[\frac{3}{4}+\frac{1}{12}]=\frac{5}{6}m{v}_{0}l$

Thus the value of n is 6.