question_answer

A thin uniform rod of mass m & length \[\ell \] is free to rotate about its upper end. When it is at rest, it receives an impulse J at its lowest point, normal to its length. Immediately after the impact, mark the correct statement (s)

A)  the angular velocity of the rod is \[\frac{3J}{m\ell }\]

B)  the angular momentum of the rod is\[J\ell \]

C)  the linear velocity of mid point of the rod

D)  all the above

Correct Answer: D

Solution :

From angular impulse - momentum theorem \[J\times \ell =I\omega =\frac{m{{\ell }^{2}}}{3}\omega \]\[\Rightarrow \]\[\omega =\frac{3J}{m\ell }\] Angular momentum, \[L=l\omega =J\ell \] Velocity of mid point of rod is \[v=\frac{\ell }{2}\omega =\frac{3J}{2m}\]