Question

A uniform rod AB which is free to swing in the vertical plane about a horizontal axis through A, is hanging freely. A particle of equal mass strikes the rod with a velocity $V_0$ and gets stuck to it.Find the angular velocity of the combination immediately after the collision.

Answer 1

Initial angular momentum of the system is

$L_i={mV}_{0}L$

Rod is at rest and particle is moving with velocity ${ V }_{ 0 }$

After impact the angular momentum is

$L_i={mV}_{0}L{L}_f=(I_{rod}+I_{particle})w=(\frac{1}{3}m{L}^2+m{L}^2)w=\frac{4}{3}m{L}^{2}w$

According to conservation of angular momentum

$L_i=L_f$

$\frac{4}{3}m{L}^{2}w={mV}_{0}L$

$w=\frac{3V_0}{4L}$