Question

A smooth uniform rod of length $L$ and mass $M$ has two identical beads of negligible size, each of mass $m$, which can slide freely along the rod. Initially, the two and the system is rotating with an angular velocity ${\omega }_0$ about an axis perpendicular to the rod and passing through the midpoint of the rod. There is no external force. When the beads reach the end of the rod, the angular velocity of the system is ______

Answer 1

from Law of conservation of momentum ,

$I_{w_0}=I_{w_1}+2m{\left(\frac{L}{2}\right)}^2{w}_1$

$\frac{M{L}^2}{12}{w}_0=(\frac{M{L}^2}{12}+\frac{m{L}^2}{2})w_1$

$w_1=\frac{M}{M+6m}{w}_0$