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 beads are at the centre of the rod and the system is rotating with an angular velocity ${\omega }_o$ about an axis perpendicular to the rod and passing through the midpoint of the rod (see figure). There are no external forces. When the beads reach the ends of the rod, the angular velocity of the system is..............

Answer 1

$Byconservingangularmomentum{J}_{in}=I{\omega }_0=\frac{M{L}^2}{12}{\omega }_0{J}_f=(\frac{M{L}^2}{12}+2m{\left(\frac{L}{2}\right)}^2)\omega \Rightarrow (\frac{M{L}^2}{12}+2m{\left(\frac{L}{2}\right)}^2)\omega =\frac{M{L}^2}{12}{\omega }_0\Rightarrow \omega =\frac{M{\omega }_0}{M+6m}$