Question

A smooth uniform rod of length $L$ and mass $M$ has 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 road and the system is rotating with angular velocity ${\omega }_0$ about an axis perpendicular to the rod and passing through the midpoint of the rod., There are no external forces. When the beads reach the ends of the rod, the angular velocity of the rod would be

• A. $\frac{M{\omega }_0}{M+2m}$
• B. $\frac{M{\omega }_0}{M+4m}$
• C. $\frac{M{\omega }_0}{M+6m}$
• D. $\frac{M{\omega }_0}{M+8m}$

Answer 1

The correct option is C $\frac{M{\omega }_0}{M+6m}$

Since there are no external forces so angular momentum will be conserved.

Let angular velocity is $\omega$

Angular momentum $L=I\omega$

$\Rightarrow \frac{M{L}^2}{12}{\omega }_0=(\frac{M{L}^2}{12}+\frac{2m{L}^2}{4})\omega$

$\Rightarrow M{\omega }_0=(M+6m)\omega$

$\Rightarrow \omega =\frac{M{\omega }_0}{M+6m}$