Question

A disc of mass $m$ and radius $R$ is free to rotate in a horizontal plane about a vertical smooth fixed axis passing through its centre. There is a smooth groove along the diameter of the disc and two small balls of mass $m/2$ each, are placed in it on either side of the centre of the disc as shown in the figure. The disc is given an initial angular velocity ${\omega }_0$ and released.

The angular speed of the disc when the balls reach the end of the disc is

• A. $\frac{{\omega }_0}{2}$
• B. $\frac{{\omega }_0}{3}$
• C. $\frac{2{\omega }_0}{3}$
• D. $\frac{{\omega }_0}{4}$

Answer 1

The correct option is B $\frac{{\omega }_0}{3}$

Initial angular momentum
$L_i=(I_d+I_{ball})\omega$
where, $I_{ball}=0$
$L+i=\frac{M{R}^2}{2}{\omega }_0$ ...(1)

Final angular momentum
$L_f=(I_d+I_b\times 2){\omega }_f{L}_f=\frac{M{R}^2}{2}+\frac{M}{2}{R}^2\times 2$
$L_f=\frac{3}{2}M{R}^2\omega$ .....(2)

By momentum conservation,
(1)=(2)
$\frac{M{R}^2}{2}{\omega }_0=\frac{3}{2}M{R}^2\omega \omega =\frac{{\omega }_0}{3}$