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 speed of each ball relative to the ground just after they leave the disc is

• A. $\frac{R{\omega }_0}{\sqrt{3}}$
• B. $\frac{R{\omega }_0}{\sqrt{2}}$
• C. $\frac{2R{\omega }_0}{3}$
• D. None of these

Answer 1

The correct option is C $\frac{2R{\omega }_0}{3}$

Initial energy $E_i=\frac{1}{2}{I}_d{\omega }_0^2$
Where, $I=\frac{M{R}^2}{2}$
so, $E_i=\frac{M{R}^2}{4}\times {\omega }_0^2$

Final energy $E_f=\frac{1}{2}{I}_d{\omega }^2+1(\frac{1}{2}\times \frac{M}{2}\times v^2)$
where, $\omega =\frac{{\omega }_0}{3}$

$E_f=\frac{1}{2}\times \frac{M{R}^2}{2}\times \frac{{\omega }_0^2}{9}+\frac{M{v}^2}{2}$

using energy conservation, $E_i=E_f$
$\frac{M{R}^2}{4}\times {\omega }_0^2=\frac{1}{2}\times \frac{M{R}^2}{2}\times \frac{{\omega }_0^2}{9}+\frac{M{v}^2}{2}v=\frac{2R{\omega }_0}{3}$