Question

A horizontal disc rotates freely about a vertical axis through its centre. A ring, having the same mass and radius as the disc, is now gently placed on the disc. After some time, the two rotate with a common angular velocity. Then :

• A. Some friction exists between the disc and the ring
• B. The angular momentum of the 'disc plus ring' is conserved
• C. The final common angular velocity is $\frac{2}{3}rd$ of the initial angular velocity of the disc
• D. $\frac{2}{3}rd$ of the initial kinetic energy changes to heat

Answer 1

Answer: (A), (B), (D)

Some friction exists between the disc and the ring
The angular momentum of the 'disc plus ring' is conserved
$\frac{2}{3}rd$ of the initial kinetic energy changes to heat

In the question it is mentioned that finally both have same angular velocity and didn't mention about any external force ,

$\Rightarrow$ there exists friction between those two bodies .

As friction is an internal force , Angular momentum about COM axis is conserved.

$\frac{m{r}^2}{2}\omega =\frac{m{r}^2}{2}{\omega }_2^2+m{r}^2{\omega }_2^2$ {let ${\omega }_2$ be final angular velocity of system.}

$\Rightarrow {\omega }_2=\frac{\omega }{3}$

Energy converted into heat=intial energy of system - final energy of system

$\Rightarrow$Energy converted to heat = $\frac{1}{2}{I}_1{\omega }^2-\frac{1}{2}{I}_2{\omega }_2^2$

Energy converted to heat=$\frac{1}{3}m{r}^2{\omega }^2=\frac{2}{3}$ of intial kinetic energy.