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 ${\left(\frac{2}{3}\right)}^{rd}$ of the initial angular velocity of the disc
• D. ${\left(\frac{2}{3}\right)}^{rd}$ of the initial kinetic energy changes to heat

Answer 1

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

The correct options are
A some friction exists between the disc and the ring
B the angular momentum of the 'disc plus ring' is conserved
D ${\left(\frac{2}{3}\right)}^{rd}$ of the initial kinetic energy changes to heat

Let ${\omega }_1$ = the initial angular velocity of the disc ${\omega }_2$= the final common angular velocity of the disc and the ring.
For the disc,
$I_1=\frac{1}{2}m{r}^2$
For the ring.
$I_2=m{r}^2$
By conservation of angular momentum,
$L=I_1{\omega }_1=(I_1+I_2){\omega }_2$
${\omega }_2=\frac{I_1{\omega }_1}{I_1+I_2}=\frac{{\omega }_1}{3}$
Initial kinetic energy $=E_1=\frac{1}{2}{I}_1{\omega }_1^2$
Final kinetic energy $=E_2=\frac{1}{2}(I_1+I_2){\omega }_2^2$
Heat produced = loss in KE
Ratio of heat produced to initial kinetic energy $=\frac{(E_1-E_2)}{E_1}=\frac{2}{3}$