Question

A horizontal disc rotates freely about a vertical axis through its central. A ring, having the same mass and radius as the disk is now gently placed on disc. After some time, the two rotates with a common angular velocity.Which options are true?

(a)Some frictional force 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 $2/3rd$ of the initial angular velocity of the disc

(d)$2/3rd$ of the initial kinetic changes to heat.

Answer 1

Let ${\omega }_1=$ the initial angular velocity of the disc and the ring.For the disc $I_1=\frac{1}{2}{mr}^2$For the ring $I_2={mr}^2$By conservation of angular momentum $L=I_1{\omega }_1=(I_1+I_2){\omega }^2$ or ${\omega }_2=\frac{{\omega }_1{I}_1}{I_1{I}_2}=\frac{{\omega }_1}{3}$Heart produced =loss in kinetic energy $=E_1-E_2$Ratio of heat produced to initial kinetic energy $=\frac{E_1-E_2}{E_1}=\frac{2}{3}$$\therefore \left(a\right),\left(b\right)$ and $\left(d\right)$