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

Options

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 `(2/3)`rd of the initial angular velocity of the disc
D. `(2/3)`rd of the initial kinetic energy changes to heat

Answer

A, B, D

Explanation

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)=1/2mr^(2)`
For the ring
`I_(2)=mr^(2)`
By coservation of angular momentum,
`L=I_(1)omega_(1)=(I_(1)+I_(2))omega_(2)`
`omega_(2)=(I_(1)omega_(1))/(I_(1)+I_(2))=(omega_(1))/3`
Initial kinetic energy `=E_(1)=1/2I_(1)omega_(1)^(2)`
Final kinetic energy `=E_(1)=1/2(I_(1)+I_(2))omega_(2)^(2)`
Heat produced `=` loss in `KE`
Ratio of heat produced to initial kinetic energy `=(E_(1) -E_(2))/(E_(1))=2/3`