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A horizontal disc rotates freely about a...

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 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 energy changes to heat

Text Solution

Verified by Experts

The correct Answer is:
A, B, D

Let `omega_(1)` = the initial angular velocity of the disc and the ring.
For the disc `I_(1) = (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) = (I_(1)omega_(1))/(I_(1) + I_(2)) = (omega_(1))/(3)`
Initial kinetic energy `= E_(1) = (1)/(2) (I_(1) + I_(2)) omega_(2)^(2)`
Final kinetic energy, `E_(2) = (1)/(2) (I_(1) + I_(2))omega_(2)^(2)`
Heat produced = loss in kinetic energy `= E_(1) - E_(2)`
Ratio of heat produced to initial kinetic energy `= (E_(1) - E_(2))/(E_(1)) = (2)/(3)`
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