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. 

(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 1

Correct Answer is: (a, b, & d)

Let ω₁ = the initial angular velocity of the disc.

ω₂ = the final common angular velocity of the disc and the ring.

For the disc, I₁ = 1/2 mr².

For the ring, I₂ = mr².

By conservation of angular momentum,

L = I₁ω₁ = (I₁ +I₂) ω₂ or ω₂ = I₁ω₁ / I₁ + I₂ = ω₁/3.

Initial kinetic energy = E₁ = 1/2 I₁ω₁².

Final kinetic energy = E₂ = 1/2 (I₁ + I₂) ω₂².

Heat produced = loss in kinetic energy = E₁ - E₂.

Ratio of heat produced to initial kinetic energy = E₁ - E₂ / E₁ = 2/3.