Following a collision in outer space, a copper disk at 850 degree C is rotating

Question

Following a collision in outer space, a copper disk at 850 degree C is rotating about its axis with an angular speed of 25 rad/sec. As the disk radiates infrared light, its temperature falls to 20 degree C. No external torque act on the disk. Does the angular speed change as the disk cools? Explain how it changes or why it does not. What is its angular speed at the lower temperature? Given that this change in temperature took 10 seconds, what is the moment of inertia? Assume that the emissivity of copper is 0.8

Explanation / Answer

a) Yes. The copper cools so its diameter shrinks, this means its moment of inertia reduces.. To conserve angular momentum the angular speed increases (like a spinning ice-skater pulling in their arms).

b) Coefficient of linear expansion of copper = 16.6x10^-6 K^-1
For an 830°C change, the radius will reduce from R to R(1 - 830x16.6x10^-6) = 0.9862R

Initial moment of inertia, I0 = MR^2 / 2
Final moment of inertia = M(0.9862R)^2 / 2 = I0 x 0.9726

Since angular momentum is constant (no external torque), the angular speed increases by 1/0.9726
Final angular momentum = 25.0 / 0.9726 = 25.7rad/s

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