Question

A steel bicycle wheel (without the rubber tire) is rotating freely with an angular speed of 11.4 rad/s. The temperature of the wheel changes from -181 to 283 °C. No net external torque acts on the wheel, and the mass of the spokes is negligible. What is the angular speed of the wheel at the higher temperature?

Answer 1

The wheel's circumference will expand due to temperature and this will increase its moment of inertia
Take the initial radius as r
Linear coefficient of expansion of steel, = 13 x 10^-6 /oC
​Change in temperature, T = 283 - (- 181) = 464
Increase in circumference = r x[1 + T]
= r[1 + (13 x 10^-6) x 464]
= 1.006032 r

Since there is no external torque, angular momentum is conserved.
Ii i = If f
Ii is the initial moment of inertia = mr².
Initial angular momentum, i = 11.4 rad/s
Final moment of inertia, If = m (1.006032 r)² = 1.0121 mr²
​Final angular momentum, f = [Iii] / If
= [(mr²) x 11.4] / (1.0121 mr²)
= 11.4/1.0121
= 11.26 rad/s.