An Physics student finds the problem with the bicycle wheel on the last quiz ver

Question

An Physics student finds the problem with the bicycle wheel on the last quiz very interesting. He starts thinking: if I have an aluminum wheel rotating at an angular speed of 18 rads/sec…

1. Would the angular speed of the wheel increase or decrease if the wheel heats up? Why?

2. What is the angular speed of the wheel at the higher temperature if its temperature changes from -100 °C to +300 °C?

assume the tire is off, ignore the mass of the spokes, and there are no net external torques on the wheel

Explanation / Answer

given w = 18 rad/s

1. if the wheel heats up, the radisu of the wheel increases, and hence moment of inertia of the wheel increases

to keep angular momentum constant the angular velcotiy must decrease

2. thermal expansion coefficient of aluminium, k = 23*10^-6

hence

change in length dl = k*l*dT

now dT = 400 C

initial radius = Ro

hence l = 2*pi*Ro

dl = 23*10^-6*400*2*pi*Ro = 0.0578053Ro

but dl + l = 2*pi*r

1.0092Ro = r

hence

intiial moment of inertia = Io = 0.5mRo^2

final, I = 0.5mr^2 = 0.50924232mRo^2

final w = Io*wo/I' = 17.673315 rad/s

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