A track is mounted on a large wheel that is free to turn with negligible frictio

Question

A track is mounted on a large wheel that is free to turn with negligible friction about a vertical axis (the figure). A toy train of mass m is placed on the track and, with the system initially at rest, the train's electrical power is turned on. The train reaches a steady speed of 0.155 m/s with respect to the track. What is the angular speed of the wheel if its mass is 1.42m and its radius is 0.627 m? (Treat the wheel as a hoop, and neglect the mass of the spokes and hub.)

Please show all work and show how to solve algebraically.

Explanation / Answer

Given that,

speed of train, vt = 0.155 m/s

mass of train = m

mass of wheel, M = 1.42*m

radius of wheel, r = 0.627 m

Let, angular speed of wheel = w

From conservation of angular momentum,

mvr = l*w

mvr = Mr^2 * w

v = 1.42*r*w

where,

v = vt - wr

vt - wr = 1.42rw

vt = 2.42rw

0.155 = 2.42*0.627*w

w = 0.102 rad/s

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