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 (Fig. 12-40). A toy train of mass m is placed on the track and, with the system initially at rest, the electrical power is turned on. The train reaches a steady speed v with respect to the track. What is the angular speed of the wheel if its mass is M and its radius is R? (Treat the wheel as a hoop, and neglect the mass of the spokes and hub. Use m for mass m, v for speed v, M for mass M and R for radius R.)

Explanation / Answer

Angular momentum is conserved as the train speeds up. Initially, the whole set-up is at rest, so the total angular momentum is zero. Qualitatively, the train goes one way and the track turns the other way. Let the final angular speed of the track be -w. The linear speed of the track relative to the ground is -wR, and the speed of the train relative to the track is v, so the speed of the train relative to the ground is v - wR. The angular speed of the train is this divided by R, or v/R - w, and the rotational inertia of the train is mR^2.

The total angular momentum is zero if

-MR^2w + mR^2(v/R - w) = 0

where MR^2 is the rotational inertia of the track. The solution of this is

w = mv/(M+m)R

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