Two disks of identical mass but different radii (r and 2r) are spinning on frict

Question

Two disks of identical mass but different radii (r and 2r) are spinning on frictionless bearings at the same angular speed 6)0, but in opposite directions. The two disks are brought slowly together. The resulting frictional force between the surfaces eventually brings them to a common angular velocity. (a) What is the magnitude of that final angular velocity in terms of omega_0? (Use the following as necessary: omega_0.) omega_f = (b) What is the change in rotational kinetic energy of the system? (Use the following as necessary: K as the initial kinetic energy.) Delta K = Explain.

Explanation / Answer

(a) By conservation of angular momentum,

(initial angular momentum) = (final angular momentum)

[mr^2/2][-w0] + [m(2r)^2/2][w0] = [mr^2/2 + m(2r)^2/2][wf]

[r^2][-w0] + [(2r)^2][w0] = [r^2 + (2r)^2][wf]

3r^2[w0] = 5r^2[wf]

wf = (3/5)w0

(b) Change in rotational KE, deltaK

= 0.5*(mr^2/2 + m(2r)^2/2)w0 - 0.5(mr^2/2 +m(2r)^2/2)wf^2

= 0.5*(mr^2/2 + m(2r)^2/2)*(w0^2 - 9/25w0^2)

= 0.5*(mr^2/2 + m(2r)^2/2)*(16/25w0^2)

= 16/25 KE

where, KE is the "initial kinetic energy" of the two separated disks.

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