Two identical uniform disks, each of mass M and radius R , are initially separated. One disk is rotating with angular velocity ω 0 about its central axis, while the other is stationary. They are the

Question

Two identical uniform disks, each of mass M and radius R , are initially separated. One disk is rotating with angular velocity ω 0 ​ about its central axis, while the other is stationary. They are then brought into contact coaxially, and due to friction, they eventually rotate together with a common angular velocity. What is this final angular velocity?

TestHub · Accepted Answer

Angular momentum of the system of two disks is conserved.

The initial angular momentum is due to the rotating disk:

$L_{i} = I_{1} ω_{0} = \frac{1}{2} M R^{2} ω_{0}$ .

The second disk is stationary, so its initial angular momentum is zero.

When they rotate together, the final moment of inertia of the combined system is

$I_{f} = I_{1} + I_{2} = \frac{1}{2} M R^{2} + \frac{1}{2} M R^{2} = M R^{2}$ .

Equating $L_{i} = I_{f} ω_{f}$ , we have

$\frac{1}{2} M R^{2} ω_{0} = (M R^{2}) ω_{f}$ .

Solving for $ω_{f}$ : $ω_{f} = \frac{1}{2} ω_{0}$ .

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