Uneven Disk Toppling Over

A circular disk of radius RR consists of two uniform halves of equal area with masses 23M\frac{2}{3}M and 13M\frac{1}{3}M. The disk is free to rotate on an axle through its geometric center.

Initially, the disk is at rest as shown on the left such that the half of mass 23M\frac23 M is on top.

Suppose the disk is toppled by an extremely gentle nudge. At the instant the disk first reaches the orientation on the right, its angular speed can be expressed as follows:

ω=AgBπR.\omega = \sqrt{\frac{Ag}{B \pi R}}.

If AA and BB are coprime positive integers, determine A+BA + B.


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