I Rigid body mechanics, solution seems incorrect?

I've attached the problem and solution as picture. To my understanding, the gear E and the rod OB are taken together as the rotating rigid body. However, the equations of motion and (##∑F = macm##)
are applied to the center of mass of the rod, G, rather than the center of mass of the rigid body. Similarly, the moment is taken about G, and the mass of the rod is taken as m in the equations. Surely this can't be correct?
For the moment, there's an issue my my physicsforums page. The drop down menus don't function and I can't attach anything.

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