Rigid body mechanics, solution seems incorrect?

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The discussion revolves around a rigid body mechanics problem involving a gear and a rod, where the user questions the application of equations of motion. They argue that the equations are incorrectly applied to the center of mass of the rod instead of the entire rigid body. Additionally, the moment is calculated about the rod's center of mass, which raises concerns about the accuracy of the solution. The user also mentions technical difficulties with their physics forums page, preventing them from attaching relevant images. The overall consensus is that the approach to solving the problem may be flawed.
Amin2014
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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?
 
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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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