Rigid body mechanics, solution seems incorrect?

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SUMMARY

The discussion centers on the application of rigid body mechanics, specifically regarding the analysis of a rotating rigid body composed of gear E and rod OB. The participant questions the validity of applying the equations of motion, particularly ∑F = macm, to the center of mass of the rod (G) instead of the entire rigid body. Additionally, they highlight the incorrectness of calculating moments about G and using the rod's mass (m) in the equations, indicating a fundamental misunderstanding of rigid body dynamics.

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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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