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Qmechanic
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Can we prove that the "Center of mass" of a homogeneous rigid body is at its "Geometric center"?

I know that for basic shapes like a solid cube or a solid sphere the proof is simple. But if we had a homogeneous rigid body with an arbitrary shape, can we prove that the body's "Center of mass" is exactly at its "Geometric center"?

Hamed.Begloo
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