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Sometimes when I work on mechanics problems, I wonder if this analysis is always valid.

Couldn't there be some motion of a rigid body that cannot be expressed as a composition of translational motion and rotation with respect to the center of mass?

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Euler's rotation theorem states that every rigid motion that leaves some point fixed is equivalent to a rotation about an axis running through that point. It is easy to see that every rigid motion can therefore be decomposed in the prescribed manner (i.e., by first moving the centre of mass to its new location).

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