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I just learned that an unconstrained rigid body always rotates about its center of mass. But there could be many axes that could pass through the center of mass.

For example, let's consider a rod lying on a table which is given an impulse at one of its ends, from intuition we could say that the rod would rotate about an axis passing through the center of mass which is perpendicular to the table (The axis for which moment of inertia of the rod is $\frac{ML^2}{12}$ if the rod is uniform). But then the rod could've rotated about a different axis passing through the center of mass.

So my question is, among all such axes that pass through the center of mass does there always exist a definite orientation of the axis about which all unconstrained rigid bodies would rotate for a given scenario (such as all rigid bodies lying on a table, all freely falling bodies, etc)? If so, why does it always have to be in such an orientation?

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The rotation axis will always be parallel to the angular momentum of the body. The angular momentum imparted by a force or impulse can in principle be in any direction, and multiple contributions to the angular momentum add as vectors. Note that the angular momentum vector points perpendicular to the plane in which the rotation occurs.

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