There is a difficulty involved with our definition of the angular velocity vector ω, namely, that we cannot properly consider this vector to be “free” in the sense of being able to parallel translate it at will. Consider the rotations of a rigid body about each of two parallel axes. Then the corresponding angular velocity vectors ω1 and ω2 are parallel. Explain, perhaps with a figure, that even if ω1 and ω2 are equal as “free vectors,” the corresponding rotational motions that result must be different. (Therefore, when considering more than one angular velocity, we should always assume that the axes of rotation pass through a common point.)
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