Consider an isolated rigid rod with moment of inertia $I_0$ with respect to an axis perpendicular to the rod and through its centre of mass. If a linear impulse is applied to a point somewhere along the rod, both rotational and linear motion occur.
If the impulse $I$ is applied perpendicular to the rod at a distance $L$ along the rod from the centre of mass, the angular impulse is given by $J = IL$, where $J = \Delta L = I_0 \omega$. If my understanding is correct, this $\omega$ is independent of the choice of origin - that is, we can choose any arbitrary point along the rod to take moments about and end up with the same $\omega$.
Why, then, does choosing the origin at the point of application of $I$ give $\omega = 0$? Is this a consequence of the ignored linear component of motion, or is the choice of origin at the centre of mass not, generally, an arbitrary one?
Thank you.