Question: in the simple equations $\tau = r \times f$ and $L = r \times p$, what is the origin of the vector $r$? In many textbook discussions on rotation of a rigid body about a fixed axis, $r$ is taken to be the closest point on the axis of rotation. Hence, $r$ is always perpendicular to the axis of rotation and has no component in the direction of $\omega$. However, in other discussions (e.g., Halliday and Resnick 5th edition Section 12-3), $r$ originates from a constant fixed point. I've heard this latter version called "generalized torque" and "generalized angular momentum." Another way of putting this question is: do you take angular momentum about a fixed point or about an axis of rotation?
Let me make this very concrete. Consider a rigid object made of two 1kg point masses; at t=0 one point mass is at the point (2,0,0) and the other is at (2,0,4). The object rotates around the z axis with $\omega$=3 rad/s. At t=0, both thus have velocity $v=r \times \omega = (0,-6,0)$. In the "non-generalized" scheme, the angular momentum about the z axis would be $L = r_0 \times mv_0 + r_1 \times mv_1$ = $(2,0,0) \times (0,-6,0) + (2,0,0) \times (0,-6,0)$ = $(0,0,-24)$. However, in the "generalized" scheme, the angular momentum about the origin x=y=z=0 would be $L = r_0 \times mv_0 + r_1 \times mv_1$ = $(2,0,0) \times (0,-6,0) + (2,0,4) \times (0,-6,0)$ = $(24,0,-24)$. The difference between the two answers stems from using (2,0,0) for $r_1$ in the first case and (2,0,4) for $r_1$ in the second case.
So which is it? Are both correct, but in different circumstances? Is it legal to get torque and angular momentum either about a point or about an axis? Is the former more general? If so, then how come an introductory textbook like H&R mentions both, and somewhat-more-advanced textbooks (Symon's Mechanics and the Feynman Lectures) mention (as far as I've found) only computing $r$ from an axis of rotation rather than from a point?
If you really can choose the origin of the vector $r$ to be in either place, then it would seem to me that most theorems about angular momentum (of which there are many) should have two versions, which is of course not the case.
I want to tell myself that the answer is as simple as "sometimes objects are rotating around an axis and sometimes they're not, and you use generalized torque and angular momentum when objects are not just rotating around an axis." But that doesn't seem quite right, either.