John Denker has already provided an operational and intuitively
appealing answer to the original question via the hypothetical use
of gyroscopes. Nevertheless, those interested in a more rigorous
mathematical treatment may want to look at a brief and highly
accessible recent article in AJP by Nivaldo A. Lemos: "Uniqueness
of the angular velocity of a rigid body: Correction of two faulty
proofs", Am. J. Phys., 68, 668-669, (2000).
Although the author refers misleadingly IMO to the angular
velocity "of" arbitrary points attached to a rigid body (rather
than the angular velocity of arbitrary points attached to a rigid
body relative to other arbitrary points also attached to the rigid
body), the proofs may be of assistance to those who, like myself,
have struggled with similar questions.
After reading the article and thinking about the current thread,
it occurs to me that there are several sources of confusion:
1 Unfounded expectation that the center of mass, a pivot point, or
a fixed axis should, somehow, be afforded special consideration
All three of these provide appealing points of reference and are
indeed "special" in certain situations especially for dynamic
considerations. Nevertheless, one can speak unambiguously about
*the* angular velocity of a rigid body (which is a purely
kinematic quantity) without reference to anything other than
a locally nonrotating frame (as provided by JD's gyros).
2 (Related to point 1) Unfounded expectation of interactions
between the translational motion of an arbitrary point and
the angular velocity
This confusion is only partially addressed by noting that one
can always find an inertial reference frame in which an
arbitrary point of the rigid body is instantaneously at rest.
The astute student will still wonder what happens in the "next
instant". It may help to note that the angular velocity of a
rigid body is itself only an instantaneously determined quantity.
We are, perhaps, the persistent victims of our early training in
which enforced axes of rotation always fix the direction if not
the magnitude of the angular velocity. Furthermore, it's hard
to visualize and even harder to gain an intuition for free body
rotation. (Along these lines, a friend of mine who was a space
shuttle astronaut once described to me watching a large
satellite that he had knocked into a slow "tumbling
motion"--i.e., with the angular momentum *not* along a major
axis--after an unsuccessful attempt to "dock" with it and return
it to the shuttle bay. He remarked that, no matter how long he
watched, he could not develop a good intuitive sense that
allowed him to predict how the motion would develop in the next
few moments.)
3 (Related to both points 1 and 2 and, perhaps, the primary source
of all confusion) Failure to truly understand the definition of
"the angular velocity of a rigid body."
This is why I think the author of the cited article should have
exercised a more careful choice of words. Any *point* can "have"
an arbitrary angular velocity with respect to another arbitrarily
chosen point, but this is not so for any two arbitrarily chosen
points attached to the *same* rigid body.