out2lunch wrote:

(Edward Green) wrote in message om...

First, extend the mass on an arm of length R to create moment of
inertia I = mR^2 about the axis. Next, rotate the arm about the axis
with angular velocity w, angular momentum L = Iw. The rest of the
body, which has some moment of inertia I_0, counter-rotates with
angular velocity w' = -L/I_0, so that the total angular momentum
remains zero. Next, stop rotating the arm: when the arm stops
rotating about the given axis, so does the remainder of the body.

Finally, retract the arm to place the point mass on axis again. Since
the moment of inertia of the point mass is zero, we may now rotate it
back to its original orientation wrt the body (assuming we have
painted index lines on the point :-) without any torque at all. We
are finished, the body configuration is restored, and the body is
reoriented.

The key is not that the angular momentum of the second part of the
body has to vanish, but that we can divide the complete body X into
segments A and B, at least one of which has a variable moment of
inertia. Since the ratio of moments of inertia will vary, so will the
ratio of angular velocities when we twist A wrt B: hence it's possible
to twist the body, flip one of the moments of inertia and reverse the
_relative_ reorientation of the parts without cancelling the _overall_
reorientation.


I stand corrected. Very clever. The key here is that the body must
be composed of two disjoint parts that can rotate freely in opposite
directions. If I were floating by myself in space, there's no way I
could perform this maneuver.

Sure you could. For example imagine yourself floating in a
'standing posture' and start by keeping your legs straight down
but put your arms out to the sides. Now the upper part of your
body has a higher moment of inertia than the lower part. Twist
about your stomach so your legs are rotated 90 degrees to the
right relative to your arms. Most of the rotation will be of the
lower part of your body to the right with only a little rotation
of the upper body to the left due to the greater inertia of your
upper body. Now pull your arms in to your sides but extend your
legs out (like doing a split). When you untwist your stomach
most of the rotation this time will be of your upper body to the
right with only a little leftward rotation of your lower body,
which now has the greater inertia. Finally put your legs
straight down again and you'll be in exactly your original
configuration but rotated to your right.

But if I were floating with someone
else, then together we could do it. Thanks Edward.