Here's a question which has bemused me for some time.

Say at time 0, we have a non-rotating object just hanging out in
inertial space.

At some later time T we encounter the same object, still just hanging
out, but now rotated through some static angle relative to the first
orientation. In between, the object was free of external torques, but
free to vary its shape, acquire and dissipate internal kinetic energy
as needed -- like T2. Its final mass distribution is however just as
before, though rotated.

Does this violate Newtonian physics, or not?

Angular momentum is zero throughout, but does this preclude a
rearrangement of matter into a rotated version of its former
distribution? Justify assertions.

Edward Green wrote:

Here's a question which has bemused me for some time.

Say at time 0, we have a non-rotating object just hanging out in
inertial space.

At some later time T we encounter the same object, still just hanging
out, but now rotated through some static angle relative to the first
orientation. In between, the object was free of external torques, but
free to vary its shape, acquire and dissipate internal kinetic energy
as needed -- like T2. Its final mass distribution is however just as
before, though rotated.

Does this violate Newtonian physics, or not?

Angular momentum is zero throughout, but does this preclude a
rearrangement of matter into a rotated version of its former
distribution? Justify assertions.

It's been rigorously done. A mass may reactionlessly "swim" through
spacetime by deforming. A Newtonian body in freefall may reorient by
squirming - cats.

--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
"Quis custodiet ipsos custodes?" The Net!

Edward Green wrote:
Here's a question which has bemused me for some time.

Say at time 0, we have a non-rotating object just hanging out in
inertial space.

At some later time T we encounter the same object, still just hanging
out, but now rotated through some static angle relative to the first
orientation. In between, the object was free of external torques, but
free to vary its shape, acquire and dissipate internal kinetic energy
as needed -- like T2. Its final mass distribution is however just as
before, though rotated.

Does this violate Newtonian physics, or not?

Angular momentum is zero throughout, but does this preclude a
rearrangement of matter into a rotated version of its former
distribution? Justify assertions.

Do the experiment: Sit in your swivel chair (might want
to oil the bearing first) with your feet off the floor and
your hands in your lap. Put your right arm straight out in
front of you. Swing it to the right. Put it back in your lap.

Mark L. Fergerson

Uncle Al wrote in message ...

Edward Green wrote:

Say at time 0, we have a non-rotating object just hanging out in
inertial space.

At some later time T we encounter the same object, still just hanging
out, but now rotated through some static angle relative to the first
orientation. In between, the object was free of external torques, but
free to vary its shape, acquire and dissipate internal kinetic energy
as needed -- like T2. Its final mass distribution is however just as
before, though rotated.

Does this violate Newtonian physics, or not?

It's been rigorously done. A mass may reactionlessly "swim" through
spacetime by deforming. A Newtonian body in freefall may reorient by
squirming - cats.

Thanks Al and Greg. I was suspicious of the case of cats since the
opinion has been voiced that they don't really reorient completely in
mid air, but somehow manage to get their front paws around, then
complete the twist on landing. And they may have some air resistance
to play with, so aren't really torque free. But I think Greg's
example clinches it (not to mention his assurance this is a freshman
physics lab :-). I really didn't know if it were possible or not.

If it's possible in the angular case, it should also be possible in
the linear case -- which also seems screwy -- hence your swimming?
Though "spacetime" seems like the wrong word, if that's what you
meant: "space" would do fine.

Is it possible in the linear case? No ... this seems wrong. If you
move the center of mass from x1 to x2 in time t, then the average
imputed velocity of the COM is (x2-x1)/t, which implies a non-zero
linear momentum at at least some intermediate time, which violates
conservation of linear momentum if we start and stop at rest.

Now I have to figure out what's different about the angular and linear
cases which makes this trick possible in one and not the other. Is it
some question of losing mass to infinity in the case of non-cyclic
(i.e. "ordinary cartesian") coordinates. Or is it that the "non-zero
momentum at at least one intermediate time (what elementary theorem
from calculus is this?) doesn't apply in the case of angular
orientation.

Ok ... consider a body at rest, and shoot out some projectile to move
by recoil. Big whoop -- COM of combined "body" doesn't move, of
course. Now, keep projectile tethered to main body by a arbitrarily
fine string, so the thing is really a single extended body at all
times. When the main body has reached its target position, pull on
the string to stop both motions. Ok ... now we _almost_ have our goal
.... we've reactionessly (wrt outside world) repositioned almost all of
our body, and are left with a single very long projection accounting
for the fact that the velocity of the COM was required to be zero.
Now, the amount of mass ejected in the psuedopod may be arbitrarily
small provided we make the arm arbitrarily long and -- viola! -- in
the limit as ejected mass goes to zero, we have achieved our goal!
:-)

Obviously nonsense, but mathematically correct nonsense. The catch
is, for any finite but small ejected mass we haven't repositioned the
undeformed body, and the thing blows up in the limit because we are
required to invest the projectile with arbitrarily high energies to
reposition the parent in fixed finite time for arbitrarily small
ejected mass. In the angular case, since the system remains confined
to a small region of space, we are free to use a finite "ejected" mass
and also to catch and reincorporate it.

I think your "swimming through spacetime" must indeed refer to
something intrinsically GR after all.

(Edward Green) wrote in message m...
Here's a question which has bemused me for some time.

Say at time 0, we have a non-rotating object just hanging out in
inertial space.

At some later time T we encounter the same object, still just hanging
out, but now rotated through some static angle relative to the first
orientation. In between, the object was free of external torques, but
free to vary its shape, acquire and dissipate internal kinetic energy
as needed -- like T2. Its final mass distribution is however just as
before, though rotated.

Does this violate Newtonian physics, or not?

Angular momentum is zero throughout, but does this preclude a
rearrangement of matter into a rotated version of its former
distribution? Justify assertions.

Other posters have claimed it's possible, but, from a Newtonian
standpoint, I don't see how it could happen. Sounds like angular
momentum is appearing out of nowhere and then disappearing again as
the object rotates and then stops. Can anyone explain this?

In article ,
Edward Green wrote:
Uncle Al wrote in message
...

Edward Green wrote:

Say at time 0, we have a non-rotating object just hanging out in
inertial space.

At some later time T we encounter the same object, still just hanging
out, but now rotated through some static angle relative to the first
orientation. In between, the object was free of external torques, but
free to vary its shape, acquire and dissipate internal kinetic energy
as needed -- like T2. Its final mass distribution is however just as
before, though rotated.

Does this violate Newtonian physics, or not?

It's been rigorously done. A mass may reactionlessly "swim" through
spacetime by deforming. A Newtonian body in freefall may reorient by
squirming - cats.

Thanks Al and Greg. I was suspicious of the case of cats since the
opinion has been voiced that they don't really reorient completely in
mid air, but somehow manage to get their front paws around, then
complete the twist on landing. And they may have some air resistance
to play with, so aren't really torque free. But I think Greg's
example clinches it (not to mention his assurance this is a freshman
physics lab :-). I really didn't know if it were possible or not.

If it's possible in the angular case, it should also be possible in
the linear case -- which also seems screwy -- hence your swimming?
Though "spacetime" seems like the wrong word, if that's what you
meant: "space" would do fine.

It's all conservation laws. You can rearrange the geometry linearly, but
the center of mass doesn't change position--momentum is conserved. In the
rotating case you're actually doing both, because when you extend your
hands you'll push away the rest of your body.

The "swimming" thing is something to do in a curved spacetime if you want
to lower your freefall acceleration by about an atom diameter per
second^2.

Now I have to figure out what's different about the angular and linear
cases which makes this trick possible in one and not the other. Is it

I feel certain that symmetry comes into it somehow--a rotation of 360
degrees returns you to your initial state, but only a translation of zero
will return you to your initial state.
--
"Is that plutonium on your gums?"
"Shut up and kiss me!"
-- Marge and Homer Simpson

(out2lunch) wrote in message . com...
(Edward Green) wrote in message m...
Here's a question which has bemused me for some time.

Say at time 0, we have a non-rotating object just hanging out in
inertial space.

At some later time T we encounter the same object, still just hanging
out, but now rotated through some static angle relative to the first
orientation. In between, the object was free of external torques, but
free to vary its shape, acquire and dissipate internal kinetic energy
as needed -- like T2. Its final mass distribution is however just as
before, though rotated.

Does this violate Newtonian physics, or not?

Angular momentum is zero throughout, but does this preclude a
rearrangement of matter into a rotated version of its former
distribution? Justify assertions.

Other posters have claimed it's possible, but, from a Newtonian
standpoint, I don't see how it could happen. Sounds like angular
momentum is appearing out of nowhere and then disappearing again as
the object rotates and then stops. Can anyone explain this?

Yeah ... I think I grok it after Mark Fergerson's simple example.

What's appearing out of nowhere is not angular momentum, which is
rigorously conserved ("rigorously" is a nice buzz-word to stick in
there, don't you think? :-), but moment of inertia. This is what
distinguishes the linear and angular cases: we are stuck with (linear)
inertia, but we can manipulate moment of inertia.

To take Mark's example to the limit, suppose we start with an extended
body with a point mass initially on an axis of rotation.

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 still find it kind of screwy. It would have been fun for the
shuttle to execute this kind of manuever to turn itself over in orbit
-- it even comes equipped with an arm -- as proof of concept.

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

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.

Peter wrote in message news:qA0Qa.46717$ye4.35832@sccrnsc01...
out2lunch wrote:

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.

You speak truth. I'm converted!

The weird thing is that by flapping your arms and legs and twisting
your stomach back and forth quickly, you can spin in space, yet never
have any angular momentum! The lesson for me here is that an overall
nonzero angular velocity does not necessarily imply a nonzero angular
momentum. If the spinning person doubts this, all he has to do is
stop flapping and twisting and, lo and behold, he stops spinning. No
momentum!

Thanks guys. I love it when I'm wrong, which means I enjoy life
almost constantly!