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We know that if an imaginary astronaut is in the intergalactic (no external forces) and has an initial velocity zero, then he has is no way to change the position of his center of mass. The law of momentum conservation says: $$ 0=\overrightarrow{F}_{ext}=\frac{d\overrightarrow{p}}{dt}=m\frac{d\overrightarrow{v}_{c.m.}}{dt}$$

But I don't see an immediate proof, that the astronaut can't change his orientation in the space. The proof is immediate for a rigid body (from the law of conservation of angular momentum). But the astronaut is not a rigid body.

The question is: can the astronaut after a certain sequence of motions come back to the initial position but be oriented differently (change "his angle")? If yes, then how?

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The astronaut can change his or her orientation in the same way that a cat does so whilst falling through the air. After the transformation, the astronaut is still and angular momentum is conserved. There is a rather beautiful way of understanding this rotation as an anholonomy i.e. a nontrivial transformation wrought by the parallel transport of the cat's (or astronaut's) state around a closed loop in cat configuration space. I'll write a bit more about this when I have some more time, but for now, one can give a simple explanation with an idealized "robot cat" (or astronaut) which I made up for the thought experiment:

A Simplified Robot Cat

Above I have drawn a simplified cat. I am a very aural person, so this is good enough for me so long as I can imagine it mewing!

Now our "cat" comprises two cylindrical sections: the "forecat" (F), "hinder-cat" (H) and two legs (L) which can be drawn in so that they are flush with the hinder-cat's surface. With the legs drawn in, the forecat on one hand and hinder-cat + legs assembly on the other have the same mass moment of inertia about the axis of the body. Here is how the cat rotates:

  1. Deploy legs symmetrically, i.e. spread them out as shown in the drawing. Now the hinder cat + legs has a bigger mass moment of inertia than the fore cat. Note that, if the legs are diametrically opposite and identical and are opened out symmetrically, the cat undergoes no motion;
  2. With an internal motor, the forecat and hinder cat exert equal and opposite torques on one another to accelerate, then stop. Owing to the differences between the moments of inertia, the forecat undergoes a bigger angular displacement than the hind cat;
  3. Pull the legs. Again this begets no motion if done symmetrically;
  4. Use the internal motor again with an acceleration / deceleration sequence to bring the forecat and hinder cat back to their beginning alignment (i.e. with the line along the cylinders aligned). Now the two halves have the same mass moment of inertia, so when the cat is aligned again, the rotation angles are equal and opposite.

Since the rotation angles are different in step 2, but the same in step 5, our robot cat's angular orientation has shifted.

If you want to know more about the "Berry phase" explanation and the anholonomy of the cat configuration space before I get around to expanding on this, see Mathematics of the Berry Phase by Peadar Coyle. This is not peer reviewed, but looks sound and is in keeping with similar treatments along these lines that I have seen.

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    $\begingroup$ @David Thanks. Be sure to take a look at the link that QuantumMechanic just posted: shows another (and probably more realistic) way a cat rotates physics.stackexchange.com/q/24632/2451 $\endgroup$ – WetSavannaAnimal Nov 29 '13 at 10:53
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    $\begingroup$ There is no step 5. Or is that "profit"? :) $\endgroup$ – a CVn Nov 29 '13 at 15:21
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    $\begingroup$ @David, there's a video by Youtuber SmarterEveryDay about this. Check this out $\endgroup$ – Pranav Hosangadi Nov 29 '13 at 16:18
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    $\begingroup$ I'm guessing "hinder-cat" (C) is supposed to be H ? $\endgroup$ – Izkata Nov 29 '13 at 17:21
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    $\begingroup$ In this paper cited by Wikipedia, there is an interesting diagram (page $18$ pdf, paragraph $6.1$), about the evolution of the 2-part cat, at constant total angular momentum. $\endgroup$ – Trimok Dec 4 '13 at 9:52
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For those that are cat-challenged, here's an alternative explanation and demonstration you can try at home! This demonstration was taught to me by my math lecturer. All you will need is:

A swivel chair

swivel chair

and a heavy object (e.g. a big textbook)

textbook

Stand on the seat of the chair (watch your balance now) holding the heavy object. Extend your arms forward with the object. From top-down, you look something like this (please excuse my poor drawing skills):

enter image description here

(the triangle thing is your nose; it shows which direction you are facing)

Holding the object, swivel your arms to the left.

enter image description here

Notice that your body (and the chair) rotates clockwise in response to this motion. Then pull the object towards yourself.

enter image description here

Still holding the object close to you, move it to your right.

enter image description here

Notice that your body and chair rotate anti-clockwise in response, but not nearly as much as when you had your arms extended.

You can continue repeating these motions...

enter image description here

Congratulations! You are now freely rotating in the swivel chair, without any bracing.

Whilst this is a very inefficient way of rotating yourself, the principle is exactly the same as the cat rotation example.

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    $\begingroup$ Lots of kids do this automatically when they sit in one of these chairs, by swinging their legs in a circle - extended, their legs do the same thing as when the book is held out $\endgroup$ – Izkata Nov 29 '13 at 17:24
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    $\begingroup$ While in principle correct, I fear that friction in the chair shaft negates this as a useful demonstration. By doing either part of the cycle slow enough it is possible to use the chair's static friction to prevent movement, and thus - without changing the cycle's direction - to turn either with or against the book's direction. $\endgroup$ – Emilio Pisanty Nov 29 '13 at 18:36
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    $\begingroup$ @EmilioPisanty: Yeah, but it's about as close an approximation as you can get without actually going into space. In practice, those chairs tend to have pretty low friction (at least if well maintained), so unless you do the exercise at a snail's pace, it can be mostly neglected. (As a double check, you can try holding the object at a constant distance, but moving it left and right at different speeds. If you don't observe any significant net rotation, that means the effect of friction was negligible.) $\endgroup$ – Ilmari Karonen Nov 29 '13 at 20:18
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    $\begingroup$ @IlmariKaronen For the regular office chair I just tried this out with, it is easy to produce net rotation at a constant distance. If you do the demonstration right, then it will do the job; however, if kids have a go at it they may discover the other mechanism, and one needs to be careful to explain what it is and where the differences are. $\endgroup$ – Emilio Pisanty Nov 29 '13 at 20:38
  • $\begingroup$ You can hang on the end of a rope and do this quite easily. $\endgroup$ – mmesser314 Aug 3 '14 at 20:57
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There's another way to do this also, more akin to how spacecraft actually do it:

Take a weight on a string, hold it up and spin it. You'll turn in the opposite direction. When you stop it you also stop turning.

Of course this will produce an off-axis force that will be a real pain to deal with. Real spacecraft do it by means of a set of internal wheels so they can turn on any axis.

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  • $\begingroup$ I made a model of a device to do this in college - three motors with three flywheels, axis mutually perpendicular. By increasing the speed of on flywheel, the device would experience a torque in the opposite direction. The other two flywheels made predicting the outcome on the fly tricky. $\endgroup$ – user27279 Nov 29 '13 at 20:25
  • $\begingroup$ @user27279 That's why NASA has computers to figure out how far to spin each wheel to get the desired rotation. $\endgroup$ – Loren Pechtel Nov 30 '13 at 18:34
  • $\begingroup$ @LorenPechtel which spacecrafts actually use flywheels? For example the Apollo service module had lateral thrusters instead and I assumed this was always the case. $\endgroup$ – magma Apr 17 '14 at 17:21
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    $\begingroup$ @magma The first one I looked up does: The Hubble. $\endgroup$ – Loren Pechtel Apr 17 '14 at 23:45
  • $\begingroup$ A nice thing about this approach is that if the axis of the flywheel passes through the center of mass, it's clear that while the flywheel is being turned at a constant rate--something the flywheel can easily do for any arbitrary length of time--the station will counter-rotate at a constant speed. The duration of the rotation will determine the final orientation. $\endgroup$ – supercat Oct 3 '14 at 17:06
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Other answers have pointed out other ways that might be more efficient, but one very simple way to do it is as follows: start with both arms parallel to the body. Then swing them both backward, up over the head, and then back down in front of the body, leaving them back in the starting position. After this manoeuvre, the body will be oriented in a slightly different position, with the feet slightly further forward than they were before, and the head slightly further back. It can be repeated to produce a larger change in orientation, or performed in reverse to rotate in the opposite direction.

It might seem as if this shouldn't work, but if we consider the conservation of angular momentum, we can see that it has to. When the astronaut starts moving her arms, she gives them some angular momentum. This means that her body's angular momentum changes by an equal and opposite amount. Because her body has a larger moment of inertia than her arms, its angular velocity will be smaller, because $\omega = L/I$. This means that once her arms have completed a full revolution, her body's orientation will have changed by only a small (but non-zero) angle. When she stops moving her arms, angular momentum is transferred in the opposite direction, and the body's angular momentum again becomes zero.

The amount of rotation this move produces can be increased by tucking the legs into the body, reducing its overall moment of inertia. As dmckee points out in a comment, this technique is used by springboard divers to perform half-twist moves, so we know that it definitely works, and if performed correctly can be quite efficient. (Doing it effectively while encumbered by a pressure suit might be a different matter, however.)

Bonus edit: the technique is demonstrated in zero-G conditions (aboard Skylab) starting at 0:50 in the following video:

https://youtu.be/RjvmXLyrtjM

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    $\begingroup$ It will not work. When the astronaut will try to put bodies back at rest, she will come back to the same orientation. Thanks to the conversation of the angular momentum law. $\endgroup$ – David Nov 29 '13 at 10:25
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    $\begingroup$ @David Springboard divers do it all the time (though only for half-twist moves, there is a different mechanism related to unstable tumbling about $I_2$ used for high twisting). See congusbongus' answer. There is no conservation rule for angular orientation; only for angular momentum and non-rigid composite bodies can alter the phase of their rotation (i.e. their orientation if non-rotating). $\endgroup$ – dmckee Nov 29 '13 at 15:53
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    $\begingroup$ @David conservation of momentum is actually the reason why it does work. She transfers angular momentum to her arm when she starts the motion, and it gets transferred back when she stops. Since her body has a different moment of inertia from her arm, her orientation will not be the same after one revolution. $\endgroup$ – Nathaniel Nov 30 '13 at 14:18
  • $\begingroup$ I've re-written the whole thing - I think the previous version was unclear, as people were voting it down for some reason. $\endgroup$ – Nathaniel Dec 4 '13 at 3:37
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I think that the easiest way to see this is by considering a reaction wheel. This device consists in a motor with a flywheel attached. When the motor starts spinning, the wheel gains some angular momentum, equal and opposite momentum is gained the cage of the motor and its holder (a ship, a rocket, the astronaut...) that counter rotate. When the desired orientation is reached is enough to power off the motor to stop the rotation.

If the astronaut is not carrying a small reaction wheel, he can as well start moving a limb in circles so that, for instance, his hand gains some angular momentum and his body counter rotates. It will take a while since the hand cannot spin as fast as a motor and the mass of the hand is small compared to the body, but it will work. Of course there are better sequences of movements that are more efficient, see the answer of Rod Vance.

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When one watches a true martial art master wind up with a variety of arm movements combined with other torso ones, an ability to turn is plainly there when in air motionless. No motion like propulsion is seen from my experience for more than one move at a time.

I happen to know, having a bad back, that, in order to get out the bed or recliner I raise my arms straight up, swing em in the direction I DO NOT want to go, to allow my torso to not get bent as I try to get to the standing position. Then I throw weight (my arms) at the ground to lift perhaps forty pounds off my back, to stand.

Yes, sometimes it looks pretty funny, I wind up and unleash, able to sit up, but everyone thinking something big is gonna happen. Nah, just standing up with least pain.

Next time you have to get up from a recliner, make two big waves way up in the air and pull yourself up somewhat, then pull, throw arms down, you have have stood up!

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protected by Qmechanic Aug 3 '14 at 18:53

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