# Conservation of angular momentum vs. linear momentum

1. Jan 3, 2012

### Curl

From a little bit of thinking, this is what I concluded:

A system initially at rest can change its angular position without any outside torques (the final state will also be at rest).

A system initially at rest cannot change its displacement without an outside force.

In other words, cons. of linear momentum also implies that a system cannot move its CM without outside forces, however, cons. of angular momentum does not prohibit a body from changing its angular orientation without outside torques, due to the fact that rotational inertia can be changed internally.

Is this correct, or can anyone offer counterexamples?

2. Jan 4, 2012

### torquil

Only by moving through a set of configurations, all of which have vanishing total angular momentum. This is e.g. with two parallel wheels of different mass, that are able to rotate relatively to one another. But how do you define the "angular position" of such an object, analogously to the centre-of-mass for linear movements?

If you put coinciding angular marks on the wheels, and let them rotate in opposite directions, they may stop in a configuration where the marks again coincide, but at a different angular position since each wheel were of different mass, but this is due to the periodic topology of the set of angular position configurations.

In the same way as above, if the universe were periodic and the system consisted two massive object of unequal masses, they would be able to move in opposite directions at different speeds, all the while having a total of zero linear momentum. After one revolution around the periodic universe, they can stop and be at rest with respect to each other at a different linear position.

So if your linear space has a periodic topoogy, a composite object can also change the position of its centre-of-mass without breaking the law of conservation of linear momentum.

3. Jan 4, 2012

### D H

Staff Emeritus
A much better example is a cat. How do cats dropped upside down manage to land right side up? Just a couple of many publications on this problem (which turns out to have some applicability to the field of robotics):

TR Kane and MP Scher, A dynamical explanation of the falling cat phenomenon, Int'l J. Solids and Structures (1969)

R Montgomery, Gauge theory of the falling cat, Fields Inst. Commun., 1 (1993), 193-218.

Edit
For a non-technical summary of these two papers, see M Abrahams, Cat physics â€“ and we are not making this up, The Guardian, 17 October 2011, http://www.guardian.co.uk/education/2011/oct/18/improbable-research-cat-physics

Last edited: Jan 4, 2012
4. Jan 4, 2012

### Curl

I was inspired by the cat when thinking this up, however this is not a threat about cat physics. Turns out, the angular displacement feat is possible because rotational inertia can be changed arbitrarily from within the system (without the need for outside torques). However, there is no linear analog to this, that is, mass cannot be changed within an isolated system therefore a net linear displacement is impossible without an outside force if it is initially at rest. Is this correct?

5. Jan 4, 2012

### D H

Staff Emeritus
J Wisdom, Swimming in Spacetime: Motion by Cyclic Changes in Body Shape, Science 21 March 2003: 1865-1869, DOI:10.1126/science.1081406.
http://dspace.mit.edu/bitstream/handle/1721.1/6706/AIM-2002-017.pdf?sequence=2

6. Jan 4, 2012

7. Jan 4, 2012

### sophiecentaur

I think this is bordering on 'reactionless propulsion' idea, which is one of those heretical ideas. Poor Prof. Eric Laithwaite was pilloried for investigating this in his later years, despite not being just 'nutty' about it.

8. Jan 4, 2012

### Curl

I meant to ask this question in the classical, newtonian sense, not GR or QM or anything like that.