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Viewed from any constant angle, location, and velocity, an isolated system will not change how much it is spinning.

This raises the question, how does one define 'how much it is spinning'? The definition that Newton made is so: For any individual mass observed from a given frame of reference, the amount of 'angular momentum' is the product of the following three quantities:

  1. The object's mass
  2. The object's displacement vector from the center of the frame of reference (the origin), and
  3. The velocity of the object.

For those last two, use a cross product. This has the effect of eliminating all vector components of #3 which are not orthogonal to #2 (and vice versa). For the rest of us, that means that it only matters how much the object is moving to the side, not how much it's getting closer or further away. Also, it means that spin has a direction, along the axis of spin. Earth, for example, has rotational angular momentum pointing along its axis, in the direction pointing from the south pole to the north.

To get the angular momentum of a system larger than one mass, simply add up the angular momentum of each component object. Make sure you add them up using vector addition. To get the angular momentum of a continuous distribution of mass (any real object), you must take an integral.

Once you have determined the angular momentum of a system from a viewpoint, it will not change so long as it does not interact with anything else. That is the law of conservation of angular momentum.

Often, the dependence on where you look at it strikes one as odd. Does a car zipping by have angular momentum? Imagine that you stuck your hand out and let the car hit it. You would begin to spin as a result of the collision (among other less pleasant things). This is a hint (not proof) that it has angular momentum from your point of view.

Lastly, quantum mechanics includes a notion of 'spin' which is capable of absorbing angular momentum. This is not conventional since even point particles and particles without mass have spin. Nonetheless, the total amount of this newly redefined angular momentum does not change.

A quick note about "where" this idea of conservation of angular momentum.

We assume that no matter which direction our system is orientated, the laws of physics are the same so if we run an isolated experiment east to west, we would produce the same results if the system were oriented north to south. That is, we assume that the universe is isotropic. This is just an assumption based on the fact that we have not had any experimental evidence to dispute this. Using quantum mechanics, we can show that if a system remains unchanged by rotating either the system (an active transformation) or the coordinate system (a passive transformation) then angular momentum is conserved.

Noether's Principle states that for any symmetry in physics there is an associated conserved quantity. This is a specific example of this. (A conserved quantity is any quantity which does not change in an isolated system)

Let's hand-wave a little:

For example, let us consider a hydrogen atom in it's own little universe and the laws of physics are isotropic. The semi-classical electron orbits the proton happily with constant angular momentum. We then throw a switch which cases the electromagnetic interaction to change slightly in a certain direction (let's say Coulomb's constant increases - all other things constant, a higher Coulomb constant means a greater force) thus breaking the isotropy of the universe. Then the Coulomb force will increase, causing the potential energy to become more negative thus forcing (by conservation of energy) the kinetic energy of the electron to increase, thus increasing the speed, thus increasing the total angular momentum of the universe.

Note that we do not mean to imply that if it's just the force that is different in different directions that angular momentum is not conserved, but rather if the rules which describe the force are different in different directions then angular momentum cannot be conserved. Nor does this argument suggest that if space is isotropic that angular momentum is conserved, just that if space is anistropic then angular momentum is not conserved. A proof of this would case a bit of sophistication (usually shown in undergrad quantum mechanics).

It goes something like this:
The earth spins west to east
Once every 23 hours and 56 minutes.
So, if I spin really, really fast
From east to west,
Then maybe, just maybe,
I could prolong 7:48 in the evening
Of April 2, 2011.

I think they call this 'conservation of angular momentum'.

So correct me if I'm wrong:
If the sudden movement of the crust
(no more than 30 miles thick)
Could shorten the day by 1.8 microseconds,
Then I have a lot of spinning to do
To push back the dawn, even a little.

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