[CliffordK]

Don't we have any rocket scientists on this board?

Or...  are they all using armchair rockets?

If you read about the moon, the issue is that the moon orbits around the earth about once a month.  The earth rotates once a day.  The tides created slowly push the moon into a faster orbit (closer to the once a day rotation), which in effect will push the moon to a higher orbit, and perhaps one day the Earth will loose its moon (probably right before its orbit destabilizes and it crashes into the Earth).

Perhaps that is part of the fallacy though.  The faster the Earth pushes the moon in absolute velocity with respect to space.  The higher orbit it achieves.  The slower it moves in velocity with respect to a point on Earth (angular velocity).  So, while the Earth pushes the moon towards a 1 day per orbit, it actually pushes it towards a longer orbit...  30 days...  60 days... 

Thinking of the skater.
Consider the skater that holds 2 barbels in her hands.
Pull the barbells close to the body and the skater appears to spin faster.
Let the barbells open away from the body and the skater appears to spin slower.

However, the circumference of a circle when the barbells are held away from the body is much greater than the circumference of the circle when the barbells are close to the body.

So, if the actual velocity around the circle is maintained constant, the skater will appear to spin faster with the barbells held close to the body as the distance traveled in each revolution is much less.

Somehow centripetal force enters in to the equation.  The force of gravity varies with distance.  The skater can vary the force she exerts on the barbells at will.

[JP]

First off, I think angular momentum is just going to complicate this problem.  It's certainly conserved when you consider everything that's interacting: the tool belt and the air, but since we don't know how the air reacts, this isn't a very useful approach.

As Matthew pointed out, you can also consider the air to be a net torque which slows down the tool belt's orbital velocity, but the problem is a bit more complex than that.  It's not only orbiting with some angular velocity, but it has a separate velocity with which it's falling down toward the earth.  In a stable orbit, the angular velocity keeps it from ever actually hitting the earth. 

Early on, it's in a relatively stable orbit and the drag is going to be much higher in the orbital direction than in the "falling down" direction, so it will keep falling at the same speed, but slow down it's orbit.  But if it's traveling with a slower orbital velocity, it can't maintain it's orbit, so it starts to fall.

It's orbit will start decaying and it will be moving downwards faster and faster, but it's orbital and angular velocity will be slower and slower since there is nothing acting to speed it up.  (And you can't apply conservation of angular momentum to just the bag, since the air isn't included in your calculation.)  At some point the drag in the falling down direction will be high enough to complicate things further, but the basic result is that your bag's orbital velocity slows down and it falls towards the earth's surface.

[Geezer]

Thanks JP. I see where I was mucking it up now. I was seriously underestimating the angular momentum of the tool kit because I had it's moment of inertia all wrong  [:I]

[imatfaal]

Thinking of the skater.
Consider the skater that holds 2 barbels in her hands.
Pull the barbells close to the body and the skater appears to spin faster.
Let the barbells open away from the body and the skater appears to spin slower.

However, the circumference of a circle when the barbells are held away from the body is much greater than the circumference of the circle when the barbells are close to the body.

So, if the actual velocity around the circle is maintained constant, the skater will appear to spin faster with the barbells held close to the body as the distance traveled in each revolution is much less.

Somehow centripetal force enters in to the equation.  The force of gravity varies with distance.  The skater can vary the force she exerts on the barbells at will.

JP having nailed the argument on orbital vel - can I just take a bit of issue with above.  Clifford, your example works iff the skater herself is massless (and I know these girls are slim but...)

If you model the skater as a cylinder and the barbels(and arms) as point masses a distance x apart

 [ Invalid Attachment ]

it shows that fat lump like me wouldnt speed up as much as the sylph-like skaters

[Geezer]

I've noticed my moments of inertia seem to be getting longer and longer. Something to do with age I suspect.

"Yes dear. I'll empty the dishwasher after I've had a moment of inertia."