A revolving wheel in a vacuum and removed from Gravitational Sources

[geordief]

I cannot think of a way the circular symmetry could be broken, unless the wheel itself starts asymmetrical.

So ,if the wheel has (evenly distributed) mass ,would this mass simply serve to slow the rate of expansion of the system as a whole-or the increase in distance of elements from the CoG?

[Bender]

The mass would indeed slow the rate of expansion, but not evenly across the radius, because each part is only affected by the gravity of the mass closer to the cog.

[Janus]

So ,if the wheel has (evenly distributed) mass ,would this mass simply serve to slow the rate of expansion of the system as a whole-or the increase in distance of elements from the CoG?

Let's assume that we have a disk of thickness H and density p. If we assume that H is very small, we can use the following simplification*.

 

The mass inside any radius R will be

 

[math]M= \pi R^2 Hp[/math]

 

And for a disk with a given angular velocity the distance from the axis of rotation for a particle moving at just the speed to maintain a circular orbital velocity will be

 

[math]R= \frac{G \pi Hp}{\omega^2}[/math]

 

Any particle closer to the axis than this will be moving slow enough for gravity to keep it from flying off even after the magnets release.

 

It isn't until you get to a radius of twice this that the particles will be moving at escape velocity. If moving at exactly escape velocity they will leave on a parabolic path at further radii they will leave on hyperbolic paths.

 

Of more interest is what happens to particles between these two radii, they are moving too fast for circular orbits but less than escape velocity. They enter elliptic orbits with apoapsides further out than their starting radii.

 

So if we consider the ring of particles at one of these distances we would see the following: The ring would expand slowing its angular velocity as it does, reach a maximum size and then shrink again while speeding up. You get a "pulsating: ring.

 

Now if we move out a bit, we get another pulsating ring, however for this one, the range of size change will be greater and the pulsation rate slower. The slower pulsation adds another complication. Since the maximum size of our original "inner ring" is larger than the minimum size of the "outer ring" and the periods won't match, you will get times when the "outer" ring will be inside of the "inner" ring.

 

Even ignoring possible collisions between particles as the orbits cross, this will put more mass inside of the "inner" ring at times than there was before, which will alter the orbits of the particles it is comprised of, lowering their periapsides. For particles that started just above the circular orbit radius, they will now graze the part of the disk that remained intact, where they could either stick or knock other particles loose.

 

Exactly how things eventually settle down would be a complex many body problem. The central mass may change, or you could end up with a ring system like that of Saturn or some combination of both.

 

* Once you start assuming a thicker disk, you have to consider additional factors, such as any given particle's distance from the bisecting plane of rotation. For example, a particle at radius R and distance from the bisecting plane of d would be a distance of sqrt{R^2+d^2) from the center of the mass while still moving at the same speed as every other particle at that R. Its resulting orbit will be different in shape and inclination. And as you can see, even just considering a very thin disk is complicated enough.

[swansont]

The mass would indeed slow the rate of expansion, but not evenly across the radius, because each part is only affected by the gravity of the mass closer to the cog.

That's true of a radially symmetric sphere, but not necessarily of other geometries.

[Bender]

I was indeed thinking of a sphere and hastily assumed it would be similar for a disk. Apparently it isn't. The gravity inside the ring would still be reduced, and in the particles might even be drawn towards the closest part of the ring rather than the cog.