KaleLetendre said:
Im suprised i missed that, thank you. Il have to figure that out tonight
There are some fundamental problems with the problem definition, especially as regards to its mass. If you have, say, a steel disk, and you spin it up, the disk will expand, because it's not rigid. This will affect the energy of the disk, and hence it's rest mass - you'll have to put more work into the disk to spin it up to speed as it expands, this affects the moment of inertia of the disk, for instance, and so the amount of energy you put into it to spin it up. There's also mechanical energy stored by the expansion itself (as in a spring). The mass of the disk (in this case, regardless of whether you mean the relativistic mass or the invariant mass) will be equal to its energy/c^2 as it will have zero total momentum in the lab frame, so the question of how much energy the disk has is the same as the question as to its mass.
You have the additional problem that steel won't be strong enough to spin a disk up to relativistic speeds, it'll disintegrate. You can take a brief look at this problem and find some interesting criterion for strentgh/weight ratios, and decide that perhaps a carbon nanotube disk would likely be the best - but, even their optimistic projected values for this stronger material aren't high enough to spin a disk up to relativistic speeds.
So, you might think to get around these issues by assuming the disk is rigid. Unfortunately, the most common standard of what it means for an object to be rigid in special relativity (Born rigidity) only applies to non-rotating objects. It's impossibe to spin up a disk in a Born-rigid manner, the short version is that one or more of the circumference and/or the radius will change - it's impossible to have them both unchanged. For details of this, see the discussion of the Ehrenfest paradox, which is related to the other part of your problem, finding the circumference of the disk.
One might then gets sucked into an interesting and rather theoretical discussion of exactly what one might mean by a "rigid disk" if one doesn't mean Born rigidity. I recall seeing some papers with proposals on the issue. However, it wouldn't be reasonable to assume the reader knew which rigidity criterion you might mean if it wasn't Born rigidity (which is impossible), so to have a meaningful discussion you'd need to describe which rigidity criterion you meant. You also have the issue that the problem becomes something that you can't really test - any physical disk you could actually make would fly apart, so you're really asking a question that can't be tested experimentally. But at least, if you have the theoretical defintion of what rigidity criterion you are using, you have a well defined problem.
If you drop the mass part of the problem, and assume the disk deforms as it spins up, things will be a lot easier. You then get a choice of how you want the disk to deform - do you want its circumference to change, or do you want it's radius to change? You could have both change, of course, but it's probably easier to keep one constant. Once you make this choice, the rest of the problem will be relatively straightforwards (though a bit tricky even so - but it will be well-defined enough to have a unique answer, rather than the debate being about what the problem is). It'll still be theoretical (due to the strength issues) of course.
This may be optimisitc - the literature on the Ehrenfest shows that there was (and perhaps is) a bit of confusion about what "circumference" means in the literature, but as a practical working matter I would say that there is a broad agreement on what "circumference" means. Proving a consensus gets a bit difficult, though, one gets into side issues of finding the "impact value" of various papers to figure out which of the published papers represent the consensus view, and which do not.
