Calculating Radius and Mass of Disc for Velocity 0.75C

[KaleLetendre]

Imagine a disc witch has a radius of R=1000m and a mass M=1000kg, this disc sits on an cube that is considered static it has no velocity in any direction whatsoever. There is a light clock with the length L=17.45240644m (approximately the same as the arc length for 1 degree on the disc θ=1) siting at the edge of the disc (point R=1000m) it sits as parallel as possible to the edge of the disc. The Disc is spinning at a velocity of V=0.75C (C=3x10⁸). Due to the fact that its velocity is constant (disregarding direction) We may say that at any point in time V=0.75C. Imagine another light clock that will be identical to the first. This new light clock sits on the cube completely static but in the perfect position that when the first clock passes they are parallel. if the second light clock measures T= 100s than due to T=T'/γ (where γ is the Lorenz contraction equation) T' = 66.14s (T' is measured by the clock on the disc) because as i said before the disc is moving at a constant velocity so we may say that the change in time is 100s and the velocity is 0.75C. Applying the same rules i equate that M'=1511.857892kg and L'= 11.5436818 going further this would cause the circumference to contract witch would then cause the radius to shrink Cc= circumference Cc'=2πR' 2γD/2π = R' ( D= 360/θ as i said before L is approximately equal to the arc of θ) R'= 661.4042475m and Rγ= 661.4042475m so R'=Rγ

I am eager to read any input given to me, Cheers.

 

[DiracPool]

Imagine a disc witch has a radius of R=1000m and a mass M=1000kg, this disc sits on an cube that is considered static it has no velocity in any direction whatsoever.

So it has a two-dimensional radius and a mass? No 3D mass density? Is it possible to have an infinitely thin disc with a mass of a 1000 Kg and 1000 meter radius?

 

[KaleLetendre]

So it has a two-dimensional radius and a mass? No 3D mass density? Is it possible to have an infinitely thin disc with a mass of a 1000 Kg and 1000 meter radius?

Good point, we could always just imagine it has a width of X, it still should not effect my calculation

 

[Nugatory]

Google for "Ehrenfest Paradox". There are some subtle and very interesting issues in this problem, and you'll have a lot more fun with them if you know what's already been done. It's worth the effort of reading beyond the wikipedia article and into some of the serious papers on the subject - Oyvind Gron's is a good start.

Your mass calculation is pretty much bogus because not all parts of the disk are moving at the speed of the edge. However, that's easily corrected if you consider a ring instead of a disk.

 

[KaleLetendre]

Google for "Ehrenfest Paradox". There are some subtle and very interesting issues in this problem, and you'll have a lot more fun with them if you know what's already been done. It's worth the effort of reading beyond the wikipedia article and into some of the serious peer-reviewed papers on the subject.

Your mass calculation is pretty much bogus because not all parts of the disk are moving at the speed of the edge. However, that's easily corrected if you consider a ring instead of a disk.

Im suprised i missed that, thank you. Il have to figure that out tonight

 

[pervect]

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.

 

[KaleLetendre]

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.

You seem very smart, thanks for your reply. It appears the mass and length changes are much more complex than i originally assumed. Although if we had a theoretical disc that could sustain this velocity and we had two light clocks one at R=1000 and the other at R=500. Would it be correct to say that they will experience different values for time

 

[Nugatory]

It appears the mass and length changes are much more complex than i originally assumed.

Actually, it's the rotating disk that makes the problem complicated. Have you tried that google search for "Ehrenfest Paradox" yet? You've only begun to explore the subtleties of this problem.

 

[pervect]

You seem very smart, thanks for your reply. It appears the mass and length changes are much more complex than i originally assumed. Although if we had a theoretical disc that could sustain this velocity and we had two light clocks one at R=1000 and the other at R=500. Would it be correct to say that they will experience different values for time

As others have pointed out, there are other ways of making clocks move in a circular orbit besides attaching them to a disk, though there's no harm in imagining them attached to a disk if you're careful not to make other assumptions that may be harmful.

So let's imagine that one clock, which we'll call Tm_1000, moves just above a circle that we draw and measure in the lab frame with a radius of 1000 meters with a constant angular velocity, and that it periodically passes directly over another clock, Tg_1000, that's on the ground. Then we will note that the period of the orbit measured by the moving clock, Tm_1000, will be lower than the period of the orbit measured by the ground clock, Tg_1000.

The details of the clock don't matter, but it's easier to imagine using an atomic clock, as one can make an ionized atom move around in a circle very quickly at speeds approaching that of light. And ionized atoms can be used to make excellent clocks.

You can imagine the same thing happening with Tm_500, a clock moving above a 500 meter circle, and Tg_500, a clock on the ground on the 500 meter circle.

We can synchronize Tg_500 and Tg_1000, since they are on the ground - they will tick at the same rate as measured in the lab frame. If we adjust the orbital periods properly, we can make Tg_500 = Tg_1000, so that the period of the orbit of the moving clocks as measured by the ground frame can be made the same.

If we do this, we will find that Tm_1000 < Tm_500 < (Tg_1000 = Tg_500).

i.e. informally, we can say that the clock over the 1000 meter circle will tick the slowest, the two clocks on the ground will tick the fastest, and the one over the 500 meter circle will tick somewhere in between.

 

[KaleLetendre]

I took a look at Ehrenfest paradox and the radius is still confusing me. If the circumference changes due to SR why can we not say that C=2piR and C'=2piR' so R does not = R'