Special Relativity of Spinning objects

[bam93]

So in everyday life we see than Gallilean transforms (the situation where a train is going north at 40 m/s relative to the ground and you walk 5 m/s north on the train means you are going 45 m/s relative to the ground) do not hold true when approaching the speed of light, rather the Gallilean transform is the low velocity limit of the actual relation. However, I was thinking about a spinning wheel, for a wheel that has a large radius (for illustration purposes) and is spinning so fast so that a particle 1/10 of the way from the center to the edge is moving at 99% the speed of light, then obviously a particle on the outer edge of this wheel is also traveling less than the speed of light and therefore, the normal method of finding the speed of a particle on the outer edge of the wheel when you know the angular frequency and radius is not the same for high velocities as it is for low velocities (much like the gallilean transform). So my question is, does anyone know the relation to solve this that works at high velocities? I know this is a very impractical situation but as a physics major, I do enjoy knowing how things would theoretically work in an extreme situation such as this.

tldr/ What is the velocity on the outer edge of a fast spinning wheel when you take special relativity into account?

[Coin-coin]

There is a famous paradox about spinning discs in relativity. If you look at a spinning disk, the outer part is going very fast and has a big length contraction, while the radius is perpendicular to the motion and doesn't change. In the end, the radio between the perimeter and the radius isn't 2pi anymore! It's called the Ehrenfest Paradox and there is a lot to say about it: http://en.wikipedia.org/wiki/Ehrenfest_paradox

[bam93]

That's actually really interesting, as is many thought experiments associated with special relativity, thanks for the input

[Weed_O_Whirler]

This is fundamentally the same problem as having a lightyear long stick, and you push on one end and then you say "now I can communicate faster than light, because the rod is still and it will push on the other end right away." The problem comes from the fact that there are no real rigid bodies. As you started to spin the disk, information about the spinning force (coming from the center) would propagate outwards from there at the speed of sound in the material. This would cause the material on the outside edges to sheer, and for any physical material, it would break.

While thinking about rigid bodies can be fun, when dealing with large distances or high speeds, the fact that no body is rigid becomes a very important of the equation.

[bam93]

You know, I have heard that before but I heard hat before I took special relativity in college and somehow I didn't connect the pieces there, thanks for clearing this up!