system
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human
Trying to understand near-light-speed travel; how would a spinning disk fare? 

Suppose a flat disk is spinning on the xy-axes, i.e. the perimeter lies on the xy-axes. If it was then traveling linearly along the x-axis and it neared light-speed. How would this affect the disk? 

I'm imagining a single point on the disk that travels around in a circle as the disk spins. Would the point travel faster and then slower as it increases and decreases along the x-axis?
gpt
The important things to remember about relativity is that there are no locally weird things that happen when traveling at a constant velocity and the speed of light is the same in every inertial reference frame.

From the perspective of the spinning disk there is nothing unusual going on, the center of the disk is at rest and the edge of the disk rotates at a tiny fraction of the speed of light. That's relativity, there are no preferred reference frames.

Now, if we switch reference frames to one where the disk is traveling at very close to the speed of light then how things appear between the two reference frames will be different. One of the most important things to wrap your mind around when it comes to relativity is that different inertial reference frames have differing definitions of time and space. Specifically, even though the disk in its own relatively stationary reference frame may appear to be perfectly circular, within a reference frame where the disk is traveling at near c along the x-axis it will be squashed along the forward direction and will appear to be an oval. Also, the amount of time it takes the disk to complete one rotation will seem different from within the disk's frame of reference and to the external observer. Within the disk's frame it may seem to have a rotation rate of, say, 100 rpm. But in a reference frame where the disk is travelling at 0.99c it would seem to have a rotation rate of 14 rpm. This is because time is slowed the closer you get to the speed of light (thinking about this too much can hurt your brain). The end result is that the faster the disk goes the more increasingly ovally "smooshed" the disk will get and the slower time will progress, and if you go to the trouble of working out the physical interactions you'll see that even in a squished state and slowed down the disk would still rotate just fine, it would just seem to do so in a funny way from the perspective of an outside observer.

I hope this makes sense. It's a very counter-intuitive result but relativity has been extremely well verified.