I'm sure it can spin a bit faster. Why can't it spin so fast that the final wheel visibly turns? Well the first wheels would melt from friction or be torn apart before then, and where are you going to get that much energy to turn them so fast? At some point there's the speed of light as a limit as well.
There are 10^80 electrons in the universe, Eddington thinks.
Even if you managed to make a contraption that turned one electron into one rotation of the first wheel, and you fed the entire universe to your contraption... you’re coming up a few dozens order of magnitude short to make that full turn :^)
After an hour, the 10th gear hasn't perceptibly moved. If you went 10x faster, the 11th gear wouldn't have perceptibly moved. At 100x faster, and you still won't see motion in the 12th gear. For every gear you want to see move, you'll need to spin the first gear exponentially faster!
Even if the first gear's teeth were moving at supersonic speeds, there still would not be any perceptible motion of the last 90 gears or so. If the first gear's teeth were moving at relativistic speeds, there would still be no apparent motion in about 80 of the gears. The speed of light is only about a million times higher than the speed of sound, and this contraption loses a factor of a million after six gears.
I did not know this, but according to what I was just reading, pretty much any physical material must rupture when it spins close to the speed of sound in that material.
If you imagine a relativistic spinning disk anyway, then it gets interesting because due to length contraction, the circumference changes relative to the diameter. I'm not up to the math, but it seems like it might change the effective gear ratio depending on your perspective and the closeness of the speed to C.
