There have been some threads on relativistic wheels/balls from the view of a road the wheel/ball is rolling on, with out slipping. So I decided to research them. I found some nice movies and saw some nice drawings that help explained the movement/shape of the wheel. It is an ellipse that is length contracted such that (at high speed close to c) 7 of the 8 spokes are near the top (say in the top quarter of the wheel) while one spoke slides around the wheel and when that spoke gets close to the top the next spoke in turn starts to move around the wheel. This was reinforced when I thought about the fact that the top of the wheel can not move faster the c, i.e. if the axle is moving at .9 c then the top of the wheel can not move at 1.8c but rather a velocity very close to c. This also supports the idea of the spokes bunching up near the top.(adsbygoogle = window.adsbygoogle || []).push({});

But now here is my beef. I noticed that in order for the wheel to work the spoke that travels from the front of the bunch to the back of the bunch must travel faster then c. Here is how I came to this conclusion, doing all my calculations from the roads view.

First of all the radius R of the wheel is contracted along the x-axis, but not the y-axis, so that when a particle at the end of the spoke travels from the front of the spoke line to the back it must travel a distance of at least 2R, from the front most point to the ground to the back most point. Admittedly, the spoke’s distance traveled is greater then 2R, but 2R will suffice for this example.

Now, in order for the wheel to remain in equilibrium When the axel moves forward 1/8th of the circumference (2 pi R) a spoke must travel from the front of the bunch to the back, so that when the wheel has moved like this 8 times every spoke is back to where it started and the wheel has turned 1 revolution after moving forward the circumference of the wheel.

Here in lies the problem, if speed is distance/time and we set R = 1. Then while equation for the speed of the wheel is (1/8 2R pi)/t or (pi/4)/t, were t is some time interval in the roads frame of reference. The equation for the speed of the particle is (2R)/t or 2/t, where t is the same as in the prior equation because they derived from the same frame of reference. Now if you evaluate these 2 equations you find that the speed of the particle is at least 8/pi faster then the speed of the axel which we already said was moving close to c. This of course leads to the particle moving faster then c.

I might have a geometric solution to this problem, but it is late and I need sleep. I am also perfectly willing to accept that I have made an error somewhere, please point it out to me. I have been told before that the spokes do not move so that when the axel moves forward 1 circumference the spokes will not have made 1 revolution, but I can not see this because it would make the wheel slip.

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# Particle speed on a relativistic wheel?

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