Physics Thought Experiment. rolling wheel speed –> c

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In the thought experiment the wheel is only turning fast enough to propel the bike at some fraction of 20 MPH.

ruveyn is correct that all rotating wheels must experience internal stress. If they were all disparate particles, they would fly apart, so the internal stress is what's holding them together.

It's possible for a very weak material object, like a piece of balsa wood, to accelerate very fast and still maintain structural integrity,
if by doing so there is no significant internal stress.
Say I threw a piece of balsa wood in a very fast elliptical orbit passing very close to a neutron star (a super-heavy object).
It would experience tremendous acceleration as it passes by, but won't shatter because all the atoms are being accelerated by the same gravitational force (unless the piece of balsa is very wide so that one side is closer to the star).

For a fast-rotating wheel it's different. The outside has an acceleration vector which points toward the center,
while the center of the hub has essentially zero acceleration.
Every part of the wheel has a different acceleration vector (and also slightly different forces),
so every part of the wheel is under some stress.
You can use conventional means (lathes, drill presses, etc) to spin a balsa wheel fast
enough to make it rupture under centrifugal forces without seeing any effects of the distortion of rotating space.

Pointing out that no real-world material can withstand the stresses of the thought experiment (with very fast rotational speeds) is a valid point.
However, to say they cannot be spun "arbitrarily fast" due to mass is a bit misleading. If the circumference of the wheel is moving at sub-light speed then the mass is finite, not infinite. We cannot accelerate any part of the wheel to the speed of light or beyond the speed of light. Nothing in the theory of relativity tells us we can't get very very close.

As for the argument that the wheel (of unbreakable and fictionally rigid material) is not deforming when spun very fast, but just existing in a deformed space, consider the following.
According to the page I cited before:
"An ant living on the rim of the wheel would then measure a circumference that is not 3.14 times, but 8.5 times the diameter of the wheel: the intrinsic geometry of the wheel defined in this way is not Euclidean."
This means that if we gradually slow this wheel's rotation until it's not spinning, the rim will become much too long, so it will either compress uniformly or it will twist around itself like the edge of those frilly Dutch collars (or perhaps into some other shape. I don't know.) That's one reason infinitely rigid materials gives us logical paradoxes: Material scientists would end up with zero and infinity if they tried to solve equations to predict the deformation of that wheel by the Mohr circle or Castigliano's theorem or other mathematical means. For our thought experiment to make any sense (still in a fictional context, mind you), we have to assume an extremely high but finite rigidity, like a modulus of elasticity of a googol newtons per meter squared. It's still a physically impossible thought experiment, but not a logically impossibly thought experiment.

(If you don't like physically impossible thought experiments, I recommend NOT reading the rest of the Mr Tompkins book.)

To illustrate one point, lets assume the wheel in question has a circumference of one meter.
In the thought experiment the local speed of light is 32 kph, so to propel the bike at light speed, that wheel would turn 32,000 times over the period of one hour.
Dividing this figure by 60 ( 60 minutes to the hour ) gives us 533 RPM.
A wooden wheel could be assumed to turn safely at this speed, or is there something else in this that I am missing?

To illustrate one point, lets assume the wheel in question has a circumference of one meter.
In the thought experiment the local speed of light is 32 kph, so to propel the bike at light speed, that wheel would turn 32,000 times over the period of one hour.
Dividing this figure by 60 ( 60 minutes to the hour ) gives us 533 RPM.
A wooden wheel could be assumed to turn safely at this speed, or is there something else in this that I am missing?

Relate the tangential velocity to the angular velocity. It all depends on the radius, doesn't it.

How about a bicycle wheel 20 million miles in diameter With the same angular velocity the tangential velocity would be much higher.

ruveyn

On a suprising tangent the idea to a a light speed of 30kph as been used in a anime for educational purpose, and they travel in bicycle. They don't go as far to think to what would happen to the wheels though. www.dailymotion.com/video/x4rzs9_ordy-tout-savoir-sur-la-lumiere-3-p_fun?search_algo=2 (In french here translated from japanesse, I don't think a english version exist. At 5m30s )

Could someone please figure this thing out?
The suspense is killing me.

If you're looking for a movies of the wheel, try these links:
approaching –> http://www.spacetimetravel.org/filme/ra ... 40x480.mpg
side view –> http://www.spacetimetravel.org/filme/ra ... 40x480.mpg
receeding –> http://www.spacetimetravel.org/filme/ra ... 40x480.mpg
You'll have to imagine the cyclist for yourself. The motion of the legs might be tricky to visualize.
Here's what the city might look like from Mr Tompkins' perspective once he gets on a bike and starts riding:
http://www.spacetimetravel.org/filme/tue2/tue2.mov

(I think it would be awesome if Pixar did a short film about that chapter. Oh well. I can still dream. )

That is only showing flexion of the spokes, not deformation and compression of the rim.
At side view the wheel will be compressed due to an algorithm calculating the different velocities compared to the bicycles direction of travel. A distorted egg shape slightly reclined, or possibly flattened to concave in one place.
This animation does not allow for the different velocities of the top and bottom of the wheel even.