Physics Thought Experiment. rolling wheel speed –> c

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Fatal-Noogie
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17 Nov 2012, 10:56 pm

Hey there physics lovers. I think you'll like this one.

In the book Mr Tompkins,
an absolutely brilliant piece of scientific fiction written by physicist George Gamow,
in Chapter 1, the main character, Mr Tompkins, awakens in a world where
the speed of light, c, is very very slow: about 32 km/hr or 20 mph.
He sees a bicyclist riding and getting compressed in the direction of travel.
Objects approaching the speed of light appear to get compressed relative to a subjectively "stationary" coordinate system.
When Mr Tompkins steals a bike and rides to catch up, the buildings and streets appear compressed relative to his perspective.

Now here's the riddle. What happens to the shape of the bicycle wheel?
as observed from a bystander? as observed by the rider?
Imagine looking at a white patch of tape on the tire.
As it rolls to the top of the wheel, its speed is twice that of the bicycle,
so it is compressed in length.
When it rolls to the bottom and momentarily comes in contact with the ground,
its velocity becomes zero, and it momentarily returns to default length (relative to a stationary bystander).
Will this cause a bystander to see the wheel as wider on the bottom and narrower on the top?

Relative to the cyclist's perspective, all parts of the tire are moving at equal speed,
so does the compression make the wheel smaller, forcing the bike's frame to sink lower to the ground?
OR
would the extreme centrifugal forces still force an infinitely rigid wheel to expand anyway,
cancelling out the angular compression and keeping it at the same diameter?

Think about it.


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ruveyn
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18 Nov 2012, 10:08 am

Hint: The spokes or the material disk of the wheel are not rigid. There is no such think as a totally rigid body.

ruveyn



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18 Nov 2012, 5:05 pm

Lets pretend (for the sake of the thought experiment)
that we're dealing with a fictional material with
a modulus of elasticity that is effectively infinite
under static conditions. In other words, if you hang
five million metric tons from the rim it will not stretch,
but when spun from the hub, it will take time for the rotational torque applied
to the hub to propagate out to the rim: a time equal to r/c.
In other words, it has a fictionally rigid property, but it can deform.
What shape would it deform too?
We also assume no failure mode.

Of course, any real material would fly apart from the
centrifugal forces long before the wheel can begin to approach c.
If it were rolling it would absorb tremendous thermal energy
from the rolling friction, and have to radiate it out
to keep from melting, so we have to discount that also
and assume it stays in a solid state of mater.

All that was omitted for brevity.

Admittedly, with those caveats applied, it is perhaps,
no longer a physics problem,
and instead a geometric mathematical problem.


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Fatal-Noogie
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18 Nov 2012, 8:20 pm

Oh I think I see the problem with rigidity now as it relates to energy.

The max speed c is related to the vertical asymptote of the energy
required to accelerate it to that speed. We can't deal with physics
if we discard that energy. Every material needs an elastic modulus
to absorb deformation energy needed to accelerate it, and it must
show up even in static trials, because if it didn't, any deformation
due to kinematic acceleration would represent an infinite
amount of deforming energy entered into the wheel.

Maybe I'll rephrase the question entirely (and re-post on a physics forum)
to get to the core quandary.

Imagine a very strong, flat, circular disk spinning in outer space
so that the radius is, say 3/4 c. The disk in in steady-state equilibrium,
with conservation of rotational inertia.
Suppose we pass by this disk in our fictional space ship at 3/4c past the disk.
After we account for the fish-eye lensing distortion of our ships camera,
will the disk appear to be shaped like an ellipse, or does the side
spinning in the opposite direction of our ships travel appear compressed,
making it look like the profile shape of an egg?


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ruveyn
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24 Nov 2012, 1:32 am

Imagine your grandmother had balls. Does this make her your grandfather.

ruveyn



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24 Nov 2012, 2:11 am

What do I look like to you? A genealogist?


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RocketPeacock
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26 Nov 2012, 12:52 am

I think the issue here is that, even if light traveled at 32 km/h, we would still have the same difficulties approaching the speed of light as we do in this universe, and thus, nothing would go the speed of light.

These difficulties include physics breaking down.



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27 Nov 2012, 2:04 am

I found a link to the chapter:
http://www.iafe.uba.ar/e2e/phys230/Gamo ... rland.html
His writing style is quite concise.

Gamow accounts for this difficulty of approaching c
by explaining that the cyclist feels increacingly more
and more tired, but never gets anywhere close to c.
He only gets fast enough for the effects to become
noticeable to the senses.

The issue I have is with his illustration with the shortened cyclist.
(I tried posting the image here and it wouldn't work, so you'll have to click the chapter link.)
The spokes should appear closer to eachother on top than on the bottom,
shouldn't they?


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ripped
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07 Jan 2013, 12:01 am

Fatal-Noogie wrote:
Hey there physics lovers. I think you'll like this one.





...Relative to the cyclist's perspective, all parts of the tire are moving at equal speed,
so does the compression make the wheel smaller, forcing the bike's frame to sink lower to the ground?
OR
would the extreme centrifugal forces still force an infinitely rigid wheel to expand anyway,
cancelling out the angular compression and keeping it at the same diameter?

Think about it.


Dude, it is the space that deforms, not the wheel itself.
"Objects approaching the speed of light APPEAR to get compressed..."



Fatal-Noogie
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07 Jan 2013, 1:04 am

I forgot to update this thread after finding new resources a few months ago.
Someone addressed the same question I had by designing their own software to render it.
The question of apparent shapes from different perspectives is answerable.
Image
Follow the link to see their 3d animated version:
http://www.spacetimetravel.org/tompkins/node8.html
or read about their project from the beginning here:
http://www.spacetimetravel.org/tompkins/node1.html


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07 Jan 2013, 1:15 am

Here's an excerpt from one of the pages that explains how the wheel is actually deformed, not just a subjective illusion.

Quote:
Gamow's cyclist rides at 93% of the speed of light as can easily be inferred from the length contraction of the wheels in Figure 1. For the motion of a single wheel this means: The point on the wheel momentarily in contact with the street is at rest, the hub moves at v=0.93c (c the speed of light) and the point on the rim on top of the wheel moves at 0.93c with respect to the hub, i.e. with respect to the street at according to the relativistic velocity addition.
However, if one wants to set a wheel from rest into rotation at nearly the speed of light, a serious mechanical problem arises: The rim that moves in the direction of its circumference will be length contracted, at a rim speed of v=0.93c by a factor of . But the motion of the spokes is perpendicular to their axes so that the spokes are not shortened.
Without going into technical details we therefore equip the bicycle with wheels that are assembled in rotation. This is done in such a way that in stationary rotation the wheels have the geometric shape of ordinary wheels at rest (Figure 11a).
An ant living on the rim of the wheel would then measure a circumference that is not times but 8.5 times the diameter of the wheel: the intrinsic geometry of the wheel defined in this way is not Euclidean [9].

Edit: The equations did not copy properly.
See link:
http://www.spacetimetravel.org/tompkins/node7.html


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ripped
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11 Jan 2013, 1:26 am

Fatal-Noogie wrote:
Here's an excerpt from one of the pages that explains how the wheel is actually deformed, not just a subjective illusion.


Einsteins equations demonstrated that it is the geometry of time and space ( therefore everything in it as well ) which compressed. So the question of the elastic properties of the material the wheel is made out of is irrelevant.

The part of the wheel in contact with the ground ( at 6 o clock ) is stationary, therefore no deformation.
As the wheel turns to 9 o clock it has a relative velocity of some fraction of the local speed of light, and therefore some compression.
When the wheel hits 12 o clock, it is traveling at twice the speed of the bike, maximum compression.
As it revolves around to 3 o clock its velocity has slowed, and it will have less compression.

What the hell this looks like I don't know.

My point was that even if the wheel was made out of wood, it would have exactly the same deformation in its geometry relative to a stationary observer without any danger of splintering or breaking.

There is no mechanical stress upon the wheel implied by the deformation of the space in or around the wheel.



ruveyn
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11 Jan 2013, 10:27 am

Hint. There is no such thing as a rigid wheel.

Aside from the breakup of a wheel rotating fast enough (by centrifugal force) the outer part of the while will gain mass as its tangential velocity increases so it cannot be spun arbitrarily fast.


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11 Jan 2013, 8:11 pm

ruveyn wrote:
Hint. There is no such thing as a rigid wheel.

Aside from the breakup of a wheel rotating fast enough (by centrifugal force) the outer part of the while will gain mass as its tangential velocity increases so it cannot be spun arbitrarily fast.


ruveyn


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



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12 Jan 2013, 4:44 am

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.)


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