Spinning wheel and light speed

[haynewp]

What if the center part of a bicycle wheel was spinning at almost the speed of light. What would keep the outside part of the rim from spinning faster than the speed of light?

 

[GOD__AM]

What if the center part of a bicycle wheel was spinning at almost the speed of light. What would keep the outside part of the rim from spinning faster than the speed of light?

The fact that nothing other than light (em) can travel at that speed. The center can only spin at a rate which has the outside approaching light speed but not reaching it.

 

[Zanket]

For an finitely rigid wheel, the spin rate would vary with the distance from the center. The spokes would make a spiral to keep the rim spinning at less than the speed of light.

 

[JesseM]

Also, due to Lorentz contraction the wheel will want to shrink in radius as its velocity increases, although depending on how compressible the spokes are they may oppose this.

 

[Zanket]

It's the circumference that is length-contracted in this case (from the perspective of a stationary observer), not the radius. Presumably the rim is intact regardless.

 

[JesseM]

It's the circumference that is length-contracted in this case (from the perspective of a stationary observer), not the radius. Presumably the rim is intact regardless.

Yes, but in flat space it's impossible for the circumference of a wheel to shrink without its radius shrinking by the same amount. The spokes will not be Lorentz-contracted, but if they're compressible they'll be pushed down by the shrinking rim.

 

[wangyi]

I think we can explain it like this: Once an object moves faster, it is heavier, more difficult to move. So if the outside part moves fast enough, and the inside part wants it to move faster, it will take the latter more and more force, and at last it apporches
infinity. So if the inner part want to move fast, the wheel will be broken, and if the wheel is not broken, the whole wheel can not move so fast.

I don't think Lorentz contration can help, because, first, for the person lives on the wheel and rotates with it, there is no velocity relative to the wheel, and it can be him, not the people standing on the Earth accelearate the wheel. second, if you consider
rigorous on this question, it is not at a free fall frame, and the Lorentz contration can not be used to measure the contration of a rotating object.

regards
wangyi

 

[Neitrino]

When tangent velocity of rim approaches c (speed of light), radius decrises and actually all effort made to accelerate wheel rotation goes to reduce the radius of the rim and at last you are getting poit (when the nearby parts' tangent velocity approach c). And spokes (I think beeing spirally wounded in the begining, they also begin to shorten (since their outer parts are obtaining tangent shape).

Regards
Neitrino

 

[russ_watters]

It's simpler than all that: inertia.

 

[wangyi]

In many posts, people say that it is Lorentz contraction that forbid the outer part of wheel to reach c. but as i posted, i still don't think this is the main reason. suppose the edge of the wheel has been running nearly at c, then the half-length from the center is running at c/2, then the length from the center to the half-length point does not contract to zero, so Lorentz contraction can not be the main reason.

regards,
wangyi

 

[Zanket]

Yes, but in flat space it's impossible for the circumference of a wheel to shrink without its radius shrinking by the same amount. The spokes will not be Lorentz-contracted, but if they're compressible they'll be pushed down by the shrinking rim.

That sounds reasonable. What could alternatively happen is the rim could stretch or break. A rotating wheel is sometimes the (poor, IMO) example used by relativity books to explain the concept of excess radius; even Einstein's, in which case the rim stretches.

 

[pervect]

Imagine a non-rotating wheel. If you imagine applying a torque to it, the wheel will start rotating. However, no matter how much torque you apply, or how long you apply it, the rim of the wheel will never reach the speed of light.

Detailed calculations of the angular momentum of a rotating disk are actually quite messy, however - for instance, the sci.physics.faq on the rigid rotating disk in relativity does not write them down.

http://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html

The analogous case with applying a linear force in a constant direction to a particle is simpler to analyze and yields the same result. If you accelerate a particle with a constant force (or a constant proper acceleration) in the same direction, it will also never reach the speed of light. The calculations in this case are simpler and better documented - see for example

http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

 

[JesseM]

I don't think Lorentz contration can help, because, first, for the person lives on the wheel and rotates with it, there is no velocity relative to the wheel, and it can be him, not the people standing on the Earth accelearate the wheel.

If you consider things from the point of view of someone on the wheel, you're dealing with a non-inertial coordinate system where the laws of physics (including those relating to Lorentz contractions) won't work the same way they do in an inertial coordinate system. You could also just consider the inertial frame where the guy on the wheel is instantaneously at rest at a given moment, but at that same moment other parts of the wheel will be in motion, so different sections of the rim will be contracted by different amounts in this frame. I'm not saying Lorentz contraction is the only thing you need to explain why the rim can never exceed the speed of light--the most basic explanation is just that it would take an infinite amount of energy to accelerate any point on the wheel to FTL speeds, and that there can be no such thing as a perfectly rigid object in relativity.

second, if you consider
rigorous on this question, it is not at a free fall frame, and the Lorentz contration can not be used to measure the contration of a rotating object.

Just consider the free fall frame where the center of the wheel is at rest (though still rotating). In this frame, every point on the rim is moving with the same velocity.