# Relativistic wobble?

yuiop
Wheel experiment:

A balanced wheel has two large masses attached to the rim opposite each other with respect to the axis of the wheel. The wheel is spun up to high speed but remains balanced to an inertial observer at rest with the non spinning axle of the wheel.

To an observer in an inertial reference frame that has velocity relative to the axle of the wheel, the wheel appears to be elliptically shaped. The two masses are opposite each other when both masses are exactly orthogonal to the linear motion of the wheel. Call those positions 12 O'clock and 6 O'clock. When the rim masses are at 9 O'oclock and 3 O'clock simultaneously in the rest frame they are somewhere near 10 O'clock and 2 O'clock in the moving frame. It is obvious that the centre of mass of the wheel in the moving frame is not in the centre of the wheel and that is a fairly well known fact of relativity. What is puzzling is that the centre of mass is constantly moving as the wheel rotates in the moving frame so superficially it seems impossible for the wheel to be balanced in the moving frame and should oscillate up and down.

Is this an example of De Broglies hypothesis that a particle with linear motion has a characteristic wavelength?

Is the assumed "wobble" corrected by the fact that the acceleration vector is not always parallel to the the vector of the force causing it in relativity?

Is the assumed "wobble" corrected by stress in the spokes being an effective mass according to the stress energy tensor?

## Answers and Replies

Wizardsblade
Did you take into account the change in mass of the particlees as they move faster and slower around the wheel. I believe this is correct the center of mass and the wobble problem =)

yuiop
Did you take into account the change in mass of the particles as they move faster and slower around the wheel. I believe this will correct the center of mass and the wobble problem =)

I did not mention it, but yes I did take the changing masses into account. For example if the apparent mass of the of the weight at the 12 O'clock position is greater than the weight at the 6 O'clock position the centre of mass is somewhere on the line joining the two masses but above the centre in the moving frame. When you look at the wheel in more detail and draw lines connecting the two rim masses at various intervals of time in the moving frame you find that there is no commom point that the wheel rotates about.

Of course, my assumption that the centre of mass of the two rim masses is always on a line connecting the instantaneous centre of mass of each rim weight might not be the root of the "problem".

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Wizardsblade
From the example you just gave I am not sure you understand what I was suggesting. The relativistic change in mass. I.e. The wheel spins fastest at 6 Oclock, so it will have the highest relativistic mass and I believe that this will counter the increase in mass due to the change in posistion of the spokes.

yuiop
From the example you just gave I am not sure you understand what I was suggesting. The relativistic change in mass. I.e. The wheel spins fastest at 6 Oclock, so it will have the highest relativistic mass and I believe that this will counter the increase in mass due to the change in posistion of the spokes.

I think we have different "pictures" in our heads of the experiment. I was imagining a wheel rolling along a road. To make the imagery clearer imagine the wheel belongs to a car and the car is moving at 0.8c relative to the road. The top of the wheel (12 O'clock position) will have a velocity of 0.98 c by velocity addition and the bottom of the wheel (6 O'clock position) will have an instantaneous velocity of zero in the road frame. The effective mass of a particle at the top of the wheel will be much greater than the effective mass of a similar particle at the bottom which has momentarily come to rest in the road frame. Also imagine the car is going from left to right to an observer standing on the road so the rear most part of the wheel is 9 O'clock and the front part is 3 O O'clock.

So yes, I agree the relatavistic masses will make a difference to where the effective centre of mass of the wheel is to the observer on the road but I am not yet sure if that completely cancels out any apparent wobble. (The calculations are non trivial). I am fairly certain that the centre of mass of the rotating wheel is not at the axle of the wheel according to the road observer. This is similar to the discussion of the rolling ball that going on currently in another thread, but analysing a rolling ball is a nightmare so I am trying to keep the analysis to that of two point particles on the rim of a simple disk, so that we might make progress and maybe even come up with an equation.

I'm not seeing a large difference in this and the rolling sphere.
All you have done is collapsed a dimension to make it a vertical disc. A thin wheel.
Are you doing this to isolate the centrifugal forces on a solid disc or is there an implied infinite mass (a translating singularity ?the- point of contact ) along the directional axis for the two masses to orbit?

It's a neat visualization - thanks

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yuiop
I'm not seeing a large difference in this and the rotating sphere.
All you have done is collapsed a dimension to make it a vertical disc. A thin wheel.
Are you doing this to isolate the centrifugal forces on a solid disc or is there an implied infinite mass ( translating singularity - point of contact ) along the directional axis for the two masses to orbit?

It's a neat visualization - thanks

Hi, .. and thanks

Perhaps it should be mentioned that this thread was started before the rolling ball thread which is just an elaboration and complication of this one. I not sure of the need for the elaboration when we have not solved this simpler case yet

Yes, I am basically trying to isolate the centrifugal force of a spinning disk with relative linear motion to the observer. As far as I am aware there is implication of of infinite mass in the experiment. The point of contact is momentarily stationary to the "road observer" so its relativistic mass should be the same as its rest mass at that instant.

ok

It's the point of contact that may be giving me a problem.
I can't stop seeing it as the ol' airplane on the conveyor belt thing. What if the wheel is moving .5C and the conveyor is moving -.5C. Is the wheel edge moving at C ? No slip contact makes me think it must.
I just don't see the need for the belt or 'point of contact'. I just picture the distortion on the spinning disc itself at no forward motion. Then you can add the effect on each position of the disc to it's relative motion as it approaches C

yuiop
ok

It's the point of contact that may be giving me a problem.
I can't stop seeing it as the ol' airplane on the conveyor belt thing. What if the wheel is moving .5C and the conveyor is moving -.5C. Is the wheel edge moving at C ? No slip contact makes me think it must.
I just don't see the need for the belt or 'point of contact'. I just picture the distortion on the spinning disc itself at no forward motion. Then you can add the effect on each position of the disc to it's relative motion as it approaches C

If the velocity of the airplane is 0.5c and the velocity of the conveyor belt is -0.5c then the point of contact of the wheel is -0.5c and the top of the wheel is 1/(1+0.25)c which is less than c using the relatavistic velocity addition equation.

Wizardsblade
I have been thinking about this some more, And if you look at the indivdual particlees then It seems to me that length contraction would not yield an eliptacal wheel. Any thoughts on this?