Does a relativistic rolling ball wobble?

[Antenna Guy]

Unfortunately, it also means that my "ground" is on the top, which is a little confusing.

See post #93 - I think you started with the wrong sign on tw.

Regards,

Bill

 

[Dale]

There is no right or wrong sign on tw. One is clockwise one is counterclockwise.

Similarly there is no right or wrong sign for the velocity in the Lorentz transform, one will result in "ground" on the top and the other will result in "ground" on the top.

 

[Antenna Guy]

There is no right or wrong sign on tw. One is clockwise one is counterclockwise.

Sorry - I thought your "chunks of hoop" were rotating CCW rather than CW...

I must be mistaken.

Regards,

Bill

 

[yuiop]

There is no right or wrong sign on tw. One is clockwise one is counterclockwise.

Similarly there is no right or wrong sign for the velocity in the Lorentz transform, one will result in "ground" on the top and the other will result in "ground" on the top.

Putting the ground on top does not correct the problem because that puts the most dense part of the hoop (the fastest part of the hoop) nearest the rolling contact which does not represent "rolling without slipping".

Also, the problem is there already in equation 4 before eq6 with the roots is substituted. I have written a Java program that simulates the rolling hoop in real time. I tried to do this a long time ago and was only able to do this successfully thanks to using your equations (with the y coordinate reversed in both frames). The program works well because the equation you gave for t and t' has a numerical solution that converges fairly rapidly in about 12 iterations. Now all I have to figure out is where I can upload the program so you guys can see it. I will try and find a handy website I can put it on because I would not expect anyone to download and install a program from an unverified source. Failing that I will just have to put a video of the running program somewhere (youtube?), but the program with adjustable parameters is probably more useful.

P.S. If and when you get time, it would be nice to have equations that allow the linear velocity to different from the rotational velocity (for example to simulate the motion of the outer rim of a train wheel that has a greater radius than the contact point with the rail due to the flange) and with a variable radius so that the motion and curvature of the spokes can be simulated.

 

[Dale]

Putting the ground on top does not correct the problem because that puts the most dense part of the hoop (the fastest part of the hoop) nearest the rolling contact which does not represent "rolling without slipping".

Also, the problem is there already in equation 4 before eq6 with the roots is substituted.

As far as I can tell, flipping the roots of phi corrects all the problems and implies that the ground is on the top. There is no slipping and the fastest part of the hoop is on the bottom, away from the ground.

I don't think there is any problem with eq4.

 

[yuiop]

As far as I can tell, flipping the roots of phi corrects all the problems and implies that the ground is on the top. There is no slipping and the fastest part of the hoop is on the bottom, away from the ground.

I don't think there is any problem with eq4.

You might be right, but flipping the sign of the y coordinate in equations 1 and 4 puts the ground back where it should be, on the bottom. (unless you are in Australia :P)

By doing that I have produced an applet simulation using your equations that behaves perfectly well with the hoop rolling from left to right in the frame S' with the ground on the bottom and the hoop rotating clockwise in both frames. I am just looking for a friendly webpage that supports java applets to upload to so that you can check it out. The source code for the applet will be available so that people can check there is no "trickery" going on. The applet would not have been possible without your excellent work in deriving the equations ;)