Passionflower said:
The pace of a helix is defined as the length of a coil it is not the rate of rotation.
OK, it's not familiar terminology for me. Doing a google search for "pace" and "helix" doesn't turn up anyone else using "pace" in these sense in the first two pages of results, are you sure this is standard usage? I also didn't know why you put (2*pi*c) after the word "pace", although now that you quote the mathworld page I see it does give parametric equations for the helix in which z=ct, and where "r is the radius of the helix and 2*pi*c is a constant giving the vertical separation of the helix's loops" (note that the c here has nothing to do with the speed of light, so it was a bit confusing for you to write 2*pi*c in the context of a relativity discussion without explaining this!)
Anyway, if the length of a coil is 2 light-seconds in the frame where the speed of the sphere is 0.6c, then the time for a rotation in this frame must be 2/0.6 = 3.333... seconds. So, in the sphere's rest frame the rotation time must be 3.333...*0.8 = 2.666... seconds. That means the tangential speed in the sphere's rest frame is (2*pi*0.3)/2.666... = 0.70686c...where did you get a tangential speed of 0.57c?
And if the height of a coil is 2 light-seconds, then the length along a single coil must be sqrt(2^2 + (2*pi*0.3)^2) = 2.7483 light-seconds, for a total length of 6*2.7483 = 16.49 light-seconds. Thus if the time is 20 seconds in the frame where the sphere moves at 0.6c, the speed of the clock in this frame must be 16.49/20 = 0.82c.
Passionflower said:
I took the radius and the height (pace) of a coil divided by 2pi. See for instance:
http://mathworld.wolfram.com/Helix.html
Where on that page do you see anything about the length of the helix? The
page I quoted said the length of a single coil with height H and radius R is given by L = sqrt(H^2 + (2 pi R)^2), do you think this formula is incorrect? It's not hard to see that this formula is correct, since you can imagine the helix as a curve drawn on a cylinder, and then if you slice the surface of cylinder on an axis parallel to the central axis and "unwrap" it (which is possible since a cylinder has zero intrinsic curvature, just like a flat plane--see
here), you'll have a rectangle with width 2*pi*R, and the vertical height of a single coil will be H.