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.... We label each point on the rotatingI may be wrong (wouldn't be the first time) but it looks to me
circle with a value of theta (where 0 <= theta < 2*pi) the Cartesian
coordinates of these points are given as:
x' = R*cos(theta + omega*t') and y' = R*sin(theta + omega*t')
like these expressions build in from the very beginning the
condition that the rotation have no effect on the shape
of the disk since you are assuming that an observer in a state of
translational (but not rotational) rest with respect to the disk sees
a circular disk. If that is true, then it is not surprising that we
end up concluding that the rotation has no effect when the observer
is *not* at translational rest either -- we are basically deriving
what we assumed to be true to begin with -- that rotation has no
effect of the apparant "circularity" of the disk.
It is not clear to me that this should be valid. It would certainly
not be true if we replaced the disk with a discrete set of small
boxes travelling in circles -- the individual boxes would be
Lorentz-distorted. Isn't the disk just a limiting case? Glue all the
boxes together and (assuming the flywheel doesn't blow up as Leigh
and others suggested) wherever they are Lorentz-contracted the disk
must get pinched.