The limits of rotation

Let D be a disk rotating in K, at angular velocity ω. Then, the radius r of D has to be smaller than rc = c / ω, otherwise the particles on the periphery would have a speed of vr = r · ω ≥ rc · ω = c.

This sounds counter-intuitive, because from classical mechanics we expect that the magnitude of the centripetal force acting upon a particle of mass m > 0 at the periphery of radius rc would be Fc = m · c2 / rc, which is a finite quantity. Thus, for an observer on D it would definitely seem possible to gradually extend D until it reaches any target radius r ≥ rc, since the centrifugal force to overcome during the process would be limited.

However, according to special relativity, in a momentarily comoving inertial frame of a particle that is on the periphery of D, the magnitude of the particle's acceleration is a' = a / (1 – vr2 / c2), where a = vr2 / r is the magnitude of the centripetal acceleration observed in K. So the magnitude of the centripetal force "felt" by the particle is F' = m · a', which converges to infinity if r → rc.

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