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I have thought that kinetic energy does *not* add to the rest energy
in the relativistic perspective as the rest mass is defined in the
rest frame. Rotational energy can be viewed as the sum of kinetic
energies of "small" parts of the wheel. Hence, it would not add to the
rest mass...or just mass as the qualifier "rest" is unnecessary as
eplained by Taylor & Wheeler in their splendid book Spacetime physics
(1992).