Consider a disc of radius ##R## with a uniform mass distribution and total mass ##M## rotating in its own plane. In its COM's inertial rest frame it has angular velocity ##\omega## about its center. In the obvious polar coordinates in that frame the kinetic energy of a small part of the disc at radius ##r## is $$dK=\frac{M}{\pi R^2}\left(\frac 1{\sqrt{1-\omega^2r^2}}-1\right)r\,dr\,d\phi$$in units where ##c=1##. I just integrate to get the total KE of the disc. The ##\phi## integral is...