The moment of inertia is defined as: $$I=\int_{}^{}r^{2}dm=\int_{}^{}r^{2}\rho dV$$
While it is obvious from the experience that the father away the mass from the axis of rotation the more it adds to the inertia, it's not obvious at all that this dependence should follow square law specifically. Why is it $r^{2}$, and not just linear $r$ or $r^{3}$? Is it possible to derive this from more fundamental laws? Or is it solely based on empirical evidence?
