 Effective mass of a rolling rigid body

For rolling without slipping with contact radius $r$, substitute $\omega =v/r$ into translational plus rotational kinetic energy:The coefficient acts as an effective inertia for a translation coordinate. It is not additional gravitational mass. A quadratic potential energy $K{x}^2/2$ then gives small-oscillation frequency ${\omega }_0=\sqrt{K/M_{\mathrm{eff}}}$ when this coefficient is constant.