Effective mass of a rolling rigid body ($M_{\mathrm{eff}}=M+I/r^2$)

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.

A hanging mass $m$ pulls the centre of a uniform disc of mass $M$ up an incline of angle $\theta$, using a light string and light frictionless pulley. If the disc rolls without slipping and the string is parallel to the slope, its effective translational mass is $M+I/a^2=3M/2$. With downward hanging acceleration $q$, $q=g(m-M\sin \theta )/(m+3M/2)$. Static friction has signed downslope component $Mq/2$ and supplies the rolling torque. The acceleration threshold is unchanged from a sliding block, although rotational inertia reduces the magnitude.