 Rolling-time factor on a straight incline

For a rigid body rolling from rest down a straight incline with fixed rolling radius $R$ and axial moment of inertia $I$, translation and rotation give $(M+I/R^2)\ddot{s}=Mg\sin \theta$. A smooth slide has $\ddot{s}=g\sin \theta$. Travel over the same distance from rest therefore gives the displayed time ratio. Static friction must suffice for rolling without slipping; its contact point is at rest, so this ideal friction does no work. A uniform solid cylinder has $I=M{R}^2/2$ and ratio $\sqrt{3/2}$.