The following bodies, (1) a ring (2) a disc (3) a solid cylinder...

Mgsin\thetaR = \left(m{k}^{2} + m{R}^{2}\right)\alpha$</p><p>$\alpha = \dfrac{Rg sin\theta}{{k}^{2} + {R}^{2}} ⇒ a = \dfrac{gsin\theta}{1 + \dfrac{{k}^{2}}{{R}^{2}}}$</p><p>$t = \sqrt{\dfrac{2s}{a}} = \sqrt{\dfrac{2s}{gsin\theta}\left(1 + \dfrac{{k}^{2}}{{R}^{2}}\right)}$</p><p>for least time,$k$should be least and we know$k, is least for the solid sphere.