A uniform solid sphere and a uniform solid cylinder, made of the exact same dense material and relea

Question

A uniform solid sphere and a uniform solid cylinder, made of the exact same dense material and released from the same height, roll down a rough inclined plane without slipping. Which one will reach the bottom first?

Professor PC Thomas Classes · Answer

(a) : The downward linear acceleration of a rigid body rolling without slipping on an inclined plane is given by $a = \frac{g\sin	heta}{1 + K^2/R^2}$. Notably, this acceleration is completely independent of the mass and radius of the object; it depends exclusively on the geometric shape factor $K^2/R^2$. For a solid sphere, $K^2/R^2 = 2/5 = 0.4$. For a solid cylinder, $K^2/R^2 = 1/2 = 0.5$. Since the sphere has a smaller shape factor, it undergoes greater acceleration and strictly reaches the bottom first.