A hollow sphere, a solid cylinder, and a hollow cylinder, all having the exact same mass and radius,

Question

A hollow sphere, a solid cylinder, and a hollow cylinder, all having the exact same mass and radius, are released simultaneously from rest at the top of an inclined plane. Assuming they all roll down without slipping, the order in which they reach the bottom is

Professor PC Thomas Classes · Answer

(a) : The acceleration of a body rolling down an incline is $a = \frac{g \sin	heta}{1 + K^2/R^2}$. The body with the smallest shape factor $K^2/R^2$ will have the largest acceleration and reach the bottom first. For a solid cylinder, $K^2/R^2 = 1/2 = 0.50$. For a hollow sphere, $K^2/R^2 = 2/3 \approx 0.67$. For a hollow cylinder, $K^2/R^2 = 1.0$. Since $0.50 < 0.67 < 1.0$, the solid cylinder is the fastest, followed by the hollow sphere, and lastly the hollow cylinder.