Question

State the expression for the speed of a circularly symmetric body rolling without slipping down an inclined plane. Hence deduce the expressions for the speed of

(i) a ring 

(ii) a solid cylinder 

(iii) a hollow sphere 

(iv) a solid sphere, having the same radii.

Answer 1

Consider a circularly symmetric body, of mass M and radius of gyration k, starting from rest on an inclined plane and rolling down without slipping. Its speed after rolling down through a height h is

(i) Ring : I = MR², so that \(\beta\) = 1

(ii) Solid cylinder or disc : I = 1/2MR²,  so that \(\beta\) = 1/2.

(iii) Spherical shell (hollow sphere) : I = 2/3MR²,so that \(\beta\) = 2/3.

(iv) Solid sphere : I = 2/5MR², so that \(\beta\) = 2/5.

[Note : If the inclined plane is ‘smooth’, i.e., there is no friction, the bodies will slide along the plane without any rotation. They will then have only translational kinetic energy, undergo equal acceleration and all three would arrive at the bottom at the same time with the same speed.]