Question

Prove the result that the velocity v of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height h is given by

Using dynamical consideration (i.e. by consideration of forces and torques). Note k is the radius of gyration of the body about its symmetry axis, and R is the radius of the body. The body starts from rest at the top of the plane.

Answer 1

A body rolling on an inclined plane of height h,is shown in the following figure:

m = Mass of the body
R = Radius of the body
K = Radius of gyration of the body
v = Translational velocity of the
body h =Height of the inclined
plane g = Acceleration due to gravity
Total energy at the top of the plane, E₁= mgh

Total energy at the bottom of the plane, Eb=KE_not+KE_trans

From the law of conservation of energy, we have:​