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,

But

From the law of conservation of energy, we have:

Hence, the given result is proved.