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 v2 = 2gh/ [1 + (k2/R2) ].
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.
A body rolling on an inclined plane of height h
,is shown in figure below:
m = Mass of the body
= 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, E
Total energy at the bottom of the plane, Eb
and ω = v
From the law of conservation of energy, we have ET
Hence, the result.