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 inclied plane of a hight h is given by `v^(2) = (2gh)/((1 + k^(2)//R^(2))` using dynamical consideration (i.e. by consideration of forces and torque). Note k i sthe radius of gyration of the body about its symmentry axis, and `R` is the radius of the body. The body starts from rest at the top of the plane.

Explanation

When a body rolls down an incline of hight h, we apply the principle of conservation of energy. `K.E.` of translation `+ K.E.` of rotation (at the bottom) `= P.E.` at the top, Fig.
i.e. `(1)/(2)mv^(2) + (1)/(2)I omega^(2) = mgh`
`(1)/(2)mv^(2) + (1)/(2)(mk^(2)) omega^(2) = mgh`
As `omega = (v)/(R ) :. (1)/(2)mv^(2) + (1)/(2)m (k^(2))/(R^(2)) v^(2) = mgh`
or `(1)/(2)mv^(2) (1 + (k^(2))/(R^(2))) = mgh`
`v^(2) = (2gh)/((1 + k^(2)//R^(2)))` which was to be proved.