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 `v^(2) = (2gh)/((1 + k^(2)//R^(2))` using dynamical consideration (i.e. by consideration of forces and torque). 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.

Explanation

To prove the result that the velocity \( v \) of translation of a rolling body at the bottom of an inclined plane of height \( h \) is given by

\[ v^2 = \frac{2gh}{1 + \frac{k^2}{R^2}} \]

we will use the principle of conservation of energy and the relationship between linear and angular motion.

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