A small steel sphere of mass m and radius r rolls without slipping on the fiictionless surface of a large hemisphere of radius R{R gt gt r) whose axis of symmetry is vertical. It starts at the top from the rest, (a) What is the kinetic energy at the bottom ? (b) What fraction is the rotational kinetic energy of the total kinetic energy at the bottom? (b) What fraction is the rotational kinetic energy of the total kinetic energy? (c) What fraction is the translational kinetic energy of the total kinetic energy? (d) Calculate the normal force that the small sphere will exerton.the hemisphere at its bottom. Howthe results will be affected if r is not very sjnall as compared to R.

Explanation

`KE_("translation") = (1)/(2) mv^(2)`
`KE_("rotation")=(1)/(2)((2)/(3)mr^(2))(omega^(2))" "(because v = r omega), KE_("rotation")=(1)/(3) mv^(2)`
`rArr" "KE_("total") = ((1)/(2) + (1)/(3)) mv^(2) = (5)/(6) mv^(2)`
(a) `(KE_("trans"))/(KE_("total")) = ((1//2)mv^(2))/((5//6)mv^(2)) = (3)/(5)`
(b) `(KE_("rotation"))/(KE_("total"))=((1//3)mv^(2))/((5//6)mv^(2))=(2)/(5)`