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 ?
(c) What fraction is the rotational kinetic energy of the total kinetic energy?
(d) What fraction is the translational kinetic energy of the total kinetic energy?

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)`