A ball rolls without slipping. The radius of gyration of the ball about about an axis passing through its center of mass is K. If radius of the ball be R, then the fraction of total energy associated with its rotational energy be

Options

A. `(K^(2)+R^(2))/(R^(2))`
B. `(K^(2))/(K^(2)+R^(2))`
C. `(K^(2))/(R^(2))`
D. `(R^(2))/(K^(2)+R^(2))`

Answer

b

Explanation

`E_("rot") = (1)/(2) I omega^(2) = (1)/(2)MK^(2) ((V)/(R))^(2)`
`E_("rot") = E_("tran") + E_("rot")`
`= (1)/(2) mv^(2) + (1)/(2) MK^(2) ((V)/(R))^(2) = (1)/(2) MV^(2) (1 + (K^(2))/(R^(2)))`
`(E_("rot"))/(E_("tot")) = ((1)/(2)mv^(2) ((K^(2))/(R^(2))))/((1)/(2)mv^(2) (1 + (K^(2))/(R^(2)))) = (K^(2))/(K^(2) + R^(2))`