A ball rolls without slipping. The radius of gyration of the ball about an axis passing through its centre of mass is `k`. If radius of the ball be `R`, then the fraction of total energy associated with its rotation will be.

Options

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

Explanation

Rotational kinetic energy,
`K _(rot) = (1)/(2) I omega^(2) = (1)/(2) MK ^(2) (v ^(2))/( R ^(2))`
(where, K is radius of gyration)
Translation kinetic energy `K _("trans") = (1)/(2) M v ^(2)`
THus totla energy,
`E = K _(rot) + K _("times")= (1)/(2) MK ^(2) ( v ^(2))/( R ^(2)) + (1)/(2) Mv ^(2)`
`= (1)/(2) Mv ^(2) ((K ^(2))/( R ^(2)) + 1) = (1)/(2) (Mv ^(2))/( R ^(2)) (K ^(2) + R ^(2)) `
Hence, `(K _(rot ))/( K _("trans"))=( K ^(2))/( K ^(2) + R ^(2))`