A solid sphere is in rolling motion. In rolling motion a body prosseses translational kinetic energy `(K_(t))` as well as rotational kinetic energy `(K_(r))` simutaneously. The ratio `K_(t) : (K_(t) + K_(r))` for the sphere is

Options

A. `2 : 5`
B. `7 : 10`
C. `10 : 7`
D. `5 : 7`

Answer

D

Explanation

(d) Translational kinetic energy of sphere
`K_t = (1)/(2) mv_c^2` …(i)
Summation of translational and rotational kinetic energy
`K_t + K_r = (1)/(2) mv_c^2 + (1)/(2) I_0 omega^2` …(ii)
Hence `(K_t)/(K_t + K_r) =(mv_c^2)/(mv_c^2 + I_0 omega^2)`
For solid sphere, `I_0 = (2)/(5) mr^2` and in case of rolling
`v_c = omega r`
It means `(K)/(K_t + K_r) = (m omega^2 r^2)/(m omega^2 r^2 + (2)/(5) mr^2 omega^2) = (1)/(1 + (2)/(5)) = (5)/(7)`.
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