Two metallic spheres `S_1 and S_2` are made of the same material and have got identical surface finish. The mass of `S_1` is thrice that of `S_2`. Both the spheres are heated to the same high temperature and placed in the same room having lower temperature but are thermally insulated from each other. the ratio of the initial rate of cooling of `S_1` to that of `S_2` is
`(a)1/3 (b)1/(sqrt3) (c) (sqrt3)/1 (d) (1/3)^(1/3)`

Options

A. `1//3`
B. `(1//3)^(1//3)`
C. `1//sqrt 3`
D. `sqrt 3 // 1`

Answer

B

Explanation

(2) Rate of cooling ltbRgt `R = (dT)/(dt) = (e sigma A)/(mx) (T^(4) - T_(0)^(4)) = (e sigma . 4 pi r^(2))/(ms) (T^(4) - T_(0)^(4))`
`m_(1) = rho. (4)/(3) pi r_(1)^(3), m_(2) = rho . (4)/(3) pi r_(2)^(3)`
`(m_(2))/(m_(1)) = ((r_(1))/(r_(2)))^(3) = (r_(2))/(r_(1)) = ((1)/(3))^(1//3)`
`(R_(1))/(R_(2)) = (r_(1)^(2))/(r_(2)^(2)) . (m_(2))/(m_(2)) = [(3)^(1//3)]^(2) . (1)/(3) = (1)/(3^(1//3)) = ((1)/(3))^(1//3)`