A concrete sphere of radius `R` has cavity of radius `r` which is packed with sawdust. The specific gravities of concrete and sawdust are respectively `2.4 and 0.3` for this sphere to float with its entire volume submerged under water. Ratio of mass of concrete to mass of swadust will be

Options

A. 8
B. 4
C. 3
D. Zero

Answer

B

Explanation

Let specific gravities of concrete and saw dust are `rho_(1)` and `rho_(2)` respectively
According to pricnciple of floation weight of a whole sphere = upthrust on the sphere
` 4/3 pi(R^(3)-r^(3))rho_(1) g +4/3 pir^(3) rho_(2) g = 4/3piR^(3)xx1xxg`
`implies R^(3)rgho_(1) -r^(3)rho_(1) +r^(3)rho_(2)=R^(3)`
`implies R^(3) (rho_(1)-1) = r^(3)(rho_(1)-rho_(2)) implies (R^3)/(r^3) = (rho_(1)-rho_(2))/(rho_(1)-1)`
`implies (R^(3)-r^(3))/(r^3)` = (rho_(1)-rho_(2)-rho_(1)+1)/(rho_(1)-1)`
`implies ((R^(3)-r^(3))rho_(1))/(r^(3)rho_(2)) = ((1-rho_(2))/(rho_(1)-1))(rho_1)/(rho_2)`
`implies (Mass of concrete)/(Mass of saw dust) = ((1.03)/(2.4-1) xx (2.4)/(0.3) = 4`.