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

Answer

4

Explanation

Let R be the radius of the whole sphere. Let `rho_1, rho_2` be the specific gravity of concrete and saw dust respectively. Ac cording to principle of floatation, weight of whole sphere = upthrust `(4)/(3) pi (R^3 - r^3) rho_1 +(4)/(3) pi r^3 rho_2 = (4)/(3)pi R^3 xx1`
or `R^3 rho_1 - r^3 rho_1 + r^3 rho_2 = R^3`
or `R^3 (rho_1 - 1) =r^3(rho_1- rho_2)`
or `(R^3)/(r^3) = ((rho_1 -rho_2))/((rho_1 - rho_1))`
`or (R^3 - r^3)/(r^3) = ((rho_1 - rho_2) - (rho_1 - 1))/((rho_1 - 1))`
or `((R^3 - r^3) rho_1)/(r^3 rho_2) = ((1 - rho_2)/(rho_1 - 1))xx (rho_1)/(rho_2)`
`:.("Mass of concrete")/("Mass of saw dust") = ((4)/(3)pi (R^3 - r^3)rho_1)/((4)/(3)pi r^3 rho_2)`
`=((R^3 - r^3)rho_1)/(r^3 rho_2) = ((1 - rho_2) rho_1)/((rho_1 - 1)rho_2)`
`=((1 -0.3)/(2.4 - 1))xx(2.4)/(0.3) = 4`