A concrete sphere of the radius R has a 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 underwater, the ratio of the mass of concrete to the mass of sawdust will be

Step-by-step solution

Let specific gravities of concrete and sawdust are ρ 1 and ρ 2 respectively. According to principle of floatation, weight of whole sphere = upthrust on the sphere 4 3 π R 3 - r 3 ρ 1 g + 4 3 π r 3 ρ 2 g = 4 3 π R 3 × 1 × g ⇒ R 3 ρ 1 - r 3 ρ 1 + r 3 ρ 2 = R 3 ⇒ R 3 ρ 1 - 1 = r 3 ρ 1 - ρ 2 ⇒ R 3 r 3 = ρ 1 - ρ 2 ρ 1 - 1 ⇒ R 3 - r 3 r 3 = ρ 1 - ρ 2 - ρ 1 + 1 ρ 1 - 1 ⇒ R 3 - r 3 ρ 1 r 3 ρ 2 = ( 1 - ρ 2 ρ 1 - 1 ) ρ 1 ρ 2 ⇒ Mass of concrete Mass of saw dust = ( 1 - 0 . 3 2 . 4 - 1 ) × 2 . 4 0 . 3 = 4