Consider a solid sphere of radius R and mass density ρ r = ρ 0 1 - r 2 R 2 , 0 < r ≤ R . The minimum density of a liquid in which it will float is:

Step-by-step solution

The density of any point inside the sphere is ρ = ρ 0 1 - r 2 R 2       0 < r ≤ R weight = buoyant force mg = B m g = ρ L V g ∫ ρ 4 π r 2 d r g = ρ L 4 3 π R 3 g ∫ 0 R ρ 0 1 - r 2 R 2 4 π r 2 d r = ρ L 4 3 π R 3 ∫ 0 R ρ 0 4 π r 2 - r 4 R 2 d r = ρ 0 4 π r 3 3 - r 5 5 R 2 0 R = ρ L 4 3 π R 3 2 5 ρ 0 = ρ L