Shown in the figure are two hollow cubes C₁ and C₂ of negligible mass partially filled (depicted by darkened area) with liquids of densities ρ₁ and ρ₂, respectively, floating in water (density ρ_w). The relationship between ρ₁, ρ₂ and ρ_w is 

CONCEPT:

Buoyancy and Floating Bodies

• An object floats when its average density is less than the fluid it is placed in.
• For floating equilibrium:

  Weight of displaced fluid = Weight of the object

• If an object is more submerged, it has a higher average density than one that is less submerged.
• For hollow containers of negligible mass, the density of the liquid filled inside determines how deep they float.

EXPLANATION:

• Two cubes, C₁ and C₂, are floating in water and partially filled with liquids of densities ρ₁ and ρ₂ respectively.
• From the diagram:
  • C₂ is more submerged than C₁, which means its average density is higher.
  • Since both cubes are of negligible mass, the density of the cube depends on the density of the liquid inside.
  • Thus, ρ₁ > ρ₂.
• Also, both cubes float, so their average density must be less than the density of water (ρ_w).
• This gives the final relation: ρ₂ < ρ₁ < ρ_w

Therefore, the correct relationship is: ρ₂ < ρ₁ < ρ_w