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 ρw). The relationship between ρ₁, ρ₂ and ρw is
[amp_mcq option1=”ρ₂ < ρw < ρ₁" option2="ρ₂ < ρ₁ < ρw" option3="ρ₁ < ρ₂ < ρw" option4="ρ₁ < ρw < ρ₂" correct="option4"]
This question was previously asked in
UPSC NDA-2 – 2024
For Cube C₁:
Weight of liquid inside = (Volume of liquid inside) * ρ₁ * g = (A * H₁_liquid) * ρ₁ * g.
Buoyant force = (Volume submerged) * ρw * g = (A * h₁) * ρw * g.
Since it’s floating, (A * H₁_liquid) * ρ₁ * g = (A * h₁) * ρw * g, which simplifies to H₁_liquid * ρ₁ = h₁ * ρw, or ρ₁ = (h₁ / H₁_liquid) * ρw.
From the figure, the submerged height h₁ is significantly less than the height of the liquid inside H₁_liquid. Therefore, (h₁ / H₁_liquid) < 1, which implies ρ₁ < ρw. For Cube C₂: Weight of liquid inside = (A * H₂_liquid) * ρ₂ * g. Buoyant force = (A * h₂) * ρw * g. Since it's floating, (A * H₂_liquid) * ρ₂ * g = (A * h₂) * ρw * g, which simplifies to H₂_liquid * ρ₂ = h₂ * ρw, or ρ₂ = (h₂ / H₂_liquid) * ρw. From the figure, the submerged height h₂ is significantly greater than the height of the liquid inside H₂_liquid. Therefore, (h₂ / H₂_liquid) > 1, which implies ρ₂ > ρw.