A rectangular bar of soap has density 800 kg/m3 floats in water of density 1000 kg/m3. Oil of density 300 kg/m3 is slowly added, forming a layer that does not mix with the water. When the top surface of the oil is at the some level as the top surface of the soap. What is the ratio of the oil layer thickness to the soap’s thickness, x/L?

Engineering

Physics

Buoyancy

Forces and Types of Forces

Free Body Diagram

Question

A rectangular bar of soap has density 800 kg/m³ floats in water of density 1000 kg/m³. Oil of density 300 kg/m³ is slowly added, forming a layer that does not mix with the water. When the top surface of the oil is at the some level as the top surface of the soap. What is the ratio of the oil layer thickness to the soap’s thickness, x/L?

$\frac{2}{7}$

$\frac{3}{10}$

$\frac{3}{8}$

$\frac{2}{10}$

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Force of buoyancy
F_B = A[xρ₀ + (L – x)ρ₀]g
Weight W = LAρ_sg
For equilibrium F_B = W
⟹ LAρ_sg = A[xρ₀ + (L – x)ρ₀]g
⟹ Lρ_s = xρ₀ + (L – x) ρ_w

$\Rightarrow ρ_{s} = \frac{X}{L} (ρ_{0} - ρ_{w}) + ρ_{w}$

$\Rightarrow \frac{x}{L} = \frac{ρ_{w} - ρ_{s}}{ρ_{w} - ρ_{0}} = \frac{1000 - 800}{1000 - 300} = \frac{2}{7}$ .