A body of volume V and density ρs, floats with volume v inside a liquid of density ρL. Show that v/V = ρ_s/ρ_L

Volume of body = V

The volume of body submerged in liquid = v

Density of body = ρ_s

Density of liquid = ρ_L

Let weight of the body be W. W = volume of the body x density of the body x g = V ρ_s g

The weight of the liquid displaced by the body will be equal to the upthrust. Let it be F_B.

F_B = volume of the liquid displaced × density of the liquid × g = v ρ_L g

From the principle of floatation,

W = F_B

⇒ V ρ_s g = v ρ_L g

⇒ {v\over V} = {ρ_s\over ρ_L}

Hence proved.