Question

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

Answer 1

Let V be the volume of a body of density ρ_s

Let the body be floating with its volume v immersed inside a liquid of density ρ_L

Then , weight of the body,

W = Volume of body × density of body × g

or , W = V ρ_s g

Weight of liquid displaced by body or upthrust,

F_B = Volume of displaced liquid × density of liquid × g

or , F_B = v ρ_L g

For floatation , W = F_B

i.e. , V ρ_s g = v ρ_L g

or , `"v"/V = ρ_s/ρ_L`

Thus , `"Volume of immersed part of body"/"Total volume of body" = "Density of body"/"Density of liquid"`