Question

For your science exhibit, you want to make a block of wood float at the interface between a

layer of oil of unknown density 𝜌oil, and water (𝝆𝐰𝐚𝐭𝐞𝐫=𝟏𝟎𝟎𝟎 [kg/m3]) inside a container as shown. The density of the wood is 𝝆𝐰𝐨𝐨𝐝=𝟖𝟎𝟎 [𝐤𝐠𝐦𝟑⁄]. What must be the density of the oil so that only 𝟑𝟎.𝟎% of the wood is submerged in water?

Answer 1

A fraction k of block volume must be submerged in water.

The force of gravity acting on the wood is

$$F=mg=\rho gV.$$

The buoyancy force in water:

$$F_1=\rho_1 g·kV.$$

The buoyancy force in oil:

$$F_2=\rho g·(1-k)V.$$

The two buoyancy forces prevent the block from sinking:

$$F_1+F_2=F,\\\space\\ k\rho_1+(1-k)\rho_2=\rho,\\\space\\ \rho_2=\frac{\rho-k\rho_1}{1-k}=714.29\text{ kg/m}^3.$$