Oil having a density of 923 kg/m3 floats on water. A rectangular block of wood 3.72 cm high and with a density of 961 kg/m3 floats partly in the oil and partly in the water. The oil completely covers the block. How far below the interface between the two liquids is the bottom of the block?

1.83 cm below the interface

Explanation:

There are three forces acting on the block of wood here:

- The weight of the block, downward, of magnitude

where

is the density of the block (wood)

V is the total volume of the block

is the acceleration due to gravity

- The buoyant force on the part of block immersed in the water, upward, of magnitude

where

is the water density

is the fraction of block immersed in the water

- The buoyant force on the part of block in the oil, upward, of magnitude

where

is the oil density

is the part of block immersed in the oil

Since the block is in equilibrium, we have

We also know that

Since the block is completely immersed in the two liquids; so we can re-arrange the equations and find:

This means that the fraction of volume in water is 0.974 times the fraction of volume in oil.

We also know that the volume of the block in each medium is proportional to the height of the block in each medium, so

(1)

We also know that the total height of the block is

so

And  substituting into (1), we find