Why does a wooden cube float in water?

Several forces act on a cube in a liquid:

the force of Archimedes, directed upwards;
downward gravity;
the forces of hydrostatic pressure acting on the side faces.

Due to symmetry, the forces acting on the side faces are balanced, and they can be ignored.

Gravity

FТ = m * g,

where m is the mass of the cube, g is the acceleration of gravity. Considering that

m = ρ * V,

the expression for gravity can be written:

FТ = ρ * V * g.

The strength of Archimedes is expressed:

FА = ρж * V0 * g.

In order for the body to be in equilibrium, it is necessary that the sum of the forces acting on it be equal to zero. Let us choose the positive direction of the axis, which coincides with the direction of action of the Archimedes force, and write down the equation for the forces acting along this axis.

FA – FT = 0.

Let us substitute the expressions for the forces FT and FA into the equation:

ρzh * V0 * g – ρ * V * g = 0.

Let us simplify the expression by canceling it by g and transfer the negative term to the right side of the equation:

ρl * V0 = ρ * V

We divide both sides of the equation by the product ρж * V:

V0 / V = ​​ρ / ρl.

Since V0 <V, the ratio V0 / V <1, hence

ρ / ρl <1.

Multiplying both sides of the inequality by ρ, we obtain:

ρ <ρzh

The density of water ρw is 1000 kg / m3, and the density of most types of wood does not exceed 700 kg / m3. Therefore, wooden cubes float in the water.