Now we address the question of what happens when a solid object is submerged in a liquid. For simplicity, let’s consider a body with a rectangular base area $A$, as sketched on the right-hand side of the original figure.
The bottom of the object is at a depth $h_2$ below the liquid surface. Therefore, the pressure at the bottom is:
$$
p_2 = \varrho g h_2
$$
On the top surface of the object, at depth $h_1$, the pressure is:
$$
p_1 = \varrho g h_1
$$
Since the column of water above is smaller at the top than at the bottom, the pressure $p_1$ is less than $p_2$. The pressure difference, using $\Delta h = h_2 - h_1$, is:
$$
\Delta p = \varrho g \Delta h
$$
Multiplying this equation by the cross-sectional area $A$, we obtain the buoyant force acting on the body:
$$
F_\mathrm{A} = \varrho V g
$$
or, more simply:
$$
\boxed{F_\mathrm{A} = F_\mathrm{Fl}}
$$
The buoyant force is therefore equal to the weight of the displaced liquid. This is known as the Archimedes' Principle, named after Archimedes, who was not necessarily the first to discover it, but was the first to describe it in detail.
Although the formula was derived for a specific shape, it can be shown mathematically to hold true for all shapes. This is due to the isotropy of the acting pressure in a fluid. Therefore, to calculate the buoyant force, it is sufficient to know the volume of the object and the density of the liquid.
For an object to float in a liquid — that is, remain neutrally buoyant and neither sink nor rise — the buoyant force must be equal to the weight of the object. In other words, the weight $F_\mathrm{Fl}$ of the displaced fluid must equal the weight $F_\mathrm{K}$ of the object:
$$
F_\mathrm{K} = F_\mathrm{Fl}
$$
Substituting both forces with $F = \varrho V g$, and canceling the volume and gravity, we get the floating condition:
$$
\boxed{\varrho_\mathrm{K} = \varrho_\mathrm{Fl}}
$$
So the density of the object must exactly equal the density of the fluid. If the object’s density is greater than that of the fluid, it sinks. If it’s smaller, the object rises until it protrudes above the surface just enough for the buoyant force to balance the weight.
The same behavior can be observed in non-mixing liquids. For example, if you pour some oil into a container of water, the oil will float on top due to its lower density.