State and prove Archimedes' principle.

(N/A) Archimedes' principle states: "When a body is partially or completely immersed in a fluid, the buoyant force acting on it is equal to the weight of the fluid displaced by the body, and it acts in the upward direction through the center of gravity of the displaced fluid."
Proof:
Consider a solid cube of height $h$ and cross-sectional area $A$ immersed in a liquid of density $\rho$ at a depth $x$ from the surface, as shown in the figure.
The forces acting on the left and right sides of the body are equal and opposite, so they cancel each other out.
The pressure on the upper surface of the body is $P_{1} = x \rho g$.
The pressure on the lower surface of the body is $P_{2} = (x + h) \rho g$.
The force on the upper surface is $F_{1} = P_{1} A = x \rho g A$ (acting downwards).
The force on the lower surface is $F_{2} = P_{2} A = (x + h) \rho g A$ (acting upwards).
The buoyant (resultant) force $F_{b}$ acting on the body is:
$F_{b} = F_{2} - F_{1}$
$F_{b} = (x + h) \rho g A - x \rho g A$
$F_{b} = h \rho g A$
Since $A h = V$ (the volume of the body), and the volume of the body equals the volume of the displaced liquid:
$F_{b} = V \rho g$
Since mass $m = V \rho$, we have:
$F_{b} = m g$
This force acts in the upward direction. Here, $m$ is the mass of the displaced liquid. Thus, the buoyant force is equal to the weight of the displaced liquid. This proves Archimedes' principle.