# Stevin’s Theorem

Cylinder containing homogeneous liquid

When a fluid is in equilibrium, it does not tend to slip. In this way, the interactions it maintains together with other objects or bodies are always normal to the contact surfaces. It is worth remembering that liquids do not pull objects or bodies with which they come into contact. Therefore, we say that the normal interaction to the surface always happens in the sense that the liquid pushes the contact area of the body. Thus, the pressure exerted by a fluid on the walls of any container is always a positive quantity.

In order to analyze how the pressure in a liquid varies in the vertical direction, let’s consider a cylinder that contains a certain amount of homogeneous liquid, as shown in the figure above.

For the liquid equilibrium condition in the vertical direction, we have:

*F _{B} = F _{A} + P _{liq} and P=mg*

*as V=Ah, we have: P= μ.Ahg*

Based on these principles, we can say that:

*F _{B} = F _{A} + μ . THE . H . g *

To simplify the equation, we can divide it by the surface area of the cylinder, thus we have:

*p _{B} = p _{A} + μ . H . g*

This result, which gives the pressure difference between two levels inside a liquid, in the presence of gravity, that is:

*p _{B} – p _{A} = μ . H . g*

it is called Stevin’s **Theorem** or Stevin **‘s Law** in honor of Simon Stevin (1548-1620). In the above equation, * p _{A}* is the pressure exerted by the atmospheric air at the location. In his theorem, Stevin says that:

*– When two points of the same portion of the same liquid in equilibrium are at the same level, it means that they are subjected to the same pressure.*

*– The difference in pressure between two points of a homogeneous liquid in equilibrium is given by the pressure exerted by the column of liquid between them.*