13.5: Pascal's Law
In 1653, the French philosopher and scientist Blaise Pascal published "Treatise on the Equilibrium of Liquids," which discussed the principles of static fluids. A static fluid is a fluid that is not in motion. When a fluid is not flowing, we say that the fluid is in static equilibrium. If the fluid is water, we say it is in hydrostatic equilibrium. For a fluid in static equilibrium, the net force on any part of the fluid must be zero; otherwise, the fluid will start to flow. Pascal observed that a change in pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid and to the walls of its container. Moreover, Pascal's principle implies that the total pressure in a fluid is the sum of the pressures from different sources. For instance, the pressure at a certain depth in a fluid depends on the depth of the fluid and the pressure of the atmosphere.
In an enclosed fluid, since atoms of the fluid are free to move about, they transmit pressure to all parts of the fluid and to the walls of the container. Any change in pressure is transmitted undiminished. Note that this principle does not say that the pressure is the same at all points of a fluid—which is not true since the pressure in a fluid near the Earth varies with height. Rather, this principle applies to the change in pressure. Suppose some water is added in a cylindrical container of height (H) and cross-sectional area (A) that has a movable piston of mass m. Adding weight (mg) at the top of the piston increases the pressure at the top by mg/A, since the additional weight also acts over the area (A) of the lid.
According to Pascal's principle, the pressure at all points in the water changes by the same amount, mg/A. Hence, the pressure at the bottom also increases by mg/A. Thus, the pressure at the bottom of the container is equal to the sum of the atmospheric pressure, the pressure due to the fluid, and the pressure supplied by the mass.
This text is adapted from Openstax, University Physics Volume 1, Section 14.3: Pascal's Principle and Hydraulics.