The Pressure in a Fluid Under Gravity Forces

Consider an element of a fluid in the form of a vertical cylinder, located as shown in Fig. 1.

If we draw a free-body diagram of this element in equilibrium, we will have acting the forces shown; a force acting vertically downward on the top of the element, an upward force acting on the bottom of the element, and a gravity force equal to the weight of the fluid in the element. Using the notation:

the density of the fluid the mass per unit volume the specific weight of the fluid the weight per unit volume, where is the acceleration of gravity

the weight of the fluid in the element is: For equilibrium, we have: From this relationship we can determine the variation of pressure with depth in an incompressible fluid, for which is a constant: hence the difference in pressure at two points varies directly with the difference in the depth of the two points.