Physics Complete: Variation of Pressure with Depth

Pressure is the force acting on a body per unit surface area.It varies with depth as explained below.

Consider a fluid at rest in a container. In figure below point 1 is at height h above a point 2. The pressures at points 1 and 2 are P1 and P2 respectively. Let us Consider a cylindrical element of fluid having area of base A and height h.

As the fluid is at rest the resultant horizontal forces should be zero and the resultant vertical forces should balance the weight of the element. The forces acting in the vertical direction are due to the fluid pressure at the top (P1A) acting downward, at the bottom (P2A) acting upward. mg is weight of the fluid in the cylinder we have (P2 − P1) A = mg

if ρ is the mass density of the fluid, mass of fluid to be m = ρV= ρhA so that P2 − P1= ρgh

Pressure difference depends on the vertical distance h between the points (1 and 2), mass density of the fluid ρ and acceleration due to gravity g. If the point 1 is shifted to the top of the fluid (say water), which is open to the atmosphere, P1 may be replaced by atmospheric pressure (Pa) and we replace P2 by P.

P = Pa + ρgh

The pressure P, at depth below the surface of a liquid open to the atmosphere is greater than atmospheric pressure by an amount ρgh. The excess of pressure, P − Pa, at depth h is called a gauge pressure at that point.

The area of the cylinder is not appearing in the expression of absolute pressure. Thus, the height of the fluid column is important and not cross sectional or base area or the shape of the container. The liquid pressure is the same at all points at the same horizontal level (same depth).

The result is appreciated through the example of hydrostatic paradox. Consider three vessels A, B and C [Fig below] of different shapes. They are connected at the bottom by a horizontal pipe. On filling with water the level in the three vessels is the same though they hold different amounts of water. It is because water at the bottom has the same pressure below each section of the vessel.