Obtain an expression for the pressure exerted by liquid column.

Consider a liquid of density ρ in a vessel as shown in figure.


To find: Pressure difference between two points A and B separated by vertical height h.

Consider an imaginary cuboid of area of cross-section a of liquid with upper and lower cap passing through A and B respectively in order to evaluate the pressure difference between points A and B. 

Volume of the imaginary cylinder is, V = ah 

Mass of liquid of imaginary cylinder, m = ρah

Let, P₁ and P₂ be the pressure on the upper and lower face of cylinder.

Forces acting on the imaginary cylinder are:

(i) Weight, mg = ρahg in vertically downward direction.

(ii) Downward thrust of F₁ =P₁a on upper cap.

(iii) Upward thrust of F₂ = P₂a on lower face.

As the imaginary cylinder in the liquid is in equilibrium, therefore the net force on the cylinder is zero. 

             $$\begin{gathered} \mathrm{i}.\mathrm{e}., {\mathrm{F}}_1 + \mathrm{mg} = {\mathrm{F}}_2 \\ {\mathrm{P}}_1\mathrm{a} + \mathrm{\rho ahg} = {\mathrm{P}}_2\mathrm{a} \\ \Rightarrow {\mathrm{P}}_2 - {\mathrm{P}}_1 = \mathrm{\rho gh} \end{gathered}$$ 

Thus, the pressure difference between two points separated vertically by height h in the presence of gravity is ρgh.

Note: In the absence of gravity this pressure difference becomes zero.