Explain how pressure varies with depth.

Explanation

Variation of pressure with depth: Consider a fluid a rest in a container. Let Point 1 is at a height 'h' above the point 2 as shown in the figure. 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 (`P_1 A`) acting downward, at the bottom (`P_2A` acting upward.

IF 'mg' is weight of the fluid in the cylinder, we have
`(P_2-P_1)A=mg...........(1)`
Now, if `rho` is the mass density of the fluid, then mass of fluid, m=`rho`v
`implies m= rho hA......(2)`
`therefore` From (1) and (2)
`P_2-P_1=rhogh.........(3)`
Pressure difference depends on the vertical distance 'h' between the points (1 and 2) mass density of the fluid `rho` and acceleration due to gravity 'g'.
If the point 1 under discussion is shifted to the top of the fluid, which is open to the atmosphere, `P_1` may be replaced by atmospheric pressure (Pa) and we replace `P_2` by P, then eq.(2) becomes
`P-Pa=rho g h implies P=Pa +rho gh`
Thus, the pressure P , at depth below the surface of a liquid open to the atmosphere is greater than atmospheric pressure by an amount pgh.