Mechanics of Fluids and Pressure - Part III

Derivation of expression for a pressure in a liquid

We have seen in the experiments above that pressure is the same at the same depth in the liquid. At a point B in the liquid, the pressure will depend on the weight of the liquid column above it.

Thus
Total thrust or force at level B    =    weight of the liquid column above B

                                              = Mass of the liquid column (m) x g         
                                                                                            (g : gravitational acceleration 9.8 m/s²)

                                              = volume x density                   ( density = volume/mass)

                                              = area x h x x g

                                 force (or thrust)

     Pressure P =            = h x x g
                                       area

Thus pressure at a point inside the liquid is given by its depth or height, density and g, the gravitational acceleration. The liquid pressure on the surface is zero as h = 0. But the liquid surface is also exposed to atmospheric pressure, pressure of the surrounding air on the surface.

Total pressure at any point in a liquid = atmospheric pressure + h x x g

When dams are built to stop and store water, it becomes necessary to make the base of the dam broader than the top. The base has to sustain greater pressure from the pressure of the water column as compared to the pressure exerted by the water on the surface.

Why liquid seeks it own level
If you fill a container with liquid, you will notice that the liquid will form a uniform level. The same does not happen with solid particles; solid level can remain uneven.  It is interesting to observe how liquid level is adjusted when you pour coloured water in one of the vessel arrangements shown below.

Irrespective of the shapes of the container, the water will stand at the same level in all the vessels.  Thus liquid seeks its own level. The reason behind this is the pressure exerted by the liquid inside itself. If the height of the liquid level is low in one vessel, the pressure experienced by its column is atmospheric pressure + height of the column. In another vessel, where the height of the liquid is larger, the pressure experienced by the water column will be more, as the height is more. To equalize the pressure, liquid will flow from the high-pressure region to the lower pressure region thereby making the heights of liquid in each vessel the same.

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