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The pressure at any point in a liquid is equal to the specific weight of liquid times the height of liquid above the point plus the atmospheric pressure.
Let
γ = specific weight of liquid,
h = depth of the point under consideration from the free surface of liquid
patm = atmospheric pressure.
Absolute Pressure at any given point is then given by,
P = γh + patm
This is called as the hydrostatic law of pressure variation. The actual pressures we normally measure in gauges and piezometers give us the relative pressures in the liquids, ignoring atmospheric pressure. The atmospheric pressure exerted on all the things on earth, hence can be omitted for practical purposes.
Gauge pressure = Absolute pressure - Atmospheric pressure
Or p = P - patm
Further, France’s Blaise Pascal stated that the pressure at any point in a fluid at rest is the same in all directions. This is known as Pascal’s law.
Since, the pressure at a point in fluid is dependent only on the depth of the fluid, vertical pressures on a horizontal line would always be same. If there is a difference in horizontal pressure along a horizontal line, there will be motion of fluid.
Hence, in static fluids or fluids at rest, the pressure is equal in all directions. This is mathematically proved.
Now, let us consider the curious case of ‘Hydrostatic Paradox’. Consider a few vessels of different shapes and sizes but essentially have same base area and filled with water to a same height.
If the water height is ‘h’ and base area, ‘A’, the force on the base of each container is the same at ‘γAh’.
If you take funnel shaped container with base area, A and a cylinder with same base area, the funnel vessel obviously holds more water but the force on the base of vessel is same in both cases. Similarly, if you take a conical shaped vessel with decreasing cross section upwards but with same base area, it holds less amount of water but it gets same force on its base!
Well, take a very shallow vessel of same base area covered at top and fitted with a long slender tube. Fill it with water to same height, h. it takes much less water compared to funnel vessel but exerts same force on base.
This apparent contradiction between pressure force on base and weight of water in vessel, came to be known as ‘Hydrostatic Paradox’. Centuries after Pascal demonstrated it with his famous ‘Pascal Vases’, it continues baffle many people still today. Then what does it mean, do they defy laws?
No, certainly not! On the other hand it is we, who fail to comprehend the mechanics. The walls of funnel vessels take the weight of extra liquid on the sides. The top cover of vessel with tube and sides of conical vessel experience upward thrust from water inside (like buoyancy), which gives additional balancing push on base, equal to the weight of missing liquid on sides of tube.
In effect as long as the base area is same and filled to same height, as the pressure is dependent only on the height of liquid, all vessels get same pressure force on base. Hence, a low level town service reservoir holding millions of gallons of water stored up to a height 10 m gives a pressure of 10 tons on every square meter of base or 1 kg/ Sq.cm. If you fix a 10 m long piezometer tube with one Sq.cm. cross sectional area to a wall, fill it with water and block its lower end with your thumb. You will experience the same pressure on your thumb as the reservoir base, even if the tube needed only one liter to fill it!
That’s the ‘Hydrostatic Paradox’!
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