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notes
Variation of pressure with depth

Consider a cylindrical object inside a fluid; consider 2 different positions for this object.

The fluid is at rest therefore the force along the horizontal direction is 0.

Forces along the vertical direction:

Consider two positions 1 and 2.

Force at position 1 is perpendicular to crosssectional area A, F_{1}= P_{1}

Similarly F_{2}=P_{2}

Total force F_{net}= F_{1}+F_{2} as F_{1} is along negative yaxis therefore it is –ive. And F_{2} is along the +ive yaxis.
 F_{net} = (P_{2}P_{1})A

This net force will be balanced by the weight of the cylinder(m).
Therefore under equilibrium condition:
F_{net}=mg=weight of the cylinder = weight of the fluid displaced.
=ρVg where ρ=density=volume of the fluid
=ρhAg where V=hA(h=height and A= area)
Therefore (P_{2}P_{1}) A=ρhAg
`therefore`P_{2}P_{1} = ρhg
Therefore the difference in the pressure is dependent on the height of the cylinder.
Consider the top of the cylinder exposed to air therefore P_{1}=P_{a}(where P_{a}= P_{1} is equal to atmospheric pressure.)
Then P_{2}=P_{a}+ ρhg
Conclusion: The pressure P, at depth below the surface of a liquid open to the atmosphere is greater than atmospheric pressure by an amount ρhg.
The pressure is independent of the crosssectional or base area or the shape of the container.
Hydrostatic Paradox:

Consider 3 vessels of very different shapes (like thin rectangular shape, triangular and some filter shape) and we have a source from which water enters into these 3 vessels.

Water enters through the horizontal base which is the base of these 3 vessels we observe that the level of water in all the 3 vessels is the same irrespective of their different shapes.

This is because the pressure at some point at the base of these 3 vessels is the same.

The water will rise in all these 3 vessels till the pressure at the top is the same as the pressure at the bottom.

As pressure is dependent only on height therefore in all the 3 vessels the height reached by the water is the same irrespective of the difference in their shapes.

This experiment is known as Hydrostatic Paradox.