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Pressure - Variation of Pressure with Depth

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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 cross-sectional area A, F1= P1

  • Similarly F2=P2

  • Total force Fnet= -F1+F2 as F1 is along negative y-axis therefore it is –ive. And F2 is along the +ive y-axis.     

  • Fnet = (P2-P1)A
  • This net force will be balanced by the weight of the cylinder(m).

Therefore under equilibrium condition:

Fnet=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 (P2-P1) A=ρhAg

`therefore`P2-P1 = 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 P1=Pa(where Pa= P1 is equal to atmospheric pressure.)

Then P2=Pa+ 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 cross-sectional 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.

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