Hydrostatic Paradox

What is Hydrostatic Paradox?

In fluid dynamics, Hydrostatic Paradox speaks about the liquid pressure at all the points at the same depth(horizontal level). It is defined as:

โ€œThe pressure at a certain horizontal level in the fluid is proportional to the vertical distance to the surface of the fluid.โ€œ

Hydrostatic Paradox Expression

Hydrostatic Paradox is mathematically expressed as:

P โˆ h

Examples of Hydrostatic Paradox

The concept can be appreciated through the following example:

Illustration of Hydrostatic Paradox: Three-vessel X, Y, Z of different shape, containing a different volume of liquid, but all exert the same pressure(P) at all points at the same horizontal level.

Consider three vessels X, Y, Z of the different shape. They are connected to the common base by a horizontal pipe. On filling it with liquid, we can observe that; although the shape of the vessel varies, the horizontal liquid level in all vessel remains the same. The reason behind this mechanism is, the liquid pressure is the same at the bottom or in general, the fluid pressure is the same at all the points at the same depth.

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Fluid Pressure Formula

The pressure at the depth h below the surface of any fluid is given by-

P= Pa + ๐†gh

Where,

  • P is the pressure at depth h from the surface of the liquid/fluid.
  • Pa is the atmospheric pressure.
  • ๐† is the mass density of the fluid/liquid.
  • g is the acceleration due to gravity.
  • h is the verticle height from the surface and the point.

Pascal(Pa) is the SI unit of pressure

One can calculate the trend of variation of liquid pressure according to depth with the help of the above-mentioned formula. Let’s concrete this paradox through an example.

Water Pressure at Depth

What is the pressure on a swimmer at 10 m below the surface of a

  1. Rectangular shaped lake?
  2. Conical shaped lake?
  3. Regular water well?

Given:

  • h = 10 m
  • g = 9.8 m/s2
  • ๐† = 1000 kg/m3
  • Pa = 1.01ร—105 Pa

Formula used:

P = Pa + ๐†gh

Solution:

  1. Theย pressure on a swimmer at 10 meters below the surface of a rectangularly shaped lake isP= Pa + ๐†gh= 1.01ร—105 Pa + 1000 kg/m3 ร— 9.8 m/s2 ร— 10 m

    = 199000 Pa

    โ‰… 2 atm

  2. The pressure on a swimmer at 10 meter below the surface of a conically shaped lake isP= Pa + ๐†gh= 1.01ร—105 Pa + 1000 kg/m3 ร— 9.8 m/s2 ร— 10 m

    = 199000 Pa

    โ‰… 2 atm

  3. The pressure on a swimmer at 10 meter below the surface of regular water well isP= Pa + ๐†gh= 1.01ร—105 Pa + 1000 kg/m3 ร— 9.8 m/s2 ร— 10 m

    = 199000 Pa

    โ‰… 2 atm

We can observe that, although the lakes are of different shapes, the pressure a swimmer experience at 10 meters below the surface remains the same. This gives a clear insight that only the height of the fluid column matters on measuring liquid pressure and not the shape of the container or volume of the container or the cross-sectional area, or base area.

In the above example, the pressure at 10 m depth of water column is 100% more as compared to the surface. At the depth of 1km, the pressure will be 100 atm. Thus, the submarines are designed to withstand such enormous pressure!

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A metal has a large number of electrons but a non-metal has only a small number of electrons.