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:
“The pressure at a certain horizontal level in the fluid is proportional to the vertical distance to the surface of the fluid.“
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Hydrostatic Paradox Expression
Hydrostatic Paradox is mathematically expressed as:
\(\begin{array}{l}P \propto h\end{array} \)
Where P is the pressure at depth “h” from the surface of the liquid/fluid and h is the vertical height from the surface to the point.
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Examples of Hydrostatic Paradox
The concept can be appreciated through the following example:
Illustration of Hydrostatic Paradox: Three-vessel X, Y, and Z of different shapes, containing a different volume of liquid, but all exert the same pressure (P) at all points at the same horizontal level.
Consider three vessels of different shapes, X, Y, and Z. They are connected to the common base by a horizontal pipe. On filling it with liquid, we can observe that; although the vessel’s shape varies, the horizontal liquid level in all vessels remains the same. The reason behind this mechanism is that 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 + 𝝆gh |
- P is the pressure at depth “h” from the surface of the liquid/fluid.
- Pais the atmospheric pressure.
- 𝝆 is the mass density of the fluid/liquid.
- g is the acceleration due to gravity.
- h is the vertical height from the surface to 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 formula mentioned above. 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
- Rectangular shaped lake?
- Conical shaped lake?
- Regular water well?
Given:
- h = 10 m
- g = 9.8 m/s2
- 𝝆 = 1000 kg/m3
- Pa = 1.01×105 Pa
Formula used:
| P = Pa + 𝝆gh |
Solution:
- The pressure on a swimmer at 10 meters below the surface of a rectangularly shaped lake is P= Pa + 𝝆gh= 1.01×105 Pa + 1000 kg/m3 × 9.8 m/s2 × 10 m
= 199000 Pa
≅ 2 atm
- The pressure on a swimmer at 10 meter below the surface of a conically shaped lake is P= Pa + 𝝆gh= 1.01×105 Pa + 1000 kg/m3 × 9.8 m/s2 × 10 m
= 199000 Pa
≅ 2 atm
- The pressure on a swimmer at 10 meter below the surface of regular water well is P= Pa + 𝝆gh= 1.01×105 Pa + 1000 kg/m3 × 9.8 m/s2 × 10 m
= 199000 Pa
≅ 2 atm
We can observe that, although the lakes are of different shapes, the pressure a swimmer experiences at 10 meters below the surface remains the same. This gives a clear insight that only the height of the fluid column matters in measuring liquid pressure and not the shape of the container or volume of the container, cross-sectional area, or base area.
In the above example, the pressure at the 10 m depth of the water column is 100% more than the surface. At a depth of 1 km, the pressure will be 100 atm. Thus, the submarines are designed to withstand such enormous pressure!
Frequently Asked Questions – FAQs
Where is hydrostatic paradox used?
What fact is acknowledged to be a hydrostatic paradox?
What is Hydrostatistics?
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