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Hydrostatic Equilibrium





MATHEMATICAL CONSIDERATION


For a volume of a fluid which is not in motion, Newton's Laws state that it must have zero net force on it - the forces up must equal the forces down.

For a Cuboid volume element of a fluid, the force downwards on the top of the fluid from the Pressure , P, of the fluid above it on the top surface area of the volume element, A, is
:F_{top} = P_{top} \cdot A
And on the volume element from the fluid below is
:F_{bottom} = - P_{bottom} \cdot A
The minus sign comes from the directon: this force supports the volume element, rather than pull it down.

Finally, the Weight of the volume element must be considered. If the density is ρ, the volume V and the Acceleration Due To Gravity g, then:
:F_{weight} = ho \cdot g \cdot V
We can split volume into the area of the top or bottom, times the height.
:F_{weight} = ho \cdot g \cdot A \cdot h

By balancing these forces:
:F_{top} + F_{bottom} + F_{weight} = P_{top} \cdot A - P_{bottom} \cdot A + ho \cdot g \cdot A \cdot h = 0
This is zero if the gas isn't moving. If we divide by A,
:0 = P_{top} - P_{bottom} + ho \cdot g \cdot h
Or,
:P_{top} - P_{bottom} = - ho \cdot g \cdot h
By saying that the change in h is very small, which means the change in P is very small, we can write this in Differential notation:
:dP = - ho \cdot g \cdot dh

Points to note:
  • This method assumes that the volume element isn't moving

  • Density changes with pressure, and gravity changes with height, so the equation would be:

  • : dP = - ho(P) \cdot g(h) \cdot dh




APPLICATIONS



Fluids

The ''Hydrostatic equilibrium'' pertains to Hydrostatics and the Principles Of Equilibrium of Fluid s. A Hydrostatic Balance is a particular balance for weighing substances in water. Hydrostatic balance allows the Discovery of their Specific Gravities .


Physics

In .

In Physics , Hydrostatic equilibrium also explains why Earth 's atmosphere does not collapse to a very thin layer on the ground and how the Tyre s on a car or bicycle are able to support the weight of the vehicle.


SEE ALSO




REFERENCE

Strobel, Nick. (May, 2001). Nick Strobel's Astronomy Notes.