Pascal's law

In this page we are going to integrate the differential expression of the hydrostatic pressure in order to calculate the pressure in a liquid at a given depth (y coordinate).

In the figure below a liquid contained in an open vessel is shown. Since the vessel is open to the atmosphere, the pressure p2 on its surface will be the atmospheric pressure:

The previously derived differential expression of pressure is given by:

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Integrating (assuming a constant density):

And finally, isolating p1 and substituting p2 for the atmospheric pressure:

The above expression explains why the deeper you go under the sea, the greater the pressure of the water pushing down on you. More precisely, for every 10 m you go down, the pressure increases by one atmosphere (1 atm approximately equals to 10 5 Pa). To prove it, we just have to find the value of the term ρ g h, given that the density of seawater is approximately ρ = 1025 kg/m3, g = 10 m/s2 and substituting h by 10 m:

From the expression above the so called Pascal’s law (or Pascal’s principle) can be stated:

A pressure change at any point in a confined incompressible fluid is transmitted equally throughout the fluid.

An interesting application of Pascal’s law is the hydraulic press. The hydraulic lift for automobiles shown schematically in the figure below is an example of a force multiplied by hydraulic press. It consists of two connected cylinders, one significantly bigger than the other, filled with an incompressible liquid:

Since an externally applied pressure is transmitted to all parts of the enclosed fluid, a small force applied on the left cylinder (represented in green), can be multiplied on the other side.

Mathematically:

Therefore, the larger the area, the larger the force on that side of the lift.

A hydraulic brake follows the same principle. Since the brake’s area is larger than that of the cylinder containing the fluid, a small force applied on the pedal is transformed into a larger one at the brake.

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