Pascal's Principle

Stevin’s law says that, at equilibrium, pressure depends only on depth. But what happens if we change the surface pressure $P_0$, for example by pressing with a piston on the free surface of the liquid? The answer, due to Pascal, is that the change does not remain confined to where it was applied: it is transmitted to the whole fluid instantly and uniformly.

Principle — Pascal's principle

In an incompressible fluid at equilibrium, a change in pressure applied at one point is transmitted unaltered to every point of the fluid and to the walls of the container: $\ev{P(h) \to P(h) + \Delta P \quad \forall h}$

The hydraulic press

This is the most famous application of Pascal’s principle. Two communicating cylinders, with cross-sections $A_1$ (small) and $A_2$ (large), are filled with oil and closed off by two pistons. Pushing with a force $F_1$ on the small piston increases the fluid pressure by $\Delta P = F_1 / A_1$; by Pascal’s principle this same change also acts on the large piston, which feels it over its whole area $A_2$ and is pushed with a force

Law — The hydraulic press

$\ev{F_2 = \Delta P\cdot A_2 = F_1\cdot\frac{A_2}{A_1}}$

The factor $A_2/A_1$ is an amplification factor: with a large piston $100$ times bigger than the small one, $1$ kgf in produces $100$ kgf out. This is the mechanism behind garage hydraulic lifts and car brakes.

Hydraulic press: the force $F_1$ on the small piston generates a pressure which, transmitted unaltered, pushes the large piston with a force amplified by a factor $A_2/A_1$.

Energy conservation is not violated

The hydraulic press multiplies the force, but does not create energy out of nothing. To raise the large piston by a height $h_2$, we must lower the small one by $h_1 = h_2\, A_2/A_1$, because the volume of fluid displaced is conserved. The two amounts of work are then equal: $F_1 h_1 = F_2 h_2$. The force is multiplied by the factor $A_2/A_1$, but the displacement is divided by exactly the same factor.

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Topics: Fluidostatica e fluidodinamica Concepts: Principio di Pascal · Pressione

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