Pascals Law Review
Pascal's law or the principle of transmission of fluid-pressure is a principle in fluid mechanics that states that pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all directions throughout the fluid such that the pressure variations (initial differences) remain the same.
This principle is stated mathematically as:
dP = pg(dh)
dP is the hydrostatic pressure (given in pascals in the SI system), or the difference in pressure at two points within a fluid column, due to the weight of the fluid; ρ is the fluid density (in kilograms per cubic meter in the SI system); g is acceleration due to gravity (normally using the sea level acceleration due to Earth's gravity in metres per second squred);
dh is the height of fluid above the point of measurement, or the difference in elevation between the two points within the fluid column (in metres in SI).
The intuitive explanation of this formula is that the change in pressure between two elevations is due to the weight of the fluid between the elevations. A more correct interpretation, though, is that the pressure change is caused by the change of potential energy per unit volume of the liquid due to the existence of the gravitational field. Note that the variation with height does not depend on any additional pressures. Therefore Pascal's law can be interpreted as saying that any change in pressure applied at any given point of the fluid is transmitted undiminished throughout the fluid.
The pressure of the liquids in each of the
previously cited cases has been due to the weight of the
liquid. Liquid pressures may also result from application of
external forces on the liquid. Consider
the following examples. Figure 2 represents a container
completely filled with liquid. A,
B, C, D, and E represent pistons of equal cross-sectional
areas fitted into the walls of the vessel.
There will be forces acting on the pistons C, D, and E due to
the pressures caused by the
different depths of the liquid. Assume that the forces on the
pistons due to the pressure caused
by the weight of the liquid are as follows: A = 0 lbf, B = 0
lbf, C = 10 lbf, D = 30 lbf, and
E = 25 lbf. Now let an external force of 50 lbf be applied to
piston A. This external force will
cause the pressure at all points in the container to increase
by the same amount. Since the pistons
all have the same cross-sectional area, the increase in
pressure will result in the forces on
the pistons all increasing by 50 lbf. So if an external force
of 50 lbf is applied to piston A, the
force exerted by the fluid on the other pistons will now be
as follows: B = 50 lbf, C = 60 lbf,
D = 80 lbf, and E = 75 lbf.
This effect of
an external force on a confined fluid was first stated by
Pascal in 1653.
Pressure applied to a confined fluid is
transmitted undiminished throughout the confining vessel of