# Under Pressure

“Pressure pushing down on me
pressing down on you no man ask for
under pressure”

###### (lyrics from “Under Pressure”)

Pressure is the ratio of force applied per area covered:

$P= \frac{F}{A}$

The unit of pressure is the pascal:

The pascal is also a unit of stress and the topics of pressure and stress are connected.

Pressure in a uniform fluid.

• The gauge pressure in a uniform fluid at a particular depth is directly proportional to …
• the density of the fluid ρ,
• the acceleration due to gravity g, and
• the depth h.

The absolute pressure in a uniform fluid at a particular depth is given by:

$P=P_0+\rho gh$

Atmospheric pressure

Atmospheric pressure is defined as the force per unit area exerted against a surface by the weight of the air above that surface.

Boyle’s Law(after Robert Boyle, Irish scientist, around 1600)

Boyle’s law states that, at a constant temperature, the volume of a given mass of gas varies inversely with pressure. For two states of pressure (P1, P2) and two corresponding volumes (V1, V2), this is stated mathematically:

$P_1V_1=P_2V_2$

Charles’ Law (after Jacques Charles, French scientist, around 1790)

By warming the balloon up, we increase the speed of the moving gas molecules inside it. This in turn increases the rate at which the gas molecules bombard the skin of the balloon. Because the balloon’s skin is elastic, it expands upon this increased pushing from inside, and the volume taken up by the same mass of gas increases with temperature. In consequence, the density [=mass/volume] decreases with rising temperature. Cooling the balloon down again will make the balloon shrink.

Thus Charles’s law states that at a constant pressure, the volume of a given mass of gas is directly proportional to its (absolute) temperature. For two states with temperatures (T1, T2) and two corresponding volumes (V1, V2):

The Ideal Gas Law or Equation of State

The example used to illustrate Charles’s law probably does not follow Charles’s law exactly. It is very likely that, during the heating process, when the rate of collisions by the gas molecules increased, the pressure increased as well as the volume. Thus, in practical situations all three variables involved in Boyle’s and Charles’s law are linked and both principles are in action at the same time:

 Pressure: P Temperature: T Density: r = m/V

These variables describe the state of the gas at any one time and are combined in the single relationship known as the ideal gas law or the equation of state:

Dalton’s Law of Partial Pressure:

The pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone. Mathematically, this can be represented as:

PressureTotal = Pressure1 + Pressure2 … Pressuren

Pascal’s Principle:

Pascal’s principle states that a pressure applied to an enclosed fluid is transmitted everywhere in the fluid. Hence, if a pressure is applied to one side of an enclosed fluid, all the other walls containing the fluid feel the same pressure. The pressure is transmitted without being diminished.

In physics, if a pressure is applied to a compressible gas, Pascal’s principle still applies, but the volume of the gas will change. For Pascal’s principle to be useful to hydraulics, the fluid should be an incompressible liquid, which will transmit the applied pressure without changing its volume.