Pressure in Fluids

Pressure is defined as force per unit area, where the force $F$ is understood to be acting perpendicular to the surface area, $A$.

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

The SI unit for pressure is N/m2. This unit is also known as a pascal (Pa): 1 Pa = 1 N/m2.

Example: Consider a 80.0 kg person whose two feet cover an area of 500. cm2. Determine the pressure applied to the ground by his feet.

Solution: The force exerted by this person on the ground would be (80.0 kg)(9.80 m/s2) = 784 N. The area over which this force is exerted would be 0.0500 m2.

$P=\frac{F}{A}=\frac{784 \ \text{N}}{0.0500 \ \text{m}^2}=15,700 \ \text{Pa}$

It has been determined experimentally that a fluid exerts pressure equally in all directions.  In the sketch below, from any given point below the surface of the fluid, the pressure in all directions is the same. The fluid exerts the same pressure upward from this point as it does downward.

We can calculate how the pressure of a fluid varies with depth, assuming the fluid has uniform density.

Consider a gigantic tub filled with water as shown below. A column of water with a cross-sectional area of 1.00 m2 is designated. If we multiply the cross-sectional area by the height of the column, we get the volume of water in this column. We can then multiply this volume by the density of water, 1000. kg/m3, and get the mass of water in the column. We then multiply this mass by the acceleration due to gravity, $g$, to get the weight of the water in this column.

$F_{\text{weight}}=(\text{area})(\text{height})(\rho)(g)$

The pressure exerted by this force would be exerted over the area at the bottom of the column.

$P=\frac{F}{A}=\frac{\rho gh A}{A}=\rho g h$

Therefore, the pressure of a column of fluid is proportional to the density of the fluid and to the height of the column of fluid above the level. This is the pressure due to the fluid itself. If an external pressure is exerted at the surface, this must also be taken into account.

Example Problem: The surface of the water in a storage tank is 30.0 m above a water faucet in the kitchen of a house. Calculate the water pressure at the faucet.

Solution: The pressure of the atmosphere acts equally at the surface of the water in the storage tank and on the water leaving the faucet – so it will have no effect. The pressure caused by the column of water will be:

$P=\rho gh=(1000.\ kg/m^3)(9.80 \ m/s^2)(30.0 \ m)=294, 000 \ Pa$

The pressure of the earth’s atmosphere, as with any fluid, increases with the height of the column of air. In the case of earth’s atmosphere, there are some complications. The density of the air is not uniform but decreases with altitude. Additionally there is no distinct top surface from which height can be measured. We can, however, calculate the approximate difference in pressure between two altitudes using the equation  $P=\rho g \Delta h$. The average pressure of the atmosphere at sea level is 1.013 × 105 Pa. This pressure is often expressed as 101.3 kPa.

Summary

• Pressure is defined as force per unit area, $P=\frac{F}{A}$.
• The SI unit for pressure is N/m2 which has been named pascal (Pa).
• It has been determined experimentally that a fluid exerts pressure equally in all directions.
• The pressure of a column of fluid is proportional to the density of the fluid and to the height of the column of fluid above the level, $P=\rho gh$.
• The average pressure of the atmosphere at sea level is 1.013 × 105 Pa, or 101.3 kPa.