3.3.1.2. Pressure

Table of Contents

What Pressure Means

Pressure tells us how strongly a force is spread over a surface. If the same force acts on a small area, the pressure is large. If the same force acts on a large area, the pressure is smaller. This is why a sharp needle penetrates easily, while a blunt object does not, even if the applied force is the same.

Mathematically, pressure is defined as force per unit area:

$$
P = \frac{F_\perp}{A}
$$

Here, $P$ is pressure, $F_\perp$ is the component of force perpendicular to the surface, and $A$ is the area.

Pressure is the normal force per unit area:
$$
P = \frac{F_\perp}{A}
$$
Only the force perpendicular to the surface contributes directly to pressure.

Pressure is a scalar quantity. It has magnitude, but no direction of its own. The force caused by pressure on a surface does have a direction, and it acts perpendicular to the surface.

Units of Pressure

In the SI system, pressure is measured in pascals, abbreviated as Pa.

$$
1 \ \text{Pa} = 1 \ \text{N/m}^2
$$

This means one pascal is one newton of force acting uniformly on one square meter.

Since the pascal is a small unit, larger units are often used in practice.

Unit	Meaning	In pascals
$1 \ \text{Pa}$	pascal	$1$
$1 \ \text{kPa}$	kilopascal	$10^3$
$1 \ \text{MPa}$	megapascal	$10^6$
$1 \ \text{atm}$	atmosphere	$1.013 \times 10^5$
$1 \ \text{bar}$	bar	$10^5$

Atmospheric pressure at sea level is about

$$
P_{\text{atm}} \approx 1.01 \times 10^5 \ \text{Pa}
$$

Pressure from a Force on a Surface

Suppose a force is applied uniformly over a flat area. Then pressure is easy to calculate from the definition.

If a force of $200 \ \text{N}$ acts on an area of $0.50 \ \text{m}^2$, then

$$
P = \frac{200}{0.50} = 400 \ \text{Pa}
$$

If the area is reduced to $0.010 \ \text{m}^2$, then

$$
P = \frac{200}{0.010} = 2.0 \times 10^4 \ \text{Pa}
$$

The force did not change, but the pressure became much larger because the area became much smaller.

Pressure in Fluids

A fluid can be a liquid or a gas. In a fluid at rest, pressure acts in all directions. This is different from a solid, where forces may be supported in specific directions.

At any point inside a fluid at rest, the pressure is the same in all directions. If this were not true, the fluid would start moving until the imbalance disappeared.

This property helps explain why fluids push on the bottom, sides, and any object placed inside them.

In a fluid at rest, pressure at a point acts equally in all directions.

Pressure and Surface Force

Pressure can create a force when it acts over an area. Rearranging the pressure formula gives

$$
F_\perp = PA
$$

So if a fluid exerts pressure $P$ on a surface of area $A$, the magnitude of the perpendicular force is $PA$.

For example, if water presses on a surface with pressure $3.0 \times 10^4 \ \text{Pa}$ over an area of $0.20 \ \text{m}^2$, then

$$
F = PA = (3.0 \times 10^4)(0.20) = 6.0 \times 10^3 \ \text{N}
$$

This shows that even moderate pressure can produce a large force when the area is large.

Absolute Pressure and Gauge Pressure

In many situations, pressure is measured relative to atmospheric pressure. This is called gauge pressure. Car tire pressure, for example, is usually given this way.

Absolute pressure is the total pressure measured from zero pressure.

The relationship is

$$
P_{\text{abs}} = P_{\text{gauge}} + P_{\text{atm}}
$$

If a tire has a gauge pressure of $2.2 \times 10^5 \ \text{Pa}$, then its absolute pressure is approximately

$$
P_{\text{abs}} = 2.2 \times 10^5 + 1.01 \times 10^5
$$

$$
P_{\text{abs}} \approx 3.21 \times 10^5 \ \text{Pa}
$$

Do not confuse gauge pressure with absolute pressure.
$$
P_{\text{abs}} = P_{\text{gauge}} + P_{\text{atm}}
$$
Gauge pressure is measured above atmospheric pressure.

Everyday Examples

Pressure appears in many familiar situations. Snowshoes help a person walk on snow because they increase the contact area, which reduces pressure on the snow. A knife cuts well because its edge has a very small area, producing high pressure. A dam must withstand large pressure from water, which creates strong forces on its walls.

These examples all follow the same idea, pressure depends on how force is distributed over area.

Simple Diagram of Pressure on a Surface

Rendering scientific sketch…

Pressure acting on a flat surface

Pressure Inside a Fluid

Rendering scientific sketch…

Pressure acting in different directions in a fluid

Key Idea

Pressure is one of the most important quantities in fluid mechanics because it connects force and area. In fluids, pressure is transmitted throughout the material and acts on all surfaces in contact with the fluid. Understanding pressure is the starting point for studying hydrostatic pressure, Pascal's principle, buoyancy, and fluid flow.

Key formulas for pressure:
$$
P = \frac{F_\perp}{A}
$$
$$
F_\perp = PA
$$
$$
P_{\text{abs}} = P_{\text{gauge}} + P_{\text{atm}}
$$

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