11.4 Variation of Pressure with Depth in a Fluid

Fluid Pressure and Depth

Pressure as force per area

Pressure measures how much force is spread over a given area:

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

• $P$ = pressure (pascals, Pa)
• $F$ = force applied (newtons, N)
• $A$ = area over which the force acts (square meters, m²)

In fluids, pressure comes from the weight of the fluid sitting above a given point. The deeper you go, the more fluid is stacked above you, and the greater the force pushing down on each unit of area. That's why the pressure at the bottom of a 10-meter pool is noticeably higher than at 1 meter deep.

Pascal's principle states that pressure applied to an enclosed fluid is transmitted equally in all directions. This is why squeezing one end of a closed hydraulic system creates equal pressure changes throughout the entire fluid.

Pressure changes with fluid depth

In a fluid at rest (a static fluid), pressure increases linearly with depth. The pressure due to the fluid itself is called hydrostatic pressure, and it's calculated with:

$P = \rho g h$

• $\rho$ (rho) = density of the fluid (kg/m³)
• $g$ = acceleration due to gravity (9.81 m/s²)
• $h$ = depth below the fluid's surface (m)

This equation gives you only the pressure contributed by the fluid. In most real situations, the atmosphere is also pressing down on the fluid's surface. So the total (absolute) pressure at depth $h$ is:

$P_{\text{total}} = P_{\text{atm}} + \rho g h$

At the surface ($h = 0$), the total pressure equals atmospheric pressure alone. As you go deeper, the $\rho g h$ term grows while $P_{\text{atm}}$ stays constant. For example, a diver at 10 m depth in freshwater ($\rho = 1000 \text{ kg/m}^3$) experiences a hydrostatic pressure of roughly $1000 \times 9.81 \times 10 = 98{,}100 \text{ Pa}$, which is close to one additional atmosphere of pressure on top of the atmospheric pressure at the surface.

A related idea: buoyancy is the upward force a fluid exerts on a submerged object. It arises because pressure is higher at the bottom of the object than at the top, creating a net upward push.

Fluid density from pressure measurements

You can rearrange the hydrostatic pressure equation to solve for fluid density if you know the pressures at two different depths:

$\rho = \frac{P_2 - P_1}{g(h_2 - h_1)}$

• $P_1$ and $P_2$ are the pressures measured at depths $h_1$ and $h_2$

Here's how to use it step by step:

1. Measure the pressure at two known depths (or altitudes).

2. Find the pressure difference: $\Delta P = P_2 - P_1$.

3. Find the depth (or altitude) difference: $\Delta h = h_2 - h_1$.

4. Plug into $\rho = \frac{\Delta P}{g \cdot \Delta h}$.

This same approach works for gases. For instance, measuring air pressure at sea level and at a mountaintop lets you estimate the average density of the air between those two elevations. Just keep in mind that air density changes with altitude, so the result is an average over that range.

Pressure Measurement Devices

• Manometers measure pressure differences by comparing fluid column heights. A U-tube manometer, for example, contains a liquid (often mercury or water). The height difference between the two sides of the tube corresponds directly to the pressure difference, which you can calculate using $\Delta P = \rho g h$.
• Barometers measure atmospheric pressure. A mercury barometer works by inverting a mercury-filled tube into a dish of mercury. The atmosphere pushes down on the dish, supporting a column of mercury in the tube. Standard atmospheric pressure supports a column about 760 mm tall, which equals $101{,}325 \text{ Pa}$.