KAHIBARO
Table of Contents
Pressure in a Fluid at Rest
Hydrostatic pressure is the pressure inside a fluid that is not moving. A fluid at rest can be a liquid, like water or oil, or a gas, like air, provided it is in equilibrium. The key idea is that deeper parts of the fluid support the weight of the fluid above them, so the pressure increases with depth.
Pressure is defined as force per unit area,
$$
P = \frac{F}{A}
$$
In a fluid at rest, pressure acts equally in all directions at a given point. This is an essential feature of hydrostatics.
In a fluid at rest, pressure at a point has no preferred direction. It acts equally in all directions.
Why Pressure Increases with Depth
Imagine a vertical column of liquid with cross-sectional area $A$ and height $h$. The fluid in that column has weight, and the fluid below must support it. The deeper you go, the more fluid lies above, so the pressure becomes larger.
If the fluid has density $\rho$, then the mass of the column is
$$
m = \rho V = \rho Ah
$$
Its weight is
$$
W = mg = \rho Ahg
$$
The pressure due to this column is the weight divided by the area:
$$
P = \frac{W}{A} = \frac{\rho Ahg}{A} = \rho gh
$$
This is the pressure caused by the fluid itself. If there is already pressure at the top surface, such as atmospheric pressure, then the total pressure at depth $h$ is
$$
P = P_0 + \rho gh
$$
where $P_0$ is the pressure at the surface.
The hydrostatic pressure formula is
$$
P = P_0 + \rho gh
$$
where $P_0$ is surface pressure, $\rho$ is fluid density, $g$ is gravitational acceleration, and $h$ is depth below the surface.
Meaning of the Formula
This formula shows several important facts. Pressure increases linearly with depth. A denser fluid gives a larger pressure increase for the same depth. Stronger gravity also gives a larger pressure increase. The shape of the container does not matter, only the depth, fluid density, and gravity matter.
For two points in the same fluid,
$$
\Delta P = \rho g \Delta h
$$
This is often the most useful form when comparing pressures at different depths.
For the same fluid at rest, the pressure difference between two depths depends only on the vertical separation:
$$
\Delta P = \rho g \Delta h
$$
Gauge Pressure and Absolute Pressure
There are two common ways to describe pressure in fluids. Absolute pressure is measured relative to zero pressure. Gauge pressure is measured relative to atmospheric pressure.
If a fluid is open to the atmosphere, then at the surface
$$
P_0 = P_{\text{atm}}
$$
and the pressure at depth $h$ is
$$
P_{\text{abs}} = P_{\text{atm}} + \rho gh
$$
The gauge pressure is
$$
P_{\text{gauge}} = \rho gh
$$
So gauge pressure tells you how much extra pressure is caused by the fluid column.
| Type of pressure | Reference level | Formula at depth $h$ |
|---|---|---|
| Absolute pressure | Vacuum | $P_{\text{abs}} = P_0 + \rho gh$ |
| Gauge pressure | Atmospheric pressure | $P_{\text{gauge}} = \rho gh$ if surface is open to air |
Pressure at the Same Depth
In the same connected fluid at rest, all points at the same depth have the same pressure. This is true no matter how wide or narrow the container is.
This explains why water seeks the same level in connected containers. If pressures at equal depths were different, the fluid would move until equilibrium was restored.
Points $A$ and $B$ are at the same depth, so their pressures are equal.
In a connected fluid at rest, points at the same depth have the same pressure.
Dependence on Density
Different fluids produce different hydrostatic pressure changes because they have different densities. Mercury produces a much larger pressure increase with depth than water, because mercury is much denser.
| Fluid | Approximate density $\rho$ in $\mathrm{kg/m^3}$ |
|---|---|
| Air | $1.2$ |
| Water | $1000$ |
| Oil | $900$ |
| Mercury | $13600$ |
For example, at a depth of $2\,\mathrm{m}$ in water, the gauge pressure is approximately
$$
P = \rho gh = (1000)(9.8)(2) = 19600\,\mathrm{Pa}
$$
At the same depth in mercury,
$$
P = (13600)(9.8)(2) \approx 2.67 \times 10^5\,\mathrm{Pa}
$$
So the pressure rise is much greater in mercury.
Visualizing Hydrostatic Pressure
A simple way to picture hydrostatic pressure is to think of a small horizontal area inside the fluid. The fluid above that area has weight, and that weight creates pressure. As the depth increases, the amount of fluid above increases, so the pressure increases.
The marked point is at depth $h$, and the fluid column above it contributes to the pressure there.
Special Case, Open Containers
If a liquid is exposed to the atmosphere, then the top surface has atmospheric pressure. In that case, the total pressure at depth $h$ becomes
$$
P = P_{\text{atm}} + \rho gh
$$
Near sea level, atmospheric pressure is about
$$
P_{\text{atm}} \approx 1.01 \times 10^5\,\mathrm{Pa}
$$
So even at the surface of an open container, the pressure is not zero. It is atmospheric pressure. The fluid simply adds more pressure as depth increases.
Simple Example
Suppose a swimmer is $3.0\,\mathrm{m}$ below the surface of a lake. Taking water density as $\rho = 1000\,\mathrm{kg/m^3}$ and $g = 9.8\,\mathrm{m/s^2}$, the gauge pressure is
$$
P_{\text{gauge}} = \rho gh = (1000)(9.8)(3.0) = 2.94 \times 10^4\,\mathrm{Pa}
$$
The absolute pressure is
$$
P_{\text{abs}} = P_{\text{atm}} + \rho gh
$$
$$
P_{\text{abs}} = 1.01 \times 10^5 + 2.94 \times 10^4
= 1.304 \times 10^5\,\mathrm{Pa}
$$
So the swimmer experiences both atmospheric pressure and the extra pressure from the water above.
Main Ideas to Remember
Hydrostatic pressure describes pressure in a fluid at rest. It increases with depth because deeper layers support more fluid above them. The basic relation is $P = P_0 + \rho gh$. Pressure differences depend only on vertical depth difference in a given fluid. Points at the same depth in the same connected fluid have the same pressure. Denser fluids create larger pressure changes with depth.
Key results for hydrostatic pressure:
$$
P = \frac{F}{A}
$$
$$
P = P_0 + \rho gh
$$
$$
\Delta P = \rho g \Delta h
$$
At the same depth in the same connected fluid, pressures are equal.