The pressure a fluid exerts when it's at rest is not random. It follows a predictable relationship that connects the fluid's properties to the depth. This relationship is captured in the fundamental equation of hydrostatics.

Essentially, this equation tells us that the pressure at any point is due to the weight of the fluid directly above it. The taller the column of fluid, the heavier it is, and the greater the pressure it exerts at the bottom.

The equation $P = \rho g h$ reveals a simple, linear relationship between pressure and depth. If you double the depth, you double the hydrostatic pressure. Go three times as deep, and the pressure triples.

Let's calculate the pressure at the bottom of a 10-metre deep swimming pool. We'll use the density of fresh water (approximately $1000 \text{ kg/m}^3$) and the standard acceleration due to gravity ($9.8 \text{ m/s}^2$).

$P = (1000 \text{ kg/m}^3) \times (9.8 \text{ m/s}^2) \times (10 \text{ m}) = 98,000 \text{ Pascals}$.

This linear increase is why divers must carefully manage their ascent and descent and why submarines are built with incredibly strong hulls to withstand the immense pressure in the deep ocean.

Our swimming pool example is an open system, meaning it's exposed to the atmosphere. In cases like this, we need to account for the pressure exerted by the air above the fluid's surface. The total, or absolute, pressure is the sum of the atmospheric pressure and the hydrostatic pressure.

A closed system, like the fluid in a hydraulic brake line, is not exposed to the atmosphere. In this case, the pressure is determined by the forces applied to the system and the fluid's properties, without the addition of atmospheric pressure. This distinction is crucial for engineering applications, from designing water towers to building hydraulic machinery.

Since pressure increases with depth, the force exerted by a fluid on a submerged vertical surface is not uniform. The pressure at the bottom of a dam's wall is much greater than the pressure at the top.

This creates a distributed load. Engineers must calculate the total force and where it effectively acts, known as the centre of pressure, to ensure structures like dams, aquarium walls, and ship hulls are strong enough to withstand the fluid's push. This is why dams are always built to be much thicker at their base than at the top.

Let's review the key formulas from this section.

Now, test your understanding of these concepts.

Understanding these equations allows us to precisely calculate the pressure and forces within fluids, a skill fundamental to countless areas of science and engineering.