9.2.2. Variation of Pressure with Depth

Consider a fluid at rest in a container. In Fig. 9.3 point 1 is at height h above a point 2. The pressures at points 1 and 2 are P1 and P2 respectively. Consider a cylindrical element of fluid having area of base A and height h. As the fluid is at rest the resultant horizontal forces should be zero and the resultant vertical forces should balance the weight of the element. The forces acting in the vertical direction are due to the fluid pressure at the top (P1A) acting downward, at the bottom (P2A) acting upward. If mg is weight of the fluid in the cylinder we have (P2 − P1) A = mg.

In a fluid that is at rest, pressure varies with depth due to the weight of the fluid above. When you consider two points at different heights within the fluid, point 1 is higher than point 2. The pressure at point 2 (deeper) is influenced by both the pressure at point 1 and the weight of the fluid column that is above point 2. The balance of forces ensures that the upward pressure force from point 2 equals the weight of the fluid column (the force acting downward). This establishes the relationship that pressure increases with depth.

Think about when you dive underwater; the deeper you go, the more pressure you feel against your body due to the weight of the water above you. This is why scuba divers must be trained about how to manage pressures at various depths.

Now, if ρ is the mass density of the fluid, we have the mass of fluid to be m = ρV= ρhA so that P2 − P1= ρgh.

Using the relationship derived from balancing the forces, we can express the pressure difference (P2 − P1) between the two points in terms of density (ρ) of the fluid and the height difference (h) between the two points. This leads us to the formula P2 - P1 = ρgh, showing that the pressure difference increases with both the density of the fluid and the height difference.

Imagine a heavy pile of books stacked on your chest. The weight of those books pressing down—representing the density of the fluid—creates a feeling of pressure. If you were to stack even more books (increasing the height), the pressure (or weight you feel) increases too, similar to how fluid pressure builds up as you go deeper underwater.

Thus, the pressure P, at depth below the surface of a liquid open to the atmosphere is greater than atmospheric pressure by an amount ρgh. The excess of pressure, P − Pa, at depth h is called a gauge pressure at that point.

When you measure the pressure underwater, you are measuring the total pressure, which includes both the atmospheric pressure at the surface (Pa) and the additional pressure due to the water column above (ρgh). The pressure that exceeds atmospheric pressure is referred to as gauge pressure, which is important in various applications, like scuba diving or engineering.

Think of how a pressure gauge on a tire works: it shows the pressure inside the tire (gauge pressure) which must be above the atmospheric pressure to keep the tire inflated. In a similar way, when submerged, the gauge pressure provides important information about the forces acting due to the fluid in addition to the normal atmospheric pressure.

The area of the cylinder is not appearing in the expression of absolute pressure in Eq. (9.7). Thus, the height of the fluid column is important and not cross-sectional or base area or the shape of the container. The liquid pressure is the same at all points at the same horizontal level (same depth).

This principle reveals the 'hydrostatic paradox' where the pressure due to a liquid depends solely on the height of the liquid column, regardless of the shape or size of the container holding it. Therefore, if two containers are filled to the same height with the same liquid, the pressure at any horizontal level will be the same, thereby illustrating that pressure is dependent on depth, not the volume or shape of the fluid.

If you fill three different vessels—one tall and skinny, one short and wide, and one odd-shaped but filled to the same height with water—each will exert the same pressure at the same depth, even though they hold different amounts of water. This is surprising and underscores the principle that pressure in fluids doesn’t care about the shape of the container, just the height.

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