Topic 8.2 Pressure: define pressure as force per unit area and apply the relation between pressure and depth in a static fluid.

What this topic is asking

The College Board (Topic 8.2) wants you to define pressure as force per unit area, and to apply the relation between pressure and depth in a static fluid, $P = P_0 + \rho g h$. You also need to distinguish absolute pressure from gauge pressure and understand that fluid pressure acts equally in all directions.

What pressure is

Pressure measures how concentrated a force is over an area. The same force spread over a large area gives low pressure; concentrated on a small area it gives high pressure, which is why a sharp knife (small contact area) cuts more easily than a blunt one. In a fluid, pressure is not directional like a force: at any point it pushes outward equally in every direction, perpendicular to whatever surface it meets.

Pressure increases with depth

This depth dependence is the heart of the topic. The deeper you go, the more fluid weight presses down from above, so the pressure climbs steadily. The result $\rho g h$ comes directly from the weight of a column of fluid divided by its base area: a column of height $h$ and base area $A$ has weight $\rho (Ah) g$, giving a pressure $\rho g h$ independent of $A$. This area independence explains the "hydrostatic paradox": containers of very different shapes but the same fluid depth have the same pressure at the bottom.

Absolute versus gauge pressure

The relation $P = P_0 + \rho g h$ separates the pressure into two parts that the exam asks you to distinguish:

• Gauge pressure is the part due to the fluid itself, $\rho g h$, measuring the pressure relative to the surrounding atmosphere. A tyre gauge or blood-pressure reading is a gauge pressure: it reads zero at atmospheric pressure.
• Absolute pressure is the total pressure, $P = P_0 + \rho g h$, including the atmospheric pressure $P_0$ pressing on the surface. It is the actual pressure a fully submerged surface experiences.

Choosing the right one is a common exam decision: a question asking "how much greater than atmospheric" wants gauge pressure, while one asking for the total force on a submerged hatch wants absolute pressure. The deeper reason pressure increases with depth, and acts in all directions, is that a fluid in equilibrium must support the weight of everything above each point; the strategic payoff is that this single relation, $P = P_0 + \rho g h$, underlies buoyancy (Topic 8.3) and the pressure terms in Bernoulli's equation (Topic 8.4). Pressure connects the density of Topic 8.1 to the forces that fluids exert on objects, completing the link from how a fluid is structured to how it pushes.

Try this

Q1. Calculate the gauge pressure at a depth of $5.0$ m in water (density $1000$ kg/m cubed, $g = 9.8$ m/s squared). [2 points]

• Cue. $P_{gauge} = \rho g h = (1000)(9.8)(5.0) = 4.9 \times 10^4$ Pa.

Q2. State whether fluid pressure at a point acts in one direction or in all directions. [1 point]

• Cue. In all directions equally.