Fluid Statics: Pressure and Buoyancy

Water does not know or care what is above it — pressure at depth $h$ is $\rho g h$ whether the container is a thimble-thin tube or a lake. That single fact lets a syringe of fluid lift a car (hydraulics) and explains why dams are thick at the bottom, not the top.

Pressure is force per unit area: $P = F/A$, measured in pascals ($Pa = N/m ^{2}$).

Hydrostatic pressure at depth $h$ below the surface:

$$P = P_0 + \rho g h$$

where $P_0$ is atmospheric pressure ($\approx 1.013\times 10^{5}$ Pa).

Pascal's principle — Pressure applied to a confined fluid is transmitted equally throughout:

$$\frac{F_1}{A_1} = \frac{F_2}{A_2}$$

This is the basis of hydraulic systems.

Archimedes' principle — The buoyant force equals the weight of displaced fluid:

$$F_b = \rho_{\text{fluid}}\,g\,V_{\text{displaced}}$$

Floating condition: An object floats when $\rho_{\text{object}} < \rho_{\text{fluid}}$. The fraction submerged is:

$$\frac{V_{\text{sub}}}{V_{\text{total}}} = \frac{\rho_{\text{object}}}{\rho_{\text{fluid}}}$$

Key insight: Pressure in a static fluid depends only on depth, not on the shape of the container. This is the hydrostatic paradox.

Common pitfall: Buoyancy depends on the displaced fluid’s weight, not the object’s. A steel ship floats because its hull displaces a huge volume of water; solid steel of the same mass, displacing only its own small volume, sinks instantly.