Pressure in a Fluid
Pressure depends on depth, not on the shape of the container.
Lesson at a glanceShow the key points
- Liquids and gases are fluids: they flow. We model liquids as incompressible (constant density) and nonviscous (no tangential forces between layers).
- Pressure at a point: P = lim F/ΔS, the normal force per unit area; for a fluid at rest it is the same in all directions (a scalar). Unit: pascal, 1 Pa = 1 N/m².
- Going up by dz, pressure falls by ρg dz: dP = −ρg dz (13.2).
- For constant density, a point a depth h below a point at pressure P₀ has P = P₀ + ρgh (13.3).
- Points at the same horizontal level in a connected fluid at rest have equal pressure, whatever the container's shape.
Dive to the bottom of a pool and your ears hurt. Nothing is touching them except water, yet the water presses harder the deeper you go. Why, and by how much?
Fluids
In solids, strong forces between molecules keep both shape and size fixed. In liquids those forces are weaker: the shape changes easily, though the volume (and density) of a given mass is hard to change. In gases the forces are tiny, and both shape and density change easily. Liquids and gases together are called fluids: things that flow.
We will mostly treat liquids, and model them as
- incompressible: the density stays the same whatever the pressure;
- nonviscous: neighbouring layers exert no sideways (tangential) forces on each other, so one part of the liquid pushes on another only perpendicular to the surface between them.
Pressure at a point
In a fluid at rest this doesn’t depend on how the little area is tilted, so pressure has a value at each point but no direction: it is a scalar. Its SI unit is the newton per square metre, called the pascal (1 Pa = 1 N/m²).
How pressure changes with height
Take a thin horizontal slab of fluid of area ΔS and thickness dz, with pressure P on its lower face and P + dP on its upper face. Three vertical forces act on it: P ΔS upward from below, (P + dP) ΔS downward from above, and its weight ρ(ΔS dz)g. In equilibrium
Going up by dz lowers the pressure by ρg dz. If the density is the same everywhere, integrating over a depth h gives
Now take two points at the same level. A thin horizontal cylinder of fluid joining them feels P₁ΔS one way and P₂ΔS the other, and nothing else horizontally. In equilibrium :
A cylindrical bucket of radius 15 cm holds water 30 cm deep. Atmospheric pressure is 1.01 × 10⁵ Pa. Find the force of the water on the bottom (g = 10 m/s², ρ = 1000 kg/m³).
- 1Fluids flow; we model liquids as incompressible and nonviscous.
- 2Pressure = normal force per unit area at a point; a scalar in a fluid at rest; unit pascal (N/m²).
- 3; for constant density .
- 4Points at the same level in a connected fluid at rest have the same pressure, whatever the vessel's shape.
Check yourself
Keep exploring
Hand-picked videos, simulations and readings that explain this idea from a different angle.