Pressure
Pressure is force per unit area in a fluid. Learn hydrostatic pressure p = ρgh, absolute vs gauge pressure, manometers, Pascal's law and hydraulic presses, forces on submerged surfaces and buoyancy.
Dive to the bottom of a pool and your ears hurt: the weight of water above you creates pressure that grows with depth. Dams are thicker at the base, submarines have thick hulls, and hydraulic jacks lift cars — all consequences of fluid pressure.
In a fluid at rest pressure acts equally in all directions and depends only on depth, not on container shape.
An analogy
A stack of books: the bottom book carries the weight of all the books above it. Fluid at depth carries the weight of the column above it.
Pressure grows with depth
Every 10 m of water adds about 1 bar (98 kPa). The shape of the container doesn't matter — only vertical depth (the 'hydrostatic paradox').
Gauge vs absolute
Gauges read relative to the atmosphere; absolute pressure includes it. Vacuum is negative gauge pressure. Thermodynamic calculations need absolute pressure; structural loads usually use gauge.
Manometers and pressure measurement
A U-tube manometer balances pressure against a liquid column: Δp = (ρ_m − ρ)g h. Mercury manometers, Bourdon gauges, piezoresistive transducers and differential-pressure cells are used in industry.
Forces on submerged surfaces
The resultant force on a plane wall = pressure at its centroid × area. It acts at the centre of pressure, lower than the centroid (at 2/3 depth for a vertical rectangle from the surface). Dam gates, tank walls and hatches are designed from this.
Buoyancy and stability
A floating body displaces its own weight of fluid. Stability depends on the metacentric height GM: if the metacentre is above the centre of gravity, the vessel rights itself. Ship designers keep GM positive but not too large (too stiff = uncomfortable roll).
Formulas
- Hydrostatic pressure:
p = p₀ + ρ g h - Absolute vs gauge:
p_abs = p_gauge + p_atm - Force on a plane surface:
F = ρ g h_c A, centre of pressure below centroid - Pascal / hydraulic press:
F₂ = F₁ · A₂ / A₁ - Archimedes:
F_B = ρ_fluid g V_displaced
Worked example: Pressure at depth
Given: 10 m of water
Find: Gauge pressure.
- p = 998 × 9.81 × 10 = 97.9 kPa ≈ 0.98 bar
Answer: 97.9 kPa gauge
Worked example: Hydraulic press
Given: Small piston Ø20 mm, force 500 N; Large piston Ø200 mm
Find: Output force.
- Area ratio = (200/20)² = 100
- F₂ = 500 × 100 = 50 kN
Answer: 50 kN (≈ 5 tonnes)
Common mistakes
- Wrong: Placing the resultant force at the centroid. — Right: It acts at the centre of pressure, below the centroid.
- Wrong: Mixing gauge and absolute pressure. — Right: State which is used; add 101.3 kPa for absolute.
Applications
- Hydraulic machinery: Excavators, presses and aircraft controls multiply force using Pascal's law at 200–350 bar.
- Dams & tanks: Walls thicken with depth because pressure (and moment) increases.
- Ships & submarines: Buoyancy, stability and hull pressure ratings.
Key takeaways
- p = ρgh — 10 m water ≈ 1 bar.
- Absolute = gauge + atmospheric.
- F = ρg h_c A at the centre of pressure.
- Pascal multiplies force; Archimedes gives buoyancy.
Going further
- Curved surfaces: Horizontal force = force on the vertical projection; vertical force = weight of fluid above (Tainter gates, domes).
- Rigid-body motion: Accelerating or rotating tanks tilt or parabolise the free surface (fuel sloshing, centrifuges).
- Atmospheric variation: Air pressure drops ≈ 12 % per km; the ISA model gives standard values for aircraft and engine ratings.