Pressure

Divide the perpendicular force in newtons by the area in square metres; the result is the pressure in pascals. Example: a person with a weight of F = 600 N stands on heels with a combined A = 0.003 m²: p = 600/0.003 = 200,000 Pa = 2 bar. The most common mistake is the area unit: 1 m² = 10,000 cm², so one square centimetre is 10⁻⁴ m². For everyday values remember 1 bar = 100,000 Pa = 1,000 hPa; air pressure is about 1,013 hPa. Crucially, only the perpendicular force component counts; slanted forces must be decomposed first.

Because for the same weight the area decides the pressure. A person of 700 N spreads the weight over about 0.25 m² with snowshoes: p = 700/0.25 = 2,800 Pa, and the snow holds. On a stiletto heel of about 1 cm² = 10⁻⁴ m² the same force concentrates to p = 700/10⁻⁴ = 7×10⁶ Pa, 2,500 times as much, and the heel punches in. Many tools work by the same principle in both directions: knives, nails and drawing pins maximise pressure through tiny areas; caterpillar tracks, skis and wide tyres minimise it through large areas. Mnemonic: force alone says nothing; only force per area determines the effect.

It exploits the Pascal principle that pressure in an enclosed liquid spreads undiminished in all directions. If you press with a small force F₁ on a small piston (area A₁), the pressure p = F₁/A₁ prevails everywhere. On the large piston (area A₂) the force F₂ = p·A₂ = F₁·(A₂/A₁) then acts; the force is multiplied by the area ratio. Example: 100 N on A₁ = 2 cm² creates p = 5×10⁵ Pa; a piston with A₂ = 200 cm² delivers F₂ = 10,000 N, enough to lift a car. The price: the small piston must travel 100 times the distance, and the work stays the same (the golden rule of mechanics). Car jacks, brake systems and excavator arms work this way.

Hydrostatic pressure arises from the weight of the liquid column above you and grows linearly with depth: p = ρ·g·h. In water (ρ = 1,000 kg/m³) about 1 bar is added per 10 m of depth: p = 1,000 × 9.81 × 10 ≈ 98,100 Pa. A diver at 20 m thus experiences about 2 bar of hydrostatic pressure plus 1 bar of air pressure from above, roughly 3 bar in total. Remarkable is what does not appear in the formula: the shape of the vessel and the amount of water. In a thin pipe the pressure at 10 m depth is the same as in the ocean (the hydrostatic paradox). Water towers rest on the same principle: the height of the reservoir sets the mains pressure in the houses.

Because it acts from all sides at once and the pressure inside your body pushes back equally; there is no net force that would crush you. Yet the load is enormous: about 100,000 N press on every square metre of your body surface, the weight of ten tonnes. Air pressure becomes noticeable only through differences: in a fast lift or an aircraft your ears "pop" because the outside pressure drops faster than the pressure in the middle ear. The classic Magdeburg hemispheres experiment (1654) also shows the magnitude of air pressure: two evacuated hemispheres could barely be pulled apart by two teams of horses because the outside air pressed them together.