Fluid Pressure: Pascal's Law and Hydrostatics

Pressure grows with depth, not with the size of the container

Submerge anything in a fluid and it feels a pressure that comes from the weight of fluid stacked above it: P = P0 + rho * g * h, where P0 is the pressure at the surface, rho is the fluid's density, g is gravitational acceleration, and h is depth. The shape and width of the container do not appear anywhere — a narrow test tube and a wide swimming pool at the same depth exert the same pressure, the hydrostatic paradox.

P(h) = P0 + rho * g * h
example: fresh water, rho ~ 1000 kg/m^3, g = 9.81 m/s^2
at h = 10 m:  P = 101325 + 1000*9.81*10 = 199,425 Pa  (~ 2 atmospheres)

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Pascal's law and the hydraulic press

Pascal law states that a pressure change applied anywhere in an enclosed, incompressible fluid is transmitted undiminished to every other point in that fluid. Push down on a small piston of area A1 with force F1, and the pressure P = F1/A1 appears everywhere, including under a much larger piston of area A2: F2 = P * A2 = F1 * (A2/A1) — a small input force becomes a large output force, at the cost of the large piston moving a proportionally smaller distance.

F1 / A1 = F2 / A2      (Pascal's law, same pressure throughout)
F2 = F1 * (A2 / A1)     (mechanical advantage from the area ratio)

Archimedes' principle: buoyancy from pressure difference

Buoyancy follows directly from pressure increasing with depth. The fluid pushes up on the bottom of a submerged object harder than it pushes down on the top, because the bottom is deeper. That net upward force equals the weight of the fluid displaced — Archimedes principle: F_buoyant = rho_fluid * V_displaced * g. If that force exceeds the object's weight it floats; if less, it sinks; if equal, it is neutrally buoyant.

Why density, not weight alone, determines whether something floats

A steel ship floats not because steel is light — it is roughly eight times denser than water — but because its hull shape displaces enough water that the buoyant force on the hollow hull equals the ship's total weight. Change the geometry (crush the hull, flood a compartment) and you change how much water gets displaced for the same submerged depth, which is why hull shape and displacement volume, not just material density, matter.

Static versus dynamic pressure

Everything above describes hydrostatic pressure — a fluid at rest. Moving fluid adds a separate contribution described by Bernoulli's principle, where faster-moving fluid has lower local pressure. The hydrostatic P = rho*g*h term still applies underneath that dynamic effect; it is the baseline pressure before adding flow effects like lift on a wing or drag on a hull.

Frequently asked questions

Does the shape of a container change the pressure at a given depth?

No. Hydrostatic pressure depends only on the fluid's density, gravity and depth below the surface — a narrow tube and a wide tank have identical pressure at the same depth. This is sometimes called the hydrostatic paradox because it defies intuition about a bigger container meaning more pressure.

How does a hydraulic press amplify force?

By Pascal's law, pressure applied at a small piston is transmitted unchanged through the enclosed fluid to a larger piston. Since force equals pressure times area, the larger piston experiences a proportionally larger force — the trade-off is that the larger piston moves a proportionally shorter distance.

Why does a steel ship float if steel sinks?

Because buoyancy depends on the volume of water displaced by the entire hull shape, not on the density of the material alone. A hollow steel hull displaces enough water to weigh as much as the ship plus its cargo, so the upward buoyant force balances the ship's total weight.

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