Physics
Fluid statics calculator: Stevin, Pascal, Archimedes
Pick the relation and clear the quantity you are after: the calculator inverts the formula for you. In a fluid at rest what counts is the depth, not how much liquid there is, and whether a body floats is decided by its density, not its weight.
Depth is what counts, not quantity
The whole of fluid statics follows from one sentence: in a fluid at rest the pressure depends on the depth and on nothing else. From there it follows that a narrow pipe ten metres tall presses on its base as hard as a lake ten metres deep, that the pressure is the same at every point on the same level, and that it pushes in every direction rather than only downwards. The p₀ in Stevin's law is whatever pressure already sits on top of the liquid: in open water that is the atmosphere, 101325 Pa, which is why the pressure ten metres down is about twice the pressure at the surface and not about the same.
The hydraulic press is the same law read backwards. The pressure applied to the small piston is transmitted unchanged through the fluid, so the large piston feels that same pressure over a larger area and the force grows in the ratio of the areas. The gain is in force alone: the volumes of liquid moved are equal, so the small piston travels as much further as its force is smaller. Anyone concluding that the machine multiplies energy has forgotten to look at the distances.
Buoyancy, in the end, is only the consequence of the bottom of a submerged body lying deeper than its top and therefore feeling a greater pressure. Added up over the whole surface, the difference comes to the weight of the displaced fluid. Hence the test that is actually used on problems: compare the densities. Denser than the fluid and the body sinks; less dense and it floats, submerged by the fraction ρ_body/ρ_fluid; equal and it stays wherever it is left.
Common mistakes
- Believing the pressure depends on how much liquid there is: it depends on the height of the column. Two containers of different shapes filled to the same level press equally on their bases.
- Dropping p₀ from Stevin's law when the problem asks for the absolute pressure. The hydrostatic term ρgh is only the part due to the liquid — the atmosphere is still above it.
- Thinking the buoyant force grows with depth. It depends on the submerged volume and the density of the fluid, not on how far down the body has gone.
- Using weight instead of density to decide whether a body floats. A steel ship weighs thousands of tonnes and floats, a steel screw weighs a few grams and sinks: what counts is the average density of the body, hollow hull included.
Frequently asked questions
Why does the pressure in a liquid not depend on the shape of the container?
Because it depends only on the depth: p = p₀ + ρgh. A narrow pipe ten metres tall and a lake ten metres deep give the same pressure at the bottom. This is the hydrostatic paradox: what counts is the height of the column, not how much liquid sits above.
What does the buoyant force depend on?
On the density of the fluid and the submerged volume, and on nothing else: not on what the body is made of, not on its mass, not even on how deep it is. It is F = ρ_f · g · V_sub, exactly the weight of the displaced fluid.
Why does ice float with nine tenths below the surface?
Because at equilibrium the submerged fraction equals ρ_body / ρ_fluid. Ice has a density of 917 kg/m³ against water's 1000, so 91.7% stays under and only 8.3% shows. In denser sea water a little more of it shows.
A hydraulic press multiplies force — does it create energy?
No. The force is multiplied by the ratio of the areas, but the travel is divided by the same factor: the small piston has to go a long way down for the large one to rise a little. The work in and the work out are equal.
What is the difference between weight and apparent weight?
The weight is mg and never changes. The apparent weight is what a scale measures while the body is submerged, that is the weight minus the buoyant force. If the body is less dense than the fluid it comes out negative, which means that left alone the body rises.
How this calculation works
Density: ρ = m/V. Pressure: p = F/A, with the force perpendicular to the surface. Stevin's law: p = p₀ + ρ·g·h, where p₀ is the pressure on the free surface and h the depth; the ρ·g·h term is the hydrostatic pressure. Pascal's principle applied to the hydraulic press: F₁/A₁ = F₂/A₂, so F₂ = F₁·A₂/A₁; the displacements are in the inverse ratio, d₁/d₂ = A₂/A₁, and the work is conserved. Communicating vessels holding two immiscible liquids, heights measured from the interface: ρ₁·h₁ = ρ₂·h₂. Buoyant force: F_A = ρ_f·g·V_sub, equal to the weight of the displaced fluid. Floating equilibrium: V_sub/V = ρ_b/ρ_f, valid only for ρ_b ≤ ρ_f. Apparent weight of a fully submerged body: P' = P − F_A = (ρ_b − ρ_f)·V·g, negative when the body tends to rise. Every calculation uses g = 9.81 m/s².