Left: connected vessels (colour depth = pressure). Right: hydraulic press (Pascal's principle).
Fluid pressure arises from the weight of fluid above a point and is transmitted equally in all directions—a principle formalized by Blaise Pascal in 1653. In a static fluid, pressure increases linearly with depth: P = P₀ + ρgh, where ρ is fluid density, g is gravitational acceleration, and h is depth below the surface. This hydrostatic pressure principle explains why dams must be thicker at the base, why deep-sea organisms withstand crushing pressures, and how manometers measure pressure differences.
Pascal's principle states that pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid and to the walls of the container. This underlies hydraulic systems: a small force applied over a small piston area creates pressure that acts on a much larger piston area, multiplying the force. Hydraulic lifts, brakes, and presses all exploit this force multiplication, with the trade-off that the large piston moves a proportionally smaller distance (conserving energy).
This simulator lets you explore pressure at different depths in a fluid column, model connected vessels of different cross-sections, and observe how pressure acts on submerged surfaces. You can verify that a U-tube manometer balances when pressure differences equal ρgh, explore communicating vessels (fluid finds the same level in all connected containers regardless of shape), and see how adding denser fluid layers changes pressure profiles.
Why does pressure increase with depth in a fluid?
At any depth, the fluid must support the weight of all the fluid above it. The greater the depth, the more fluid mass overhead, and therefore the greater the downward force per unit area (pressure). Mathematically, considering a thin horizontal slice of fluid at depth h, the pressure must equal the weight per unit area of the column above: P = ρgh. This is why the pressure at 10 m depth in water is approximately one additional atmosphere (101 kPa).
How does a hydraulic press multiply force?
A hydraulic press has two pistons connected by an enclosed fluid. Pressing down on the small piston (area A₁) with force F₁ creates pressure P = F₁/A₁. Pascal's principle transmits this pressure undiminished to the large piston (area A₂), which experiences an upward force F₂ = P·A₂ = F₁·(A₂/A₁). If A₂ is 100 times A₁, the force is multiplied 100-fold. The small piston must move 100 times farther than the large piston, so no energy is created—work in equals work out (minus friction losses).
What is gauge pressure versus absolute pressure?
Absolute pressure is total pressure measured relative to perfect vacuum. Gauge pressure is the pressure relative to local atmospheric pressure: P_gauge = P_absolute − P_atm. A tire inflated to 35 psi (gauge) actually contains air at about 50 psi absolute (35 + 14.7 atmospheric). Vacuum gauges read negative gauge pressure. Gauge pressure is more practically useful since instruments and systems typically interact with the local atmosphere as their reference, but absolute pressure is needed for thermodynamic calculations.
Pressure in a fluid is isotropic (equal in all directions) because fluid molecules are in constant random thermal motion. At any point, molecules strike surfaces from all angles with equal average momentum per unit area. Additionally, if pressure were not isotropic, fluid elements would accelerate in the direction of pressure imbalance until equilibrium was restored. This isotropy is what makes hydraulic systems work—pushing down on one side transmits pressure sideways and upward equally.
Archimedes' principle states that a submerged object experiences an upward buoyant force equal to the weight of fluid it displaces. This arises directly from the pressure distribution: the bottom of a submerged object is at greater depth (higher pressure) than the top, creating a net upward pressure force. The pressure difference between bottom and top, integrated over the object's surface area, always equals ρ_fluid · g · V_displaced, regardless of the object's shape.
Each preset button sets rho directly: water 1000 kg/m3, oil 850, seawater 1025, mercury 13600, ethanol 789. Since hydrostatic pressure is P = P0 + rho*g*h, switching from water to mercury at the same 3 m column height multiplies the hydrostatic term by 13.6x — this is why classic barometers use mercury: a column only about 76 cm tall balances one atmosphere, versus over 10 m for an equivalent water barometer.
The simulation computes hydrostatic pressure as Ph = rho*g*colH, which is a direct linear multiplication with no exponents. Doubling the column height from 0.5 m to 10 m simply doubles the added pressure term, because each additional metre of fluid overhead adds the same fixed weight per unit area (rho*g) as every metre before it — unlike, say, atmospheric pressure, which does fall off exponentially with altitude in a compressible gas.
F2 is computed as F1*(A2/A1): the input force is multiplied by the ratio of the two piston areas. With the default 10 cm2 small piston and 100 cm2 large piston, the mechanical-advantage readout shows a 10x multiplier, so a 100 N push on the small piston yields 1000 N at the large piston — increasing A2 while A1 stays fixed increases this ratio directly.
No — g is fixed at 9.81 m/s2 (standard Earth gravity) throughout the model, since it appears in every formula the sim uses (P = P0 + rho*g*h, F2/F1 = A2/A1 does not depend on g, and Fb = rho*g*V). Only fluid density, applied pressure, column height, and piston areas are exposed as sliders, keeping the focus on how those four variables interact rather than on planetary gravity.
The reference panel shows Fb = rho*g*V, using a fixed representative displaced volume of 0.1 m3 so the buoyancy formula can be read alongside the hydrostatic and hydraulic ones. Because rho is the same density slider used for the column, switching to mercury multiplies Fb by 13.6x, showing why identical objects feel far more buoyant in denser fluids even though V stays constant.