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Hydrostatic Pressure Lab

Pressure grows with depth: p = p0 + rho g d. Slide the probe down the tank, open the side jets, pump the hydraulic press and read the U-tube manometer - on Earth, the Moon, Mars or Jupiter, in water or mercury.

Liquid World

Open to air (1 atm) Let the tank drain

-probe depth

-absolute pressure at the probe

-gauge pressure (above p0)

-force on the tank floor

-speed / range of the movable jet

-manometer level difference

Hydraulic press

-force on the big piston

-load lifted?

-strokes: small down / big up

-work in / work out (ideal)

Independent checks

Each row compares a closed form with a separate calculation: a trapezoid integral of rho g dz, a Runge-Kutta flight of each jet, a scan over 400 hole depths, a step-by-step drain, and Pascal's volume balance. The mercury barometer must give about 760 mm.

The science

Hydrostatic pressure. In a liquid at rest the weight of the column above sets the pressure: p(d) = p0 + rho g d, because dp/dz = -rho g. It depends only on depth, density and gravity - not on the shape or width of the container. On the Moon the same water gives six times less pressure per metre; mercury gives 13.6 times more.

Pascal's principle. A pressure change applied to a confined liquid is transmitted undiminished everywhere. In the press p = F1/A1 pushes the big piston with F2 = p A2 = F1 A2/A1. Force is multiplied, but the liquid volume is conserved, so the big piston moves A1/A2 as far: work in equals work out (ideal, no friction or leaks).

Torricelli's law. A jet leaving a small hole at depth d has speed v = Cv sqrt(2 g d) (energy balance / Bernoulli, tank surface almost still). Leaving horizontally from height y0 = H - d it lands at R = v sqrt(2 y0 / g) = 2 Cv sqrt(d (H - d)), which does not depend on g or on the liquid. The range is largest at d = H/2, where R = Cv H. Draining a tank of area A through holes of area a gives dH/dt = -sum (a/A) Cv sqrt(2 g d).

Manometer and barometer. A U-tube measures gauge pressure from the height difference of its liquid: p_gauge = rho_m g Delta-h. Normal atmosphere 101 325 Pa supports 760 mm of mercury.

Simplifications. Incompressible liquid with constant density (real water is about 0.5 percent denser at 5 km depth), uniform gravity, tank area much larger than the holes, no viscosity or surface tension in the jets, no air drag, a fixed velocity coefficient Cv (real discharge also has a contraction coefficient), frictionless leak-free press, tube meniscus ignored. The Jupiter value is for the cloud tops; the model has no atmosphere there. Heights are drawn to scale; jets are drawn as smooth parabolas.