Physics · Mechanics · Grade 7

Pascal's law and hydraulics

How does one person lift a 1.5-tonne car at a garage?

The situation

In a garage a mechanic presses a pump handle again and again — and a car weighing a tonne and a half slowly rises. The same law lifts water from a tower to the upper floors of a house, and makes a diver feel the pressure on their ears at depth. How does a liquid transmit pressure, and what does pressure depend on: the depth, the shape of the vessel or the amount of water in it?

Topic

Pascal's law and fluid pressure. Hydraulic lift in a car workshop, communicating vessels and a water tower

Key ideas

Pascal's law
Hydrostatic pressure
Communicating vessels

Formulas

p = \frac{F}{S}

Pressure is the force acting perpendicular to a surface (Pa = N/m²).

\frac{F_{2}}{F_{1}} = \frac{S_{2}}{S_{1}}, S_{1} h_{1} = S_{2} h_{2}

Hydraulic press: Pascal's law and conservation of liquid volume (we gain in force and lose in distance).

p = p_{0} + ρ g h

Hydrostatic pressure grows linearly with depth; $p_{0}$ is the pressure at the surface (atmospheric).

ρ_{1} h_{1} = ρ_{2} h_{2}

Two different liquids in communicating vessels: the heights of the columns measured from the interface are inversely proportional to the densities.

F_{b a s e} = ρ g h S \neq ρ g V

Hydrostatic paradox: the force of pressure on the base does not have to equal the weight of the liquid in the vessel.

Theory

Pascal's law

The particles of a liquid or gas move freely, so external pressure is transmitted through them equally in all directions (Pascal's law). In “Manometer” mode turn the probe membrane up, sideways or down: at one depth the reading does not change. A solid, however, transmits force only in the direction in which it acts (the “Pressure of a solid” experiment).

Hydraulic lift

If we apply a force $F_{1}$ to the small piston, a pressure $p = F_{1} / S_{1}$ is created in the oil, and it is transmitted to the large piston just the same: $F_{2} = p S_{2} = F_{1} S_{2} / S_{1}$ . But oil is incompressible: the volume displaced by the small piston passes under the large one ( $S_{1} h_{1} = S_{2} h_{2}$ — continuity). Therefore $F_{1} h_{1} = F_{2} h_{2}$ : however many times we gain in force, we lose the same number of times in distance — look at the number of pump strokes. Real lifts are driven by an electric pump; hydraulic brakes, excavators and jacks work on the same principle. Model: the weight of the oil column and the friction at the pistons are neglected.

Hydrostatic pressure

The liquid at depth supports the weight of the column above it: $p = ρ g h$ (this is what the manometer shows), while the total pressure, together with atmospheric pressure, is $p = p_{0} + ρ g h$ , $p_{0} \approx 101$ kPa. Pressure depends only on depth and density. In a university course this follows from Euler's equation for a liquid at rest: $\nabla p = ρ \vec{g}$ , that is, $d p / d z = - ρ g$ ; integrating it gives $p = p_{0} + ρ g h$ . The difference between the pressures on the lower and upper faces of a body gives the buoyant force (the “Archimedes” experiment).

Manometer and barometer

A liquid manometer is a U-shaped tube: the pressure being measured pushes the liquid down in one arm, and the difference in levels $Δ h = p / (ρ_{m} g)$ gives the pressure. The dial (metal) gauge on a lift has a curved tube inside: when the pressure rises it straightens and turns the needle. Both show the pressure above atmospheric. Total atmospheric pressure is measured by a barometer (the “Atmospheric pressure” experiment).

Communicating vessels and the water tower

In vessels connected at the bottom, a liquid of one kind at rest stands at the same level whatever the shape: otherwise the pressures in the connecting tube would not be equal and the liquid would flow. For two immiscible liquids, for the columns measured from the interface, $ρ_{1} h_{1} = ρ_{2} h_{2}$ — the column of lighter oil is taller. A water tower and the pipes of a house are also communicating vessels: the pressure at a tap is $ρ g (H - h)$ , and water does not reach a floor above the level of the tower. Ideally a fountain shot upwards from any tap would rise to the level of the tower; in reality friction in the pipe and air resistance make it lower.

Hydrostatic paradox

In vessels with the same base and the same liquid level, the force of pressure on the base $F = ρ g h S$ is the same, while the weight of the liquid inside differs greatly. In a vessel that widens upwards, the sloping walls support part of the weight; in a vessel that narrows upwards, the walls press the liquid down and the force on the base is greater than the weight. According to legend, Pascal fitted a thin long tube into a barrel full of water, poured a few mugs of water into it and burst the barrel.

Did you know?

The hydraulic press was patented in 1795 by the English inventor Joseph Bramah.
At the deepest point of the Mariana Trench (≈ 11 km) the water pressure is more than 1000 times atmospheric pressure: $ρ g h \approx 1025 \cdot 9.8 \cdot 11 000 \approx 1.1 \cdot 10^{8}$ Pa.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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