Hydrostatics and aerostatics

Liquids and gases have no shape of their own, yet they press on the walls of their container and on objects inside them, and this pressure is transmitted in all directions. Hydrostatics studies fluids at rest: why a ship does not sink, how a hydraulic jack lifts a car, how a barometer “senses” the weather and why dams are built thicker at the bottom. This topic is the basis for “Hydrodynamics” and “Gas laws”.

§1. Pressure

Pressure is the force acting perpendicular to a surface divided by the area of that surface:

Pressure depends not on the force alone but on the area it is spread over. The same force on a small area gives a large pressure: a sharp knife edge, a needle, the point of a nail. Conversely, skis, caterpillar tracks and wide foundations reduce pressure and stop things sinking into soft ground. In practice kilopascals (1 kPa = 10³ Pa) and megapascals (1 MPa = 10⁶ Pa) are used.

Worked example

A granite column has a volume of 9 m³, a base area of 1.4 m² and ρ = 2800 kg/m³. What pressure does it exert on its foundation?

1. Mass: m = ρV = 2800·9 = 25 200 kg; weight F = mg = 252 000 N.
2. p = F/S = 252 000/1.4 = 180 000 Pa = 180 kPa.

§2. Hydrostatic pressure in a liquid

Because of gravity, a liquid presses on its lower layers and on the bottom of the container. Take a column of liquid with base S and height h: its weight ρShg is spread over the area S:

• The pressure depends only on depth and the liquid’s density, not on the shape of the container or the volume of liquid (the hydrostatic paradox): wide and narrow vessels with the same base and the same water level have the same pressure on the bottom.
• At a given depth the pressure is the same everywhere and acts equally in all directions.
• Every 10 m of water depth adds roughly 1 atmosphere (≈ 100 kPa).

Dams are thicker at the bottom because water pressure grows with depth. For the same reason submarines and bathyscaphes need very strong hulls.

§3. Pascal’s law and hydraulic machines

The law follows from the fact that the particles of a fluid can move freely. A ball with holes shows it: when the piston is pushed, water squirts out equally in all directions.

A hydraulic press consists of two connected cylinders filled with liquid, each with a piston. The pressure on the small piston is passed fully to the large one:

The force gain equals the ratio of the areas. But by the “golden rule” there is no gain in work: the liquid is incompressible, the displaced volumes are equal, S₁h₁ = S₂h₂, and the large piston moves that many times less. Hydraulic presses, car brakes, jacks and excavators work this way.

Worked example

A press has a small piston of 2 cm² and a large piston of 200 cm². A 100 N force pushes the small piston down by 10 cm. What force does the large piston exert, and how far does it rise?

1. Force gain: S₂/S₁ = 200/2 = 100, so F₂ = 100·100 = 10⁴ N.
2. Equal volumes: h₂ = h₁·S₁/S₂ = 10/100 = 0.1 cm.
3. Check: A₁ = 100·0.1 = 10 J, A₂ = 10⁴·0.001 = 10 J; the work is the same.

§4. Communicating vessels

In vessels connected at the bottom, a uniform liquid at rest settles at the same level whatever the shape and cross-section of the vessels. Water towers, teapot spouts, canal locks and water-level gauges rely on this.

With different liquids, the pressures are equal at the level of the interface:

The column of the less dense liquid is taller.

Worked example

Water is poured into a U-tube, then oil (ρ = 800 kg/m³) is added to one arm to a height of 10 cm. What is the difference in levels?

1. At the interface level the pressures are equal: ρ_oil·h_oil = ρ_w·h_w.
2. h_w = 800·10/1000 = 8 cm.
3. Difference in levels: 10 − 8 = 2 cm (the oil surface is higher).

§5. Atmospheric pressure

The Earth’s atmosphere presses on everything because air has weight. Torricelli (1643) first measured it: a tube sealed at one end and filled with mercury, turned upside down in a dish of mercury, holds a column about 760 mm high with a vacuum above it.

The same experiment with water would need a tube more than 10 m long. With height the air gets thinner and the pressure falls: near the ground by about 1 mmHg for every 12 m of rise. Atmospheric pressure is measured with a barometer (mercury or aneroid), gas pressure in a container with a manometer. The Magdeburg hemispheres experiment (Guericke, seventeenth century) showed vividly how large atmospheric pressure is.

§6. Archimedes’ force and floating

The pressure on the lower face of a submerged body is greater than on the upper face (it is deeper). This difference produces an upward buoyant (Archimedes’) force:

The buoyant force equals the weight of the fluid displaced and does not depend on the body’s density, only on the fluid’s density and the submerged volume. The same force acts in gases.

• ρ_body > ρ_liquid: the body sinks; ρ_body = ρ_liquid: it floats anywhere inside the liquid; ρ_body < ρ_liquid: it rises and floats partly submerged.
• For a floating body mg = F_A, so the submerged fraction is V_s/V = ρ_body/ρ_liquid.
• Apparent weight in water: P_water = mg − F_A.

A steel ship floats because it contains large air-filled spaces and its average density is less than that of water. Balloons and airships rise thanks to the buoyant force of air.

Worked example

Ice (ρ = 900 kg/m³) floats in water. What fraction is under water?

1. V_s/V = ρ_ice/ρ_water = 900/1000 = 0.9.
2. 90 % of the ice is under water; only 10 % shows.

Worked example

A body weighs 30 N in air and 20 N in water. What is its density?

1. Buoyant force: F_A = 30 − 20 = 10 N.
2. Volume: V = F_A/(ρ_w·g) = 10/(1000·10) = 10⁻³ m³; mass m = 30/10 = 3 kg.
3. ρ = m/V = 3/10⁻³ = 3000 kg/m³.

Check yourself

1. Hydrostatic pressure of water at 50 m depth (without the atmosphere)?

2. A press has pistons of 2 cm² and 200 cm². 100 N acts on the small one. Force from the large one?

3. A body of volume 0.002 m³ is fully submerged in water. Buoyant force?

4. Total pressure at 20 m depth (p₀ = 10⁵ Pa)?

5. Wood (ρ = 600 kg/m³) floats in water. What fraction of its volume is submerged?

6. A 500 m³ helium balloon (ρ_air = 1.29, ρ_He = 0.18 kg/m³). What mass can it lift (ignore the envelope)?

7. Does hydrostatic pressure depend on the shape of the container?

8. Iron sinks in water. What happens in mercury (ρ_iron = 7800, ρ_Hg = 13 600 kg/m³)?

Common mistakes

• Bringing the container’s shape or the volume of liquid into hydrostatic pressure.
• Using the body’s density in the buoyant force: you need the fluid’s density.
• Converting cm² to m² wrongly: 1 cm² = 10⁻⁴ m².
• Using the whole volume of a floating body: only the submerged part counts.
• Thinking a hydraulic press gains work as well as force.