Fluids at rest
The pressure in a fluid at rest depends only on depth. From this come the hydraulic press, the float and the measurement of pressure with a column of mercury.
01Pressure
A fluid at rest cannot sustain shear stresses: it only exerts forces perpendicular to any surface it touches. That force per unit area is the pressure, and it is a scalar: at a point it has the same value in all directions.
| Unit | Equivalent to | Where it is used |
|---|---|---|
| 1 Pa | 1 N/m² | International System unit |
| 1 bar | 10⁵ Pa | Pneumatics and industry |
| 1 atm | \( 101 325 Pa \approx 1.013 bar \) | Standard atmospheric pressure |
| 1 mmHg (torr) | 133.3 Pa | Medicine and vacuum |
| 1 psi | 6895 Pa | Gauges of Anglo-Saxon origin |
Most pressure gauges measure above atmospheric: that is gauge pressure. Absolute pressure is the sum of the two. A tire “at 2 bar” actually has 3 bar absolute, and that distinction matters when the gas equation is used.
02Hydrostatic pressure
In a liquid at rest the pressure increases with depth, and does not depend on the shape of the container or on the amount of liquid:
In water, every 10 meters of depth adds roughly one atmosphere. That is why communicating vessels equalize their levels, and why a mason’s water level works with a transparent hose: two points at the same pressure are at the same height.
A change in pressure applied to an enclosed fluid is transmitted undiminished to all its points. In a hydraulic press, F₁/A₁ = F₂/A₂: with areas in a ratio of 1 to 100, a force is multiplied by a hundred. What is not multiplied is the work, because the large piston travels a hundred times less.
03Archimedes’ principle
Since pressure grows with depth, the lower face of a submerged body receives more pressure than the upper one. That difference produces a net upward force:
| If… | Then | Submerged fraction |
|---|---|---|
| ρbody < ρfluid | It floats | ρbody/ρfluid |
| ρbody = ρfluid | Equilibrium at any depth | All of it |
| ρbody > ρfluid | It sinks | All of it, and rests on the bottom |
A floating body does so with exactly the fraction of its volume that equals its relative density: an iceberg, with ρ = 917 kg/m³ in seawater of 1025, leaves less than 11% of its volume above the surface.
Change the densities and see how far the cube sinks. The gauge measures the pressure at the depth you choose.
04Measuring pressure
| Instrument | Principle | Output |
|---|---|---|
| U-tube manometer | Liquid column: \( p = \rho \cdot g \cdot \Delta h \) | Visual reading |
| Mercury barometer | Column balanced by the atmosphere: 760 mm | Visual reading |
| Bourdon tube | A curved tube straightens when pressurized | Needle |
| Piezoresistive sensor | Strain gauges on a silicon diaphragm | Electrical signal |
| Capacitive sensor | The diaphragm changes a capacitance | Electrical signal |
The last two are pure electronics: a BMP280 measures atmospheric pressure with a silicon diaphragm and delivers the value over I²C, and from it the altitude is calculated. The same principle, with a different range, is in the oil pressure sensor of an engine and in a water tank sensor.
At the interface between a liquid and another medium a force per unit length appears that tends to minimize the surface. It explains spherical drops, capillarity and, in electronics, the behavior of molten solder: a well-made solder joint forms a concave meniscus because the solder wets the copper.
05What it is good for in electronics
- Level and pressure sensors. Measuring the height of a tank is measuring ρ·g·h with a sensor at the bottom.
- Barometric altimeters. A drone estimates its height with a pressure sensor with centimeter resolution.
- Sealed equipment. A submersible enclosure is designed for the pressure at its working depth, and its IP rating is tested at that pressure.
- Liquid cooling. The circuit is sized with static pressures and head losses.
- Industrial pneumatics. The actuators of an automated line are driven by solenoid valves and calculated with F = p·A.
06In the lab
Measure what fraction of a body is submerged in water and deduce its density. Compare with the value obtained by weighing it and measuring its volume by displacement.
Weigh an object in air and submerged. The difference is the buoyant force, and from it you get the volume of the body. It is the method by which Archimedes tested the crown.
With a differential pressure sensor connected to a microcontroller, measure the pressure at the bottom of a graduated cylinder at different water heights. Plot p against h: the slope is ρ·g, and from it you obtain the density of the liquid.
07Common mistakes
- Believing that pressure depends on the amount of liquid and not on the depth.
- Mixing absolute with gauge pressure.
- Thinking that the buoyant force depends on the weight of the body. It depends on the submerged volume and on the density of the fluid.
- Forgetting that the buoyant force also acts in air. It is negligible for a solid, but it is what holds up a balloon.
- Assuming the hydraulic press multiplies energy. It multiplies force and divides travel: the work is conserved.
- Using g = 10 m/s² in a precision calculation and carrying a 2% error.
08Self-assessment
What is the pressure at 10 m depth in water?
p = ρgh = 1000·9.8·10 = 98 kPa gauge, almost one atmosphere. In absolute terms, about 199 kPa.
A 700 kg/m³ body floats in water. What fraction is submerged?
70%: the submerged fraction is the ratio of densities.
What is the buoyant force on a cube with a 20 cm edge fully submerged in water?
V = 8·10⁻³ m³, so E = 1000·8·10⁻³·9.8 = 78.4 N, the weight of 8 liters of water.
In a press with A₂/A₁ = 50, what force is needed to lift 5000 N?
F₁ = 5000/50 = 100 N. The small piston must travel 50 times farther.
Why does a steel ship float?
Because its average density —steel plus the air enclosed by the hull— is lower than that of water. What matters is the density of the whole, not that of the material.
How does a barometric sensor measure altitude?
By the drop in pressure with height: near sea level, roughly 12 Pa per meter. A sensor with 1 Pa resolution can distinguish less than 10 cm.
09Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics, Volume 1. 9th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage Learning, 2014. Chapter 14 covers pressure, Pascal and Archimedes with clear examples.
- Yunus A. Çengel and John M. Cimbala. Fluid Mechanics: Fundamentals and Applications. 4th ed., McGraw-Hill, 2018 (Spanish edition, “Mecánica de fluidos: fundamentos y aplicaciones”). The engineering text: chapters 2 and 3, with real measuring instruments.
- Paul A. Tipler and Gene Mosca. Physics for Scientists and Engineers, Volume 1. 6th ed., W. H. Freeman, 2008 (Spanish edition, “Física para la ciencia y la tecnología”, Reverté, 2010). Good treatment of surface tension and capillarity.