Properties of Matter: Pressure, Fluids & Elasticity

Why does a sharp knife cut easily while a blunt one does not, even when you push with the same force? Why does a camel not sink in the desert sand, while a person in high heels does? The answer is pressure. The force pressing perpendicularly on a surface is called thrust. Pressure is the thrust acting on each unit of area: P = F / A. So for the same force, a smaller area gives a bigger pressure, and a larger area gives a smaller pressure.

Density tells how much mass is packed into a given volume: ρ = m / V. Water has a density of 1000 kg/m³ (1 g/cm³), and it is the standard for comparison. Relative density (also called specific gravity) compares a substance with water: relative density = density of the substance / density of water. It has no unit, because it is a ratio. A substance with relative density less than 1 floats on water; more than 1 sinks in water. Pure water has its maximum density at 4 °C.

Dive into a swimming pool and you feel the pressure on your ears grow as you go deeper. A liquid has weight, and the liquid above any point presses on it. So the pressure at a depth h in a liquid of density ρ is P = h ρ g (plus the atmospheric pressure on the surface). Three facts matter: pressure increases with depth, it increases with the density of the liquid, and at one point it acts equally in all directions. It does not depend on the shape or width of the vessel, only on the depth. That is why dam walls are built thicker at the bottom and deep-sea divers need strong suits.

Pascal's law (Blaise Pascal, 17th century) says: when pressure is applied to an enclosed fluid, it is transmitted undiminished and equally in all directions to every part of the fluid and to the walls of the container. Liquids are almost incompressible, so they pass the pressure on without losing it. This is the secret of every hydraulic machine: a small force on a small piston creates the same pressure in the fluid, and that pressure, acting on a large piston, gives a large force.

Air has weight, and the tall column of air above us presses on everything. This is atmospheric pressure. At sea level it equals 1 atmosphere (atm) = 101,325 Pa ≈ 10⁵ Pa = 760 mm of mercury. That is a push of about 10 tonnes on one square metre, yet we are not crushed, because the pressure of the fluids inside our body pushes outwards with the same strength. Atmospheric pressure decreases as we go up, because less air remains above us. This is why mountaineers feel breathless, nose-bleeds can occur at great heights, and water boils at a lower temperature on mountains.

The barometer measures atmospheric pressure. The simple mercury barometer was made by the Italian scientist Evangelista Torricelli in 1643. A long glass tube closed at one end is filled with mercury and inverted in a dish of mercury. The mercury column falls to about 76 cm (760 mm) and stops, because the weight of this column balances the atmospheric pressure on the dish. The space above is a near-vacuum. Mercury is used because it is very dense; a water barometer would need a tube about 10.3 m tall. An aneroid barometer has no liquid; it uses a flexible metal box that responds to pressure, so it is portable and is used in aircraft as an altimeter.

Why are we not crushed? The pressure of the air outside is balanced by the equal pressure of the blood and other fluids inside the body. If you went suddenly to a place of very low outside pressure (or into a vacuum), the inside pressure would no longer be balanced, which is why astronauts wear pressurised suits. In 1654 Otto von Guericke's Magdeburg hemispheres showed this: after the air was pumped out from between two copper halves, teams of horses could not pull them apart, because the outside air pressed them together.

Lift a heavy stone in a river and it feels light, but the moment it leaves the water it feels heavy again. The water pushes the stone upwards with a force called buoyant force or upthrust. The Greek scientist Archimedes of Syracuse found its rule: when a body is wholly or partly immersed in a fluid, it experiences an upward force equal to the weight of the fluid it displaces. The story says that he discovered this in his bath while checking whether King Hiero's crown was pure gold, and ran out shouting 'Eureka!' (I have found it).

Why does the upthrust exist? Pressure increases with depth, so the water presses up on the bottom of the body harder than it presses down on the top. The difference is the upthrust. Its size is F = V ρ g, where V is the volume of the body that is inside the fluid and ρ is the density of the fluid. So the upthrust depends on the density of the fluid and the volume immersed; it does not depend on the density or the material of the body, nor on how deep a fully immersed body is.

Why is a raindrop round? Why can a small insect walk on a pond? Why does a needle placed carefully on water float although steel is denser than water? The answer lies in the forces between molecules. Molecules of the same substance attract one another; this is cohesion. Molecules of different substances attract one another; this is adhesion. Inside a liquid every molecule is pulled equally in all directions, but a molecule at the surface is pulled inwards and sideways only. As a result the surface behaves like a stretched elastic film that tries to become as small as possible. This property is called surface tension.

Capillarity is the rise or fall of a liquid in a very narrow tube (a capillary). When a thin glass tube is dipped in water, water rises inside it because water wets glass: adhesion between water and glass is stronger than cohesion, so the water surface curves upwards (a concave meniscus) and the surface tension pulls the liquid up. In mercury the opposite happens: cohesion is stronger, the surface curves downwards (a convex meniscus) and the mercury level falls in the tube. The height of the rise is inversely proportional to the radius of the tube: h = 2T cos θ / (r ρ g). So the thinner the tube, the higher the liquid climbs.

Viscosity is the internal friction of a fluid. When a fluid flows, its layers slide over one another and each layer drags the next one, resisting the flow. Honey, glycerine and engine oil are highly viscous; water and alcohol are not. The SI unit of viscosity is pascal-second (Pa·s) (also N·s/m²); the CGS unit is the poise (1 Pa·s = 10 poise). In liquids viscosity decreases as the temperature rises (honey flows easily when warm), but in gases it increases with temperature. A body moving through a fluid feels the frictional force of the fluid, called drag. For a small sphere the drag is given by Stokes' law: F = 6π η r v. A falling raindrop or a ball in glycerine speeds up until the drag (plus upthrust) equals its weight, and then falls at a constant speed called the terminal velocity.

When water flows from a wide river into a narrow gorge, it speeds up. When you press your thumb over the end of a garden hose, the water shoots out faster. This is the equation of continuity: for a steady flow of an incompressible fluid, the volume passing every point per second is the same, so A₁v₁ = A₂v₂. Where the pipe is narrow, the fluid moves faster; where it is wide, the fluid moves slower. In streamline (steady) flow each particle follows the same smooth path; at high speeds or around obstacles the flow becomes turbulent, with eddies and whirls.

The Swiss scientist Daniel Bernoulli (1738) tied speed and pressure together. Bernoulli's principle: for a steady, non-viscous flow of an incompressible fluid, where the speed of the fluid is higher, its pressure is lower, and where the speed is lower, the pressure is higher. In formula form, along a streamline, P + ½ρv² + ρgh = constant. The principle is nothing but the conservation of energy applied to a flowing fluid: if the kinetic energy per unit volume (½ρv²) rises, the pressure energy (P) must fall.

Experiment to remember: hold two sheets of paper a few centimetres apart and blow between them. They do not move apart; they move together. Fast air between them has lower pressure than the still air outside, so the outside pushes them inwards. In the same way, two ships sailing close and parallel are drawn towards each other, which is a real danger at sea.

Stretch a rubber band and let go: it springs back. Bend a steel scale slightly and release it: it straightens. Press a lump of clay: it keeps the new shape. The first two bodies show elasticity, the property of a body by which it regains its original size and shape after the deforming force is removed. The clay shows plasticity, where the deformation stays. A body that returns completely is called perfectly elastic (quartz is the nearest example); one that does not return at all is perfectly plastic (putty, wax). No real body is perfect, but steel, quartz and glass are very close to elastic.

When a deforming force acts, the body develops an equal opposing force inside it, called the restoring force. Its value per unit area is stress: stress = F / A, with SI unit N/m² (pascal). The change it produces, measured as a fraction of the original size, is strain: strain = change in dimension / original dimension, for example ΔL / L. Strain is a ratio, so it has no unit and no dimension. Hooke's law (Robert Hooke, 1678) says that within the elastic limit, stress is directly proportional to strain. The constant of proportionality is the modulus of elasticity: E = stress / strain.

All three moduli have the same unit as stress, N/m² (Pa), because strain has no unit. Steel has a much larger Young's modulus (about 2 × 10¹¹ Pa) than rubber, so for the same stress steel stretches far less. That is why steel is more elastic than rubber: 'more elastic' does not mean 'stretches more', it means 'resists deformation more and returns better'. Among common metals the order of Young's modulus is steel > copper > aluminium > lead.

The energy stored in a stretched wire or spring is elastic potential energy. For a spring it is U = ½ k x². Per unit volume of a deformed body, energy density = ½ × stress × strain, so the total energy stored is ½ × stress × strain × volume. Two more terms often appear. Elastic fatigue is the loss of strength of a material after repeated stretching and releasing, which is why bridges and aircraft parts are checked regularly. Elastic after-effect is the delay with which a body regains its shape after the force is removed.