Illustrated Physics
physics, drawn
Concept

Archimedes principle — where it appears

The rule that a submerged body is pushed up by the weight of the fluid it displaces, which follows from pressure rising with depth. It is a theorem about a closed surface, so it fails for a body sealed to the bottom with no fluid underneath — which then feels no upthrust at all.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

Where the upward force comes from. A block submerged with its top 1.2 m down. The pressure on the bottom face (19.6 kPa) exceeds that on the top (11.8 kPa) by exactly the weight of a column of water as tall as the block, and the sideways pressures cancel in pairs. Nothing has been added to the physics of pressure to get buoyancy out of it.

The weight of the water that is not there

A submerged object is pushed up by the weight of the fluid it has displaced — not by something like it, not approximately, but exactly. The reason is that the pressures on its faces do not cancel, and the sum that survives has forgotten everything about the object except its shape.

fluids · Buoyancy Buoyancy that falls away as the body sinks. The net upward force on a body containing a little gas, against how deep it has been taken, for 3 gas fractions. The weight does not change with depth. The buoyancy does, because the gas obeys Boyle's law and the pressure rises by an atmosphere every ten metres, so a body that displaced its own weight at the surface displaces less at depth. Every curve therefore slopes downward, and that slope is the whole point: where a curve crosses zero the body is in equilibrium, and the crossing is always from above, which makes every one of these equilibria unstable. Push the body a little deeper and the force does not push back — it turns downward and grows. The crossings drawn are at 8.2 m, 10.0 m, 13.6 m, and a body sitting at one of them is balanced in the sense that a pencil is balanced on its point. This is why a diver at neutral buoyancy has to keep adjusting, why a submarine's depth is held by hydroplanes and not by ballast alone, and why a fish that loses the use of its swim bladder sinks rather than drifting.

The depth past which it must sink

A body carrying a pocket of gas can be trimmed to hang motionless in water at exactly one depth. Push it a little deeper and it does not come back — the gas compresses, the buoyancy falls, and the equilibrium turns out to have been balanced on its point.

fluids · Buoyancy A closed cabin accelerating at 3 m/s², and everything in it leaning the same way. A closed box accelerating to the right at 3 m/s², with water in the bottom and air above. Everything that can hang or float lines up with the effective gravity g − a, tilted 17.0° from the vertical towards the back: the water's surface and the isobars below it are perpendicular to it, a floating block stands along it, a plumb bob hangs along it — and a helium balloon on a string leans the other way along the same line, forward, into the acceleration. The balloon is not doing anything unusual. Buoyancy always points opposite to gravity, and in the cabin the effective gravity has a horizontal part, so the air's pressure gradient does too, and it pushes the balloon along it.

The balloon that leans the wrong way

When a car pulls away, everything loose in it swings back — except a helium balloon on a string, which swings forward. Nothing strange is acting on it. Buoyancy points against gravity, and inside an accelerating cabin gravity has a sideways part. Read that way, a lift cannot change how deep a boat floats, free fall abolishes floating altogether, and a centrifuge is Archimedes' principle with the word "up" pointing at the axis.

fluids · Buoyancy Five square logs of different densities, as they float. Cross-sections of five long square logs in their stable floating positions, with relative densities 0.1, 0.25, 0.5, 0.75 and 0.9, each drawn with its water line and its submerged part shaded, and a dot at its centre. The lightest floats flat and high; at 0.25 it has tilted to about 27°; at 0.5 it floats on its corner with a diagonal vertical and the water line across the middle; at 0.75 it tilts to the same angle as at 0.25, and at 0.9 it floats flat again, low in the water. The heavy logs are the light ones turned upside down, with air and water exchanged.

The log that floats on its corner

A long square beam of very light wood floats flat, one face up, as anyone would expect. A beam of wood half as dense as water does not: it rolls onto a corner and floats with a diagonal vertical, like a diamond. Between the two, at a quarter of water's density, it floats tilted at twenty-seven degrees, neither flat nor on edge. Heavier beams repeat the sequence in reverse, and the whole pattern is symmetric: a beam of density s floats at exactly the angle of one of density 1 − s. None of it depends on anything but a square, a density and Archimedes, and the same arithmetic decides which way an iceberg rolls when it breaks.

fluids · Buoyancy The lift of a zero-pressure balloon as it climbs. The gross lift, in tonnes, of a zero-pressure balloon whose envelope holds at most 1.1 million cubic metres, against height, for launches with 3,500 and 5,000 cubic metres of helium; the dashed line is its 3 tonnes of envelope, payload and ballast. At launch the envelope is mostly empty. As the balloon climbs the gas expands exactly as fast as the air thins, so the mass of air it displaces stays the same and the lift stays at 3.69 or 5.28 tonnes all the way up. The envelope fills at 37.5 km with the larger fill and 39.9 km with the smaller; above that, gas vents through the open base, the volume is fixed and the lift falls with the air's density. Both reach the same float, 41.3 km, where the lift has fallen to the weight. More helium only makes the climb faster; the ceiling is set by the envelope's size and the mass it carries.

The balloon that floats at a density

A helium balloon let go rises until something stops it, and what stops it depends entirely on how it is built. A party balloon or a weather balloon stretches until it bursts. An open-bottomed science balloon climbs at constant lift — its gas expanding exactly as fast as the air thins — until its envelope is full, then floats where the air is thin enough. A sealed, unstretchable balloon floats where its whole density equals the air's and bobs back if pushed away. None of them floats at a height; each floats at a density. The difference between the three is how compressible the balloon is compared with the air around it, the same comparison that decides whether a Cartesian diver sinks.

fluids · Buoyancy A floating needle and the water it displaces. A cross-section of a steel darning needle 1.36 mm thick (density 7.85) floating on water, with a contact angle of 100°, its meniscus computed from the balance of surface tension and pressure; one capillary length is 2.73 mm. The needle's centre sits 1.43 mm below the undisturbed surface, and the water meets it 117° from the top, sloping at 37°. Two regions of water have been displaced. Below the level, under the needle, a region whose weight — 21 per cent of the support — is what pressure on the needle's underside supplies. Beside it, the two dents in the surface, whose weight — the other 79 per cent — is exactly the vertical pull of surface tension along the two contact lines. Together they weigh what the needle weighs: Archimedes, counting the dent.

The water the dent displaces

A steel needle laid carefully on water floats, and the usual explanation is that surface tension holds it up instead of buoyancy, as though the two were rival mechanisms. They are not rivals. The needle presses a dent into the surface, and the upward pull of surface tension along its edges is exactly the weight of the water missing from that dent. Archimedes' principle holds without amendment: a floating needle displaces its own weight of water. Most of that water is simply not underneath it.

fluids · Buoyancy Where the level goes when a floating ice cube melts. The change in level when a 30 g ice cube floating in a glass 7 cm across has melted completely, against the density of the liquid it floats in (the liquid must be denser than ice, 917 kg/m³, or the cube sinks). In fresh water the level does not move: the meltwater exactly fills the volume the ice displaced. In seawater it rises 0.21 mm and in saturated brine 1.3 mm, because the ice was displacing a denser liquid than it melts into; in castor oil, which is lighter than water and does not mix with it, the level falls 0.32 mm as the meltwater sinks beneath the oil. The change is the meltwater's volume minus the displaced volume, m/ρw − m/ρ, spread over the glass.

The floating ice that still raises the sea

An ice cube melting in a glass of water leaves the level exactly where it was, and the reason is exact: a floating body displaces its own weight, and the meltwater has that weight. The same argument is often used to say that melting sea ice and ice shelves cannot raise the sea. It nearly works. The ice was floating in salt water and melts into fresh, which is lighter, so the meltwater overfills the hole by almost three per cent — and the ice that really fills the sea is the part of a grounded glacier above the thickness at which it would float.

fluids · Buoyancy The force to pull a foot out of packed sand. The suction that resists pulling a foot (a sole of 250 cm², 10 cm long) straight up out of sand, as water is drawn through the sand into the space the foot leaves — Darcy's law with a Kozeny–Carman permeability — against the pulling speed, on logarithmic axes, for compacted sand (fraction 0.6) of three grain sizes and for loose sand (0.41) of 0.1 mm; the dotted line is a 70 kg person's weight. The force is proportional to the speed and inversely to the square of the grain size. Pulled at a centimetre a second from compacted 0.1 mm sand, a foot resists with 2.5 kN, 4 times the person's weight; the speed at which the suction equals that weight is 2.71 mm/s. In the loose sand before it compacted, the same pull would need 15 per cent of the force; with clay filling the pores of the compacted sand, as in natural quicksand, a thousand times more, and the speed a body's weight can pull at falls to microns a second.

The sand that lets a body in and will not let it out

Quicksand cannot swallow anyone: liquefied, it is nearly twice as dense as water, and a person in it floats with their chest clear. What makes it dangerous is the opposite asymmetry. Going in is easy, because a step breaks the gel holding the grains apart and the sand turns liquid. Coming out is hard, because the sand settles and packs around whatever sank, and a trapped foot can only rise as fast as water can seep through the packed grains to fill the space it leaves. Pulling harder barely helps; letting water in does.

fluids · Rheology The height at which hot air stops being light enough. Lift per cubic metre of hot air against height in the standard atmosphere, with the inside at 100 °C and 120 °C, and (dashed) what a 2800 m³ balloon carrying 550 kg plus its envelope needs per cubic metre, 0.241 kg. As the balloon climbs, the outside air thins, and although it also cools — which improves the temperature ratio — the density wins: lift per cubic metre falls by about a tenth for every kilometre. The balloon can climb only to where the curve crosses its need: 6.0 km with the envelope at 120 °C. Above the tropopause the outside temperature stops falling and the lift falls faster still; a hot-air balloon that reaches the stratosphere needs an envelope enormous compared with its load.

The balloon a cold morning lifts

A hot-air balloon is open at the bottom, so the air inside is at the same pressure as the air outside and lighter only because it is hotter. Its lift is the outside air's density times one minus the ratio of the two absolute temperatures — a small number, a third of a kilogram per cubic metre at best — and since the fabric cannot be heated much above a hundred degrees, the lift depends as much on the cold outside as on the heat inside. That is why balloons are the size of houses, why they fly at dawn, why each has a ceiling, and why the pilot must burn for where the balloon will be half a minute from now.

fluids · Buoyancy

Named alongside it

The objects these essays reach for when they reach for this one.

BuoyancyDensityHydrostatic pressureStabilityCapillary lengthCompressibilityThe ideal gas lawPotential energyStandard atmosphereApparent weightBifurcationBoyle law

All concepts