Illustrated Physics
physics, drawn
Concept

Density — where it appears

Mass per unit volume, the quantity that decides whether a body floats and the one a compressible body changes as it descends. Comparing it with a fluid's settles flotation only when nothing else is holding the body — a surface, a seal or a wall can override it entirely.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Three vessels, one pressure. Three vessels filled to the same depth of 3 m. The pressure on each base is 29.4 kPa — identical, because pressure is set by depth — while the weight of water each holds differs by a factor of 4.7. The base of the flaring vessel carries more force than the water standing over it weighs.

The pressure that only knows depth

A litre of water and a swimming pool press equally hard on a floor at the same depth. Pressure in a still fluid is a scalar with no direction of its own, it depends on how far down and on nothing else, and the shape of the container falls out of the arithmetic entirely.

fluids · Hydrostatics 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 The phase boundary of water, from one equation. Pressure against temperature for water on a logarithmic pressure axis spanning 6.6 decades. The vaporisation curve is integrated from Clausius–Clapeyron between the triple point at 273.16 kelvin and 611.7 Pa and the critical point at 647.096 kelvin, with a single latent heat of 43.32 kilojoules per mole — the value the two published points on the curve imply. The measured latent heats are 45.05 at the triple point and 40.65 at the reference point, and the fitted value sits between them, because a constant latent heat is an average over the interval. The sublimation curve below the triple point is not measured but predicted, from the two latent heats adding where all three boundaries meet: 51.1 kilojoules per mole, which reaches 103.2 Pa at 253.1 kelvin against a measured 253.15. The melting curve is drawn at the slope Clapeyron gives it, -13.5 megapascals per kelvin, which is a volume ratio and nothing else: water's solid is 917 against 1000 kilograms per cubic metre for its liquid, so melting shrinks it and the line leans backwards. Across the whole 6.6 decades of this axis that line moves 5.2 kelvin, and one atmosphere shifts the melting point by 0.0075 kelvin. At 1 atmosphere the boundary is crossed at 373.1 kelvin, where water boils. The one place the curve fails is its top end: a constant latent heat reaches 37.4 MPa at the critical temperature where the measured critical pressure is 22.1 MPa, 70 per cent high, because the latent heat falls to zero at the critical point and this curve does not know that.

A boiling point is a pressure, not a temperature

Nothing about water names one hundred degrees; the air does. Where a liquid turns to vapour in its bulk is fixed by what pushes on it, which makes the familiar figure a coordinate on a curve — 71 °C on Everest, 119.5 °C in a sealed pot — and one latent heat draws the whole curve.

thermodynamics · Phase change Everything is decided against one line at 9.76 K per kilometre. Temperature against height for five environments, with the dry adiabat drawn heavy. A parcel lifted from the ground cools along the adiabat, at g/c_p = 9.76 K/km — a number with no meteorology in it, only gravity and the heat capacity of air. If the environment cools faster than that, a lifted parcel finds itself warmer than its surroundings and keeps going; if it cools more slowly, the parcel finds itself colder and sinks back. -5 K/km gives N² = 5.02e-4 s⁻², a period of 4.7 min; 0 K/km gives N² = 3.32e-4 s⁻², a period of 5.7 min; 6.5 K/km gives N² = 1.11e-4 s⁻², a period of 9.9 min; 9.8 K/km gives N² = -1.43e-6 s⁻², an e-folding time of 835 s; 12 K/km gives N² = -7.63e-5 s⁻², an e-folding time of 114 s. The classification is a comparison of two slopes and nothing else: no density appears in it, and the same cold air is stable under one profile and unstable under another.

The layer a parcel cannot leave

Whether a column of air overturns is not decided by its density but by a difference of two gradients — the rate the environment cools with height, and the rate a lifted parcel cools on its own. Subtract one from the other and what is left is a restoring force per unit displacement, so a stable atmosphere rings at a period of minutes and an unstable one has no period at all.

fluids · Stratification The distance that does not know how big the moon is. How close a satellite held together by its own gravity can orbit before the tide pulls it apart, in units of the primary's radius, against how much denser the primary is than the satellite. The curve is the distance at which the tidal stretch across the satellite's own body equals the satellite's surface gravity. Setting those two equal cancels the satellite's radius on both sides, so a boulder and a thousand-kilometre moon of the same material break up at exactly the same distance — the limit is a ratio of densities and nothing else, and it goes as the cube root of that ratio, measured here as 0.3333. For ice around a planet of density 687 kg/m³ the rigid limit is 1.15 radii and the limit for a body that can deform under the tide is 2.23, because a satellite pulled into an egg presents a longer body to the tide and gives way sooner. Saturn's rings end at 2.27 radii, just outside that second number, and its innermost round moon orbits at 3.08 — so the boundary between a ring and a moon falls where this calculation puts it. What this calculation leaves out is strength: a body small enough for its material strength to beat its own gravity ignores the limit entirely, which is why Phobos is well inside Mars's and still in one piece, and why the Shoemaker–Levy fragments were held together by nothing at all.

The distance that forgets the moon

A satellite held together by its own gravity comes apart if it orbits too close, and the distance at which it does contains no reference to its size. Both the tide pulling it apart and the gravity holding it together are proportional to its radius, so the radius cancels twice over and what is left is a ratio of two densities.

astrophysics · Self-gravity Melting curves, and the one that leans the wrong way. Melting temperature against pressure for water, benzene, naphthalene, each measured from its own melting point at one atmosphere, with pressure in bars. The slope of every coexistence line is the latent heat divided by the temperature and the change in volume, and the latent heat of melting is positive for everything — so the sign of the slope is the sign of the volume change, and nothing else. Almost everything expands on melting and its line leans forwards. Water's solid is less dense than its liquid, so its line leans backwards at 135 bars a kelvin: pressing on ice at just below zero melts it, and it takes 135 atmospheres to gain a single degree. The anomaly is not in the thermodynamics; it is in the fact that ice floats.

The melting curve that leans the wrong way

The slope of any coexistence line is the latent heat divided by the temperature and the change in volume. Latent heat is always positive, so the sign of the slope is the sign of the volume change — and for water the volume change is negative, which is the whole of why ice floats and why the melting curve leans backwards.

thermodynamics · Phase change A body at an interface, and the difference that holds it. The net upward force on a ten-centimetre cube straddling the boundary between two fluids, against how far its underside sits below that boundary — each curve scaled by its own largest value so that three very different cases fit on one axis. The displaced weight has to be counted twice, once with each density, so the force is linear in the displacement with a stiffness set by gravity, the cube's cross-section and the difference of the two densities — the difference, and neither of them alone. air over water holds the cube with its underside 60 per cent of the way through, at a stiffness of 97.8 newtons per metre; oil over water holds the cube with its underside 47 per cent of the way through, at a stiffness of 14.5 newtons per metre; water over mercury holds the cube with its underside 24 per cent of the way through, at a stiffness of 1229.4 newtons per metre. Both numbers are read off the drawn curve rather than substituted. The consequence is the one worth the figure: a body at an oil–water interface is held six times less stiffly than the same body at an air–water one, because the density difference is six times smaller, and the ordinary intuition that a denser fluid holds a body more firmly is exactly wrong — what matters is the contrast across the surface the body is sitting in.

The body that displaces two things

Every earlier argument has a body wetted by one fluid, so the displaced weight is a volume times a density. A body at an interface displaces two, and what holds it is the *difference* between them — so the same block is held six times less stiffly at an oil–water boundary than at an air–water one. In a continuously stratified column the neutral depth becomes stable, which is the exact opposite of the compressible case.

fluids · Buoyancy

Named alongside it

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

EquilibriumHydrostatic equilibriumPressureBuoyancyDisplaced volumeFree surfaceLatent heatPhase transitionStabilityStratificationTemperatureAdiabatic process

All concepts