Illustrated Physics
physics, drawn
Concept

Potential energy — where it appears

Energy stored by position in a field, definable only where the work done depends on the endpoints and not on the route taken between them. Its zero can be put anywhere without changing anything measurable, since only differences appear in any answer.

Named by 12 essays across 5 fields — each of them below, with the objects they name alongside it.

The pendulum's phase portrait. Angle plotted against angular velocity. Closed loops are swinging back and forth; the open curves above and below are rotating all the way round; the dashed curve between them is the separatrix.

The pendulum, and the small lie that makes it simple

A pendulum's period does not depend on how far it swings. This is one of the most useful false statements in physics, and it is worth knowing exactly how false.

mechanics · Pendulum A harmonic well. Potential energy against position, with a horizontal line at the total energy. The motion is confined to where the line lies above the curve, and the turning points are the intersections — computed by solving for them, not marked by hand.

The hill that gives it back, and the forces that do not

Potential energy turns a question about motion into a picture of a landscape. It works for gravity and springs, it fails for friction, and the difference between those two cases is the whole of what makes energy useful.

mechanics · Energy Released together on a 20° slope, 1.1 s later. 3 bodies of different shape, released from the same line on a 20 degree slope and drawn where each has reached after 1.1 seconds. The order is sphere, then disc, then hoop. Each spoke is turned by the distance that body has rolled divided by its radius.

The mass, and where it sits, which is what decides the race

Release a hoop and a marble together on a slope and the marble wins, whatever they weigh and whatever their size. Neither mass nor radius survives the arithmetic; only the arrangement does.

mechanics · Rotation Heating 1 kg of water from -20°C to 130°C. Temperature against heat added for 1 kilogram of water taken from -20 to 130 degrees Celsius. The two flat stretches are the melting and the boiling, where 334 and 2260 kilojoules go in and the temperature does not move. Melting costs as much as warming the water by 80 degrees; boiling costs as much as warming it by 541, which is 73 per cent of the whole journey.

The heat that changes no temperature, and where it actually goes

A kettle reaches a hundred degrees in a minute and takes five more to boil dry. The heat going in during those five minutes changes nothing a thermometer can see, and it is most of the energy in the whole process.

thermodynamics · Phase change The isothermal atmosphere against the real one. Pressure as a fraction of its sea-level value, against altitude. The curves are the isothermal barometric formula at 220, 288, 400 kelvin, whose scale heights are 6.4, 8.4, 11.7 kilometres. The points are the measured standard atmosphere. At 20 kilometres the 288 kelvin model is 71 per cent out, because the air up there is not at 288 kelvin.

Why the air thins with height, and why that is the same law as the speeds

The pressure of the atmosphere falls exponentially with altitude, and the distribution of molecular speeds falls exponentially with energy. These are not two results that happen to look alike. They are one statement read on two axes.

thermodynamics · Kinetic theory A wavefunction crossing a barrier it has not the energy for. An electron of 2 electronvolts meeting a 3 electronvolt barrier 0.3 nanometres wide, with the four matching conditions solved rather than sketched. Left of the barrier the incident and reflected waves add to a standing pattern; inside it the amplitude decays exponentially; to the right a travelling wave continues with amplitude 0.3912 of the incident one, so 15.3 per cent of the electrons get through. Classically none of them do.

The wall that is not quite a wall

A particle without enough energy to climb a barrier sometimes appears on the other side of it. The probability falls exponentially with the barrier's width, which is why the effect is invisible at ordinary scales and why it can be turned into a microscope.

quantum · Tunnelling A torque, and no force at all. A loop of 1 turn enclosing 20 cm² and carrying 10 A has a magnetic moment of 0.02 A m². In a uniform field of 0.05 T the torque on it is m B sin θ, drawn here against the angle between the moment and the field: zero when they are aligned, largest at 0.001 N m across, and zero again when they are opposed. The second curve is the energy, −m·B, whose minimum is the aligned position and whose maximum is the opposed one — which is why a compass needle settles one way round and not the other. The net force is zero at every angle on this axis, exactly and not approximately: the force is I dl × B summed round the loop, the sum of dl round any closed path is zero, and a constant B comes outside the sum. The inset shows the four forces on a rectangular loop; the pair across the axis is the couple, and the pair along it cancels.

The loop that behaves like a needle

Far enough away, a current going round in a circle is indistinguishable from a bar magnet, and one number describes both. That number tells a uniform field how to turn the loop and gives it no way to pull on it at all — which is why two magnets attract by the fourth power of the distance and not the second.

electromagnetism · Magnetism 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 Twice the kinetic energy, and what it equals. Twice the time-averaged kinetic energy of a bound orbit, divided by its time-averaged potential energy, against the power with which that potential depends on separation. Each point is a measurement: an eccentric orbit integrated for more than a hundred radial periods, with the two averages accumulated along it, and the radius checked to vary by at least a fifth so that the orbit is not trivially circular. The line is the exponent itself, and the points miss it by at most 6.6e-4. Two cases carry everything. At n = 2, a harmonic well, the two energies are equal — which is the ordinary equipartition statement, half a kT to the kinetic term and half a kT to the potential one. At n = −1, which is gravity and the Coulomb force, twice the kinetic energy equals minus the potential energy, so the total energy of a bound system is minus its kinetic energy. Nothing about temperature entered, and nothing about equilibrium: the relation holds for one orbit averaged over time as well as for a crowd averaged over members.

Weighing what cannot be put on a scale

Summed over every coordinate of a bound system, equipartition stops being a statement about temperature and becomes a relation between two averages: twice the kinetic energy equals n times the potential energy for a potential going as the nth power. For gravity that fixes a bound system's total energy from how fast its parts move — so a Doppler shift and an angular size return a mass, and for the Coma cluster the mass they return is fifty times the mass that shines.

thermodynamics · Equipartition A chain in which every domino is half as large again. The first 6 dominoes of a chain in which each is 1.5 times the size of the one before, drawn to scale, each standing 0.5 of its own height from the next, the first tipping onto the second. The sixth is 7.6 times the first's height and 3325 times its barrier. Each domino falls onto one it could not have been pushed over by hand at the same effort, and the chain passes on not the push but the permission to release energy each domino already held.

The push that pays only for the hill

A domino can knock over one half as large again as itself, and that one a still larger one, so that thirteen steps take a fall that began with a fingernail's flick to a slab as tall as a person. Nothing is being amplified by a machine. Each domino holds energy it was given when it was stood on its end, far more than it takes to tip it, and the push that starts it pays only for the small hill in front of that store. A chain of dominoes passes along not energy but permission — which is how a spark lights a fire and a nerve carries a signal the length of an arm without fading.

mechanics · Energy 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 How high a jumper of any size can rise. Jump height — the rise of the centre of mass after take-off, in vacuum — against body mass, on logarithmic scales. The flat line is Borelli's limit: if muscle can do a fixed amount of work per kilogram and a fixed fraction of the body is jumping muscle, every animal rises the same height, here 0.61 m. The sloping line is the limit set by power: the legs straighten in a time proportional to their length, and a small animal must deliver its energy in a fraction of a millisecond, faster than muscle can. That limit rises as the 2/9 power of mass and meets Borelli's near 80 kg. Points mark take-off speeds reported for six animals, converted to the height they would give without air: flea 0.18 m, froghopper 1.13 m, locust 0.52 m, frog 0.62 m, galago 2.22 m, human 0.49 m. Every animal well left of the crossing jumps far above its power limit, and every one of them is known to store energy in a spring before it jumps.

The jump that does not get higher with size

A flea, a locust and a person all lift their centres of mass by roughly half a metre when they jump, though their masses differ by a factor of a hundred million. Giovanni Borelli saw why in 1680: the work a muscle can do grows with its mass, and so does the work needed to lift a body, so the height cancels the size. What does not cancel is time. A small animal's legs straighten in a fraction of a millisecond, and no muscle can deliver its energy that fast — a flea would need more than five hundred times the power any muscle can give. Every small jumper reaches Borelli's height anyway, and every one of them does it with a spring.

mechanics · Energy

Named alongside it

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

StabilityArchimedes principleBuoyancyCentre of massEnergy conservationEquilibriumEquipartitionKinetic energyPhase portraitTemperatureTorqueTurning point

All concepts