Illustrated Physics
physics, drawn
Concept

Kinetic energy — where it appears

The energy a body has by virtue of its motion, frame-dependent in value and not conserved unless nothing does work on it. Different observers assign it wildly different values and all of them agree about how much a collision destroyed, because the disagreement cancels when momentum is conserved.

Named by 12 essays across 4 fields — each of them below, with the objects they name alongside it.

Two costs, and the width that balances them. The energy of a particle in a harmonic well against how tightly its wavefunction is squeezed, in units of ħω and of the width that minimises the total. Two terms compete. Squeezing the particle into a smaller region raises its kinetic energy, because the uncertainty relation makes a narrow position spread a wide momentum spread and momentum is squared in the energy; that term rises as the inverse square of the width and goes to infinity as the particle is localised. Letting it spread out raises its potential energy, since the well gets steeper away from the bottom; that term rises as the square of the width. The sum has a minimum at a width of 1.0000 in these units, where the total is 0.5000 ħω and the two terms are equal at a quarter each. That number is exactly the true ground-state energy of a quantum harmonic oscillator, obtained here with nothing but the uncertainty relation and a minimisation. What the figure shows and the formula does not is why there is a floor at all: it is not that the particle happens to keep moving, but that every way of stopping it costs more than it saves.

The motion that cannot be stopped

A particle in a well cannot sit at the bottom of it. Squeezing it into a smaller region costs kinetic energy faster than it saves potential energy, so there is a width that minimises the total — and the minimum is not zero. Helium never freezes because of it.

quantum · Uncertainty Arms in: 4.33× the rate, and 4.33× the energy. A body of 1.2 kg m² carrying two 4 kg masses on arms, spinning freely at 60 revolutions a minute with the arms out at 0.75 m, as the arms are pulled in to 0.12 m. The axis runs right to left, in the direction the arms move. Angular momentum is flat — nothing exerts a torque about the axis, and pulling inward is a force along a radius, which has no moment about the centre. The rate rises as the inverse of the moment of inertia, by a factor of 4.33 here, and the kinetic energy L²/2I rises by exactly the same factor, which is where the usual account stops and where the question starts. The fourth curve is the work done by whoever pulled the arms in, integrated from the force needed to hold each mass on its circle. It lies on the energy curve, to 1.6e-7 joules. Nothing is unaccounted for and nothing is created: the energy is bought, at full price, by pulling against the force that would otherwise fling the arms out.

The quantity that survives a change of shape

A skater pulls her arms in and spins four times faster. Angular momentum is conserved, which is the usual explanation, and it accounts for only half of what happened — because the kinetic energy has gone up by the same factor, and something had to pay for it.

mechanics · Rotation Six observers, six energies, one loss. The kinetic energy of the same collision before and after it, as measured by observers moving at 6 different speeds. No two of them agree about how much energy there was: the totals here range from 5.50 to 25.50 in the same units. Every one of them agrees about how much was lost — the gap between the two curves is 3.413 for all of them, varying by 6.2e-15. Energy is a quantity an observer owns; a change in it is not, and that is why heat, deformation and sound can be counted at all.

The energy that depends on the observer

A moving train has kinetic energy. Watched from a second train alongside it, it has none. Both statements are correct, neither can be corrected, and the whole of mechanics still works — because what conservation laws constrain is not how much energy there is but how much of it changes.

mechanics · Energy Thrust against the air it is supposed to be pushing on. The thrust of one F-1 engine against ambient pressure, in atmospheres. It is 6.78 meganewtons at sea level and 7.77 in vacuum — 14.6 per cent more with the air taken away. The line falls at exactly 9.787 newtons per pascal, which is the nozzle's exit area, because the term is (p_e − p_a)·A_e: the ambient pressure pushes on the exit plane from outside and there is nothing to push back on it. The account in which the exhaust shoves against the atmosphere makes the opposite prediction — thrust falling with the pressure and vanishing in vacuum — and it is not a small disagreement about a coefficient. It has the sign wrong. A rocket works better in vacuum than in air, and every engine ever fired has said so.

The push that needs nothing to push against

A rocket engine produces more thrust in vacuum than at sea level — fifteen per cent more, for the engine drawn here, and the extra is exactly the ambient pressure times the nozzle's exit area. The account in which the exhaust shoves against the atmosphere does not merely overstate a coefficient. It has the sign wrong, and every engine ever fired has said so.

mechanics · Momentum The energy a spin is allowed, and the only direction it can go. A body with principal moments 1.000, 3.000, 4.000 spinning with a fixed angular momentum. Every possible motion has an energy somewhere in the band drawn here, and the three marks are the three principal-axis spins: energy L²/2I, so the largest moment of inertia gives the smallest energy. The band runs from 0.1250 to 0.5000 in units where the momentum is one — a ratio of 4.0. Anything inside the body that flexes and warms takes energy out and leaves the momentum untouched, so the state can only move leftwards along this band, and there is exactly one place for it to stop. A spin about the axis of least inertia is at the far right: it is a perfectly good solution of the equations of motion, stable against small disturbances in a perfectly rigid body, and it sits at the top of a hill the smallest leak will roll it off.

The axis a leak of energy chooses

A body spinning with nothing pushing on it keeps its angular momentum exactly, and keeps its kinetic energy only while nothing inside it flexes. At fixed momentum the energy is least for a spin about the axis of greatest inertia — so any leak, however small, has a destination. The first American satellite found this out in orbit.

mechanics · Rotation Grip, then power, then air. The force a 1500 kg car can put through its driven wheels against road speed, with 100 kW at the wheels and a tyre friction coefficient of 0.9. The grip allows 13.2 kN at any speed; the engine allows its power divided by the speed, a hyperbola; the car gets whichever is smaller, drawn solid. The two are equal at 7.6 m/s, 27 km/h: below it the car is limited by friction and extra power would change nothing, above it by power and better tyres would change nothing. The rising curve is the resistance, rolling plus air, which grows as the square of the speed; it meets the drive at 61.2 m/s, 220 km/h, the top speed, solved for and checked there.

The speed at which grip hands over to power

A car's specification lists its power, and power does not limit how hard a car can push. It limits how hard it can push at a given speed, and at low speed that limit is higher than anything the tyres can transmit. So every car leaves the line as a friction problem and becomes a power problem a second later, at a crossover speed that decides which upgrade would make it faster — and at the top of its speed range a third limit, the cube of the speed, takes over from both.

mechanics · Energy 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 The frequency of a mass on a spring that weighs something. The angular frequency of a mass M on a spring of stiffness k and mass mₛ, in units of √(k/M), against the ratio mₛ/M on a logarithmic axis: exact, from β tan β = mₛ/M (solid); Rayleigh's estimate √(k/(M + mₛ/3)) (dashed); and the textbook answer that ignores the spring's mass, 1 (dotted). For a spring a tenth of the mass the exact value is 0.9836, Rayleigh's 0.9837 and the textbook's 1 — 1.7 per cent too fast. With spring and mass equal the exact frequency is 0.8603, Rayleigh's 0.8660: the third is good to 0.66 per cent where the textbook answer is 16 per cent out.

The third of itself a spring carries

Every textbook gives the frequency of a mass on a spring as the square root of stiffness over mass, and every real spring has mass of its own, which the formula leaves out. The repair is famous and oddly specific: add a third of the spring's mass to the load. The third is not a fudge. It is what the spring's kinetic energy comes to if each coil moves in proportion to its distance from the fixed end, and it is good to two-thirds of a per cent even when the spring weighs as much as the mass. Where it fails, it fails because the spring has stopped being a spring and become a column of waves, with notes of its own above the bounce — the notes that make the valve springs of an engine lose control.

mechanics · Harmonic approximation A drop that sweeps up mist settles at a seventh of g. The acceleration of a water drop falling from rest through a still mist that it sweeps up completely, in units of g, against time, for drops starting at radii of 0.2 mm, 0.05 mm and effectively zero, with no air resistance. The mist holds 10 g of water per cubic metre, twenty times a typical cloud, so that the approach takes seconds rather than minutes. Every drop starts at g, because at first it has gathered nothing, and every one settles to g/7 = 0.1429 g: after 2 s the three are at 0.377, 0.172, 0.143 g, and after 8 s at 0.144, 0.143, 0.143 g. The drop that starts from nothing is at g/7 from the start. The final acceleration contains no property of the mist, the drop or the water: a denser mist makes the drop grow faster and reach g/7 sooner, and nothing else.

The raindrop that falls at a seventh of g

A drop of water falling through a still mist, gathering every droplet in its path and with no air to slow it, does not fall at g. It settles to an acceleration of exactly a seventh of g, whatever the mist, whatever the drop, and its radius grows by the same amount for every metre it falls. Of the work gravity does on it, three-sevenths goes into the gathering and never comes back. The same reasoning gives a third of g for a chain paid out from a heap, and a single rule — g/(2n + 1) — for anything that sweeps up mass as the n-th power of the distance it falls.

mechanics · Momentum 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 The speed that stops while the energy keeps coming. The speed of an electron, as a fraction of the speed of light, against the kinetic energy given to it, from special relativity (solid) and from Newton's ½mv² (dashed); dots at the five energies Bertozzi used in 1964. Newton's speed passes the speed of light at a quarter of a megaelectronvolt and is 7.7 times it at 15 MeV. Relativity's speed is 0.863 c at 0.5 MeV, 0.967 c at 1.5 MeV, 0.9948 c at 4.5 MeV and 0.99946 c at 15 MeV. Between 1.5 and 15 MeV the energy grows tenfold and the speed by three and a half per cent. The energy does not stop arriving; the speed stops answering it.

The speed that stops while the energy keeps coming

In 1964 William Bertozzi gave electrons energies from half a megaelectronvolt to fifteen, timed them over eight and a half metres, and caught them in an aluminium disc to measure the heat they brought. By Newton the fastest should have crossed in under four nanoseconds, seven times faster than light. They took twenty-eight, and so did the slower ones, more or less. The heat, meanwhile, rose in proportion to the energy put in. The experiment is famous for showing that nothing outruns light, but its sharper result is the second half: the energy did arrive, all of it, carried at very nearly the same speed.

relativity · Relativistic dynamics The same energy in less and less whip. The speed of the moving part of a whip 2 m long and weighing 250 g, against how far the loop the arm threw into it still has to go to the tip, on a logarithmic scale, if the 15 J the arm gave it stays in the part still moving. Three whips: uniform, tapering linearly, and tapering faster, each ending in a 0.2 g cracker. At the handle every part moves at 11 m/s. Half a metre from the tip the uniform whip is at 22 m/s and the faster taper at 72; the uniform whip passes the speed of sound (dashed) only within its last millimetre, the tapers 1.6 cm and 1.5 cm from it. All three end at the same ceiling, √(2E/m) = 387 m/s, set by the cracker alone: the taper decides how gradually the speed builds, not how fast the end goes.

The crack at the end of a whip

An arm moves a whip's handle at ten metres a second, and a fifth of a second later its tip passes the speed of sound. Nothing in between is a lever or a spring. The arm throws a loop into the whip, the part behind the loop comes to rest, and the energy stays with the part still moving, which grows lighter as the loop runs out towards the tip. Energy shared by less and less mass means speed, and what caps it is the tip's own mass, the tuft of string at the end. The crack is a sonic boom, made by something the size of a shoelace's end.

mechanics · Momentum

Named alongside it

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

MomentumConservationDissipationVariable massWorkAngular momentumCentre of massEnergyMoment of inertiaPotential energyReference framesTorque

All concepts