Illustrated Physics
physics, drawn
Concept

Conservation of energy — where it appears

The rule that the total energy of an isolated system does not change, whatever form it moves between or how it is divided up. The total is different for different observers, and only the change is agreed — which is why what a conservation law constrains is a difference rather than an amount.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

A metre of tube, with and without the metal. The magnet's position against time over a 1.00 m drop, integrated from Newton's second law with the eddy-current drag included, and the same fall with the drag switched off. The braked descent takes 2.09 s against 0.45 s in free fall. After a transient of about 50 milliseconds — the mass divided by the drag coefficient, and the only timescale in the problem — the trace is a straight line, which is to say the magnet is falling at a constant 49.0 cm/s. Every joule of the 118 mJ of gravitational energy released has gone into resistive heating of the wall, and none into the magnet's kinetic energy, since that is the same at the bottom as it was a moment after the start.

The magnet that falls slowly

Drop a magnet down a copper pipe and it takes several seconds to fall a metre, drifting rather than falling. Nothing touches it, the copper is not magnetic, and there is no circuit anywhere. The pipe arrives warm.

electromagnetism · Induction The curve that makes both fusion and fission release energy. Binding energy per nucleon against mass number: how much energy would have to be supplied, per particle, to take a nucleus apart into free protons and neutrons. The curve is the semi-empirical mass formula, evaluated at whichever proton number binds most tightly for each mass number rather than at a guessed one; the points are measured values. It rises steeply at the light end, peaks at mass number 58, and falls slowly thereafter. Everything about nuclear energy follows from that shape and from nothing else. Two light nuclei joined move up the curve and release the difference; one heavy nucleus split moves up it too, from the other side. Both directions are downhill in energy because the peak is in the middle, and the peak is in the middle because two effects fight — the surface term, which penalises small nuclei for having most of their nucleons on the outside, and the Coulomb term, which penalises large ones because every proton repels every other. The energy released is the height climbed times the number of nucleons carried, and it is a million times a chemical bond for the same reason the vertical axis is in millions of electronvolts rather than in single ones.

The mass that is missing

A helium nucleus weighs less than the two protons and two neutrons it is made of. The shortfall is not an error in the weighing; it is the binding energy, converted at the going rate. One curve of that shortfall against size explains why both fusion and fission release energy.

relativity · Mass-energy The self-force matters at 6.27 × 10⁻²⁴ s, and nowhere a charge has ever been. The ratio of the radiation reaction to the applied force, which is τ divided by the time the force takes to change, on a logarithmic axis. τ = μ₀q²/6πmc = 6.266e-24 s for an electron, and light crosses 1.879 femtometres in that time — two thirds of the classical electron radius. a 3 GHz accelerating cavity: 1.9e-14; a 500 nm optical field: 3.8e-9; an electron orbiting a proton, ground state: 4.1e-8; an X-ray at 0.1 nm: 1.9e-5; over a classical electron radius: 6.7e-1. The largest of them, "over a classical electron radius", is still 1.5e+0 times too slow. So the correction is never large for any force anybody can apply, and the only regime where it would be is one in which the charge's own structure has already made the whole description meaningless.

The force a charge exerts on itself

Larmor's formula says how much an accelerating charge radiates and says nothing about who pays. Conservation says the charge does, so there is a force on it — and the equation that force produces has a free particle accelerating for ever with nothing pushing it, or else beginning to move before it is pushed. Both solutions are absurd, and the interval over which they are absurd is smaller than the electron the equation was written for.

astrophysics · Radiating charge The floor does no work and the jumper leaves the ground. A 70-kilogram person pushing off the floor: the floor's force in units of body weight against time, with the centre of mass's height and speed drawn on the same axis, each scaled. The force reaches 2.6 body weights, the contact lasts 260 milliseconds, and the take-off speed that comes out of integrating it is 1.67 metres a second — a jump of 14 centimetres. Integrating the floor's force over the centre of mass's rise gives 247 joules. The work the floor does is zero, because the patch of floor under the foot never moves and work is a force times the displacement of its own point of application. Both numbers are correct and they are answers to different questions: the first is what Newton's second law integrated over the centre of mass gives, and the second is what crosses the boundary between the floor and the person, which is nothing.

The floor that does no work

A jumper leaves the ground with three hundred joules of kinetic energy, supplied by a floor that does exactly zero work — because work is a force times the displacement of its own point of application, and the patch of floor under the foot never moves. Newton's second law integrated over the centre of mass gives the right kinetic energy and is not the work-energy theorem, and telling the two apart is what the first law of thermodynamics is for.

mechanics · Energy A wall pulled away fast leaves the energy behind. The energy of a ball bouncing in a box whose wall is moved, against the length of the box, for 3 wall speeds — each a fraction of the ball's own starting speed — with the adiabatic prediction drawn dashed. Every collision is solved for exactly rather than stepped, so nothing here assumes the wall is slow. Moved slowly, the wall takes energy from the ball at the adiabatic rate, and the energy falls as the inverse square of the length. Moved as fast as the ball is moving, it takes almost nothing: the ball cannot catch a wall retreating faster than it travels, so the collisions stop and the energy stops falling. That is the difference between a gas pushing a piston and a gas expanding into a vacuum, and it is drawn here for one particle. At the end of the range the slowest wall leaves the energy at 0.058 of its starting value and the fastest at 1.000, against an adiabatic 0.065.

The wall that moves while the ball is in flight

Energy is conserved because the rules do not depend on the time. Move the walls of a box and the rules do depend on the time, so what is inside gains or loses without limit — and how much depends entirely on how fast. Moved slowly, a wall takes energy at exactly the rate the adiabatic law says; moved faster than the ball travels, it takes none at all, and the same box is a piston or a vacuum according to a speed.

mechanics · Energy The number of collisions for heavier and heavier blocks. The number of collisions, simulated one by one, for mass ratios of 1, 10², 10⁴, 10⁶, 10⁸, on a logarithmic axis: 3, 31, 314, 3141, 31415. Each factor of a hundred in the mass ratio adds one digit of π. Nothing in the setup mentions a circle — two blocks, a wall and the conservation of energy and momentum — but energy conservation makes the velocities live on a circle, and the count is how many equal steps of a small angle fit into half of it.

The collisions that count the digits of π

Slide a heavy block towards a light one resting near a wall. The light block is batted between the heavy one and the wall until the heavy block turns round and leaves. Count the collisions. With equal masses there are 3; with one block a hundred times heavier, 31; ten thousand times, 314; a million, 3141. Nothing in two blocks and a wall mentions a circle, but conservation of energy puts their velocities on one, and each collision steps round it by the same small angle. The count is how many such steps fit in half a turn.

mechanics · Momentum The ladder that leaves the wall. A 4-metre ladder released from rest at 15° from the vertical, its foot on a frictionless floor and its top against a frictionless wall, drawn at equal intervals of time from release to the moment it lies flat, computed by integrating its motion; the dots trace its centre. While both ends touch, the centre moves exactly on a circle of radius L/2 about the corner (dashed). The top leaves the wall when it has come down to 2.58 m, two-thirds of its starting height of 3.86 m; from then on the centre's horizontal speed is fixed, it leaves the circle, and the ladder falls clear of the wall with its foot still sliding. The fall takes 1.34 seconds.

The ladder that lets go of the wall

A ladder whose foot slips on a smooth floor slides down a smooth wall — and partway down, its top comes away from the wall and the ladder falls the rest of the way clear of it. The point where it lets go of the wall is not set by its length, its mass or how steeply it started: it is always two-thirds of the height the top started at. The reason is a wall that can push but cannot pull, and the number falls out of energy alone, without a single force being calculated in advance.

mechanics · Energy 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 The force a rider feels on the way up a loop. The force of the seat on a rider, in multiples of the rider's weight, against how far the track has turned on the way from the bottom (0) to the top of the loop (1), for the 20 m circle and clothoid of the previous figure, each entered at the speed that leaves 0.5 g at the top. On the circle the force jumps the moment the car enters the loop, from 1 g on the straight to 6.5 g, and falls steadily to the top. On the clothoid it starts at 1 g, because the curvature starts at zero, and rises smoothly to 4.3 g partway round before falling. The clothoid's car also enters more slowly — 22.7 m/s against 23.2 — because its tighter top needs less speed to hold the car on.

The loop that is not a circle

A roller coaster's vertical loop looks circular from a distance and is not. A circular loop, entered fast enough for the car to stay on the track at the top, presses its riders into their seats at six times their weight at the bottom — six, whatever the circle's size — and slams that force on in an instant at the entry. The fix, found for railways a century earlier, is a curve whose curvature grows steadily with distance, the Euler spiral: the same curve that describes how light spreads round the edge of a shadow. Tight at the top, gentle at the bottom, it lets the force build instead of arriving all at once.

mechanics · Circular motion A spin set by the length of a cord. The body's spin against the cord let out, from conservation of energy and of angular momentum alone, for three values of c, one plus the body's moment of inertia over the weights' mR². Every curve passes through zero at a cord length of exactly √c radii, and a direct integration of the equations of motion, which assumes neither conservation law, lands on the same curve to a part in a thousand. Past that length the spin reverses. The starting spin does not appear in the zero's position: only the shape of the machine does. A weight on a cord cut to length therefore removes all the spin whether it was spinning at 20 or 100 revolutions a minute, provided the cord is released at that length.

The weights that stop a spinning satellite

A rocket stage is often spun up to keep it pointing straight during its burn, and the spin then has to be removed. A small cord and a weight wrapped round its rim do the whole job, with no fuel, no sensors and no moving parts besides the cord. The cord's length has no dependence on how fast the stage is spinning, so the same two weights stop it at any speed. And the cord unwinds at exactly the rate the stage was first spinning, while its spin falls away to nothing.

mechanics · Energy

Named alongside it

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

Centre of massCollisionEnergyFrictionMoment of inertiaMomentumNormal forcePhase spaceAccelerationActionActivation barrierAdiabatic invariant

All concepts