Illustrated Physics
physics, drawn
Concept

Centre of mass — where it appears

The mass-weighted average of a system's positions, which moves as though the external forces acted on the total mass gathered at that point. Its motion is unaffected by anything internal, so an exploding shell's fragments continue to straddle the parabola the whole shell was on.

Named by 21 essays across 4 fields — each of them below, with the objects they name alongside it.

A collision with restitution 0.6. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.

Collisions are easier than forces, and momentum is the reason

Nobody knows what happens inside a collision. Momentum conservation makes that ignorance irrelevant, which is the whole trick — and energy, deliberately, is not conserved.

mechanics · Momentum The same 120 N force at 3 different arms. 3 spanners of different lengths, each with the same 120 newton force applied at its end. The torque printed under each is the force times its own moment arm, so it rises with the length while the force does not change.

The same push, further out, and why that is a different quantity

A force is not enough to say whether something turns. What decides is where the line of the force passes, and the distance from the pivot to that line is the whole of the story.

mechanics · Rotation 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 A collision with restitution 0.4. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.

The point that keeps moving as if nothing had happened

Newton's third law makes every internal force cancel against its own partner, which leaves the external sum governing a single mass-weighted average of positions. In the collision below the total momentum stays at 4.00 kg·m/s while 63 per cent of the kinetic energy leaves, and the average travels at 1.00 m/s throughout, before and after.

mechanics · Momentum What puts a scale on a tilted axis. A spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0.

The invariant that survives a boost

Energy and momentum are both answers to the question "how fast is it going, and according to whom". One combination of them is not, and that combination is the mass — which is why two photons of 511 keV can be a thing of mass 1.022 MeV or a thing of no mass at all, depending only on the angle between them.

relativity · Mass-energy 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 Two pivots with one period, and the length between them. The period of a uniform bar 1 m long swung about a pivot, against the distance of that pivot from its centre of mass. Close to the centre the period runs away, because there is almost no restoring torque; far from it the bar behaves more and more like a simple pendulum. In between is a minimum at h = k = 0.2887 m, the radius of gyration, and because there is a minimum every period above it belongs to two pivots at once — here 0.4 m and 0.2083 m, whose product is k² and whose periods agree to 2.2e-16 seconds. Suspend the bar from those two knife edges in turn, adjust until the periods match, and the equivalent simple pendulum is the distance between them: 0.6083 m. Then g = 4π²(h+h′)/T² gives 9.8100 m/s², recovered from the drawing to 1.8e-16. The mass of the bar, the distribution of that mass, and the position of the centre of mass appear nowhere in the answer.

The length nobody has to measure

A pendulum's period depends on its length, so a pendulum will measure gravity — except that a real swinging bar has no single length, and finding where its mass effectively sits is far harder than timing it. Kater's answer was to build the instrument so that the awkward quantity cancels, leaving a distance between two knife edges that a rule can read to five figures.

mechanics · Pendulum What a collision has to spend, against what it is given. The energy available in a proton–proton collision, against the energy of one beam, on logarithmic axes. Against a stationary target the available energy is √(2mE) and the line has slope one half; head-on it is 2E and the slope is one. Bevatron at 6.2 GeV per beam reaches 3.7 GeV; SPS fixed target at 450 GeV per beam reaches 29.1 GeV; LEP at 104.5 GeV per beam reaches 209.0 GeV; Tevatron at 980 GeV per beam reaches 1960.0 GeV; LHC at 6500 GeV per beam reaches 13000.0 GeV. The gap is the whole architecture of the subject: the LHC's beams give 13000 GeV head-on and would give 110 GeV against a stationary proton, a factor of 118. Reaching the same 13000 GeV in fixed-target mode would need a beam of 90.1 million GeV. What the missing energy has gone into is not lost: it is the kinetic energy of the centre of mass, which every product has to carry away and which no experiment can use.

The collision that wastes most of the energy

The LHC's two beams carry 6,500 GeV each and 13,000 GeV are available. Fire one of those beams at a stationary block of copper instead and 110 GeV are available — the other 12,890 have gone into the motion of the wreckage and cannot be used for anything. The difference is a square root, and every accelerator built since 1970 is a consequence of it.

relativity · Relativistic dynamics 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 What a scale reads while a chain falls onto it. The reading of a scale, in units of the whole chain's weight, against the length of chain that has already landed, for two ways of putting the same chain down. Lowered gently, the scale reads the weight of what is resting on it and nothing else, so the reading climbs along the diagonal to one and stops. Dropped from rest with its lower end just touching, the scale reads three times that at every instant of the fall: one part is the pile's weight and two parts is the force needed to stop the links that are arriving, which is λv² with v² = 2gx and is therefore exactly twice λgx however far the fall has got. The peak, read off the drawn curve, is 3.00 chain weights. It is reached at the instant the last link lands, and the reading then falls discontinuously to one, because the momentum flux stops all at once. The discontinuity is the part a real experiment does not show — a real chain has links of a finite size and a scale has a response time — and it is the reason a chain dropped into a bucket on a kitchen scale reads high and then settles.

The pile that lands heavier than it weighs

Drop a chain onto a scale and the reading is three times the weight of the part that has landed — not approximately, exactly, all the way through the fall. The extra two parts are the force needed to stop links that are still arriving, and the same arithmetic run backwards says that picking a chain up wastes exactly half the energy it takes to get it moving.

mechanics · Momentum A loop at rest carrying 2.2e-12 kg m/s. A square loop of area 100 cm² carrying 20 amps, sitting still in a uniform electric field of 1.00 megavolts a metre. The carriers going up the field on one side cross 100 kilovolts on the way, so the ones in the top wire are less energetic than those in the bottom by that much per unit charge. The current is the same all the way round, so the same number of carriers pass per second in each wire — but they carry different energy, and momentum is energy times velocity over c². The two wires therefore contribute unequally, and the difference is 2.23e-12 kilogram metres a second, pointing across both the field and the dipole. Nothing in the picture is moving as a whole. The electromagnetic field round the loop carries exactly that momentum the other way.

The momentum of something that is not moving

A current loop sitting still in an electric field has momentum in the space around it. Nothing is moving, so something must be carrying an equal and opposite amount — and it is the loop, whose carriers on the high-potential side are more energetic than those on the low.

electromagnetism · Field energy The point that does not notice the collision. Two bodies of rest mass 1 and 2, approaching at 0.8c and -0.3c, colliding elastically and leaving at -0.6168c and 0.5969c — speeds obtained by reversing the motion in the zero-momentum frame, with the total energy and momentum checked to a part in a million million. The third line is the energy-weighted centre. It runs straight through the collision at 0.1872c, which is the total momentum divided by the total energy, and it has no kink — verified at two hundred instants. Nothing here is the centre of mass: the rest masses are unchanged by the collision but the energies are redistributed, and it is the energies that do the weighting.

The centre that is not a place

The centre of mass is replaced in relativity by the centre of energy, which moves uniformly and does everything the old point did — except be the same point for everybody. Boost a spinning body and its centre moves, so a spinning object has no centre at all.

relativity · Relativistic dynamics The one number the boost leaves alone. The mass of a lambda into a proton and a pion, reconstructed from the two products' laboratory energies and momenta alone, against how fast the parent was moving — for five different rest-frame emission angles. Every curve is the same horizontal line. The lab energies vary by more than a factor of ten across this range and the angles between the products vary from almost 180° to a few degrees; the combination E² − p² of the pair does not vary at all, to 3.2e-15, which the figure requires before drawing anything. The light curves are the energies of the two products, on the same axis and to a different scale, drawn to show how much is moving while the invariant does not. This is the whole method of particle physics. A parent that lives for 10⁻²³ seconds is never detected; what is detected is two tracks, and their invariant mass is computed and histogrammed over millions of events. A parent that exists shows up as a peak at its own mass, at the same place whatever the beam energy, which is what makes the peak believable.

The cone a decay cannot leave

A particle at rest breaks into two and they go opposite ways. Set the parent moving and the whole pattern folds forward — into a cone with a hard edge, beyond which nothing is emitted at any rest-frame angle at all. The energy spectrum that comes out is exactly rectangular, and the one number the boost leaves alone is how the parent is identified at all.

relativity · Relativistic dynamics 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 The centre of charge, and whether it moves. A pulsar of 1.46 and a white dwarf of 0.181 solar masses in a circular orbit, drawn twice about their fixed centre of mass (the cross). On the left each body carries a charge in exact proportion to its mass, as gravitational charge is under the equivalence principle: the centre of charge sits on the centre of mass and never moves, so the dipole moment is constant and nothing is radiated at dipole order. On the right the pulsar carries 60 per cent less charge per unit mass than its companion — an exaggerated mismatch, chosen so that the motion is visible at this scale — as a compact body's scalar charge would in a theory where its own binding energy couples differently: the centre of charge now circles the centre of mass at 12.6 per cent of the separation, once per orbit, and a dipole turning at the orbital frequency radiates.

The radiation a difference in falling would make

Gravitational waves start at the quadrupole because the equivalence principle removes the dipole: when every body's gravitational charge is its mass, the centre of charge is the centre of mass and cannot move. If a neutron star's own gravity fell even slightly differently, a binary containing one would radiate a dipole — louder than Einstein's quadrupole in slow orbits — and a pulsar with a white dwarf in an 8.5-hour orbit has been listening for it.

astrophysics · Equivalence principle The block has to move, and which way decides the momentum. A pulse of light crosses a free block of glass of index 1.5 and length L, with time in units of L/c. The pulse falls behind where it would have been in vacuum by 0.50 L while it is inside. The block's displacement, in units of EL/Mc², is obtained from momentum conservation alone: if the light inside carries Abraham's E/nc the block takes up the rest of E/c, moves forward, and ends 0.500 units along — Balazs's (n − 1). The sum of the pulse's lag and the block's shift, which is the centre of energy's departure from uniform motion, is then zero to within 0.0004 units at every moment. If instead the light carried Minkowski's nE/c, the block would have to recoil backwards and the centre of energy would end 1.25 units behind a straight line, which no isolated system can do. For a one-joule pulse through a kilogram of glass ten centimetres long the real shift is 5.6 × 10⁻¹⁹ m.

The momentum light carries into glass

A pulse of light in glass has two momenta, nE/c and E/nc, and for a century the argument was about which one is right. Both are measured. A free block that the light crosses must move forward, which only Abraham's value explains; a Cherenkov cone exists at all, which only Minkowski's allows. They answer different questions, and the difference between them is momentum carried by the glass.

electromagnetism · Field energy Where to strike so the pivot feels nothing. The impulse the pivot has to supply, as a fraction of the blow, against where along the body the blow lands, measured from the pivot, for a uniform rod pivoted at one end and for a 0.84 m wooden bat held 15 cm from its knob, with the blow's position as a fraction of the length beyond the grip. For the rod the reaction is 1 − 3q/2L, positive for blows nearer the pivot, negative beyond, and zero at 0.6667 of the length. For the bat it is zero 51.9 cm from the grip, 17.1 cm from the end of the barrel. A blow there sets the body turning about the pivot as though the pivot were not needed.

The point that takes the blow without a jolt

Hit a stick held at one end and the hand feels a jolt — forwards if the blow lands near the hand, backwards if it lands near the tip, and nothing at all if it lands two-thirds of the way along. That point is the centre of percussion, and it is exactly where a simple pendulum with the same period would end: two different questions about a swinging body with one answer.

mechanics · Pendulum 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 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 spring falls from the top down. Snapshots of the falling spring, every fifth coil, at 0.00, 0.05, 0.09, 0.14, 0.18, 0.23, 0.26, 0.29 seconds after release, for a spring that hangs 1 m; the dashed line is the height of the bottom coil before release. The coils collapse onto one another from the top, and the collapsed block falls on the coils below; everything below the block hangs exactly as it did before the spring was let go, the bottom coil included. Only when the block reaches the bottom, 0.261 s after release, does the bottom begin to fall, and then the whole collapsed spring falls together.

The spring whose bottom does not know it has been dropped

Hang a soft spring from its top, let go, and its bottom end stays exactly where it was — not falling slowly, not falling at all — while the top races down to meet it. A spring that hangs a metre takes about a quarter of a second to collapse, and only then does the bottom start to fall. Nothing holds it up. The centre of mass falls freely from the first instant, as it must, and the bottom simply has not been told: the news that the top has been released travels down the spring as a front of collapsing coils, and nothing below the front can feel anything until the front arrives.

mechanics · Momentum Eight blocks leaning past the edge of a table. 8 identical blocks of length 1 stacked on a table edge (the vertical line), each set as far out as balance allows: the dot on each interface marks the centre of mass of all the blocks above it, which sits exactly on the edge of the block beneath. The top block overhangs the second by 1/2, the second the third by 1/4, the third the fourth by 1/6, and the k-th by 1/(2k); together the stack reaches 1.359 block lengths beyond the table — the top block lies wholly beyond it, since 1.359 is more than one. Every interface is balanced on its edge, so the stack stands, but with no margin at all.

The stack that leans past its own table

Stack identical books on the edge of a table, each pushed out as far as it will go without falling, and four of them are enough for the top book to lie entirely beyond the table's edge. Keep going and the overhang grows without limit — the stack can reach as far out as anyone likes — but only as half the harmonic series, so that two book-lengths need 31 books, three need 227 and five need more than twelve thousand. The arithmetic is the same that settles whether a ladder leaning on a wall slips or a wardrobe topples, and the slow growth is the price of having nothing but the stack's own weight to hold it back.

mechanics · Free-body

Named alongside it

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

MomentumMomentum conservationConservationConservation lawsEnergy conservationImpulseThe Lorentz transformationMoment of inertiaReference framesTorqueAngular momentumDissipation

All concepts