Illustrated Physics
physics, drawn
Concept

Torque — where it appears

The turning effect of a force, being the force times its lever arm about a chosen axis, which changes angular momentum as a force changes momentum. Only the component of the force perpendicular to the arm counts, so a force through the axis has no effect however large.

Named by 17 essays across 4 fields — each of them below, with the objects they name alongside it.

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 Equipotentials, with the field lines that cross them. Contours of constant potential, traced by marching squares, with field lines traced along the gradient of the same potential. The two families meet at right angles everywhere, which is a consequence of the field being the gradient rather than a property of the drawing.

The attraction that needs no charge

Gauss's law says nothing comes out of a neutral molecule, and yet water's field one nanometre away reaches 1.1 × 10⁸ V/m. What survives when the monopole vanishes is a separation, and every step down the tower of falloffs below it is paid for with one more order of cancellation.

electromagnetism · The field concept Every path the body can take, at one angular momentum. The angular momentum vector, drawn in the body's own frame on the sphere its length confines it to, for a body whose principal moments are 3.068e-3, 6.817e-3 and 9.750e-3 kg m². Each closed curve is one motion, traced by integrating Euler's equations rather than by solving for the intersection of the sphere with the energy ellipsoid, so a curve closes only if the physics closes it. The low-energy curves circle the greatest-moment axis and the high-energy ones circle the least; both sets are small loops that stay near their axis, which is what stability looks like. Between them is the one curve that is not a loop at all — four arcs, drawn heavier, meeting at the intermediate axis and leaving it again. A body spun about that axis is balanced on the crossing point of paths that go somewhere else, which is the whole of why it does not stay.

The axis that will not hold

A book spun about its long edge keeps spinning about it. Spun about the axis through its covers, it keeps spinning about that. Spun about the third axis, it flips end over end, again and again, with nothing touching it. Three numbers decide, and what matters is only their order.

mechanics · Rotation Which failure comes first. The dividing line between a block that slides and one that tips over, for a horizontal push applied at the top. Sliding needs a force of μ_s times the weight; tipping needs the push's moment about the leading bottom edge to beat the weight's, which is the weight times half the width. The weight appears in both and cancels, so the boundary is the curve aspect ratio = 1/2μ_s and nothing else: not the mass, not how hard the block is pushed, and not what it is made of except through μ. At μ_s = 0.5 the dividing shape is as tall as it is wide; at μ_s = 0.2 it is 2.5 times as tall. Of the 5 objects marked, 3 sit above the line and go over rather than sliding: a paperback, standing, a full filing cabinet, a pint glass.

Slide or topple

Push a wardrobe and it goes over; push a brick and it skids. Both are held by the same friction and both are pushed by the same hand, and which of the two failures arrives first has nothing to do with how hard the push is. The floor decides it, by shifting where it pushes back.

mechanics · Free-body 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 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 A 120 g top at 3000 rpm, precessing once every 1.92 s. A disc of radius 30 mm spinning at 3000 revolutions a minute on a shaft 45 mm long, tilted 30° from the vertical. The weight acts at the centre of mass and the pivot holds the bottom, so the torque about the pivot is horizontal and at right angles to the plane containing the axis and the vertical. Angular momentum points along the axis; a torque at right angles to a vector turns it without changing its length, so the axis sweeps round the dashed circle instead of falling. The precession rate is Mgl divided by I₃ω₃ to leading order, which is 3.269 radians a second here, or one turn every 1.92 seconds — slower the faster it spins.

The push that comes out sideways

Push down on a spinning wheel's axle and it swings horizontally. Nothing about that is mysterious once angular momentum is a vector — but the steady precession every demonstration shows is a solution nobody's initial conditions select, and a top released from rest does something else first.

mechanics · Rotation Angular momentum that was in nothing at all. A ring carrying 10⁻⁶ C on a freely pivoted disc, with a solenoid on the axis threading 0.002 Wb through it, switched off over 1 second. Nothing is turning at the start and nothing has been touched. The collapsing flux drives a circumferential electric field round the ring, the ring is torqued, and the disc ends up spinning with 3.183·10⁻¹⁰ kg m²/s of angular momentum. Where was it? The two curves are the field's share, ε₀∫r × (E × B), and the matter's share integrated from the torque — computed by different routes and summing to a constant to 4.2e-15 of the total throughout. So the angular momentum was there before the switch was thrown, in a static electric field crossed with a static magnetic one, in a room where nothing whatever was moving. It is qΦ/2π, it does not depend on the radius of the ring or the shape of the solenoid, and it is the plainest demonstration available that the field is not a bookkeeping device for forces between distant charges.

The angular momentum that is in nothing at all

A charged ring and a solenoid, both at rest, with nothing moving anywhere. Switch the solenoid off and the ring starts to turn. Angular momentum is conserved, so it was there before the switch was thrown — and it was not in the matter, because nothing was moving. It was in the field, and it is qΦ over 2π whatever the geometry.

electromagnetism · Field energy The same top, let go four ways. The path traced by the top of the axis, seen from directly above, over 1.2 precession periods. The dashed circle is the tilt the top was released at and the outer circle is 46.8° from the vertical. Released from rest the axis falls, and the fall is what generates the sideways motion: the path comes to a cusp each time it returns to the starting tilt, because at that instant the precession rate is momentarily zero. Launched at exactly the steady rate the path is a circle and the nutation is absent. Launched slower it waves; launched faster it loops: at 0× the steady rate the path comes to cusps, at 0.45× the steady rate the path waves, at 1× the steady rate the path stays a circle, at 1.9× the steady rate the path waves. Every one of these is the same equation with the same top and the same spin.

The top that nods before it settles

A spinning top let go from rest does not begin to precess. It falls, catches itself, and comes back up, over and over, at a frequency that has nothing to do with gravity — and the steady precession every textbook draws is what is left after friction has removed the nod.

mechanics · Rotation 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 The pressure under a wheel rolling on a soft surface. The pressure under a rigid wheel of radius 30 cm carrying 3 kN per metre of width as it rolls over a rubber-like bed, against position along the contact (ahead of the axle to the right), scaled to the largest pressure and the widest contact, at three speeds. The bed relaxes with times of 2 and 20 ms. At 1 mm/s the pressure is symmetric and its resultant passes through the axle. At 1.0 m/s the material ahead is being squeezed faster than it can flow and pushes back harder than the recovering material behind, which lags and lets go early: the resultant sits 6.39 mm ahead of the axle. At 1000 m/s the bed has no time to relax at all and behaves as a stiffer elastic solid, symmetric again. A wheel is held back by where its support pushes, not by any sliding.

The push a millimetre ahead of the axle

A wheel rolling freely on a flat road slows down, and nothing in it slides. What holds it back is where the road pushes on it: not directly under the axle but a little ahead, because the material in front of the contact is being squeezed and pushes back harder than the material behind, which is recovering and lags. A car tyre's push is about three millimetres ahead of its axle, a train wheel's about two-thirds of a millimetre, and that offset, divided by the radius, is the whole of rolling resistance. It peaks at the speed where the time under the wheel matches the time the material takes to relax, and the same loss, at a different frequency, is what gives a tyre its grip.

mechanics · Friction One lever, two frames, two different torques. A right-angled lever pivoted at its corner, with arms of length L, held still by a force F at the end of each arm, perpendicular to it. Left, in the lever's own frame: torque FL anticlockwise from the horizontal arm's force, FL clockwise from the vertical arm's, and they cancel. Right, the same lever seen moving to the right at 0.8 of the speed of light: the horizontal arm, along the motion, is contracted to L/γ = 0.60L, and the force on it, which is across the motion, is reduced to F/γ = 0.60F; the vertical arm and its force, along the motion, are unchanged. The torques are now 0.36FL and FL, and the lever is twisted clockwise by 0.64FL — the square of its speed as a fraction of light's. It does not turn.

The lever that is twisted and does not turn

A right-angled lever held still by two equal forces has no net torque on it. Watch the same lever go by at high speed and one arm is shorter, one force is weaker, and the torques no longer cancel — yet the lever goes by without turning. Nothing is wrong with either frame's arithmetic. The forces are doing work on the moving lever, the energy they put in is flowing through its arms, and a flow of energy carries momentum that points across the lever's motion: a body that moves straight with a momentum that does not.

relativity · Mass-energy The level at which a hinged gate swings open. The height of the centre of pressure on a vertical gate, as a fraction of its height H, against the depth of water above the gate's bottom (solid), approaching the centroid at half the height (dotted) as the water deepens; the dashed lines are hinges at 0.3H, 0.4H, 0.45H. A gate hinged on a horizontal axis and resting against a sill on the downstream side is held shut while the centre of pressure is below its hinge and swung open the moment it passes above. A hinge at 0.3H opens when the water is 0.90H deep, before it reaches the gate's top; at 0.4H, at 1.33H; at 0.45H, at 2.17H. A hinge at or above the centroid never opens at any depth, and the opening level grows steeply as the hinge approaches it — the setting is a choice of hinge position, made once, with no mechanism, sensor or power.

The floodgate that opens at a chosen depth

Hang a flat gate on a horizontal hinge across a channel and it can be made to stay shut while the water rises, press harder against its seat for a while, and then swing open on its own at one particular level — with no float, no motor and no sensor, only the position of the hinge. The trick is that the water's push on a submerged surface acts below the surface's middle by an amount that shrinks as the water deepens, so the point the push acts at climbs as the flood rises. Put the hinge on that climb and the gate opens where the climb crosses it.

fluids · Hydrostatics Where the thread's line meets the floor. A spool resting on the floor on its rims of radius R, its thread wound round a hub of radius r = 0.5R and leaving the hub's underside tangentially, pulled up and to the right at 20°, 60° and 78° above the horizontal; the dashed lines continue each thread's line back down to the floor. The spool turns about the contact point, and the sense of the turn is set by where the thread's line meets the floor. At 20° it meets the floor behind the contact point, on the far side from the hand, and the spool rolls towards the hand. At 60°, where cos θ = r/R, it passes exactly through the contact point; the pull has no turning effect there, and the spool stays still while the hand pulls. At 78° it meets the floor in front of the contact point, between the spool and the hand, and the spool rolls away from the hand, paying the thread out.

The spool that rolls towards the hand pulling it

Pull gently on the thread of a cotton reel lying on its side on a table, with the thread coming off the underside of the hub, and the reel rolls towards the hand, winding the thread up as it comes. Raise the thread and at one particular angle the reel will not move at all however hard it is pulled; raise it further and the reel rolls away, paying the thread out. Which way it goes is decided by a single line — where the thread's direction, extended, meets the table — and the same line decides which way a bicycle goes when its lower pedal is pulled backwards.

mechanics · Free-body 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.

Angular momentumMoment of inertiaStabilityCentre of massConservation lawsEquilibriumFree-body diagramMoment armWorkCentre of pressureConservationDissipation

All concepts