Illustrated Physics
physics, drawn
Concept

Rotating frame — where it appears

A set of coordinates turning with respect to an inertial one, in which Newton's law acquires exactly three extra terms. They are the centrifugal, Coriolis and Euler terms, and each is an artefact of the coordinates rather than an interaction — no third body exerts any of them.

Named by 21 essays across 5 fields — each of them below, with the objects they name alongside it.

Velocity and acceleration in uniform circular motion. Velocity drawn tangent to a circular path and acceleration drawn toward its centre, at eight points around the circle. The speed never changes and the acceleration is never zero; the two vectors are perpendicular everywhere.

Turning is an acceleration, and constant speed does not help

An object going round a circle at unchanging speed is accelerating hard, all the time, toward a point it never reaches. The construction that shows this needs two arrows and no calculus.

mechanics · Circular motion One puck, two frames. A puck slides outward from the centre of a turntable at 1.4 m/s across a table of radius 1 m, which turns through 0.55 of a revolution in the 0.71 seconds the crossing takes. On the left, in the room: a straight line, because no force acts along it. On the right, on the turntable: a spiral, because the coordinates turned underneath it. The two panels are the same numbers with the same clock — the second is the first with the angle rotated by −Ωt — so whatever is bending the path on the right is arithmetic, and it is still worth a name, because on a turning planet the right-hand panel is the one everybody lives in.

The forces that are not there

Writing Newton's law in a frame that is turning produces three extra terms. Nothing was added to the world to make them appear and nothing is removed by calling them fictitious — one of them flattens the planet, one of them turns the weather, and both are computable to four figures.

mechanics · Circular motion 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 The surface a spin decides. The free surface of a liquid in a dish of radius 0.5 m turning at 10, 20, 40 revolutions a minute. In the rotating frame the surface is a level set of gz − ½ω²r², so it is a paraboloid exactly and not to some approximation, and nothing about the liquid appears in its shape: the same curve is got with mercury, water or oil. The rim stands 14.0 mm, 55.9 mm, 223.6 mm above the centre at those rates. A parabola z = r²/4f has focal length f, so these surfaces are mirrors of focal length g/2ω² — 4.47 m at 10 rpm, 1.12 m at 20 rpm, 0.28 m at 40 rpm. That is checked here on the drawn curves rather than quoted: a vertical ray reflected off the surface at a quarter, a half, three quarters and the whole of the radius crosses the axis at the same height to 0.00%, which is what a mirror with no spherical aberration means. Doubling the spin quarters the focal length, and there is no other adjustment: the dish can only ever look straight up.

The surface a spin decides

Spin a dish of liquid and its surface settles into a paraboloid — exactly, with nothing about the liquid in the shape. A parabola of that form has a focal length of g over twice the spin rate squared, so a bucket of mercury turning at twenty revolutions a minute is a telescope mirror figured by a clock instead of by grinding.

fluids · Hydrostatics One expression over 3 decades of area times rate. Sagnac time difference against the product of enclosed area and rotation rate, both logarithmic. The relation Δt = 4AΩ/c² is linear in that product — the fitted slope of the drawn points is 1.0000 — and it contains no refractive index, no shape of the loop and no position of the axis inside it. a 1 km fibre gyroscope on a 10 cm coil: 1.62e-19 s, 3.14e-5 fringes; Sagnac's own ring, 1913: 4.84e-17 s, 0.0666 fringes against a reported 0.07; a laboratory turntable at one revolution a second: 6.99e-17 s, 0.0331 fringes; Michelson and Gale, 1925: 4.50e-16 s, 0.2364 fringes against a reported 0.23. Michelson and Gale's rectangle in Illinois is the one that carries the check: 0.236 fringes predicted from its own dimensions and the vertical component of the Earth's rotation at its latitude, and 0.230 reported.

The ring where the two beams disagree

Send light both ways round a closed loop on a turntable and the two beams come back at different times, by 4AΩ/c² — an expression with no refractive index in it, no shape of the loop and no position of the axis. The same number is what a set of clocks round the rim fails to close by, which is the sharper statement — on a rotating platform there is no global simultaneity to be had.

relativity · Simultaneity A circumference that is more than 2π times the radius. The ratio of a rotating disc's measured circumference to 2π times its measured radius, against the speed of the rim, together with the rate of a clock carried on the rim. Rulers laid round the rim lie along their own direction of motion and are contracted; rulers laid along a radius lie across it and are not. So the circumference takes more of them than a stationary observer counts and the radius takes the same number, and the ratio is γ: 1.091 at β = 0.4, 1.400 at β = 0.7, 2.294 at β = 0.9. The geometry a rotating observer measures is therefore not Euclidean, and it is not Euclidean by an amount that depends on where on the disc the measurement is made. That is the observation Einstein said set him on the road to describing gravity with curved geometry: here is an accelerated frame, and here is a geometry in it that no choice of Cartesian coordinates can flatten. The rim's clock runs slow by the same factor, so a rotating frame has neither a common time nor a flat space.

The disc that cannot be spun

Set a disc turning and measure its circumference with rulers carried on the rim. They lie along their own direction of motion and are contracted, so more of them fit; rulers along a radius lie across the motion and are not. The ratio of circumference to radius is therefore not two pi, in a frame where nothing is happening but rotation — and Einstein said that was what set him looking for gravity in geometry.

relativity · Length contraction The deflection is a circle, not a bend. Trajectories integrated from Newton's law in a rotating frame with the Coriolis term and nothing else — no pressure gradient, no friction, no force of any kind. 0.5 m/s at 45°, 0.3 m/s at 30°. Each path closes on itself after one inertial period, checked to a millionth of its own radius, and the radius is the speed divided by the Coriolis parameter: 4.85 km, 4.11 km. The deflection usually described as a curving of the path is a complete circle, traversed clockwise in the northern hemisphere in half a pendulum day, and a body left alone in a rotating frame goes nowhere at all.

The deflection that closes on itself

The Coriolis term is usually described as bending a path to the right. Integrated rather than described, it does not bend the path — it closes it. A body left alone in a rotating frame travels a circle of radius U/f and comes back to where it started in half a pendulum day, having gone nowhere at all, and drifting buoys in every ocean draw exactly that.

mechanics · Circular motion Pushed one way, travelling another. A parcel released from rest under a steady pressure-gradient acceleration of 2.0e-4 metres per second squared pointing east, integrated for 36 hours at 60°, 30°, 10°. Without rotation it would accelerate east indefinitely. With it, the motion is an inertial circle about a mean velocity at right angles to the push, and the mean over a whole number of inertial periods is measured off each path and matches the geostrophic value a/f to two per cent. At high latitude the loops are tight and the drift is almost purely along the isobars; near the equator the parcel travels a long way down the gradient before the rotation has had time to turn it, which is why geostrophic balance is a high-latitude statement and the tropics need a different set of approximations.

The ratio that decides whether the planet is turning

Whether the rotating terms matter is not a question about size. It is one dimensionless ratio, U over fL, and it runs from ten thousand in a teacup to a millionth in the Earth's core. Where it is small the pressure gradient stops accelerating the fluid and starts balancing a force on fluid already moving across it — so the flow runs along the pressure contours instead of down them, and a weather map is a streamline plot.

mechanics · Circular motion How large now is, on this planet. By how much a synchronisation carried around a region of the rotating Earth fails to come back to itself, against the size of that region — from a metre to the whole planet, both axes logarithmic. The three horizontal lines are what three kinds of clock can resolve, and where each crosses the curve is where that clock can detect that 'now' is not a global notion: a good wristwatch at 785 thousand km, a quartz oscillator at 25 thousand km, a caesium clock at 785 km. Carried the whole way round the equator the defect is 207 nanoseconds, which is sixty metres of light travel and is the correction every satellite-navigation system applies. The effect is not small and not exotic; it is a routine engineering term, and the reason it was not an engineering term before 1955 is that nothing could measure it. A wristwatch's now is global out past the Moon; a caesium clock's reaches about the width of a large country.

How big now is

Three earlier arguments have established that a global now is a choice, that part of the choice is convention, and that for a rotating observer no consistent global choice exists at all. What survives is a size. Every observer has a local now, and how local is computable: on the rotating Earth it is 785 kilometres to the nanosecond, and a freely falling frame is inertial over the tolerance times the distance to the centre, divided by two.

relativity · Simultaneity A closed cabin accelerating at 3 m/s², and everything in it leaning the same way. A closed box accelerating to the right at 3 m/s², with water in the bottom and air above. Everything that can hang or float lines up with the effective gravity g − a, tilted 17.0° from the vertical towards the back: the water's surface and the isobars below it are perpendicular to it, a floating block stands along it, a plumb bob hangs along it — and a helium balloon on a string leans the other way along the same line, forward, into the acceleration. The balloon is not doing anything unusual. Buoyancy always points opposite to gravity, and in the cabin the effective gravity has a horizontal part, so the air's pressure gradient does too, and it pushes the balloon along it.

The balloon that leans the wrong way

When a car pulls away, everything loose in it swings back — except a helium balloon on a string, which swings forward. Nothing strange is acting on it. Buoyancy points against gravity, and inside an accelerating cabin gravity has a sideways part. Read that way, a lift cannot change how deep a boat floats, free fall abolishes floating altogether, and a centrifuge is Archimedes' principle with the word "up" pointing at the axis.

fluids · Buoyancy A pendulum swung 30° with a sideways push, seen from above. The path of the bob of a 1 m pendulum free to swing in any direction, seen from directly above, released 30° from the vertical with a sideways push that would, for a small swing, make an ellipse 0.3 as wide as it is long. The path is an ellipse that turns: its long axis advances 5.30° every half swing, in the same sense as the bob goes round, where Airy's small-amplitude rate (3/8) ωab/L² gives 5.06°. Nothing outside the pendulum turns it — the pivot is fixed and there is no Earth rotation in this calculation. A pendulum on a turning Earth that is meant to show the Earth's rotation has this turning to contend with too.

The ellipse a pendulum turns by itself

Let a pendulum swing in any direction and give it a small sideways push, and its bob traces an ellipse. For a small swing the ellipse closes. For a larger one it turns, steadily, in the sense the bob is going round — with nothing outside the pendulum turning it. Airy worked out the rate in 1851, the same year Foucault hung his pendulum in the Panthéon, and the two effects are the same size for a pendulum a metre long whose ellipse is a twentieth of a millimetre wide.

mechanics · Pendulum Why the meander has to move west. A line of fluid columns at 45° pushed into a meander 3,000 km long and 300 km in amplitude, seen from above with north up and east to the right. Columns pushed north find the planet spinning faster beneath them and turn clockwise; columns pushed south turn anticlockwise. Each turning column drives the fluid north on one side of it and south on the other, and the arrows are that flow, from integrating the columns' spin along the line: it peaks at 2.32 m/s, a quarter of a wavelength west of each crest, so the columns just west of a crest are carried north and the crest itself is rebuilt further west. The dashed meander is the same pattern 2 days later, moved 638 km west at β/k² = 3.69 m/s. Nothing in the argument depends on which way the meander was first pushed.

The wave that can only travel west

A column of air or water on a spinning planet keeps one quantity as it moves: its total spin divided by its depth. Carry it north, where the planet's own spin about the local vertical is larger, and it must turn the other way to compensate. That single bookkeeping rule makes a wave that exists nowhere else — one whose crests can only ever move west — and the same rule holds the jet stream's meanders still over the mountains, makes the ocean take decades to notice a change in the wind, and pins the Gulf Stream to the western side of the Atlantic.

mechanics · Circular motion Tadpoles and a horseshoe, drawn in the turning frame. Orbits of small bodies near the circle of a planet's orbit, integrated in the frame turning with it, with the distance from that circle exaggerated 12 times so the shapes can be seen. With Jupiter's mass ratio, 0.000954, a body started 10° and 25° ahead of L4 at rest in the turning frame circles round L4 rather than falling off the hill: it swings between 51° and 70°, and between 42° and 85° from the planet — a tadpole. With a mass ratio of 10⁻⁴, a body started opposite the planet sweeps round through L4, L3 and L5 and back, never coming within 24° of the planet — a horseshoe, the kind of orbit Saturn's moons Janus and Epimetheus follow relative to each other, exchanging places every four years. In every case the Jacobi constant, the one quantity conserved in the turning frame, is held by the integration to better than a part in a million.

The hilltop that holds the Trojans

Sixty degrees ahead of Jupiter and sixty behind it, thousands of asteroids share the planet's orbit, held at the corners of the equilateral triangles that Jupiter and the Sun make. Seen from the frame that turns with Jupiter, those corners are the tops of two hills in the effective potential — the worst places to try to keep anything still. They hold asteroids anyway, because in a turning frame a body sliding off a hill is swung sideways by the Coriolis force into an orbit round the top. The trick works only if the planet is light enough: above a mass ratio of about one in twenty-six, the hilltops let go.

astrophysics · Orbit stability A gyroscope swinging into the meridian. The direction of a gyrocompass's axis, in degrees east of true north, against time in hours, released 20° east of north at latitude 50°. The Earth's rotation tilts the horizon under the spinning axis, a pendulous weight turns the tilt into a torque, and the torque precesses the axis towards the meridian. Undamped (dashed) it swings through north and back with a period of 84.4 minutes, the compass having been built to that period. With damping it settles: after 8 hours it points within 0.03° of north. Nothing in the instrument knows where north is. It finds the direction about which the Earth turns, because only when its axis lies in the meridian does the turning horizon stop tilting it.

The compass that finds the axis the Earth turns about

A spinning wheel held so that its axis stays level will, on a turning Earth, swing its axis round to point at true north and stay there — not magnetic north, the geographic pole. Nothing in the instrument knows where north is; it finds the one direction about which the Earth's turning stops tilting it. Built to be used at sea, such a compass has to ignore the ship's lurches, and that forces its swing to have one particular period: 84.4 minutes, the period of a pendulum as long as the Earth's radius, of a satellite skimming the ground, and of a stone dropped through a tunnel to the far side of the planet.

mechanics · Rotation The band in which radiation feeds a mode instead of draining it. The speed of a mode's pattern as seen from far away, in units of the star's own spin rate, against the spin rate as a fraction of the fastest a star can turn. The shaded band, between zero and one, holds patterns that run forward across the sky and backward through the star; gravitational radiation drives any mode inside it and damps any mode outside. The r-modes with ℓ = m = 2, 3 and 4 sit at 0.667, 0.833 and 0.900 at every spin, inside the band however slowly the star turns. The dashed curves are schematic modes whose crests run backward on the star at a fixed 0.2, 0.5 and 0.9 of the Kepler rate: each is dragged into the band only once the star turns faster than its crest runs, which is why an ordinary oscillation needs a star near break-up and an r-mode does not.

The wobble that radiating makes larger

Radiation normally drains a vibration: a struck bell goes quiet because it sends its energy away as sound, and a ringing black hole settles in milliseconds because it sends its energy away as gravitational waves. A spinning star can have the opposite. A pattern that runs backward through the star's material, while the star carries it forward across the sky, has less angular momentum than the star would have without it, and every wave it radiates takes away angular momentum it did not have to give. The pattern grows. One family of such patterns, the r-modes, runs backward on every rotating star, so every spinning neutron star is unstable unless friction wins — and that may be why none has been found spinning faster than 716 turns a second.

astrophysics · Gravitational waves Rotation is a magnetic field, and a superconductor cancels it. Magnetic fields along the spin axis inside a spinning metal, in units of 2mω/e, as seen by its electrons in the frame turning with it. In that frame the Coriolis force on an electron is exactly the force a magnetic field of 2mω/e would exert, pointing against the spin for a negative charge (Larmor's theorem). A normal metal spun in no field has no real field inside, and its electrons feel the rotation's equivalent field in full. A superconductor will not let its carriers feel any field in their interior — that is the Meissner effect — so it generates a real field of exactly the same size pointing the other way, and the sum vanishes. The real field is the London moment.

The field a spinning superconductor makes of itself

Spin a lump of superconductor with no magnet anywhere near it, and a magnetic field appears inside it, pointing along the axis of spin: about eleven trillionths of a tesla for every radian per second, the same in lead, niobium or a copper-oxide ceramic. The field is not left over from anything. In a turning frame the Coriolis force on a moving charge is indistinguishable from a magnetic force, a superconductor will not let its carriers feel any field inside it, and so it makes a real field to cancel the one rotation imitates. The size depends on nothing but the carriers' mass and charge, which makes the field a balance for weighing them — and the one time it was read to a part in a hundred thousand, it gave an answer theory has not matched.

electromagnetism · Superconductivity Every path a rolling ball can take on a turntable, seen from the room. The paths, seen from the room, of four solid balls rolling without slipping on a turntable of radius 15 cm turning at 33.3 revolutions a minute (grey disc), over 6.3 seconds. A ball set down at rest relative to the room stays exactly where it is, spinning on the spot while the table turns under it. Every other ball moves on a circle, turning the same way as the table at 2/7 of its rate — once every 3.5 turns of the table — and comes back to where it started, whatever the push. A ball set down moving with the table does not ride with it: it circles on a path three and a half times the radius it started at, and the circle's centre is on the other side of the axis.

The ball a turntable will not throw off

Set a marble down on a spinning turntable and the expectation is that it will be flung off the edge. It is not. Placed at rest relative to the room, it stays exactly where it is, spinning on the spot while the table turns beneath it. Given a push, it moves in a circle, turning the same way as the table at two-sevenths of the table's rate, and comes back to where it started. Tilt the turntable and the marble does not roll downhill: it wanders across the slope in a chain of loops. A rolling ball on a rotating surface obeys exactly the equation of an electric charge in a magnetic field, and the two-sevenths is how much of every push goes into turning the ball rather than moving it.

mechanics · Circular motion The drift is the area under the height. Height against time for the two balls: the throw (dashed curve) rising to 100 m in 4.52 s and falling back in as long again, and the drop (solid) falling from 100 m. For vertical motion the east–west velocity is always −2Ω cos λ times the height above the starting point, so the east–west drift is −2Ω cos λ times the area between the path and the starting height. The throw's area (lower shading) is (4/3)·h·t = 602 m s, above its start, so it drifts west; the drop's (upper shading) is h·t/3 = 151 m s, below its start, so it drifts east. The ratio of the two areas is four.

The ball thrown up that lands to the west

Drop a ball from a tower a hundred metres high and, because the Earth turns, it lands a centimetre and a half east of the point directly below. Throw a ball straight up so that it just reaches the same height, and it lands six centimetres west of where it was thrown — four times as far, the other way. The asymmetry has a one-line explanation that turns a problem in rotating frames into a problem about areas: for anything moving straight up or down, the sideways drift is proportional to the area under its graph of height against time. A dropped ball spends its flight below where it started, a thrown one above, and the thrown one's area is four times as large.

mechanics · Projectile A gas pressed against the wall of a spinning tube. The density of uranium hexafluoride gas in a rotor spinning at a steady rate, relative to its density on the axis, on a logarithmic scale, against the distance from the axis as a fraction of the radius, for wall speeds of 300, 450 and 600 m/s at 300 K. In the turning frame the gas sits in the centrifugal potential −½mω²r² and settles into the same exponential that thins the atmosphere with height: n ∝ exp(mω²r²/2kT). At 600 m/s the density at the wall is e^25.4 — about 10¹¹ — times that on the axis; at 300 m/s, e^6.4. Almost all the gas lies in a thin layer against the wall, and the middle of the rotor is close to vacuum.

The spin that sorts by a difference of mass

In a spinning tube a gas settles against the wall the way the atmosphere settles towards the ground, with an exponential whose scale is set by the rotor's speed rather than by gravity. Heavier molecules settle more steeply, and the ratio of two isotopes changes from axis to wall by a factor that depends on the difference of their masses, not on the ratio. That one change from a ratio to a difference is why a spinning rotor separates uranium-235 from uranium-238 more than fifty times as strongly per stage as the porous barriers that first did it — and why the history of the machine is a history of how fast a material can turn before it flies apart.

mechanics · Circular motion A ring turns, and its vibration turns less. Four snapshots of a thin ring vibrating in its lowest flexural mode, squashing alternately into two ellipses, while the whole ring is turned steadily about its axis; the deformation is exaggerated. The dot marks one point of the ring; the dashed line, the axis of the vibration's greatest swing. As the ring turns through 30°, 60° and 90°, the pattern of vibration turns through 18°, 36° and 54° — 60 per cent of the ring's angle. Measured from the ring, the pattern has slipped back 36° by the quarter turn. The vibration is neither carried along with the material nor left fixed in space. It follows the turning by a fixed fraction that depends only on the mode's shape, which is the effect G. H. Bryan described in 1890 from the note of a turning wine glass.

The vibration that turns part of the way

Strike a wine glass and turn it on the table while it rings, and the note beats: the pattern of the vibration does not turn with the glass. Nor does it stay still in the room. For a ring vibrating in its lowest mode it follows exactly three-fifths of the rotation and slips back the other two-fifths, a fraction that depends only on the shape of the vibration. That fraction, worked out by George Bryan in 1890, is how some of the most precise gyroscopes ever built measure an angle directly, with nothing spinning in them at all.

mechanics · Rotation Three throws in a spinning room. Balls thrown at 5 m/s from the floor of a station 10 m in radius, spun at 0.99 rad/s (9.5 turns a minute) so that its floor presses with one g, as seen by someone standing in it: at 45° spinward, at 45° antispinward and straight up; with the Earth's parabolas for the two 45° throws dashed. Distances along the floor and heights in metres. On the Earth both 45° throws land 2.55 m away. In the station the spinward throw lands 1.52 m away and the antispinward one 3.06 m away, and the ball thrown straight up comes down 1.30 m ahead of the thrower, spinward. Nothing acts on any of the balls in flight. Each flies straight across the station while the floor turns beneath it, and what the thrower sees as gravity bending the path is the room rotating.

The throw a spinning station bends

A space station spun fast enough presses its floor up against its crew with the force of the Earth's gravity, and from the inside it can feel like home. Throw a ball, and the illusion fails. Nothing acts on the ball once it leaves the hand: it flies in a perfectly straight line across the station while the floor turns beneath it, and what the thrower sees as a parabola is a chord of a circle seen from a turning room. In a station ten metres across, a ball thrown up a metre comes down a metre ahead of the hand, a ball thrown with the spin falls short and one thrown against it carries long, and a dropped spanner lands a metre behind the point beneath it. How big a station must be to feel like the Earth is a question projectiles answer.

mechanics · Projectile

Named alongside it

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

Coriolis forceReference frameAngular momentumCentrifugal forceFictitious forceFoucault pendulumLatitudePrecessionCoriolis effectEquivalence principleFree fallGyroscope

All concepts