One puck, two frames
At its defaults it draws 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.
rotating-frame is one function in lib/figures/mechanics.js —
motion, force, energy and rotation. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
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.
Every path a rolling ball can take on a turntable, seen from the room
The options are the ones The ball a turntable will not throw off passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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 same four balls, seen from the turntable
The options are the ones The ball a turntable will not throw off passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The same four paths drawn in the turntable's own frame, over the same 6.3 seconds. Here the ball set down at rest in the room circles backwards round the axis once a turn; the others trace loops — circles at 2/7 of the table's rate, seen from a frame turning at the full rate, which is the difference of the two, and look nothing like the simple orbits a person standing beside the table sees.
On a tilted turntable a ball goes across the slope, not down it
The options are the ones The ball a turntable will not throw off passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The path of a solid ball released from rest, seen from the room, on a turntable tilted by 0.2° and turning at 33.3 rpm, drawn to the same scale in both directions; downhill is down the page. On a table that does not turn the ball would roll straight down the slope, 10 cm in 2.9 s (dashed). On the turning table it rolls a little way down, is turned across the slope, and drifts sideways in a chain of loops at an average 2.5 cm/s — 5g sinα/2Ω — never getting more than 4.9 cm downhill in 16 s. It is the drift of a charged particle in crossed electric and magnetic fields, with gravity for the electric field and the table's rotation for the magnetic one, and it runs perpendicular to the push for the same reason.
The rate of the circle is set by where the ball keeps its mass
The options are the ones The ball a turntable will not throw off passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The rate at which a ball rolling on a turntable circles, as seen from the room, as a fraction of the table's own rate, against the ball's moment of inertia divided by ma², κ: Ω/(1 + 1/κ). A solid ball, κ = 2/5, circles at 2/7 of the table's rate, 0.286; a thin hollow ball, κ = 2/3, at 2/5; a ball with its mass concentrated towards the centre more slowly, and a ball with no rotational inertia at all would not turn — a frictionless puck goes straight. No rolling ball can reach the table's own rate: that would need infinite rotational inertia. The rate measures how much of the push from friction goes into turning the ball rather than moving it.
How much grip the circling needs
The options are the ones The ball a turntable will not throw off passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The friction force a solid ball needs to keep rolling on a turntable at 33.3 rpm, as a fraction of its weight, against its speed seen from the room: (2Ω/7)v/g, since the only horizontal force on the ball is friction and it is what turns the ball's velocity. With a coefficient of friction of 0.3 the ball rolls cleanly up to 3.0 m/s, and with 0.1 up to 0.98 m/s; faster than that it skids, and the circles are replaced by a sliding path that heats the contact. For any push a hand gives a marble the circling needs only a small fraction of its weight, which is why the effect is so easily seen.
What checks it
physicscheck asserts something about rotating-frame that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
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.
MechanicsThe 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.
MechanicsThe 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.
MechanicsThe governor that needed friction to work
James Watt's steam engines were kept at a steady speed by two heavy balls on hinged arms, spun by the engine itself: run faster and the balls fly out and close the steam valve; run slower and they drop and open it. The balls stand at a height set by the speed alone. As engines grew and governors were made more sensitive, many began to hunt — speeding and slowing in a growing rhythm, the governor always too late. In 1868 James Clerk Maxwell wrote the equations down and showed why: a governor with no friction cannot settle at all, and the more precisely it is meant to hold the speed, the more friction it needs.
MechanicsThe 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.
MechanicsThe 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.
MechanicsThe 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.
FluidsThe 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.
MechanicsThe 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.
MechanicsTurning 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.