Which axis is unstable, and by how much
At its defaults it draws which axis is unstable, and by how much. The three linear exponents for a torque-free rigid body, against the value of its intermediate principal moment, with the outer two held at 3.068e-3 and 9.750e-3 kg m². Two of the curves are oscillation frequencies: a perturbation of a spin about the least or the greatest moment goes round and comes back. The third is a growth rate, and it belongs to the intermediate axis alone. It reaches zero at both ends of the range and is positive everywhere between, so the instability is a property of the ordering rather than of any shape — it disappears exactly when two moments become equal. Its largest value is at 6.409e-3 kg m², the midpoint of the outer two; this body sits at 6.817e-3, giving a growth rate of 0.606 per turn of the spin, or 2.425 s⁻¹ at 4 rad/s — an e-folding time of 0.41 s.
rigid-rotation 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.
The three linear exponents for a torque-free rigid body, against the value of its intermediate principal moment, with the outer two held at 3.068e-3 and 9.750e-3 kg m². Two of the curves are oscillation frequencies: a perturbation of a spin about the least or the greatest moment goes round and comes back. The third is a growth rate, and it belongs to the intermediate axis alone. It reaches zero at both ends of the range and is positive everywhere between, so the instability is a property of the ordering rather than of any shape — it disappears exactly when two moments become equal. Its largest value is at 6.409e-3 kg m², the midpoint of the outer two; this body sits at 6.817e-3, giving a growth rate of 0.606 per turn of the spin, or 2.425 s⁻¹ at 4 rad/s — an e-folding time of 0.41 s.
A 120 N force at 0.30 m and 90°
The options are the ones Force multiplied, and nothing gained passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
A spanner on a bolt. A force of 120 newtons is applied at 0.30 metres from the pivot, at 90 degrees to the shaft. The force is square to the shaft, so the moment arm is the full length of the spanner and the torque is 36.0 newton metres.
The energy a spin is allowed, and the only direction it can go
The options are the ones The axis a leak of energy chooses passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
One quantity held, one quantity spent
The options are the ones The axis a leak of energy chooses 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 run, with the two quantities that decide it plotted against time in units of their starting values. The angular momentum is flat to 1.8e-7 — no external torque acts, and the internal damping torque is constructed to be perpendicular to the momentum so that it cannot change it even in principle. The kinetic energy falls from its starting value to 25 per cent of it and stops. The floor is not an accident of the damping model. At fixed angular momentum the energy of a spin about a principal axis is L²/2I, so the lowest available energy is the one belonging to the largest moment of inertia, and the sink runs until it is reached and then has nothing left to remove — the damping torque is proportional to the part of the angular velocity not parallel to the momentum, and on a principal axis there is none. The end state is where the dissipation switches itself off, which is the general shape of every relaxation problem.
A pencil spin, given time and a small leak
The options are the ones The axis a leak of energy chooses 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 three components of the angular velocity of a body started spinning about its axis of least inertia with a tilt of 1.1°, integrated with an internal energy sink. Nothing pushes on the body from outside: the angular momentum holds to 1.8e-7 over the whole run, checked rather than assumed, while the kinetic energy falls monotonically at every step. The wobble grows exponentially at first — an amplitude too small to see, doubling steadily — and then the motion turns over entirely, ending as a flat spin about the axis of greatest inertia at 25 per cent of the energy it began with. Nothing was done to it. The initial state was a solution of the equations, stable in a rigid body against small disturbances, and it was destroyed by the body being made of matter.
How long a pencil spin lasts
The options are the ones The axis a leak of energy chooses 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 time for the body to lose half the energy available to it, against the strength of the internal leak, both on logarithmic axes. The points fall on a line of slope minus one: halving the leak doubles the time and changes nothing else, which is checked here across the whole decade drawn rather than read off the plot. A leak of 0.004 takes 3276; A leak of 0.01 takes 1310; A leak of 0.02 takes 655; A leak of 0.04 takes 328 in units where the body spins once in about six. That scaling is what makes the effect a design problem rather than a curiosity. The leak in a spacecraft is small — a whip antenna flexing, a few grams of propellant sloshing — and the time is correspondingly long, but it is a time and not an immunity. Explorer 1 was spun about its long axis at 750 revolutions a minute with a moment-of-inertia ratio of order a hundred, and its four flexible antennas took it into a flat tumble within hours of launch. Every spin-stabilised satellite since has been spun about its axis of greatest inertia.
What checks it
physicscheck asserts something about rigid-rotation 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.
Force multiplied, and nothing gained
A push on a small piston becomes a much larger push on a large one, in the ratio of their areas, with no machinery in between except the liquid. What the liquid will not do is give anything away — the distances shrink by the same factor the forces grow by, and the product is untouched.
MechanicsThe 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.
MechanicsThe 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.
MechanicsThe 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.
MechanicsThe disk that spins faster as it stops
A coin spun on a table settles with a whirr that rises in pitch and then stops dead. Everything about it is losing energy, yet the rattle gets faster to the very end. The point where the rim touches the table races round faster and faster as the coin tips flatter, while the coin's face, and the head on it, turns more and more slowly. Two plausible mechanisms of loss both predict that the tilt reaches zero, and the rattle infinity, at a definite instant — a singularity a coin on a kitchen table runs into every time.
MechanicsThe egg that keeps turning after it is stopped
Spin an egg on a table, touch it for a moment to stop it, and let go. A cooked egg stays still. A raw one starts turning again by itself, as if something inside had kept going — which it has. A raw egg is two bodies, a thin shell and a liquid that the shell can turn only through the slow grip of viscosity, and the kitchen test that tells them apart is a measurement of how long that grip takes. The same two-body arithmetic says that a raw egg wastes most of the energy of a flick, starts out spinning six times faster than a cooked one given the same twist, and keeps a record of its past rotation in its interior for a second or two after its shell has forgotten.
MechanicsThe 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.
MechanicsThe 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.
MechanicsThe 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.
MechanicsThe 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.
MechanicsThe 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.
MechanicsThe 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.
MechanicsThe wobble that is slow because the Earth gives
Euler worked out in 1765 that a spinning, flattened Earth whose axis is knocked slightly off its figure should wobble, the pole wandering in a small circle once every ten months. Astronomers looked for a ten-month period for a century and found nothing. In 1891 Seth Chandler found a wobble of fourteen months instead. The four extra months are not an error in Euler's arithmetic. They are the Earth yielding: its bulge partly re-forms round the displaced axis, weakening the torque that drives the wobble, and the lengthening measures how soft the planet is.