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.

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 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.

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 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.

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.

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.

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.