A 120 g top at 3000 rpm, precessing once every 1.92 s

At its defaults it draws 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.

gyro-top 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.

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.

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 two steady precession rates a top of this shape has at 30° from the vertical, against how fast it is spinning. Setting the tilt's second derivative to zero gives a quadratic in the precession rate, not a single value, so there are two answers at every spin: a slow one that falls as the inverse of the spin and is the one every demonstration shows, and a fast one that rises in proportion to it and is almost never seen because it takes a deliberate launch. They meet at 1245 revolutions a minute, where the quadratic's discriminant reaches zero and below which no steady precession exists at all — the top simply falls. At 6000 rpm the slow root is 1.578 radians a second and the fast one is 144.

The two steady precession rates a top of this shape has at 75° from the vertical, against how fast it is spinning. Setting the tilt's second derivative to zero gives a quadratic in the precession rate, not a single value, so there are two answers at every spin: a slow one that falls as the inverse of the spin and is the one every demonstration shows, and a fast one that rises in proportion to it and is almost never seen because it takes a deliberate launch. They meet at 680 revolutions a minute, where the quadratic's discriminant reaches zero and below which no steady precession exists at all — the top simply falls. At 6000 rpm the slow root is 1.566 radians a second and the fast one is 484.

The path traced by the top of the axis, seen from directly above, over 1.15 precession periods. The dashed circle is the tilt the top was released at and the outer circle is 35.3° 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 0.55× the steady rate the path waves, at 1× the steady rate the path stays a circle, at 2.5× the steady rate the path loops. Every one of these is the same equation with the same top and the same spin.

The path traced by the top of the axis, seen from directly above, over 2.2 precession periods. The dashed circle is the tilt the top was released at and the outer circle is 33.2° 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 1× the steady rate the path stays a circle, at 2.5× the steady rate the path loops. Every one of these is the same equation with the same top and the same spin.

physicscheck asserts something about gyro-top 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.