Illustrated Physics
physics, drawn
Series

Rotation — the series

12 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The same 120 N force at 3 different arms. 3 spanners of different lengths, each with the same 120 newton force applied at its end. The torque printed under each is the force times its own moment arm, so it rises with the length while the force does not change.

    The 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.

    part 1 · mechanics
  2. Released together on a 20° slope, 1.1 s later. 3 bodies of different shape, released from the same line on a 20 degree slope and drawn where each has reached after 1.1 seconds. The order is sphere, then disc, then hoop. Each spoke is turned by the distance that body has rolled divided by its radius.

    The 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.

    part 2 · mechanics
  3. 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.

    part 3 · mechanics
  4. Arms in: 4.33× the rate, and 4.33× the energy. A body of 1.2 kg m² carrying two 4 kg masses on arms, spinning freely at 60 revolutions a minute with the arms out at 0.75 m, as the arms are pulled in to 0.12 m. The axis runs right to left, in the direction the arms move. Angular momentum is flat — nothing exerts a torque about the axis, and pulling inward is a force along a radius, which has no moment about the centre. The rate rises as the inverse of the moment of inertia, by a factor of 4.33 here, and the kinetic energy L²/2I rises by exactly the same factor, which is where the usual account stops and where the question starts. The fourth curve is the work done by whoever pulled the arms in, integrated from the force needed to hold each mass on its circle. It lies on the energy curve, to 1.6e-7 joules. Nothing is unaccounted for and nothing is created: the energy is bought, at full price, by pulling against the force that would otherwise fling the arms out.

    The 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.

    part 4 · mechanics
  5. 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.

    The push that comes out sideways

    Push down on a spinning wheel's axle and it swings horizontally. Nothing about that is mysterious once angular momentum is a vector — but the steady precession every demonstration shows is a solution nobody's initial conditions select, and a top released from rest does something else first.

    part 5 · mechanics
  6. The same top, let go four ways. The path traced by the top of the axis, seen from directly above, over 1.2 precession periods. The dashed circle is the tilt the top was released at and the outer circle is 46.8° 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 0.45× the steady rate the path waves, at 1× the steady rate the path stays a circle, at 1.9× the steady rate the path waves. Every one of these is the same equation with the same top and the same spin.

    The top that nods before it settles

    A spinning top let go from rest does not begin to precess. It falls, catches itself, and comes back up, over and over, at a frequency that has nothing to do with gravity — and the steady precession every textbook draws is what is left after friction has removed the nod.

    part 6 · mechanics
  7. The energy a spin is allowed, and the only direction it can go. 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 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.

    part 7 · mechanics
  8. The wobble slows as the body yields. The period of the Earth's free wobble against how far the body yields to its own rotation, measured as the Love number k over its value for a fluid Earth, kₛ = 0.941, computed from the Earth's flattening. A rigid Earth, at zero, wobbles in 303.6 days, Euler's period. The observed 433 days requires k/kₛ = 0.299, an effective k of 0.281: the Earth yields about 30 per cent as much as a fluid would. The period grows without limit as the body approaches a fluid, because a fluid body's bulge re-forms round any new spin axis and leaves nothing to push the axis back.

    The 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.

    part 8 · mechanics
  9. 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.

    part 9 · mechanics
  10. 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.

    part 10 · mechanics
  11. A tilt that reaches zero at a definite time. The tilt of the disk against time from 10°, when the losses are set to end the motion after 100 s, for two ways of losing energy: viscous air squeezed out of the thin wedge under the rim (red), which takes energy faster the smaller the tilt, so that α³ falls in a straight line; and a steady rolling resistance at the contact (blue), which takes energy in proportion to the contact rate, so that α to the power 3/2 does. Both fall slowly at first and steeply at the end, and both reach zero tilt at a definite time rather than approaching it for ever: at 99 s, one second before the end, the tilt is 2.15° under the air model and 0.46° under rolling resistance. The disk's energy, (3/2)Mgaα, runs out in a finite time, and with it the tilt.

    The 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.

    part 11 · mechanics
  12. Stopped for half a second, a raw egg starts again by itself. Two eggs spinning at 3 revolutions a second, each touched with a finger at t = 4 s so that its shell is held still for 0.5 s and then released. The cooked egg (solid) stops and stays stopped. The raw egg's shell (dashed) stops too, but its contents (dotted) are still turning — slowed to 2.15 revolutions a second during the hold, by viscosity working against the held shell — and once the finger lifts they drag the shell back into rotation, reaching 1.81 revolutions a second, 60 per cent of its speed before the touch. The table's friction is left out here, so that nothing else slows either egg. It is the oldest kitchen test of whether an egg is cooked, and the whole of the physics is a liquid's reluctance to stop turning.

    The 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.

    part 12 · mechanics

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