# The Intermediate Axis Theorem

Early in Matt's video, a learned doctor points out the flaw in the explanation Veritasium reports – the argument used to explain the instability of the difficult axis can just as well be applied to the long axis, so why doesn't that have the same problem ? So I paused to think about it.

Now I really need to sit down and write up, in my own terms, an account of the orthodox description of angular momentum, but for now it suffices to say that the behaviour of any body, that'll exibit this effect, will be tolerably well modelled by a simple plane sheet of some light (i.e. negligible mass) material (whose stiffness roughly corresponds to that of the body it's to model) with four weights attached to it in two pairs on peprendicular straight lines through a central point of the sheet. Each pair is of equal masses at equal distance each side of the central point along its straight line. This leaves some choice, in how far from centre each pair of masses is, coupled to how big those masses are, enough to let us arrange that the two pairs' distances from centre are equal. When we do that, the case we're interested in has one pair of masses bigger than the other. The line through the centre-point, where the two lines in the sheet meet, perpendicular to the sheet is known as the major axis, the line through the two smaller masses as the minor axis and the line through the two larger masses as the intermediate axis. The body can in principle spin about any of these axes, so the rest of our analysis is about the stability of the motion in each of these cases. The discussion of that is naturally done by considering how small perturbations of the motions of the masses will affect what's going on. In each case, I'll describe what happens from the point of view of a spinning frame, in which the body's momentum and angular momentum are zero.

In such a frame, centrifugal and Coriolis forces really are forces, just as much as gravity is. Indeed, in Einstein's General Theory of Relativity (which has been orthodox for more than a century, now), this is a perfectly legitimate (non-inerital) frame and these forces are gravitational. That's the principle of equivalence, identifying gravitational forces in a frame as encoding its departure from being an inertial frame. (I do wish folk would get over the it's not a force, it's an artefact of your frame of reference hang-up – exhibited by both Matt and Veritasium – that's a hold-over from the old Newtonian orthodoxy.) The centrifugal force pulls radially away from the axis of spin. The Coriolis force acts on any body moving (with respect to this rotating frame) at least partially perpendicular to that axis; the force is proportional to the body's mass, the rate of rotation and the component of its velocity prependicular to the axis; the force is perpendicular to both the axis and the body's velocity.

## Major axis

When the body spins about its major axis, all the masses are rotating in the plane of the sheet. There's tension in the sheet, stronger parallel to the intermediate axis but still significant parallel to the minor axis, countering the centrifugal forces on the masses.

Small perturbations of the positions of the masses parallel to the spin axis don't incur any Coriolis force but bend the sheet off the perpendicular to the axis, so that the tension isn't parallel to the centrifugal force, leading to a restoring force that'll tend to straighten up the sheet and keep it perpendicular to the axis. Perturbations in position within the plane of the sheet shall tend to be countered by tension in the sheet, keeping them tiny.

Any circumferential perturbations will produce radial Coriolis forces, that'll eithe rcounter or reinforce the tension in the sheet, either causing it to contract or extend slightly, a tiny radial movement; any radial movement shall likewise produce circumferential Coriolis forces, tending to produce small movements that'll cause tension in the sheet that opposes this movement; in each case, the induced movement caused by the Coriolis force is in the direction that produces a secondary Coriolis force opposing the original movement that induced the initial Coriolis term.

So the sheet shall tend to flap in small osscilations and the masses shall tend to move around their nominal positions in tiny epicycles but buth kinds of motion shall tend to die out in so far as the sheet isn't perfectly elastic.

## Minor axis

When the body spins about its minor axis, through the two larger masses, these two masses experience centrifugal force only to the extent that they are perturbed off the axis, while the smaller masses experience steady centrifugal force inducing tension parallel to the minor intermediate axis (on which they lie); so there's significant tension parallel to the intermediate axis and only small tension parallel to the minor axis, about which we're spinning.

Small perturbations of these smaller masses parallel to the major axis, flexing the sheet, will cause the tension to be misaligned with the centrifugal force, so that the difference is a restoring force tending to bring the small masses back into the plane of the sheet; however, this time, this perturbation is perpendicular to the spinaxis so also induces a Coriolis force, albeit smaller than the centrifugal one. This Coriolis force will be parallel to the intermediate axis, so either increasing or decreasing tension in the sheet, leading to radial movement, on which Coriolis will induce a force opposing the original motion parallel to the major axis. So the flapping of the small masses will now be an oscillation, as before, but it'll be accompanied by a little radisl stretching and shrinking.

Radial perturbations, likewise, feed into the same loop, inducing Coriolis perturbations that flapt the sheet, whose Coriolis consequence counters the original perturbation. The centrifugal effect of this will actually alternate between reinforcing and countering the radial perturbation, but is fighting the tension in the sheet, so that the radial perturbation only slightly perturbs the flapping. The small masses can also move parallel to the spin axis, inducing tensions in the sheet that counter this movement, without any Coriolis or centrifugal impact, so the rigidity of the sheet will tend to limit such motions to tiny oscillations.

The large masses are (approximately) on the axis, so experience (approximately) no centrifugal force, so there is little tension in the sheet along the axis. Any movement they make along the axis shall tend to be countered by tension or compression in the sheet; and such movement incurs neither centrifugal nor Coriolis forces. Movement in the two directions perpendicular to the axis will, however, incur both kinds of force; Coriolis from the movement iself and centrifugal from being off the axis. Since the distance from axis is small, the Coriolis effect isn't dominated by the centrifugal, so will tend to turn movement away from the axis into movement around the axis (in additino or opposition to the over-all rotation that our choice of frame is hiding from us) in the direction that, again, Coriolis will tend to turn into movement towards the axis. None the less, while it's off axis, it's experiencing centrifugal force which can counter that secondary Coriolis term. However, all of these forces shall be small compared to the tension in the intermediate axis, due to the centrifugal force on the smaller masses, which shall thus strongly resist rotation within the plane of the sheet.

Written by Eddy.