The Torque Free Ratio of Ellipsoid Spin to Precession equation computes the torque free spin-to-wobble (aka spin-to-precession) ratio for an ellipsoid. This ratio can be used as a measure of the stability of the motion of an ellipsoid spinning about the major axis.

A common example of this ratio can be used to understand why the mechanics of a football pass dictates that putting spin on the ball has the effect to cause the ball to travel smoothy with little wobble.

Derivation

Eq. 1: `"spin"/"precession" = omega_3 / dotphi = (I_1*cos(theta)) / I_3 = 1/2 (1 + a^2/b^2) cos(theta)`, where

• `I_1` is the moment of inertia about the cross axes,  `"axis"_1`.  (Note that  `"axis"_1` = `"axis"_2`.)
• `I_3` is the moment of inertia about the long axis of the ellipsoid, `"axis"_3`.
• `a` is the semi-major axis of the ellipsoid (half the length of the ellipsoid).
• `b` is the semi-minor axis of the ellipsoid (half the length of the cross dimension). Note `b = c` because the cross section is a circle and therefore `b` & `c` are radii.
• `theta` is the angle between the moment of inertia direction and the symmetry axis (`"axis"_3`).

Usage

Since the ellipsoid is a good approximation for the shape of a football, this computation is a good approximation of the spin-to-wobble ration of a well thrown pass.  This indicator of stability (the spin-to-wobble ratio) can predict  the amount of wobble one might expect when throwing a spiral pass.

From Eq. 1 we see that the wobble-to- spin ratio (the inverse of the spin-to-wobble ratio) when length `a` approaches length `b`.

`cos(theta) = ((2 * "spin")/"precession" ) / (1 + a^2/b^2)` =>

`theta = arccos (((2 * "spin")/"precession" ) / (1 + a^2/b^2))`

This implies the angle of the wobble (precession) is minimized when `a` approaches `b`. In other words, the wobble  decreases as the ellipsoid comes closer to being a sphere.  This also implies the angle of the wobble increases as spin tends toward zero.  This is kind of intuitive, as a football passed without spin tends to wobble a lot.

1. ^ On the Flight of the American Football