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