[Part 2 of ?]

In the previous part, we introduced a spinning oblate spheroid, and showed that a distant point mass *M* at displacement **d** will exert a torque on the spheroid that can be approximated as

. Now, suppose that **d** is in the *xy*-plane of the inertial frame. Further, suppose that our point mass is orbiting our spheroid in a circular orbit of angular frequency *Ω* (or that our spheroid is orbiting our point mass in a circular orbit of angular frequency *Ω*; the model will turn out the same). Then in the inertial frame. Suppose then at a time *t* the *x* and *x’* axes coincide. Then, if we let the angle between *z* and *z’* be *θ*_{0}, we see that **d** has components in the body coordinates of .

Now, if the density of our spheroid is sufficiently symmetric about its axis, then we will have *I*_{x’}=*I*_{y’}, and the moment of inertia tensor in the body coordinates will be . Using our results from part one, we find the torque in this situation;

,

.

Now, we presently have and , so we can rewrite the above in a way independent or our choice of *x’* and *y’* axes:

,

giving torque:

Supposing that this torque is small enough that any precession produced is of frequency much slower than *Ω*, we can then average the torque over time; recalling the time average of trigonometric functions and their products, we get average torque

Recalling that our object has angular momentum along the *z’* axis, we see then that our average torque is perpendicular to our angular momentum, and to our *z* axis (as it is along the cross product ). Thus, as in here, we have precession of our spheroid’s rotation about the *z* axis (so *θ*_{0} is constant), and from our previous work on torque-driven precession, we see that the precession has angular frequency

.

(The negative sign indicates that the direction here is opposite the sense of the rotation *ω*_{z}‘.)

Now, supposing our spheroid has total mass *M*_{E}, then Kepler’s third law for our spheroid-point mass orbit tells us that the period *T* of the orbit is . Since , we find , which lets us rewrite the precession frequency in terms of the orbital frequency and the ratio of the masses :

.

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Tags: Body Frame, Ellipsoid, Friday Physics, Gravitation, Inertia Tensor, Moment of Inertia, Oblate Spheroid, physics, Precession, Principal Axis, Rotation, Torque

This entry was posted on March 13, 2009 at 9:03 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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March 19, 2009 at 11:40 pm |

[…] Friday 64 By twistedone151 [part 3 of 3] In part 2, we showed how if the distant point mass M orbits our oblate spheroid (or our ellipsoid orbits it) […]