[Part 1 of ?]

Let us consider a rotating oblate spheroid with equatorial radius *a* and polar semi-axis *c*. We choose an inertial frame (*x*,*y*,*z*) and body coordinate system (*x’*,*y’*,*z’*), both with origin at the ellipsoid’s center of mass. Suppose we have a rotation about a principal axis with angular momentum , where *I*_{z’} is the moment of inertia along the *z’* principal axis and is the *z’* unit vector).

Now, let us put a point mass *M* at a displacement **d** from the center of our ellipsoid. Then a mass element of our ellipsoid at position **r** ( is the density at that point in the object, not necessarily uniform). experiences a gravitational force due to the mass *M* of

Integrating the torque element due to this force over the volume *V* of the ellipsoid gives a total torque of

.

Now, suppose our point mass is far from our body, compared to its size, so that , where and . Now using

, we see that approximating to first order in *r*/*d*,

, so we can in this situation approximate our torque by:

.

Now, to find the first integral on the right, we note that

.

For a an object in volume*V* of total mass *m* and density , the center of mass has position (vector)

(see here); as our origin is the center of mass, we thus see that the integral , and the first integral in our approximation is the zero vector, and

.

Consider the moment of inertia tensior **I** of our object (see here):

, where **E**_{3} is the 3×3 identity matrix and is the outer product of **r** with itself. Applying the tensor to **d**, we use and to get

.

Next, consider the cross product of **d** and this vector:

,

which is our remaining integral in the torque approximation:

.

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Tags: Body Frame, Center of Mass, Ellipsoid, Friday Physics, Gravitation, Inertia Tensor, Inertial Frame, Moment of Inertia, Oblate Spheroid, physics, Rotation, Tensor, Torque

This entry was posted on March 6, 2009 at 9:28 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 13, 2009 at 9:13 am |

[…] Friday 63 By twistedone151 [Part 2 of ?] In the previous part, we introduced a spinning oblate spheroid, and showed that a distant point mass M at displacement d […]