Posts Tagged ‘Body Frame’

Physics Friday 63

March 13, 2009

[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 θ0, 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 Ix’=Iy’, 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 θ0 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 ωz.)
Now, supposing our spheroid has total mass ME, 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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Physics Friday 62

March 6, 2009

[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 Iz’ 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 volumeV 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 E3 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:
.