Is there an exact solution to the length of the axes of an ellipsoid of inertia?

Question

If I have a body with a known mass M and principle inertias I1, I2, and I3 (or Ixx, Iyy, and Izz), is there an exact formula for the lengths of the three major axes of a uniform solid ellipsoid
X^2/A^2 + Y^2/B^2 + Z^2/C^2 = 1
which has the same inertias? In other words, knowning M, I1, I2, and I3, can A, B, and C be calculated exactly? [Or vice-versa: If A, B, and C (plus the density) are known, can I1, I2, and I3 be calculated?]

Answer 1

In fact, for any ellipsoid (centered at the origin) the translational inertia vectors act on the center of mass and that center is *always* at the origin.

Note that if a force is applied that does **not** go thru the center of the origin then all bets are off (even with a sphere) since there will be a rotational component to the resulting motion as well as a translational motion.


Doug

Answer 2

What? (I just wanted to sound dumb for once in my life)