Moment of Inertia--Ellipsoid -- from Eric Weisstein's World of Physics

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Moment of Inertia--Ellipsoid

For an ellipsoid, Eric Weisstein's World of Math let C be the moment of inertia along the minor axis c, A the moment of inertia about the minor axis a, and B the moment of inertia about the intermediate axis b. Consider the moment of inertia about the c-axis, and label the c-axis z. Then in Cartesian coordinates, Eric Weisstein's World of Math

Making the substitutions

then transforms the ellipsoid Eric Weisstein's World of Math V into the unit sphere Eric Weisstein's World of Math , with

where

is the Jacobian. Eric Weisstein's World of Math Transforming to spherical coordinates Eric Weisstein's World of Math then gives

Plugging in

then gives

By symmetry, the other two axes have similar expressions, so the final result is

Moment of Inertia, Moment of Inertia--Earth, Moment of Inertia--Sphere, Routh's Rule