For a uniform sphere, the cross-terms disappear and the moments of inertia are all the same, leaving , for . Since any three orthogonal vectors can serve as eigenvectors of the moment of inertia tensor, we have that, for a uniform sphere, any three orthogonal axes can be chosen as principal axes.
For a cylinder that is not spinning about its axis, we similarly obtain two uncoupled equations , for , given (no spin). Note, however, that if we replace the circular cross-section of the cylinder by an ellipse, then and there is a coupling term that drives (unless happens to cancel it).
Young's Modulus as a Spring Constant
Euler's Equations for Rotations in the Body-Fixed Frame