#### Examples

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).

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Young's Modulus as a Spring Constant

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Euler's Equations for Rotations in the Body-Fixed Frame