Examples

For a uniform sphere, the cross-terms disappear and the moments of inertia are all the same, leaving $ \tau_i=I\omega_i$, for $ i=1,2,3$. 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 $ \tau_i=I\omega_i$, for $ i=1,2$, given $ \omega_3=\tau_3=0$ (no spin). Note, however, that if we replace the circular cross-section of the cylinder by an ellipse, then $ I_1\ne I_2$ and there is a coupling term that drives $ \dot{\omega}_3$ (unless $ \tau_3$ happens to cancel it).


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Young's Modulus as a Spring Constant
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