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If rigid bodies $R_1$ and $R_2$ has exactly same total mass $M$, central of mass, and rotational inertia $I$, but different third moment of inertia $M_3$, how would they move/rotate differently? What's the observable phenomenon of the difference in their third moment of inertia?

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  • $\begingroup$ There are three independent components to the moment of inertia tensor (in the principal-axis frame). There can be a variety of effects on rotation. For example, rotation about the intermediate axis of rotation (not the smallest or largest moment of inertia) is unstable. $\endgroup$ – Ben Crowell Oct 28 '14 at 19:33
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    $\begingroup$ Can you please define the 3rd moment of inertia so we are all in the same footing. See "Be Specific" section of physics.stackexchange.com/help/how-to-ask $\endgroup$ – ja72 Oct 28 '14 at 21:17
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I assume that by $M_3$ you mean $I_{zz}$ of a diagonal inertia tensor. Nothing wrong with your notation though, I'm just clearing that up. The inertia tensor plays an analogous role to that of mass on linear motion. If we are talking specifically about $I_z$, it means that IF the body has angular velocity around this axis z, the kinetic energy associated with this motion is $T_{rot}=\frac{1}{2}I_z \omega$ So if both objects are spinning with same angular velocity you know that one has more energy than the other. If you had to put these two ojects into a spin aroudn that axis it would require more work to do it for the one with larger Iz.

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