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Consider the following question:

a-) By using the principal axis frame, calculate the torque needed to rotate a rectangular plate with sides a and b about its diagonal with a constant angular velocity $ω_0$.

(b) For the set-up in part (a), calculate the torque by using the Lagrangian formulation. Compare it with what you found in part (a).

I've solved the part a-) by directly using Euler's equation(s) $$\vec \tau = I \dot w + w \times (Iw);$$ since $\dot w = 0$ is given, we can direct find the torque needed, which turns out a nonzero in the $\hat z$ direction.

However, physically speaking, I don't understand why the plate needs an external torque to continue rotation with a constant angular velocity; the rotation axis passes through the CM, and once we provided the necessary torque to set $\vec w = \vec w_0$, the plate should not need any torque.

Question:

Why does plate need an external torque to continue its rotational motion with constant angular velocity ?

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  • $\begingroup$ I would suggest you to check what the angular momentum does when the body rotates as stated in the problem's text. I also have no intuition on why the angular momentum is not oriented like $\omega$ (important feature!) but perhaps with a bit of patience and looking at the trajectories of the particles which compose the object you can find it out. Note that for Euler's equation to give "counterintuitive" results $I$ must be a tensor. $\endgroup$ – JTS Apr 12 '19 at 16:40
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It's simply because angular momentum is conserved, not angular velocity. In components, we have $$L_i = \sum_j I_{ij} \omega_j.$$ Since the plate is rotating, the $I_{ij}$ change over time; in the absence of external torques the $L_i$ are constant, but that doesn't mean the $\omega_i$ are.

I'm not sure what would make this intuitive to you, but a simpler case might be a barbell with masses on the ends. Picture it oriented diagonally but rotating about the vertical axis -- this should look quite unnatural, and impossible if the barbell is isolated.

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