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