A rigid body having one point $ O $ fixed and no external torque about $ O $ has equal principal moments of inertia. Then the body must rotate with
(1) angular velocity of variable magnitude
(2) angular velocity with constant magnitude
(3) constant angular momentum but varying angular velocity.
My approach: The option $ (c) $ is true. Am I right ?
From the point of view of an inertial frame of reference the equation of movement is $\dot J=0$ with $J=\omega I$, with $I$ the tensor of inertia, so, with no torque, $J$ is constant and
$\dot J=\dot\omega I+\omega\dot I=0$. Nothing but equal principal moments of inertia prevents the angular velocity to vary (as the matrix I is varying). If all principal moments of inertia are equal, $I=k\mathbf 1$, with $\mathbf 1$ the identity matrix, in any reference frame. In this case,
$\dot J=\dot\omega I=0$, making $\omega$ constant. The correct answer is 2), being 1) and 3) both true of the less symmetrical cases.
