Rotation about a principal axis, zero torque and all that

Question

Consider a body pivoted at the origin and now apply a force at the origin. The body experiences zero torque but can still rotate with constant angular velocity $\vec\omega$ where $\hat {\omega}$ is a fixed principal axis.

(The above thing is true for any arbitrary origin. The only necessary conditions are that the object is pivoted there and the force is applied on this pivot.)

I have two questions regarding this:

1. What is the intuitive explanation for the fact that the body can start rotating even in the absence of torque?

2. Why does it not work for non-principal axes?

Answer 1

1. Is not true. A body cannot start rotating in the absence of torque about the center of mass. The rotational law of motion makes this clear when $\vec{\omega} =0 $, you need torque $\vec{\tau}$ to affect rotational acceleration $\dot{\vec{\omega}}$. $$\vec{\tau} = \mathrm{I}\, \dot{\vec{\omega}} + \vec{\omega} \times \mathrm{I}\, \vec{\omega} $$

2. For a rotating body, in the absence of torque, angular momentum vector $\vec{L} = \mathrm{I}\,\vec{\omega}$ is preserved in magnitude and direction. So the second term in the equation above is zero only when angular momentum and rotational velocity are parallel to each other. This happens along the so called principal axis of rotation. In all other cases, when $\vec{\tau}=0$, if the second term is not zero, then first term must be non-zero also in order to cancel out and produce zero net torque.