Rotating Rod As a conical pendulum

Question

Consider A Rigid Rod hinged at its top point Whirled around in a circle (similar to a conical pendulum). It is given that the angular velocity (and thus the semi-vertical angle) is constant. I am trying to analyse this sytem:

1. From the rotating frame, the rod is at rest. Thus, The net torque must be zero. We have to consider Pseudo forces as well. Consider a mass element $dm$ of length dx at a distance x from the hinge point. It rotates in a circle of radius $xsin(\theta)$ so it exerts a torque: $(dm)(\omega^2)(xsin\theta)(xcos\theta)$ about the suspension point. This must be balanced by the Torque of weight=$dmgxsin\theta$. Equating Both these integrals gives a relationship between $\omega$ and $\theta$.
2. However, from the ground frame , It is clear that there will be a Net torque (due to weight) about the suspension point, which made me question the fact that $\omega$ is constant. However, if you consider the axis of rotation,$mg$ cant produce a torque, since its parallel to the axis! So is that why $\omega$ is constant? What then , is the significance of the Net torque being non zero about the suspension point?
3. An (unsolved) illustration in my textbook asks for the " rate of change of angular momentum" of the rod. Since this is equal to the "torque", what should I consider? Torque about the suspension point? or about the axis? and why?

Answer 1

I think you have misunderstood the question, where the rod is not rotating about the axis, but is whirled, thus implying that it is being forced to rotate at a constant frequency, and not yet in equilibrium. Thus the assumption that the semi- vertical angle is constant is incorrect. The angular velocity is constant, but the moment of inertia about the axis changes with the angle, thus changing the angular momentum. This is probably what you have been asked to calculate (maybe instantaneous rate of change at a given angle, i don't know the exact question). You can do this using a free body diagram.