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:
- 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 $\theta$.
- 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?
- 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?