A uniform rod of length ' ℓ ' is pivoted at one of its ends on a vertical shaft of negligible…

A uniform rod of length ' ${\ell }^{\text{'}}$ is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed $\omega$ the rod makes an angle $\theta$with it (see figure). To find $\theta$ equate the rate of change of angular momentum (direction going into the paper) $\frac{m{\ell }^2}{12}{\omega }^2\sin \theta$ about the centre of mass (CM) to the torque provided by the horizontal and vertical forces $F_H$and $F_v$ about the CM. The value of $\theta$ is then such that: