When a ceiling fan is switched off its angular velocity reduces to $50\%$ while it makes $36$ rotations. How many more rotation will it make before coming to rest (Assume uniform angular retardation)

A rigid massless rod of length $3l$ has two masses attached at each end as shown in the figure. The rod is pivoted at point $P$ on the horizontal axis. When released from the initial horizontal position, its instantaneous angular acceleration will be

A rod of length $50 cm$ is pivoted at one end. It is raised such that it makes an angle of ${30}^o$ from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in $rad s^{-1}$ ) will be $\left(g=10 m{s}^{-2}\right)$

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: