Problem 1.282

The middle of a uniform rod of mass $m$ and length $l$ is rigidly fixed to a vertical axis $O O^{\prime}$ so that the angle between the rod and the axis is equal to $\theta$ (see Fig. 1.71). The ends of the axis $O O^{\prime}$ are provided with bearings. The system rotates without friction with an angular velocity $\omega$. Find: (a) the magnitude and direction of the rod's angular momentum M relative to the point $C$, as well as its angular momentum relative to the rotation axis; (b) how much the modulus of the vector $\mathbf{M}$ relative to the point $C$ increases during a half-turn; (c) the moment of external forces $N$ acting on the axle $O O^{\prime}$ in the process of rotation.