All Problems Dynamics of a solid body
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.
1.282. (a) \(M=1 /_{12} m \omega l^{2} \sin \theta, \quad M_{z}=M \sin \theta\) (b) \(|\Delta \mathbf{M}|=\) \(=1 /{ }_{12} m \omega l^{2} \sin 2 \theta\) (c) \(N=1 / 2_{24} m \omega^{2} l^{2} X\) \(x \sin 2 \theta\)