A small ball is suspended from a point $O$ by a light thread of Iength $l$. Then the ball is drawn aside so that the thread deviates through an angle $\text{θ}$ from the vertical and set in motion in a horizontal direction at right angles to the vertical plane in which the thread is located. What is the initial velocity that has to be imparted to the ball so that it could deviate through the maximum angle $\frac{\mathrm{\pi }}{2}$ in the process of motion?

A small body of mass $m$ tied to a non-stretchable thread moves over a smooth horizontal plane. The other end of the thread is being drawn into a hole $O$ with a constant velocity. Find the thread tension as a function of the distance $r$ between the body and the hole, if at $r=r_0$ the angular velocity of the thread is equal to ${\omega }_0$.

Demonstrate that the angular momentum $\overrightarrow{M}$ of the system of particles relative to a point $O$ of the reference frame $K$ can be represented as $\overrightarrow{M}=\tilde{M}+\left[r_{\mathrm{C}}p\right]$, where $\tilde{M}$ is its proper angular momentum (in the reference frame moving translationally and fixed to the centre of inertia), $\overrightarrow{r_{\mathrm{c}}}$ is the radius vector of the centre of inertia relative to the point $O, \overrightarrow{p}$ is the total momentum of the system of particles in the reference frame $K$.

A ball of mass $m$ moving with velocity $v_0$ experiences a head-on elastic collision with one of the spheres of a stationary rigid dumbbell as shown in figure. The mass of each sphere equals $\frac{m}{2}$, and the distance between them is $l$. Disregarding the size of the spheres, find the proper angular momentum $\tilde{M}$ of the dumbbell after the collision, i.e., the angular momentum in the reference frame moving translationally and fixed to the dumbbell's centre of inertia.