A ring of mass $m$ connecting freely two identical thin hoops of mass $M$ each starts sliding down. The hoops move apart over a rough horizontal surface.

Determine the acceleration $a$ of the ring at the initial instant if $\angle A{O}_1{O}_2=\alpha$

(Fig.), neglecting the friction between the ring and the hoops.

A flexible pipe of length $l$ connects two points $A$ and $B$ in space with an altitude difference $h$ (Fig.). A rope passed through the pipe is fixed at point $A$.

Determine the initial acceleration $a$ of the rope at the instant when it is released, neglecting the friction between the rope and the pipe walls.

A smooth washer impinges at a velocity $v$ on a group of three smooth identical blocks resting on a smooth horizontal surface as shown in Fig. The mass of each block is equal to the mass of the washer. The diameter of the washer and its height are equal to the edge of the block.

Determine the velocities of all the bodies after the impact.

Three small bodies with the mass ratio $3:4:5$ (the mass of the lightest body is $m$) are kept at three different points on the inner surface of a smooth hemispherical cup of radius $r$. The cup is fixed at its lowest point on a horizontal surface. At a certain instant, the bodies are released.

Determine the maximum amount of heat $Q$ that can be liberated in such a system. At what initial arrangement of the bodies will the amount of liberated heat be maximum? Assume that collisions are perfectly inelastic.

A long smooth cylindrical pipe of radius $r$ is tilted at an angle $\alpha$ to the horizontal. A small body at point $A$ is pushed upwards along the inner surface of the pipe so that the direction of its initial velocity forms an angle $\phi$ with generatrix $AB$. Determine the minimum initial velocity $v_0$ at which the body starts moving upwards without being separated from the surface of the pipe.

An inextensible rope tied to the axle of a wheel of mass $m$ and radius $r$ is pulled in the horizontal direction in the plane of the wheel. The wheel rolls without jumping over a grid consisting of parallel horizontal rods arranged at a distance $l$ from one another $\left(l\ll r\right)$.

Determine the average tension $T$ of the rope at which the wheel moves at a constant velocity $v$, assuming the mass of the wheel to be concentrated at its axle.