6. At the ends of a weightless, inextensible rod, a small body and a massive ring of the same mass are attached. The ring can move without friction along a horizontal rigid spoke in a gravitational field (see figure). At some moment, the ring and the body were given velocities $v$ and $2 v$, directed horizontally along the rod and opposite to each other (see figure). It is known that when the body rises to its maximum height, the rod makes an angle $\alpha$ with the spoke. Find the length of the rod.

![](https://cdn.mathpix.com/cropped/2024_05_06_bea1dbb251077dbc2bb6g-1.jpg?height=343&width=385&top_left_y=1704&top_left_x=1535)
The spoke is very long.
6. At the moment of rising to the maximum height, the body moves horizontally. Therefore, from the condition of the inextensibility of the rod, we conclude that the speed is the same (their projections on the direction of the rod should be the same, and the speeds of the body and the ring at the moment of the body's rise to the maximum height are directed the same). From the law of conservation of momentum in the horizontal direction, we have

$$
2 m v - m v = 2 m v_{1}
$$

where \( v_{1} \) is the speed of the body and the ring at the moment of the body's rise to the maximum height. From this, we get

$$
v_{1} = \frac{v}{2}
$$

Now, using the law of conservation of energy, we obtain

$$
\frac{m v^{2}}{2} + \frac{m (2 v)^{2}}{2} = 2 \frac{m v_{1}^{2}}{2} + m g l (1 - \sin \alpha)
$$

where \( l \) is the length of the rod. From this, we find

$$
l = \frac{9 v^{2}}{4 g (1 - \sin \alpha)}
$$

Criteria for evaluating the problem (maximum score for the problem - 2 points)