A small massless ring can slide without friction along a long horizontal fixed rod. One end of a massless inextensible string is attached to the ring, and the other end is fixed at a stationary point $O$ (see figure). A massive bead can slide frictionlessly along the string. Initially, the string is slightly taut, with the bead located at the same horizontal level as point $O$ and directly above (below) the ring. The length of the horizontal segment of the string is $L$.
The bead is given an impulse and set into motion in the vertical plane containing the string. Immediately after the impulse, the bead moves away from the rod, and the string becomes taut. The acceleration due to gravity is $g$.
1
At what angle $\alpha$ to the horizontal is the bead's velocity directed immediately after the impulse?
2
Find the range of initial speeds $v_0$ for which the string stays taut while the bead is moving away from the rod.
