Problem:
A small ring of mass m is attached at an end of a light string, the other end of which is tied to a small block B of mass 2m. The ring is free to move on a fixed smooth horizontal rod. Find the velocity of the ring when the string becomes vertical.

Solution:
To solve this problem, we can use the principles of conservation of energy and conservation of linear momentum.

1. Initial Conditions:
- At the initial position, the ring and the block are at rest.
- The height of the ring from the ground is h.
- The length of the string is L.

2. When the string becomes vertical:
- At this point, the block B has fallen a distance h, and the string has become vertical.
- The ring has also moved horizontally a distance x.

3. Conservation of Energy:
- At the initial position, the total mechanical energy of the system is given by:
E1 = mgh
(where g is the acceleration due to gravity)
- At the final position, the total mechanical energy of the system is given by:
E2 = mgh + (1/2)mV^2 + (1/2)(2m)V^2
(where V is the velocity of the ring when the string becomes vertical)
- According to the conservation of energy, E1 = E2:
mgh = mgh + (1/2)mV^2 + (1/2)(2m)V^2

4. Conservation of Linear Momentum:
- At the initial position, the total linear momentum of the system is zero, as both the ring and the block are at rest.
- At the final position, the total linear momentum of the system is given by:
P2 = mV + 2mV
(where P2 is the total linear momentum of the system when the string becomes vertical)
- According to the conservation of linear momentum, P2 = 0:
mV + 2mV = 0

5. Solving the Equations:
- From the conservation of linear momentum equation, we get:
mV + 2mV = 0
3mV = 0
V = 0

- Substituting V = 0 in the conservation of energy equation, we get:
mgh = mgh + 0 + 0
mgh = mgh

6. Conclusion:
The velocity of the ring when the string becomes vertical is zero. This means that the ring comes to rest at the moment the string becomes vertical.

Summary:
- The velocity of the ring when the string becomes vertical is zero.
- This is because of the conservation of energy and conservation of linear momentum principles.
- The ring comes to rest at the moment the string becomes vertical.

It's 18mg