Motion of the Bead on the Fixed Ring

The motion of the bead on the fixed ring can be analyzed by considering the forces acting on the bead at different positions. Let's break down the analysis into different stages:

Stage 1: Initial Velocity at Point C
- At the beginning, the bead is given a small velocity at point C.
- As there is no friction and the bead is on a frictionless surface, it will continue to move in a circular path along the ring.

Stage 2: Tension in the Strings
- As the bead moves along the ring, the tension in the strings connected to nails A and B comes into play.
- The tension in each string will always be directed towards the center of the ring (point O).
- The tension in the strings will provide the necessary centripetal force to keep the bead in circular motion.

Stage 3: Length of the Strings
- As the bead moves, the strings connected to nails A and B will start to unwind from the nails.
- The length of each string will increase as the bead moves away from the respective nail.
- However, the relaxed length of each string is negligible compared to the radius of the ring, so the change in string length can be ignored for simplicity.

Stage 4: Velocity and Acceleration
- As the bead moves, its velocity will change due to the tension in the strings.
- The centripetal force provided by the tension in the strings will cause the bead to accelerate towards the center of the ring.
- The acceleration of the bead can be calculated using the equation: a = v^2 / r, where v is the velocity of the bead and r is the radius of the ring.

Stage 5: Bead's Path
- The bead will continue to move along the ring, following a circular path.
- The path of the bead will depend on the initial velocity given at point C and the tension in the strings.
- The bead will keep moving until one of the strings strikes its respective nail.
- At that point, the tension in that string will become zero, and the motion of the bead will be affected.

Predicting the Subsequent Motion
- Since the relaxed length of each string is negligible compared to the radius of the ring, the bead will continue to move in circular motion until one of the strings strikes its respective nail.
- The direction of the bead's motion will depend on the initial velocity given at point C.
- If the initial velocity is such that the bead is moving in the clockwise direction, then the bead will continue to move in the same direction until one of the strings strikes its respective nail.
- If the initial velocity is such that the bead is moving in the counterclockwise direction, then the bead will continue to move in the same direction until one of the strings strikes its respective nail.
- Once a string strikes its respective nail, the tension in that string becomes zero, causing the bead to move in a straight line tangent to the point of contact.
- The subsequent motion of the bead after one of the strings strikes its respective nail will depend on the initial velocity, the tension in the remaining string, and the angle at which the bead strikes the nail.

Conclusion
- The bead will continue to move in circular motion on the fixed ring until one