A

To solve the problem, we can apply the principles of energy conservation and circular motion.
1. As the particle slides down from point A, it converts gravitational potential energy into kinetic energy.
2. The height difference between point A and point B can be found using the radius R and angle θ: height = R(1 - cos(θ)).
3. The potential energy at point A is given by PE = mgh = mgR(1 - cos(θ)).
4. The kinetic energy at point B is KE = \frac{1}{2}mv^2, where v is the velocity at point B.
5. By conservation of energy: \( mgR(1 - cos(θ)) = \frac{1}{2}mv^2 \)
6. Rearranging gives us v = \sqrt{2gR(1 - cos(θ))}.
The angle θ at which the particle breaks off will be determined by the balance of forces acting on it, which includes gravitational and normal forces. Setting up the equations for circular motion leads us to express whether the normal force becomes zero at point B. Thus we can find an expression for θ.
Overall, using these principles should lead to the appropriate computation for angle θ.