Motion in a vertical circle
The cases of a bead on a wire which is curved into the shape of a vertical circle, and that of a particle moving inside a horizontal cylindrical pipe, a particle on a string, and a particle on the top of a smooth hemisphere, are all essentially the same.
When thinking about the reaction force on the particle on a wire this can be positive or negative. The only difference with the other cases is that the constraints don't allow the force to change sign.So the point where the particle leaves the hemisphere or pipe, or where the string becomes slack, are defined by the point where the reaction force is 0.
In the case of the particle inside a smooth pipe, you would analyse the reaction force as pointing inwards. If this ever becomes negative, then this means that the particle has lost contact with the surface, as the surface can't pull the particle towards it. In the case of the particle on top of a hemisphere, then you would analyse the force as pointing outwards, and again a negative value means the particle has left the hemisphere. If the particle is on a string, then a negative tension would mean that the string is pushing the particle, which again it cannot do.
Step 1 to find velocity as a function of position you use conservation of energy
,If the circle radius is r, the mass is m, velocity v, and the angle measured from the vertical is θ, then we have:
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