In a vertical circular motion, a particle of mass m moves in a circular path in a vertical plane. The motion is sustained by the tension in the string or rope that keeps the particle moving in a circular trajectory.
If the velocity of the particle is increased, there are two critical points at which the string could potentially break:
At the top of the circular path: When the particle is at the highest point of its motion, the tension in the string is at its minimum. If the velocity is increased to a point where the tension in the string becomes insufficient to support the particle's weight, the string could break at the top.
At the bottom of the circular path: When the particle is at the lowest point of its motion, the tension in the string is at its maximum. If the velocity is increased beyond a certain threshold, the tension in the string may exceed its maximum strength and cause it to break at the bottom.
To determine which point the string will break, we need to consider the balance between the gravitational force acting on the particle and the tension in the string. If the velocity is increased gradually, the tension in the string will also increase to counteract the particle's increased centrifugal force. However, there is a limit to how much tension the string can withstand before breaking.
If the velocity is increased beyond that limit, the string will break either at the top or the bottom of the circular path, depending on the specific conditions and properties of the system, such as the strength of the string and the mass of the particle.
It's important to note that the exact point at which the string breaks will depend on various factors, including the specific values of velocity, mass, and the properties of the string. Additionally, factors such as the presence of external forces or variations in the tension of the string during the motion can further complicate the analysis. A detailed analysis of the system's dynamics would be required to determine the precise point of breakage in a specific scenario.