A particle of mass m is tied to a light string of length l having its other end fixed. The string is initially held horizontal, and the particle is projected with an upward velocity v . The particle is just able to complete a circle.

Step-by-step solution

Let the fixed end of the string be the origin. The initial position of the particle is at a horizontal distance l from the fixed end. The particle is projected upwards, so it travels a vertical distance l to reach the highest point of the circular path. For the particle to just complete the vertical circle, the tension in the string at the highest point must be exactly zero. This means the string momentarily becomes slack at the highest point. At the highest point, the centripetal force is provided entirely by the