A particle is rotated in a vertical circle by connecting it to a string of length I and keeping the other end of the string fixed. The minimum speed of the particle when the string is horizontal for which the particle will complete the circle is

Let the particle speed when the string is in vertical plane be v m/sec. Here, it is balanced by its own weight (m× g). When the particle is rotating in a circle centripetal force acts towards the center of the circle. Therefore, we have Now, let the speed of the particle when the string is horizontal be u m/sec. Total energy in horizontal plane = Similarly, total energy in vertical plane = u =

1