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Derive an expression for the minimum horizontal velocity to be given to a ball hanging vertically from a point so that it is able to just complete a vertical circular path.

While pushing, we are giving the ball some kinetic energy which will be converted to potential energy as it moves upward. If the whole kinetic energy is converted to potential energy at the top most point, it will fall straight down, resulting in only a semicircle. For the ball to continue the path, it should have a centrifugal force equal to weight of the body when it is at the topmost point.

let V be the initially applied velocity and v be the velocity at the topmost point

mv²/r = mg

v²/r = g

v² = rg

v = √(rg)

Potential Energy at the topmost point , Ep = 2mgr

Ep is the difference in kinetic energy between initial point(Ei) and at topmost point(Et).

                Ei – Et =  Ep

½mV² – ½mv² = 2mgr

      ½m(V²-v²) = 2mgr

            V²-v² = 4gr

                           V² = 4gr + v²

                          V² = 4gr + rg

                V² = 5rg

                  ========

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