A small particle attached to a string of length L is given just enough velocity at its lowest point to complete a vertical circle. The magnitude of the change in its velocity as it moves from the position where the string is horizontal to the highest point of the circle is x gL . The value of x is

Step-by-step solution

For a particle to just complete a vertical circle, the tension at the highest point must be zero. This gives the critical velocity at the highest point as v_ top = gL . Applying conservation of mechanical energy between the highest point and the horizontal position: 1 2 mv_ horiz ^2 = 1 2 mv_ top ^2 + mgL v_ horiz ^2 = gL + 2gL = 3gL So, the speed at the horizontal position is v_ horiz = 3gL . The velocity vector at the horizontal position is directed vertically (say, j ), and at the highest point, it is directed h