A body of mass m is tied with a string and is rotated in a vertical circle of radius r. Find the minimum velocity at lowest point that should be given to the mass so that it completes the loop in vertical circle without slackening of the string.

Let at lowest point, the mass be given the velocity u.
The velocity V of mass when string subtends an angle of 0 with vertical is,

The condition for the mass to complete the vertical loop without slackening the string is that the tension in the string at every point should be greater or equal to zero.
The tension in the string is minimum at highest point of loop.
Therefore, tension at highest point should be greater or equal to zero.
i.e.                 
                                 ...(1) 
At highest point,  therefore velocity at highest point is,
                 
                                 ...(2) 
Substituting in equation (1), we get 
                     
                     
                              
                              , is the minimum velcity given to the mass at the lowest point.