Question

show that minimum velocity required for a particle to loop a loop while going in verticle circle of radius r is underroot 5gr

Answer 1

for a particle to revolve in a vertical loop, the minimum velocity that the particle has at the top of the circle must be just 0.  The tension in the string attached to the particle must be just above 0. .  The centripetal force of the particle  at the top of the circle must be just equal to the weight.
         m g = m v2² / r         ,  v2 = velocity of particle at the top

KE of the particle at the bottom of the circle = initial KE = 1/2 m v²
         v = initial velocity
 applying the conservation of energy at the top of the circle:
    1/2 m v² =  m g (2 r) + 1/2 m v2²
               = 2 m g r + 1/2 m g r

  simplyfying, we get :  v²  = 5 g r    and      v = √(5gr)