The motion in the vertical circle involves the concepts of centripetal force ,conservation of energy and the kinetic energy. its all about the constraint that is in simple words a system of parameters that the system must obey. eg. a stone sliding down through mountain. so motion in a vertical circle means that constraint of motion in a vertical circle must be equal to the centripetal of remained circle and when the mass moved down it must be satisfied with the demands of conservation of energy as gravitational potential energy when converted to kinetic energy .

Velocity and tension in string when body moves in Vertical Circle.

Energy at Point P = Energy at Point L

1/2 m v sq. + m g h =1/2 m u sq.

Hence v sq. + 2gh = u sq.

Hence v sq. = u sq. - 2gh

Hence v = underroot u sq. - 2gh ...............................( 1 )

Hey Mate..

Consider the centripetal force at Point T

T - mg Cos0 = mv sq. / r ......... ( 2 )

Hence mg Cos0 + mv sq. / r

Cos0 = ( r - h ) / r

Subsituting in eq ( 2 )

T = mg ( r - h ) / r + mv sq. / r

Hence T = m / r ( g ( r - h ) + v sq. )

But v sq. = u sq. - 2 gh

T = m / r ( gr - gh + u sq. - 2 gh )

T = m / r ( u sq. - 3 gh + gr )