Consider the motion of a small body tied to one end of the string and whirled in a vertical circle. At any time, the body is at point P with the angular position θ. The forces acting on the body are

• Weight mg of the body acting vertically downwards
• Tension T of the string acting along PO

The weight mg can be resolved into two rectangular components T cos θ opposite to T and T sin θ along the tangent to the circle at P.

Net force on the body at P, acting along PO = T – mg cos θ

This is provided by the centripetal force which is mv² /r. Hence, T – mg cos θ = mv² /r; T = mv²/r + mg cos θ

T will be minimum when θ = 180⁰ , the body is at the highest point and  T_min = mv²/r – mg

Looping the circle condition:

The body will loop the circle only if T_min >= 0

                                        mv_A ²/r – mg >= 0

mv_A ²/r >= mg which means v_A >= (gr)^1/2

The minimum value of velocity at the highest point is (gr)^1/2

Similarly, at the lowest point L, applying the conservation of energy ½ mv_B² = ½ mv_A² + mg (2r)

Solving the above, the minimum value of velocity at the lowest point is  we get v_B >= (5gr)^1/2

Oscillation over the arc:

Here, the velocity becomes zero before T vanishes. Hence, the body oscillates.

½ mv_B² <= mgr or v_B <= (2gr)^1/2

The value of θ should be between 0 and 90⁰ and 0 < v_B <= (2gr)^1/2

Leaving the vertical circle:

The value of θ should be between 90⁰ and 180⁰ which is given by 90⁰ < θ < 180⁰

At this point, Tension is zero but velocity is not equal to zero.

(2gr)^1/2 < v_B <= (5gr)^1/2