A classical mechanics problem by Miraj Shah

Classical Mechanics Level 3

Consider a body, attached to a string of length $r$ , to be initially at rest in the vertical plane. It can be proved that if we give the body an initial velocity of $v>\sqrt{5gr}$ , then, the body can make complete revolutions in the vertical plane about the point of suspension.

Let us consider that the body, initially at rest, is given a velocity $v<\sqrt{5gr}$ . If at this value of $v$ the body eventually hits the point of suspension then we can write:

$v^2 = (a+\sqrt{b})gr \; .$

Find $a\times b$ .

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