A classical mechanics problem by Miraj Shah

Classical Mechanics Level 4

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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