A stone of mass 1 kg tied to a light inextensible string of length L = 5 3 ~m is rotating in a circular path of radius L in a vertical plane. If the ratio of maximum tension in the string to the minimum tension in the string is 3 , the speed of the stone at the highest point of the circle is ( g = acceleration due to gravity)

Step-by-step solution

The forces acting on the stone at the highest point yield the minimum tension, satisfying T_ + mg = mv_H^2 L , where v_H is the speed at the highest point. Rearranging gives T_ = mv_H^2 L - mg . At the lowest point, the centripetal force requires T_ - mg = mv_L^2 L , giving T_ = mv_L^2 L + mg . By conservation of mechanical energy, taking the lowest point as zero potential energy, 1 2 mv_L^2 = 1 2 mv_H^2 + 2mgL . Multiplying by 2/m yields v_L^2 = v_H^2 + 4gL . Substituting into the expression for T_ : T_ = m(v_H^2