Halliday, Resnick & Walker — Potential Energy and Conservation of Energy: Problem 87_SSM

A massless rigid rod of length \(L\) has a ball of mass \(m\) attached to one end (Fig. 8.51). The other end is pivoted in such a way that the ball will move in a vertical circle. First, assume that there is no friction at the pivot. The system is launched downward from the horizontal position \(A\) with initial speed \(v_0\). The ball just barely reaches point \(D\) and then stops. (a) Derive an expression for \(v_0\) in terms of \(L\), \(m\), and \(g\). (b) What is the tension in the rod when the ball passes through \(B\)? (c) A little grit is placed on the pivot to increase the friction there. Then the ball just barely reaches \(C\) when launched from \(A\) with the same speed as before. What is the decrease in the mechanical energy during this motion? (d) What is the decrease in the mechanical energy by the time the ball finally comes to rest at \(B\) after several oscillations?