4.36 A metal ball (mass m) with a hole through it is threaded on a frictionless vertical rod. A massless string (length 7) attached to the ball runs over a massless, frictionless pulley and supports a block of mass M, as shown in Figure 4.27. The positions of the two masses can be specified by the one angle . (a) Write down the potential energy U(0). (The PE is given easily in terms of the heights shown as h and H. Eliminate these two variables in favor of 0 and the constants b and 1. Assume that the pulley and ball have negligible size.) (b) By differentiating U(0) find whether the system has an equilibrium position, and for what values of m and M equilibrium can occur. Discuss the stability of any equilibrium positions.

4.36 A metal ball (mass m) with a hole through it is threaded on a frictionless vertical rod. A massless string (length 7) attached to the ball runs over a massless, frictionless pulley and supports a block of mass M, as shown in Figure 4.27. The positions of the two masses can be specified by the one angle . (a) Write down the potential energy U(0). (The PE is given easily in terms of the heights shown as h and H. Eliminate these two variables in favor of 0 and the constants b and 1. Assume that the pulley and ball have negligible size.) (b) By differentiating U(0) find whether the system has an equilibrium position, and for what values of m and M equilibrium can occur. Discuss the stability of any equilibrium positions.