Chapter 12, Problem 45P

A rectangular block of mass m measures w × w × L, where L is the longer dimension. It’s on a horizontal surface, resting on its long side, as in the left-hand block in Fig. 12.8. (a) Taking the zero of potential energy when the block is lying on its long side, find an expression for its potential energy as a function of the angle θ that the long dimension of the block makes with the horizontal, starting with θ = 0 in the left-hand configuration of Fig. 12.8 and continuing through the upright position shown at the right (θ = 90°). (b) Use calculus to find the angle θ where your function has a maximum, and check that it agrees with the answer to Problem 37. (c) Use calculus to show that this is a point of unstable equilibrium.