A uniform rod of length 2a has one end smoothly pivoted at a fixed point O. The other end is connected to a fixed point A, which is a distance 2a vertically above O, by a light elastic spring of natural length a and modulus ½ mg. The rod moves in a vertical plane through O. Show that there are two equilibrium positions for the rod, and determine their stability.
[The vertically upwards position for the rod would compress the spring to zero length and is excluded.]