A wire lying in the vertical plane ins bent in the shape of a cardioid given by the equation where a>0, and where is the angle between the downward vertical and the radius vector. A small ring of mass m slides along the wire, and is attached to the origin by an elastic string of natural length a and modulus 2kmg, where k is a positive constant. The ring is released from rest when the string is horizontal, that is, when .

a) Write down the total energy of the system

b) Show that is an equilibrium position and determine its stability.

c) Show that when k=2, there exists another equilibrium position, which is stable. Find its period.

At the moment, I am still stuck on a). I have written down the conservation of energy equation, that is :

but cannot think of what to make F(r). I have also looked at b) and have the same problem. Any help?