Think of it in terms of tangents, at point III, the lowest point with respect to the diagram, the centripetal force points towards the center of mass of the flywheel, but the "centrifugal" force points outwards and wants to push the ant outwards. To keep the ant on the flywheel, the weight and the "centrifugal" force have to be added together, and that force keeps the ant on the flywheel at point III. So you have mg + (mv^2)/r
