Vertical Circular Motion

Beyond horizontal circular motion, many real situations instead involve VERTICAL circular motion, and these split into two genuinely different categories. In a CONTROLLED vertical circular motion -- a giant wheel or a similar fairground ride -- the speed is either deliberately kept constant by a motor, or is at least not governed purely by gravity alone. The second, more interesting category is vertical circular motion controlled ENTIRELY by gravity: here some initial kinetic energy is supplied once, typically at the lowest point of the path, and from that point onward the entire subsequent motion is dictated purely by the gravitational force, with kinetic energy and gravitational potential energy continuously converting back and forth into each other as the object rises and falls around the circle. This second category -- explored for a bob on a string in section 1.4.1, for the circus 'sphere of death' in section 1.4.2, and for a vehicle cresting an over-bridge in section 1.4.3 -- is genuinely more subtle than the horizontal cases of section 1.3, because unlike uniform circular motion, the SPEED here keeps changing throughout the motion (it is fastest at the bottom and …