CONCEPT MAP

The concept map for this unit traces one continuous chain of reasoning. It starts from work, defined as the dot product of force and displacement, and shows this is equivalent, via the work-kinetic energy theorem, to a change in kinetic energy; for the special class of conservative forces, work done is also equivalent to a change in potential energy (gravitational or elastic). Kinetic energy and potential energy together make up the total mechanical energy, whose conservation for an isolated system is the single most powerful tool in the unit -- applied here to the specific, classic case of motion in a vertical circle, where energy conservation combines with Newton's second law to fix the minimum speeds needed at the top and bottom of the loop.

A second branch defines power as the rate of doing work, $P=dW/dt=\vec F\cdot\vec v$ -- capturing not how much energy is transferred, but how fast. …