In order to understand why Work corresponds to a negative change in Potential Energy, we can construct an closed system, and show the formula holds. We can use different forms of potential energy, and still end up with the same result, which is proving that:
- = Work done by a conservative force
- = Change in potential energy
The tricky part is understand that energy is a relative concept. Kinetic energy is defined to be zero when a body is at rest (i.e. ). From this rest motion, where , any movement results in a change in velocity, and therefore . What about potential energy? When is potential energy defined to be 0?
We need to also ensure our system strictly involves potential energy, so we need to ensure the object moves, but does not accelerate. Using Newton’s second law of motion: , we need to ensure so that no acceleration, and therefore no kinetic energy happens in the system.
Some simplifications
From Newton’s third law of motion, all forces inside a system exist as force-pairs, so forces will exert on all objects in the system. However, for this derivation, we’re going to treat one of the bodies as static, which means it is unaffected by forces.
Secondly, you’ll see further below that we have configurations of systems in which bodies are displaced. Since they started at rest, there is some velocity and as such, some Kinetic Energy in the system. However, for our purposes, assume the velocity is , i.e. a miniscule amount, enough to cause displacement, but have