What you Already Know
In the last chapter we did essentially three things:
1. We defined kinetic energy which is energy due to motion:
2. We defined work:
3. We established the work-kinetic energy principle:
In reading the last chapter, one thing that I hope came across, is that when dealing specifically with gravity, that the work done by gravity is a path-independent quantity. To remind you what this means, we calculated the work that gravity would do in three cases:
1. When an object is dropped vertically by a distance h.
2. When an object slides without friction down a ramp and loses height h.
3. When an object descends along a helical path and loses height h.
Do you recall how those three results compared? If you don't, it might be a good time now to revisit those calculations. If you don't feel you have the time to do that right now, here's the answer: They are all the same!
The only thing that matters when we want to know work done by gravity is the initial height and final height of the object. The work done by gravity was always The answer does not depend on the path taken. This path-independence certainly begs a question: Why should I ever go through the trouble of integrating along some path as we did in the last chapter when I keep getting the same result of The answer is that there is no good reason!
Gravity, however, is a special kind of vector field. If you recall the last chapter, we specifically called it a conservative vector field. Other conservative vector fields exist in nature - like the electrostatic force field that holds electrons and protons together, and some other forces we'll discuss.
We will see in this chapter that it is more efficient to associate conservative forces with potential energy expression than to keep doing work integrals that always return the same expression. The potential energy really is just a way of accounting for the work integral that keeps giving the same result regardless of path.
Last comment: It is important to note that some forces and associated fields in nature are not conservative. Examples are air resistance, sliding friction, magnetic fields, etc. We will see that it is not possible to define a potential energy for such forces.