Derivation of the Work/Energy Principle

In my previous post on methods for particle dynamics, I stated that, by integrating Newton’s Second Law over a distance, we could make use of ‘Work/Energy’ methods. This is true as far as it goes, but is certainly not the whole truth. It’s important to realize what’s actually happening, lest we fall into a trap of some sort.

Newton’s Second Law, is, of course, a vector equation:

 So I have to be a little more specific when I say 'Integrate over a distance.' Start tossing around vectors willy-nilly, and you'll end up with results that don't make any sense. Anyway, there are two main questions regarding what needs to happen here:

Acceleration, Energy, and Momentum

In any Dynamics class, 3 main methods of analysis will be taught -- Acceleration, Energy Methods, and Impulse/Momentum. Within the class, it’s pretty easy to figure out which method the professor wants: If we’re in Chapter 4, it must be Energy, right?

What about later on? While I’m still only through the first-level Dynamics course in my curriculum, I do assist others with the class occasionally (I’m a walk-in tutor for some of the 200/300 level engineering courses here), and haven’t yet developed a good set of rules to delineate between the methods. Here’s a first effort, hopefully I can add more in the future.