Conservative Forces and the Area Under the curve of a (x,F) graph

This slide shows how to get the work done (and therefore the potential energy) of a conservative force: If you can plot a (distance, Force) graph, all you do is take the area under it. This is what in calculus is called an integral.

 

This slide finds spring potential energy using two different methods- a simple geometric method (which works since this area is just a triangle) and the integral of a polynomial function.

 

Note that the left part of this slide also shows that the change in potential energy is the negative of the work done by the conservative force. This often confuses students. It is because the energy added to the system doesn't come from the conservative force- it comes from whoever is pushing the thing against that force. If it weren't for the conservative force, then this energy would go into kinetic energy (see next slide.) The conservative force doesn't allow the thing to move faster, it turns the additional energy into stored (potential) energy and hence the negative.

 

Note the distinction that this slide is careful to make between U- the potential energy at any one point on how far a string is stretched and deltaU, the change in potential energy as a spring stretches. The calculus makes this distinction pretty clear. We'll see soon that the change is much more important than the energy in any one location.

 

Thanks to a student in 3rd Hour in 2015-16 who took this photo

Done