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Energy of place -- potential energy (2013)

Page history last edited by Ben Dreyfus 8 years, 11 months ago

 Working ContentEnergy: The Quantity of Motion

 

Prerequisites

 

Although the term "potential energy" is used freely in many science classes and in standard introductory physics textbooks, there are subtle issues with the concept that make it confusing and difficult to understand. These have to do with the question, "Who does the potential energy belong to?" The fact that we sometimes talk as if "energy" refers to a single object and sometimes to a collection of objects (or to the universe as a whole) can mislead us as to what we are really talking about.

 

Energy and work

In our discussion of Kinetic energy and the Work-Energy Theorem, we focused on a single object and looked at the implications of Newton's 2nd law for changing the speed of our object. Our result was the work-energy theorem:

 

It was sensible in that it said if we were considering an object's speed, then only the parts of the forces it felt along (or against) the direction it was moving mattered. The more technical result was that a reasonable quantity to describe the amount of an object's motion (without consideration of direction) was its kinetic energy -- ½mv2. And what changed the kinetic energy was the work done on the object by the forces it felt -- the component of the force times the change in position.

 

Now our next step is to see some of those forces-times-distance as a new kind of quantity -- an energy of place or potential energy.  (Potential energy can be represented by the symbol U.)  It's called potential energy because it is a way of storing motion in a way that it can be retrieved later. For example, if we have a ball rolling on a horizontal track, it has kinetic energy.  If we now let it run onto a part of the track that curves up, as the ball heads up the track it slows down, thanks to the force of gravity. If the track goes high enough, the ball will stop. If we hold it there at rest, we can later recover the motion by letting it roll down. We have three of our forces that we can do this with: our forces of gravity, electricity, and springs. We'll figure out how to do this quantitatively in the follow-on pages, but first let's be more explicit about what we are doing.

 

Potential energy for a one-object system

In order to get the hang of how potential energy works and how we can think of energy transformations in terms of a conservation law, we typically begin by considering the motion of a one-object system subject to external forces. For the three forces for which it works, we will rewrite the work that the object experiences as a new kind of energy -- a potential energy. This makes it look like the potential energy belongs to the object we are talking about.  And we'll sometimes find that a useful language.

 

But we need to be very careful not to get confused! Every force is an interaction between two objects and therefore every potential energy really belongs to the interaction of the objects -- not to one of them!  We will keep coming back to this point.

 

When we have simple forces where we can essentially ignore the other object's motion, then we can get away with treating the PE as if it belongs to a single object. This works fine for flat-earth gravity, since the Newton's 3rd law pair of the objects we consider on the earth don't move the earth significantly. It works pretty well for springs if our springs don't have much mass and we can ignore any kinetic energy they might have. And for some electrical systems we can get away with it when a lot of fixed charges are producing the forces we are looking at. Later, when we look at chemical interactions of atoms, we will have to be careful to realize that our potential energies belong to the interaction, not to one or another of the interacting objects.

 

The way it works

The W-E Theorem looks at a change -- the change in KE -- and shows how it is produced.  This means that we have an initial situation and a final situation.  The KE only looks at the initial and final velocity:

 


 

In our three special cases, the work done by a force looks like a change in something also. This is because the total work done by these forces only depends on where you started and where you ended. The work looks like a change in some function that only depends on the starting and ending point -- like this: (we put a minus sign in our definition of U so the final result looks nicer)

 

 

then the WE Theorem takes on a very nice form:

 

This is a conservation law! Something stays constant when our object moves.  This turns out to be immensely valuable in figuring out lots of stuff -- and extends our concept of energy.

 

In the follow-ons we look at just how this actually plays out in our three cases and why what seems so strange in the abstract is actually very natural.

 

Follow-ons:

 

Joe Redish 10/30/11

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