Energy of place -- potential energy

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:

$$\Delta \big({1 \over 2} \;  m_A v_A^2 \big) = \overrightarrow{F}_{net\;on\; A} \cdot  \overrightarrow{\Delta r}_A$$

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  ${1 \over 2} \;  m v^2$. And what changed the kinetic energy was the work done on the object by the forces it felt — the component of the force in or against the direction of motion 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.  We'll represent potential energy can be represented by the symbol $U$.  It's called potential energy because it is a way of storing motion in the locationa 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 can do this with three of our forces:  gravity, springs, and electricity. 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 an external force from another fixed object. (You might think of a mass moving in response to the earth's force of gravity.)

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. 

When we can consider one of the objects fixed, it makes it look like the potential energy belongs to the moving object.  And we'll sometimes find that a useful language. But 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! It is sometimes useful to talk as if "potential energy" refers to a single object ("the gravitational potential energy of a falling body" or "the potential energy of an electron in an atom") and sometimes to a collection of objects (or to the universe as a whole). This can mislead us as to what we are really talking about if we are not careful.

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:

$$\Delta \big({1 \over 2} \;  m v^2 \big) = {1 \over 2} \;  m v^2_{final} - {1 \over 2} \;  m v^2_{initial} $$

In our three special cases, the work done by a force looks like a change in something — a function $U(\overrightarrow{r})$ [or $U(x,y,z)$] that depends on the position of the object (better, on the relative position between the two interactingobjects). 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)

$$\overrightarrow{F}_{B \rightarrow A} \cdot  \overrightarrow{\Delta r}_A = -\Delta U = U(\overrightarrow{r}_{initial}) - U(\overrightarrow{r}_{final})$$

then the WE Theorem takes on a very nice form:

$$\Delta \big({1 \over 2} \;  m v^2 \big) = -\Delta U$$

$$\Delta \bigg({1 \over 2} \;  m v^2  +\Delta U \bigg) = 0$$

or, since there is no change

$${1 \over 2} \;  m v^2_{initial}  + U(\overrightarrow{r}_{initial}) =  {1 \over 2} \;  m v^2_{final}  + U(\overrightarrow{r}_{final}) $$

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. See the various examples attached to the specific definitions of potential energy in the follow-ons.

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.

Joe Redish 10/30/11