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Energy of place -- potential energy (2012)

Page history last edited by Ben Dreyfus 10 years, 7 months ago

Class content > Energy: 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 consider to describe the amount of an object's motion (without consideration of direction) was its kinetic energy -- ½mv2. And that what changed that was the work done on the object by the forces it felt -- the component of the force times the change in position.

 

Our next step is to see some of the work done by those forces as a new kind of quantity -- an energy of place or potential energy.  It's called potential energy because it is a way of storing motion in a way that it can potentially 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 be letting it roll back down. We can do this with three of the forces that we we have considered so far: our forces of gravity, electricity, and springs. We'll figure out quantitatively how to do this 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 re-write 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  involves an interaction between two objects. So when we look at object A for example in the system schema on the right, it experiences a force due to a contact interaction with object B, and another force due to a contact interaction with object C.  Potential energies can't be assigned to a single object - they have to be assigned to interaction arrows.  In other words, in the system schema on the right you might be able to calculate a potential energy for the interaction of A and B, and a potential energy for the interaction of A and C.  The system schema then shows us why the connection between potential energy and forces can be tricky:  The potential energy for the interaction between object A and B is connected to the interaction between A and B, and so connects BOTH with the force that B exerts on A, and also the force that A exerts on B. So a a single potential energy could generate forces and motion for two objects! (Think, for example, of a spring connecting two masses.)


 

There are some very important cases where we can essentially ignore the other object's motion. In that case the PE would only generate motion for one of the two objects!  Then we can calculate the potential energy (PE) in a way that sometimes makes us forget that potential energy is based on a pair of objects: 

  • flat-earth gravity --  In this case the earth does not move significantly but still its important to remember that the earth is the second object.  
  • fixed springs -- in this case the spring is the second object, but if one end is fixed to a large object like a wall and if the spring has a small mass, we may ignore its motion and assign the spring PE to the moving object attached to the free end. 
  • some electrical systems -- In some cases charges that are fixed in some way are producing the forces we are looking at so we don't have to consider their motion. 


In these three cases the PE is also easier to calculate because the second object does not move. 

 

Later, when we look at chemical interactions of atoms, we will have to be careful to realize that the second object can move as well.  In that case the distinction we are learning now will be very important:  potential energies belong to the interaction, not to one or another of the interacting object.  So the PE generates motion for both objects and each of the objects will move less than it would if it could take all the PE for itself!

 

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, (where the other objects do not move) the work done by a force that also looks like a change in something. 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)

 

 

If forces of this type are the only work we have to consider, then the WE Theorem takes on a very nice form:

 

This is a conservation law! Something -- the kinetic plus potential energy -- 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. We refer to this (limited) form of the conservation of energy principle as the conservation of mechanical energy. We will consider later under what conditions this limited (and useful) form of energy conservation holds.

 

In the follow-ons we look at just how this actually plays out in our three cases.

 

Follow-ons:

 

Joe Redish 10/30/11

Vashti Sawtelle 11/16/12

Wolfgang Losert 11/17/12

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