The Work Energy Principle is one of the big ideas in introductory physics courses. It's so big that the textbook presentation can get a little confusing - but it doesn't have to be that way.
How Do Textbooks Introduce the Work-Energy Principle?
I haven't looked in all the introductory textbooks, but it seems like they all follow a similar style. Oh, this is for the algebra-based physics course. That means no integration, no dot products.
Here is how they do it (roughly).
Conservation of Energy. Many texts start off with some type of statement like "energy is neither created nor destroyed".
Types of Energy. There are many different types of energy: kinetic, potential, thermal energy, chemical energy.
Definition of Work. Work is defined as the ability to change energy. I know that sound silly, but sometimes textbooks make a circular definition like that. They then go on to define work in some way. Usually, it will look something like this:
Just so you know, that's a fine definition of work.
Non-Conservative Work. This is the part that most textbooks strive for. This is the version of the work-energy principle.
Non-conservative work is a work that depends on the path. Conservative work is path independent. A great example of non-conservative work is the work done by friction. Suppose I push a block along a surface with friction from point A to point B along the two paths shown.
The work done along path 2 will be greater than path 1. However, if this was work done by gravity (no friction) then the work done along the two paths would only depend on the starting and ending points. Gravity is conservative, friction is non-conservative. Why does this matter? Well, it turns out that for any conservative forces (like gravity, springs, electrostatic) you could make that work a potential instead of a "work done by". That's usually how a textbook explains it - perhaps it's not the best description.
Special Cases. What about special cases where the work (non-conservative work) done is zero? In these cases, we can just say there is constant energy. Pick any two points in space and the following would be true:
This isn't wrong, but it is just for the special case where the work is zero.
A Different Approach
Why do we need a different approach? I think the above presentation is a little bit disjointed and confusing. Here is the way I present it in class. First, two notes. My views on work-energy are heavily influenced by the Matter and Interactions textbook (which I think is awesome). Second, it can cause a small problem when your approach isn't the same as the textbook.
What is energy? Energy is just a way to view the world. The work-energy principle is a mathematical tool that works very well at predicting and explaining real world phenomena. That's it. The work-energy principle is just something that works (pun intended).
The simplest version of the work-energy principle is for a single point particle. The above definition of work is still fine, but in the case of a point particle the work-energy principle is:
That's it. A point particle can only have kinetic energy. Note: in Matter and Interactions, this would be W = ΔE where is the energy of a particle. This version is different in that it includes an energy definition that works at relativistic speeds also.
It's all about the system. If you want a potential energy, you need to pick a system that includes more than just a mass. Let's consider a ball released from rest near the surface of the Earth that falls a distance h.
If I choose a system just consisting of the ball (which is sort of like a point mass) then I can look at the work done on this ball as it falls. What forces are acting on the ball? Just the gravitational force (mg). Since the gravitation force is in the same direction as the displacement, the angle between these two is zero. I can write:
From here, you could solve for the velocity at the bottom position. Not too difficult.
What if I change my system to include both the ball AND the Earth? In that case, I can subtract the work done by the gravitational force from both sides of the equation. I would get this:
Algebraically, this is the same equation as before. However, this says that there is no work done on the system and instead we have a change in gravitational potential energy (U). The change in potential is then defined as the negative of the work done by that force. This is technically the gravitational potential energy of the ball-Earth system. In the end, you would get the same expression as before (with the system of just the point particle).
Be careful. You can't have work done by gravity AND a change in gravitational potential energy. You have to do it one way or the other.
This means that the most important step in solving work-energy problems is choosing a system. For internal forces (like gravity) in a system, you will have a potential energy term.
What about those special cases of conservation of energy? Yes, they can be useful at times - but you have to be careful to realize that they are just special cases.
Summary
As I read over this post, it seems like I am that guy from Spinal Tap that tries to explain why his amplifier is better because the dial goes to 11. Yes, it might seem like I am basically saying the same thing as the textbooks. Let me emphasize the key points:
- If you are talking about work but not a system, you are missing something.
- You can do just about all of the basic problems in intro physics by picking a point particle as your system and having all the forces on that particle do work. You don't even need potential energy.
- If you try to use energy = constant for some situation, be very careful. This is only true for some cases (not always true).
