December 04, 2015 Filed in: PER Corner

Imagine a child, perhaps “Dennis the Menace,” who has blocks which are absolutely indestructible, and cannot be divided into pieces. Each is the same as the other. Let us suppose that he has 28 blocks. His mother puts him with his 28 blocks into a room at the beginning of the day. At the end of the day, being curious, she counts the blocks very carefully, and discovers a phenomenal law—no matter what he does with the blocks, there are always 28 remaining! This continues for a number of days, until one day there are only 27 blocks, but a little investigating shows that there is one under the rug—she must look everywhere to be sure that the number of blocks has not changed. One day, however, the number appears to change—there are only 26 blocks. Careful investigation indicates that the window was open, and upon looking outside, the other two blocks are found. Another day, careful count indicates that there are 30 blocks! This causes considerable consternation, until it is realized that Bruce came to visit, bringing his blocks with him, and he left a few at Dennis’ house. After she has disposed of the extra blocks, she closes the window, does not let Bruce in, and then everything is going along all right, until one time she counts and finds only 25 blocks. However, there is a box in the room, a toy box, and the mother goes to open the toy box, but the boy says “No, do not open my toy box,” and screams. Mother is not allowed to open the toy box. Being extremely curious, and somewhat ingenious, she invents a scheme! She knows that a block weighs three ounces, so she weighs the box at a time when she sees 28 blocks, and it weighs 16 ounces. The next time she wishes to check, she weighs the box again, subtracts sixteen ounces and divides by three. She discovers the following:

number of blocks seen + (weight of box − 16 ounces)/3 ounces = constant

There then appear to be some new deviations, but careful study indicates that the dirty water in the bathtub is changing its level. The child is throwing blocks into the water, and she cannot see them because it is so dirty, but she can find out how many blocks are in the water by adding another term to her formula. Since the original height of the water was 6 inches and each block raises the water a quarter of an inch, this new formula would be:

number of blocks seen + (weight of box − 16 ounces)/3 ounces + (height of water - 6 inches)/¼ inch = constant.

In the gradual increase in the complexity of her world, she finds a whole series of terms representing ways of calculating how many blocks are in places where she is not allowed to look. As a result, she finds a complex formula, a quantity which has to be computed, which always stays the same in her situation.^{2}

Now let’s break down Feynman's story and explore a new conceptual framework for the teaching of energy.

At this level of physics we don't worry about heating, radiation, and so on, but adding in another term would be a simple extension. This mathematical representation of energy has the form of a continuity equation, which is a very practical way to track energy and is pedagogically very desirable. It helps to connect mechanical work to the system energy, which was often a challenge for my students in the past. I remember them being able to perform separate calculations for work and system energy, but not having a clear idea whether the system was gaining or losing the energy (whether the energy was flowing in or out): they usually guessed. The work-energy equation with the block model helps students to visualize this: they can picture blocks of energy moving from object to object, or passing in or out of the system.

The diagram contains a lot of information, more than is typically needed when solving a problem, but provides the representational power to illustrate thorny situations like energy transfers due to sliding friction (coming up later, just wait!).

Notice how they are coded for the successive moments in time — a flow diagram can be created for any pair of moments. Being clear and precise about the selection of moments is critical for an accurate accounting of energy and its representation. This is not a trivial issue — just like knowing whether to close your fiscal books on December 31

One side of the bar chart represents the energy stored in system objects at the first moment in time, the middle bar represents the work done on system objects by external forces, and the second side represents the energy stored at moment two. Students construct bars for the relevant storage mechanisms and use these to decide how to construct a work-energy equation. This has been a great support for my students’ mathematical work. In the past they would haphazardly include mathematical terms in their work-energy equations or just memorize one instance of it and apply it to every situation (sound familiar?). The icing on the energy picture cake is that bar charts and flow diagrams allow students to cross-check their understanding of the energy processes that are present.

We later connect this to the mathematical definition of work, noting the angle between the force of the hand and displacement of the car. We approach the topic of energy just like all the other topics in physics: we introduce the concepts first and then construct the mathematical framework around it.

½

This statement reveals that we are tracking the energy stored in the book and Earth's gravitational field, as shown more carefully in the diagrams below.

These are the objects of the unspoken system. Regardless how clear the traditional approach is in our own heads, it is confusing for our students when we do not clearly state the system and take it seriously! (In my own experience, I now understand that the traditional approach was not clear, even to myself: I had never bothered making sense of the energies because I was getting mathematically correct answers. Mathematics can be a great impediment to understanding if the right conceptual tools are never taught.) I can remember when I taught the old way, my students always asking which side of the equation the term for work should go on. Without a clear sense of the system that underlies the equation, they were usually just guessing. Returning to the book example, suppose the book is moving at a constant velocity. In this case there is work done on the system, but the kinetic energy is not changing. This is an example where the net work-kinetic energy theorem is not valid since the system is not a single point particle. In my current teaching practice, I rarely use the net work-kinetic energy theorem — it is just a special case of the work-energy equation and does not deserve much emphasis.

As an interesting comparison, we can choose the book alone to be the system.

In this case there are two external forces doing work on the book, the applied force of the hand and the force of gravity of Earth. One force does positive work, the other negative work, yielding a total work of zero. The kinetic energy of the book does not change showing that the net work-kinetic energy theorem is valid. Notice the arrows in the two flow diagrams are the same as the previous for diagram with the original system (all the flows are the same). But note that the only system energy being tracked is that of the book. The gravitational interaction must be shown as negative work, rather than an interaction energy of the system.

½

Our understanding of forces tells us that the friction force between the car and the road surface causes the car to stop. The “energy equation” above suggests that the amount of work done is equal to the original amount of kinetic energy, but this is not correct. This equation is derived from Newton’s second law and is based on force ideas rather than energy ideas. To properly describe the flow of energy out of this system we must fully account for all the energy in the system (which Newton’s laws cannot do) by constructing a proper work-energy equation based on the conservation of energy.

½

The result looks similar, but we introduce an important term for the thermal energy that is present in the car the moment it stops. This equation tells us something very surprising: that the amount of work done by friction is less than the amount of kinetic energy (

The amount of work done by a friction force cannot be easily found from the definition of work unless you know true microscopic details of the surface particles in contact. As a result, there is no simple way to predict how much energy actually flows out of the system due to the friction force. In situations such as these, I instruct my students to include the surface as part of the system so in our accounting of energy, all the energy remains in the system (

Even though we can't find the work done by friction, we can find the transfer to thermal energy due to the friction force:

It is challenging to teach in a conceptually new way and has taken me many years to fit all these pieces together. You can take the shortcut: take a look at my grade 11 and 12 lessons in our physics handbooks and check out the videos of these lessons in action. All of this can be found on my website: www.meyercreations.com/physics. The door of my classroom of always open and you are most welcome to drop by and talk to my students about energy, just send me an email and we can set up a day.

2 Feynman, Richard P., Robert B. Leighton, and Matthew Sands.

3 https://en.wikipedia.org/wiki/Conservation_of_energy#History

4 Swackhamer, Gregg. "Cognitive resources for understanding energy." (2005).

5 https://en.wikipedia.org/wiki/Conservation_of_energy#First_law_of_thermodynamics

6 Scherr, Rachel E., et al. "Representing energy. II. Energy tracking representations."

7 https://en.wikipedia.org/wiki/Conservation_of_energy#History

8 Daane, Abigail R., et al. "Energy conservation in dissipative processes: Teacher expectations and strategies associated with imperceptible thermal energy."

9 Daane, Abigail R., Stamatis Vokos, and Rachel E. Scherr. "Goals for teacher learning about energy degradation and usefulness."

10 Chabay, Ruth W., and Bruce A. Sherwood. Matter and interactions. John Wiley & Sons, 2015.

11 Sherwood, Bruce Arne. "Pseudowork and real work."

12Sherwood, Bruce Arne, and W. H. Bernard. "Work and heat transfer in the presence of sliding friction."

13 Serway, Raymond, and John Jewett.

14 Jewett Jr, John W. "Energy and the confused student I: Work."