One of the most important ideas I want my students to recognize is that so much of Physics (and, indeed, science) can be understood by thinking deeply about energy. Today's lesson moves us from thinking about charges and electrostatic forces to the role that energy plays in this realm. In addition to being a central idea in my course, the concept of energy is sprinkled throughout the NGSS and, by keeping it foremost in our minds, it will open doors to several other ideas (electric potential energy, DC electronics, etc.) and to a major investigation later in this unit.
It is, however, a bit of a challenge to introduce these ideas at this moment in the course and, to be successful, I leave the micro-world for some time and build up some understanding at the macro-level. After checking in with a warmup that features electrostatic forces, I spend the rest of this lesson developing ideas about the concept of "work."
As a follow-up to the previous class, I present my students with a warmup problem that simply requires them to apply Coulomb's Law. The task is intentionally simple and provides me with an opportunity to address any issues with scientific notation numbers or misunderstandings of the formula. Indeed, the solution shows how I call attention to the scientific notation aspect to the problem. Though I want students to be able to use their calculators to properly manipulate these numbers, I also want them to realize that one can use the exponent rules and some simple mental math to arrive at a very reasonable estimate. I show my estimate (60 Newtons), elaborating on my use of the exponent rules, then ask students for the more precise answer (57.6 Newtons).
Though the task is easy, the implications of this particular problem are profound. The electric forces acting upon the protons are huge and imply that the nucleus of the Helium atom (and, by extension, all atoms) should be blowing itself apart due to repulsive electrical forces. This result allows me to share some thoughts with students about the strong and weak nuclear forces and to make broader points about forces in the universe.
My goal today is to get students to understand the concept of "work." My plan is to develop this idea over some time, introducing a simple version today and ending with a discussion of how to handle a more complex problem by the end of class.
I want to continue to use the language of "energy" as we begin so I ask my students to consider an image showing two charges and to imagine how much energy it would take to move them away from one another. This is meant to be qualitative and may include comments about distance, strength of charges, and the possibility that, as the charges are moved further apart, one would need less and less force. This is a complex situation about abstract properties (charges are essentially invisible and the idea of moving them is doubly mysterious) so, before dealing with it, I want to have students work with a more familiar situation - working against gravity. So, I introduce a simpler problem: one featuring gravitational forces.
Though this is a major shift, my intention is to increase comfort with "work" by moving from the micro-world of charges to the macro-world of rocks, apples, etc. In addition, as students consider the energy needed to lift these macro-world objects, the language of "work" becomes more obvious - on some level students already know that one has to do work to lift an object. Finally, it's easier to have students appreciate the need for replenishing a person's energy should the work of lifting continue for some time.
All of the above ideas are featured in a large-group discussion about lifting an object. I supply students with a general definition of work as the product of forces and distance, then specify how to calculate that force at the Earth's surface. Naturally, there are several new ideas associated with this but it's my hope that, by making the situation more concrete and recognizable, students will be able to absorb this new information quickly. To test that assumption, I provide a set of practice problems in the next segment of class.
I provide a set of practice problems for students to try. I'm looking for comfort with applying the definition of work. I circulate and "teach between the desks" providing individual students with precise and targeted feedback. In particular, I want students to recognize two important ideas. First, I want them to recognize the difference between masses and forces - the first two problems LOOK very similar though one is provided information about a mass in one problem and a force in the other. Second, I want them to realize that, based on today's lesson, if one knows both the force applied and the distance covered, one can very simply calculate the work.
This is an exercise that one might call "assessment FOR learning." The actual calculations are trivial but the real learning happens as students decode the problem to identify the critical bits of information. After 10-15 minutes, I ask students to review answers with me at the board. This is an excellent time to provide affirmation for those who are successful and clarification for those who struggle.
The final problem presents students with a dilemma. The force is not constant and requires a different approach. I attempt to elicit student thoughts about it. That problem, and student strategies, drives the discussion of the last segment of class.
In this part of the lesson, we take time to consider the very last problem of the practice set - the force is now a function of distance and not a constant. I ask students to draw the function and ask for a volunteer to add this sketch on the board, then add in a few prompting questions to stimulate the discussion about the challenges of this scenario. An important outcome of this discussion is that students confront the question of what force should be used in the work definition (W = F * d). Once students appreciate the idea that the force is not constant (an underlying assumption of all previous problems) we move on to a brief set of notes explaining the geometric approach to work as the area under a force-vs.-distance graph.
Typically, I do not derive this result. Historically, what my students appreciate is that a new idea extends a previous one. In this case, for example, we can see that the area idea works for both constant and non-constant forces. In the coming lessons, we will build upon this geometric understanding of "work as area." By introducing this concept now, I open up many opportunities for mathematical reasoning and creative problem-solving.
To complete this segment and the day's lesson, we return to the final problem of the practice set and show how to apply the idea of work as area. Students recognize that the shape created by the function is a triangle and we can readily calculate the area. As a follow-up, I create an extension problem: a new force function for students to consider. In this case, I ask only that students sketch the function and suggest ways to think about the area. We end the class with a quick conversation about how the area can be viewed, without actually calculating the area. In this student-generated solution, one can see that this student saw the shape as a combination of a triangle and a rectangle.