Our students come into physics expecting a frustrating math course. Then many get even more frustrated -- not only do they have to solve math problems, but they have to create their own problems to solve, to boot! Guh.
In an honors or AP level course, it's important early in the year to make a big show of separating the physics from the math in problem solving. Firstly, here are some facts, concepts, and a routine that will set you on the path to a solution; then, here's how you know that the problem is set up appropriately, that doing ninth-grade algebra will in fact lead to a solution. I go so far as to write, in big capital letters, PHYSICS IS DONE. Students do the same, initially to poke some fun at me, but then as a way of communicating their problem solving.
The canonical technique for recognizing mathematical solvability is to write a relevant equation, then to identify known and unknown variables. Once we have a single equation with a single unknown, the problem is solvable; similarly, two equations and two unknowns is solvable. But don't underestimate how intimidating the actual mathematical solution process to a two-equation system is to a high school student. They may have passed algebra 1, but I trust my students to get accurate solutions even less than I trust the evil bastards of the TSA to get me to my gate in a timely, convenient, and comfortable manner.
Very early in the school year, I assign the hanging stoplight problem. You know, an object is suspended by two strings, each at a different angle; determine the tension in each rope. The solution requires algebraic manipulation of a full-scale two-variable-two-equation-system. Those of you who have assigned this problem and observed your students can probably verify my report that many of those students spend 30-60 minutes doing math, often getting lost along the way. A significant fraction get so frustrated that they simply give up, or follow a friend's solution blindly.*
* I know this because quite often that friend's solution is itself incorrect.
Here's a great chance to make my point about the separation of physics and math. By this point, in class we've emphasized over and over and over the three-step approach to equilibrium problems:
1. Draw a free body diagram
2. Break angled forces into components, if necessary
3. Write (up forces = down forces) and (left forces = right forces)
The majority of the students who spent the better part of an hour on this problem didn't follow these three physics steps carefully; they got too worried about the forthcoming mathematics.
So, why not give a quiz in which students are given explicit instructions not to solve the two-variable system?
See the quiz below. I find that it relieves much anxiety from those who got lost in the mathematics. It sends an important message to those who didn't follow the process, because they see just how quickly they could have gotten to the answer by, well, listening to the teacher and following his advice.
Finally, note that the AP Physics 1 exam will not ask students to solve a true two-variable system of equations, ever; but "write two equations which could, together, be used to solve" is a legitimate form of AP question.
Two ropes support a 33 kg stoplight, as shown above. The goal of this problem is to find the tension in each rope, as on last night's homework problem.
I am NOT asking you to solve the problem completely in this quiz; rather, I want to see that you can quickly and accurately follow our four step procedure for solving equilibrium problems.
- Draw a complete free body diagram of the traffic light, including descriptions of each force.
- Redraw the diagram, breaking force vectors into components where necessary. Express components in terms of the given angles; i.e. do not simply write “Tx”, include the angle in your expression.
- Write two equations. Circle the unknowns. DON’T SOLVE.