In intro physics, I’ve been teaching N2nd law problem without introducing any formal algebraic solutions. Instead, students have been asked to use multiple representations to organize their thinking and then to reason about net force, and Newton’s 2nd Law using their representations to help them think through their work.

On Friday, students were working through a set of of the challenges, and I had one group struggling with one of the static situations. It was a situation where a 1.0 kg object was partially supported from below via a force sensor and partially supported by a rubber band (suspended above). Students could read the bottom force sensor, and were asked to predict the reading on the top force sensor.

The students had correctly drawn a free-body diagram, but were having a hard time thinking about to go from there. I have an undergraduate physics major in the class, who must have ended up helping the students, because their work had become very an algebraic.

N + T – W = 0

T = W- N

And students got the right answer. When I came over, I recognized that it was likely the undergraduate student had helped them, and that it was likely that there solution didn’t really make sense to them.

So decided to asked them some questions, the first of which was about the net force. It was not obvious to most the students what the answer to that should be. However, one student, who doesn’t often speak up, finally said that the net force must be zero because the object wasn’t moving. I helped them to add that information to their free-body diagram, since in our class, we follow Knight’s procedure of drawing a separate Fnet vector next to your free-body diagram. I pointed to the individual forces on their diagram and added, “So that means, these three individual forces need to come together to act as if there was a total of zero force. This question is essentially asking us to figure out, what must each individual force be doing to end with a result of zero net force.”

I then asked the students about the value of each forces, and asked them add those values to their free-body diagram values as they figured them out, but I left them to work it out, and walked away. When I came back, I asked them to tell me about their new solution and whether it agreed with the first, and you could just immediately tell from tone of voice and body language (especially from one of the students) that their new solution made a whole lot more sense.

To help them summarize their work I asked them a sequence of questions

- Which forces did you determine from a direct measurement? (normal in this case)
- Which forces did you determine indirectly from knowledge you have about how specific forces behave? (weight in this case)
- Which force did you have to reason about? How did the diagram help you to organize your reasoning? (tension in this case)

I then helped them connect their second solution to their first solution. I probably could have asked more questions rather than telling. My main goal was to help them see two features of how they are related– how the direction of arrows related to the algebraic signs, how the net force being zero was related to the RHS of the equation, and how once they rearranged their equation, both solution methods concluded that subtracting the normal force from the weight force would give you the amount of force remaining for the tension to hold up.

So I have two this is my goal for this week: To progress toward algebraic solutions and to progress toward problems that non-orthogonal forces. I’m inclined to start with the vector stuff, and then later get more formal with the algebraic.

Here’s my thinking so far about how to get this started:

I’m going to set up thrre Situations where a 500 gram mass is in static equilibrium. The first one has only vertical forces. The second one has vertical and horizontal forces. The third one has one vertical force, one horizontal force, and one angled force.

I’ve set it up so that each of the tension forces can be read with a scale. I’ve set all the purely horizontal forces to be equal.

I might ask students to draw the FBD first without seeing those measurement values, but I’m not sure yet. Either way, I want students to compare and contrast these situations in terms of how the individual forces “work together” to hold the mass at rest.

I definitely want them to compare and contrast the first two situations. How are they similar? How are they different?

Then I want them to compare the 2nd situation with the 3rd situation. How are they similar how are they different?

I’m really curious how this will stir up students’ thinking, especially about the 3rd situation and its relationship to the 2nd. While we have done vector components (with projectiles), we haven’t yet done so with forces.

Somethings we might think about / question?

- In each case, 5N of vertical support force is needed. And it’s fairly obvious how the first two situations do that. How does the 3rd situation accomplish this
- In each case there is a net horizontal force of zero. This is trivial in the 1st, fairly straightforward in the second, but less obvious in the 3rd
- Students might try ask how the value of the angled force compares to the horizontal and vertical forces… they should note that they don’t add. Students might go to Pythagorean?
- I’m curious if students’ ideas will provide an opportunity to think something like… in the 2nd situation, does this mean we could replace two of the forces with a single force of the strength and angle of the 3rd?
- Will stuff about angles come up ? Will any of that link to trig? Or maybe it will take the form, how will changing the angle of the 3rd force effect the force reading.
- This 3rd situation is easily “seeable” as vector addition, and while we’ve done some of that, I’m doubtful that will come up, but maybe?

- One possibility is we start investigating empirically “How does the angle of the 3rd one effect force reading?”
- Another possibility is that students will have ideas about how this works that they want to test… like, “Will the reading on the angled force always be a Pythagorean result?”
- I’m curious if a good time will occur to introduce a situation where both force are angled (probably the symmetric case)… but I’m also wondering if I should include this as one of my examples…. like the 4th example.
- I’ll have to do some direct instruction, just given the pace of the course, and I think after discussion and some chance to observe more and/or test out ideas, I’ll move to formalizing these ideas under the umbrella of vectors and vector components. Probably some clicker questions and exercises, before some problem solving? Not sure

I have some time to think about this, since our next meeting is a test.

Next year, I want to leverage the relationship between demo 2 and 3 demo to talk about equivalent substitutions. That is we can “think” of the 3rd situation in the following way: you can substitute the single angled force with one vertical and one horizontal force. But you have to pick the precisely right vertical force and right horizontal force, other wise it won’t be an equivalent substitution. Then later in problems, I would make students draw 2 FBD, the “actual” FBD and the “equivalently” substituted diagram. The second is essentially the components. We were definitely wading in that territory but it was not explicit as it needed to be. 2nd thing I want to remember is to let student solve for the values of components first (if that makes sense)… that is to solve for the “equivalent” forces, then put that back together to find the magnitude …. Rather than do the more standard thing of using algebra and trig to directly solve for the magnitude. I think that process makes more sense to students, as I’ve written before, AND for these problems in helps students have “insight” into the problem.

Even though it seems to be leading toward components, I’m super energized right this moment about the idea of setting this up so that the third one works out to be 3 N, 4 N, and 5 N and have that right angle, still. (I’m not sure how easy or realistic it is.) If I could set that up correctly, I think the kids could come up with the idea of using the vector addition diagram to solve these kinds of things. Just wanted to comment that while I’m still excited and reality of making this happen hasn’t set in yet! 🙂

(Hi. I’m working through this set of posts on forces now because I’m writing my balanced forces packet for our brand new 10th grade physics course.)

I bet you could get the 3,4,5 thing going. Yes, I do go into components with it, but I think the idea of seeing equivalent substitutions is powerful, and this set up (or an improved one) seems promising for stirring up ideas.

It might just be HS students, but nothing seems to scream triangle more than 3, 4, and 5, right? 🙂

It might just be HS students, but nothing seems to scream triangle more than 3, 4, and 5, right? 🙂