Quantitative Circular Motion Recap
We had a pretty awesome day in physics. 1 hour long warm-up. 1 hour long “preparation for future learning” (making predication, analyzing data) and direct instruction. 1 hour of problem-solving.
Long Warm Ups: Applying the Concept of Centripetal Force to Tricky Cases
We spent WAY more time on the warm-ups, then I was anticipating. But it was really needed, and engagement was high so I it took up basically an hour to do 4 clicker questions.
The two we spent the most time on was Car Cresting the Hill, and Roller Coaster Cresting the Top of the loop. Partially it took long because I framed our goal as two fold:
1. To answer the questions on the basis of what we have learned about circular motion the previous class.
2. To decide whether the answer made sense. Students had to vote on the FBD that they thought was correct (A,B,C,D), but also had to vote on a 1-5 scale of how much that answer made sense.
I’m having a hard time remembering what the details of our conversations were, but they touched upon lots of stuff… apparent weight, normal force, object of interest, weight force, velocity vs. force, knowing that the rules say the normal force lessens, vs. figuring out how (what would make) the normal force lessen, trying to connect that to the sensation of… which free body diagram would be correct if you were hitting the bottom of a hill instead of cresting the top.
For the roller coaster, we were pretty split again among many choices, but about 75% of the students picked one that had a central force (with weight down and a second upward force). After brief small group chat and whole class discussion, I re-voiced ideas to navigated us to use what we had learned about circular motion to rule out all but 2 FBDS. Normal and Weight down, or Weight down and an upward force (less than weight). After going to back to small groups to decide which of those were correct, everyone but one person voted weight and normal down. So many people changed their mind to correct answer. Weeks ago, that would have signaled to me we were ready to hear an explanation or two, and then move on. But today was different. I asked the one student, if they felt comfortable telling us why they were sticking with their original choice, and then since so many people had changed their mind, they could tell us what had convinced them to change their mind. The hold out student voiced some good ideas, but the main idea he had was that if the only forces acting on the ball were down forces then the ball would get to the top and fall straight down. I helped make that idea clear, and then truthfully left the classroom while students discussed. I told them I had to go to the bathroom (which was true and urgent, because I had like 4 cups of coffee), and that they should continue the conversation with out me. I came back a few minutes later and they were still passionately discussing the question. (After class the physics major who is undergraduate teaching assistant in my class told me that right when I left, a student who is usually pretty quiet immediately spoke up). Anyway, some of the arguments I heard and others I didn’t, and we gave the holdout student the final say.
Transition to Quantitative: Preparation for Future Learning and Direction Instruction
The transition to quantitative went smoother than I thought, even though I stream lined it because of the longer than expected warm up. I reintroduced the pendulum that we had studied before, and asked the question, about what we could change. Students suggested increasing mass, increasing the speed, and strong agreement that would make the force stronger. There was not agreement on what effect changing the length of the string would have… between taking more net force, taking less net force, and taking more net force. I modeled using the pendulum how I actually did these experiments, made groups pin down predictions with reasoning, and then looked at the data– graphs. Students were asked to describe which graph went with which experiment, and patterns in the data, and whether it agreed or disagreed with their conclusions. I thought it might be too abstract to look at the data without actually exploring, but students had enough contexts from our extended discussion to understand the data without actually playing the equipment.
With short time, this led to met doing about 5 minutes at most of direct instruction showing how the patterns from the graphs can be integrated as mv^2/r, and how comparing with Newton’s 2nd Law, this means a = v^2/r.
Students did a quick job of the ranking tasks. A few students needed guiding questions on some on the ones that were tied.
Problem-Solving: Building the Problem
We did have time to work both problems and to “build” the problems together. I had students tell me what they wanted to me measure and how to predict the tension. I told students the list of must haves. They did a good job of “reasoning” through the problem using representations, and not mindlessly using equations. Honestly, even with thoughtful discussion, good diagramming, they answered the question in about 15 minutes or less. We checked our answer against the demo, and then we were on to the second problem.
We watched the you tube video of car sliding out the exit ramp. I motivated the question of, “So how fast can you go without sliding off?” I asked students to tell me what information they would want to know:
Students said mass of the car, radius of the circle… we had a short debate about whether we needed to know kinetic or static friction. I guided here strongly, by demo-ing. We haven’t studied rolling without slipping specifically, and so I didn’t want to linger here too long, just show the difference between sliding vs. rolling (without any sliding). We talked about needing to know conditions and we talked about ice, wet, and dry. We also talked about assumptions we might need to make: they suggested we should assume constant speed, and I suggested we should assume a flat surface (even though most are banked).
I gave some canned data rather than have them estimate or research. Everyone did the exit ramp nearest to our highway, where I estimated from google maps it was about 100m radius.
Two groups did icy road conditions: One a 1000 kg car, the other 2000 kg car.
Two groups did wet conditions: One a 1000 kg car, the other a 2000 kg car.
Two groups did dry conditions: 1000 kg / 2000 kg.
Students were asked to predict which in their pair would be able to go at higher speed. Students struggled through this a little more, having to recall things they knew about friction. Some groups struggled with which direction the static friction force points. I used good questioning to help here: “How can you decide which way the forces must point based on what we’ve learned about circular motion during the past 2 days?” “How can you make sense of that answer?” (e.g,. “we need net force toward center of the circle.” and “that makes sense because that force is preventing you from sliding out, not causing you to slide out.”)
Students also needed help with what perspectives to draw things. I could have modeled this, but I ended up just asking each group to make a top view, side view, and a head on view of the car. And to decide which of those three would be useful for representing what kind of information. Now would be a good time to introduce the x and . notations.
Students were pretty surprised to see that the speed didn’t depend on the mass! As we wrapped things up, I had a chance to chat with groups on a one-on-one basis about this. Some groups could explain why pretty well, others needed a bit more direction. It was key to say that heavier one still needed more force, but that is got this greater force by gripping the ground better (normal –> friction).
I’m really happy to NOT be making these problems super difficult in terms of angles and such. Students were really doing problem-solving as a sense-making thing, applying new knowledge to practiced ways of representing and reasoning. I literally have not done an example problem since early February, and it’s never felt like we’ve fallen into the pit of “blindly trying to work the example problem”. Still, students work is sophisticated and not haphazard. High standards are maintained through “must haves” and culture, and most of the high expectations are around representing the work carefully and meaningfully, not in carefully structuring algebra.