Step 1: Get students thinking and talking about their experiences
Uniform Circular Motion was first introduced by thinking how it “feels” to be a rider on a swing carousel.
Students described what it feels like differently:
- You feel weightless, like you are not that heavy, almost floating
- You feel heavier than normal, feeling pressed into the seat.
- You feel like you are being thrown outward.
- You are leaning in, or tilting into, to not be thrown out.
Next we had a clicker question about where a rider would go if at a specific point the cables suddenly broke.
Choices were varied, about 1/3 saying it would go off tangential (B), 1/3 radial (E), and 1/3 something between tangential and radial (C/D). There were very few who felt it would continue curving along A (e.g., circular impetus). We had some discussion about each of the answers, and why. Some felt the velocity (or force) would throw you out, others felt like velocity (or force) would be tangential, and yet others felt like the two would compromise to curve slightly outward while going around.
Step 2: Help re-establish and practice applying what we already know
We next did a review of what we already know about forces and motion. Students were asked to use meterstick to exert forces on the hover puck in order to
- to get the hover puck speeding up
- to get the hover puck moving with constant speed
- to get hover puck to slow down.
I summarized these ideas at the board with motion diagrams and force vectors. We then did a clicker question about what the hover puck would do with a constant force that starts 90 degrees from the velocity. This one was hard, for students. Besides solving projectile motion problems (before forces), we hadn’t really talked about this. After discussion, we watched a simulation and tried it out with the hover pucks to see that it curves while speeding up. I helped them link this observation to motion diagrams and force diagrams for free-fall in 2D.
Step 3: Students explore what we don’t yet know
I told students this was a good review of what we currently know about forces– we know how forces can maintain velocity, speed up, slow down, and turn while changing speed. BUT we didn’t know how to turn without changing speed, and that’s what we needed to investgiate
Next students were tasked with exploring out how to get the hover puck to move pretty much in a circle at pretty much constant speed using the meter sticks. Students were asked to discuss a few questions: how getting the puck initially started moving in a circle was different than keeping it moving in a circle, and questions to direct them to think about how they would describe the direction of the force. Then we gathered consensus around the following video of me doing it:
— Brian Frank (@brianwfrank) April 3, 2017
I then ended up introducing the demonstration where the hover puck moves also moves in a circle in using a string. I had originally wanted them to do this (which may have been better), but I/we were feeling the need to pick of some momentum with class. We watched the demo in class, and then I showed them these pictures, which helped identify what is meant by “uniform” circular motion.
Step 4: Clarify the idea that a central force seems to be required
I then drew a “top view” motion diagram that showed both the velocity vectors and the forces we were exerting. I put forth the idea that we had now seen 2 situations where an inward force was needed to keep an object moving in a circle – the inward pull of the tension from the string and the inward pushes of the meter stick.
[Note: Old me would have definitely been inclined to think that we are pretty good at this point. But I know better, and know better how else to keep us moving along]
Step 5: Press at students’ understanding of the new idea
Then was a clicker question about, “Why is a inward force needed?” I had four statements
A. An inward force is needed to balance the force that the puck experiences outward.
B. An inward force is all that is needed to turn the puck around the circle.
C. In addition to an inward force, a force around the circle is needed to keep it moving as it goes around.
D. There are actually three forces: One force that keeps the puck going around the circle, and two forces that are balanced to keep the hover puck from being thrown out of the circle.
I gave students time to think and talk with their small group. We were widely and almost evenly split between all choices. We spent a little time talking in whole class, to clarify what was meant by each idea. Instead of having them talk in small groups, I had them get together with people with similar answers. They had some time to chat, and then elected a spokesperson to make the initial presentation. I allowed questions aimed at clarifying, and added clarifying comments myself, but didn’t allow back-and-forth until each group had a chance to present. Then, we started talking more freely with ideas bouncing around.
- The inward only group focused on explaining that all you are doing is tapping inward, so that’s the only force, and that while they agree there’s a velocity around the circle, that doesn’t count as a force. They agreed that you initially needed some velocity around the circle to get it started, but the question was about “keeping it in the circle”, and they didn’t think it needed any force around the circle to do that.
- The inward + force around the circle went next and said that a force was needed to get it started and to keep it going. They came back to the spinny ride, saying that the ropes have two jobs, to pull you in and pull you around the circle. There were a few who said after hearing group 1, they weren’t sure whether it was just velocity of force around the circle.
- The next group talked about the in and out forces needing to balance. During discussion it became clear that some in the group thought the “balance” force was the velocity, that made it want to go out, and others thought there really was some outward force. Through this conversation, it became clearer to us all that issues around whether something was a force (or velocity), and whether or not these were pointing around the circle or outward from the circle was an crucial difference among the ideas (both within and across groups).
- The last group, made a compelling argument for why an inward force was needed. Without an outward force to balance the inward force, the inward force would “win” and make the circle spiral inward. We referred to this as the “death spiral”. To maintain a circle, the argument goes, a balance of forces would be needed. This was pretty convincing to lots of folks.
During the more open conversations time here are some of the ideas that came up:
A. A big question came to identifying what the outward force was. We’ve been pretty picky all semester about identifying forces. Someone eventually suggested that it might be the force pair. I helped flesh out what the student meant by this, by identifying the forces specifically. But I initially didn’t press for the implications of that idea. I wanted that idea alone to be important. Yeah, what is that force? Maybe it could be this force, the force pair. I probably could have asked, “Do people have others ideas about what the outward force could be?” to further value that line of reasoning.
B. Another idea was that a force can’t be needed around the path of the circle, because a force around the circle would mean that force and velocity were parallel. Our rules state that when force and velocity are in the same direction, speeding up occurs. Through discussion and re-voicing, I helped them to articulate a possible new rule that they were seeming to suggest: “If force stays 90 degrees from velocity, all you do is change direction, without changing speed.”
C. Someone argued that they don’t think that an inward force necessarily means a spiral inward. They were trying to think of what would make something spiral in or out, and they came up with situations where spiraling in or out would be associated with a change in speed. People gave examples of twirling keys with a lanyard, and letting it swing around your finger vs. wrap around your finger. Since we were talking about uniform speed, not spiraling should happen unless you speed or slow the object.
D. There were also more conversation about getting it started vs. keeping, that were helpful, but I can’t quite remember, and ‘m sure there were other ideas that I’m forgetting.
Step 6: Help clarify the different possibilities
Anyway, we took a break, and I chatted with some more students. During break, I talked with several students about the “force pair” idea, and helped connect that to our previous learning about how force pairs shouldn’t be on the same FBD. So that if the outward force was the force pair, it shouldn’t be included in the diagram that shows forces that act on the puck (or rider). I also talked with the two students who brought up the spiral in idea and the counter proposal.
After break, I tried summarizing some of the big ideas, and helped the whole class in on two ideas:
- That if the outward force is the force pair, then it’s a force on another object, and thus shouldn’t be included, and
- that there are two different ideas that I see as similar. I told them that the “inward” only group thinks that what the puck will do without a force is “go straight” and that an inward force is needed to “bend” the puck so that it moves around with a constant spiral (not spiraling in or out). The “balanced force” idea seems to focus on the fact that the puck is already moving in a circle, so that any extra force would “bend” the puck into a tighter circle. I emphasized that both groups agreed that inwards forces cause “bending in”, but they disagreed about the detail.
There was a bit more conversation after I introduced those ideas, and students had some time to be back at their groups to rethink.
Step 7: Introduce a Testing Experiment
I then introduced a new experiment that might help us decide. I showed them a metal ball going around inside of a metal ring. I told them that I could suddenly pull the ring upward, and that it would be like “breaking” the cable on the ride.
I introduced what we were about to do as very different than what we did before. Before, we were shopping around for ideas, experiences, arguments, and that it was kind of OK to change our mind. But now I wanted us to stop letting our minds think whatever we want. I want to find out not “what we think”, but what our different ideas imply. I asked students to go back to their groups and draw 3 diagrams:
A free-body Diagram (just before the ring is pulled away), based on whether you think the forces are “inward only”, “balance in and out”, or “also around the circle”
A free-body Diagram (just after the ring is pulled away), based on what forces should disappear when the ring disappears.
We agreed that each diagram should be very clear about whether something is a force or a velocity, but that it was OK to not specify exactly what the outward force might be, since we were still not sure. We also agreed that the object of interest for our diagrams was the ball, and that only forces that act on the ball should be included.
The last diagram was to show a path about where the ball goes. I spent a lot of time telling them that this was not where “you think” the ball should go, but rather what the diagram has to say about where the ball should go based on a rules:
- No Net Force -> Constant Velocity
- Force in Direction of Velocity –> Speeding Up
- Constant Forces in 2D –> Curved trajectory
Groups split up and worked on their diagrams, and then we circulated around looking at different diagrams. Several groups went in the direction of “inward only”, one group did balanced in and out, and another group did balanced in and out (with maybe force in the direction of motion).
We concluded that a balanced forces implies that the loss of ring would leave the ball with only an outward force… and that this would mean the ball curves slightly outward as it leaves the circle.
We concluded that the inward forces only idea implies that ball will go off tangentially, moving not only straight but with constant velocity. This is because since the only force was the inward normal force from the ring, once this force is gone, the particle is subject to zero net force.
I sent students off to make the observations at their tables.. Because it’s hard to see, students spend a fair amount of time on this. In small groups, I talked about “confirmation bias” and whether or not they think it was possible to see whatever you want, like those that want to see it as straight can see it that, and those that don’t, can claim they see a slight curve.
Step 8: Help students connect what they observed to what it means
Good thing we have the slow motion camera:
— Brian Frank (@brianwfrank) April 2, 2017
I did some summarizing here (should have made them do more of that work), and connected what we were learning to textbook notes and diagrams. This is also something I should have them do. I should have had them open their textbooks to the passages and diagrams, and ask them to do this work. This is like in my previous post about discourse, where I took too much responsibility for discourse 2 and 3, where I should have been the one to support them in engaging in the discourse.
Step 9: Practice Applying the Idea in progressively more challenging situations
The way I did it was ask them to engage somewhat in the 3rd type of discourse was by getting practice with applying the ideas, which is better than nothing. We did a few clicker questions about identifying the “force” that plays the role of the centripetal force. We did free body diagrams for penny on a turntable, the swingers on the ride, and then finally free body diagram at the bottom of pendulum swing.
Students did fairly well with the first two. I emphasized how in the first case we might think of just one force (friction) as playing the role of a (net) centripetal force, and how in the second we needed to think of the just the horizontal component of tension as playing that role. But the third one was a really interesting trouble spot.
In the pendulum swing, students had to decide which of the two individual forces was larger, weight or tension. Maybe one or two students said weight would be stronger than tension (probably focused on “outward throw” still), but most groups picked that tension and weight would be equal. What was interesting is that many of the people who said that tension and weight would be equal were the also the ones who were most adamant for the inner force only in the previous activity. I thought that was interesting because now they were saying that balances forces were needed. Some of these students even argued that if the T were greater than the weight than the mass would like “rise up” at the bottom (making a smaller circle). Essentially, the spiral in argument resurfaced, but not from the same students. It made me wonder if “having the right” idea the whole time made them more vulnerable to being tricked later.
Anyway, the situation is hard for several reasons (one it’s not uniform circular motion, vertical forces are involved, and there are two forces in the radial direction). I think a lot of students reasoning was actually trying to borrow from “independence of motions in projectile motion”… Something like Like the velocity is horizontal, and so it should have no impact on the forces vertical. Thus the vertical forces should be balanced. It’s an interesting case of saying “no velocity vertical” means “no vertical force”.. which is “force~velocity” reasoning just resurfacing with knowledge of components.
In discussion, however, groups were able to be pretty convincing that the tension should be larger than the weight. Groups did a good job of making it clear that what we had talked about, seen, and learned today was the basis for saying that the forces cannot be balanced if it’s moving in a circle, and that rather a surplus of force pointing upward toward the center of the circle was needed. I’m glad the class did this (without my help), but I could have done a better of job of pushing questions in small groups, like… “How did you apply what we learned today to help inform your answer?” or “How is your answer consistent with what we learned about the forces necessary to keep an object moving in a circle?” I might have even asked a question like, “So any time you have velocity that is purely horizontal, the forces vertically must be balanced?” The conversation was good, but I definitely see ways I could have been a better at “pressing for disciplinary connections” toward the end of the lesson.
We ended the day by observing the force sensor data. With five minutes left in class, I asked students to take some time to reflect, write, summarize what they learned today. This is also a practice I need to use more, using the last 5 minutes for reflection. All and all a good day. On Friday, we get more quantitative with “how much force exactly is needed” –> this will build off nicely from the spiral in /out question… we want just the right amount of force to keep the circle (not let it widen or tighten)… We will investigate what factors are involved.