[Sorry for the brain dumps, but…]
Day 2: Energy and Frequency Relations in SHM
- Clicker Question Review of SHM Graphs: Day 2 started with a review of what was learned in Day 1. We quickly ran through two clicker questions that showed a position vs time graph for a horizontally oscillating object and students had to identify the sign of the net force and velocity at different points along the graph. In discussing where velocity = 0, we had two nice ways of talking about this-one being that at a turn around v= 0 (e.g., like free fall), and two that the slope of the graph was zero. Similarly, in a location with negative velocity, we could describe particle as moving from right to left or point to slope as being negative. Students had a harder time (at first) identifying the sign of Force correctly from the graph.
- Direction Instruction with PhET Energy Skate Park: I pulled up PhET simulation “Energy Skate Park” that begins with the skater in the bowl. I had previously set y=0 to be the bottom of the bowl (so that E = Kmax= Umax). I quickly oriented students to the simulation as an example of an oscillation that at least approximates SHM.
- Then I posed the question, “On this skater’s path, could you point to a location where the skater has maximum kinetic energy? Where could you point for maximum potential energy? Where could you point where the skater has both kinetic and potential?” As usual I emphasized that students should explain how they know / can tell. Students in pairs discussed for a bit. In bringing it back to discussion, I reminded them of the terminology “energy transformation” and the importance of choosing a system to do energy analysis. Since I was projecting the simulation on the whiteboard (not screen), I also marked various locations with “Umax”, “Kmax”, and “U+K” on top of the simulation
- The next thing I brought up was the Energy vs. Position graph on the PhET simulation. I oriented students to what the axes were plotting, but didn’t explain anything about the graph. I asked them to turn to their partner and “see if they could make sense of what this graph was showing about the energy, and see if they could explain it in a way that an smart 8th grader would understand”. In circulating, there was a lot of peer teaching going on. Back in whole class discussion, my job was to emphasize a few key ideas, using graph as away to anchor it. Total Energy is a constant; and at the extremes, the Total Energy was all Potential (E = U max); and at the equilibrium position, the total energy was all kinetic (E = Kmax). Anywhere, in between the energy E = K + U.
- The last thing I did was connect the discussion to the reading and in doing so quickly formalizing some of this to quantitative relationships in the text.
- Energy Problem Solving: I opted not to do an example whiteboard problem. Students had already learned about solving energy problems last semester, and now it was just a matter of applying it to a new situation. So they just had to jump into a problem–one that asked them to determine the energy of mass-spring system and also to find the location(s) where the spring had half its maximum speed. While students were quick to find the total energy, many groups struggled with the second part (applying COE). For some, it was a matter of not knowing how to embed half the maximum speed into the problem. For others, it’s because they didn’t approach the problem as energy conservation–several pairs of students tried solving the problem by setting K = U, Rather than E = K + U. [We see this in 1st semester two, maybe from too many problems where something falls through gravitational field and we ask about the speed at the bottom]. It was helpful that the simulation was still up, so we could point on the skater, “Where do you think skater has about half his max speed? Is that a location where it’s all K, all U, or a combination of the two.” This helped some groups get a correct expression, but not all groups. Next time, if I were to do this problem, I would first do a qualitative clicker question, “At which of these positions do you think the mass has half it’s maximum speed?” with some choices that are maybe not even have any numbers, just general locations. The interesting thing is that the mass reaches half the maximum speed in traveling a very short distance from returning from the extremes, due to how strong the force and thus acceleration are when the spring is stretched by a large amount. I tried to infuse these discussion as I circulated, but it would have been better to have the discussion upfront. This would have helped to motivate / orient students to the problem, and made the problem seem about helping us to resolve the debate rather than just finding an answer to a question.
- Very Brief Direction Instruction- Deriving how Energy Depends on Frequency and Amplitude: I did a one-step derivation on the front board, which took an idea from Day 1 and an idea from Day 2 and put them together. Last time we have learned that v = (2pi)fA, and today we learned that E = 1/2 m v_max^2. Putting these together, we can see that an oscillating system’s can be energy either because it has a large amplitude or because it is rapidly oscillating. I did some silly demonstrations with my body and talked about a few examples connected to the real life. I then pointed out something using the vertical oscillating spring… that it’s fairly obvious how to change the amplitude of the vertical spring–I just grab it and pull it farther. This gives it more potential energy and so the system has more energy. But the relationship at the board suggests that the system could have also have more energy if it were to vibrate with higher frequency. We had observed last time that the amplitude did not change the frequency, so how can you change the frequency of an oscillating system?
- Qualitative Lab First: What factors effect frequency? Each pair in class was given a meter stick and some three pinch clamps. In the lab exploration, students placed the part meter stick off the edge of the table (while securing the end on the table), and plucked it so that it vibrated. Students were tasked with seeing how they could get the frequency to change. They are then strongly guided to explore how “stiffness” and “mass” effect frequency. Students vary the stiffness by changing the amount of the meter stick that is off the table. Students vary the mass by adding the pinch clamps to the end that hangs off the table. In circulating around, it’s important to ask students question about amplitude vs. frequency—the lab is about frequency and you can’t take it for granted that students have learned to “see” the two separately. It was also important to ask them about what they had found, how that made sense to them, etc. Some groups made sense of their results by linking to Newton’s Laws (stiffness creates a larger force, mass creates more inertia). Others made sense of it by linking to every day situation (guitar strings, fishing line). This was a fairly rich laboratory activity—enough room to play and enough room to think. This also helps bring in sound to the study of early, because you “hear” the metersticks as they oscillate. It’s fun to start the meter stick hanging mostly off the table and get it wobbling and then pull it shorter and shorter rapidly to see and listen to the frequency change.
- Quantitative Lab Second: With students armed with their new Logger Pro skills of fitting sinusoidal data (and using the fit to extract period/frequency info), we moved to quantitative part of the lab. Students had to set up a vertical oscillating spring again, same as day 1. While each group had a nearly identical spring (a spring constant around 15 N/m), they each had a different mass, ranging from 75g to 300g. Each group had to collect data from logger pro & motion sensor to get the period / frequency for their setup. We amassed our data together in a Desmos file to plot Period vs. Mass. The day was running out of time, so we didn’t have time to go further (model the data, link to relationships in text that relate period to spring constant and mass, etc). What I did like about the Qual –> Quan lab structure was the following: Everyone got to qualitatively explore the terrain of variables (and to make sense of it), but no group got bogged down in data collection because they each contributed just one data point. Next time I do the quantitative part of the lab, I would put up the Desmos file with ONE data point that I had collected ahead of time. I would ask them to predict where they expect their data point might fall (based on their mass), and then to predict the shape of the graph. This would help bridge the gap between the qualitative / quantitative, get them engaged in thinking about other people’s data, and help us connect with the theory better.