[Sorry for the brain dumps, but…]

Day 2: Energy and Frequency Relations in SHM

1. 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.
2. 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.
3. 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.
4. 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?
5. 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.
6. 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.

Day 1 of Oscillations (Day 1 of a Unit of Oscillations of Waves; Day 1 of the 2nd Semester physics)

1. Unit Motivation: I began this unit with a rather silly demonstration. I had a function generator send a sinusoidal signal into a speaker; an oscilloscope showed what was happening with the electrical signal. I started with the frequency set low enough so that we could see the speaker moving, but not hear it. In front of the speaker, I placed a microphone which then plotted the pressure changes. After briefly orienting students to the setup, I used a fairly standard, “What do you notice? What are you wondering?” line of questioning.  This ultimately led into a brief orientation to what we would be learning this semester (oscillations, waves, sound, electricity, etc). I also used this as an opportunity to motivate why we would spend this week studying two simple (and boring) oscillating systems (pendulum and mass-spring). With these, we didn’t need any specialized equipment to observe these systems– either to understand its mechanisms or to take careful measurements. I alluded that learning about these systems would pay off in terms of understanding more interesting things like sound, electricity, and light.
2. Observing Kinematics Graphs for SHM: As a demo, I showed students how to setup a vertical oscillating spring.  [Side Note: In this class, we don’t set up any lab equipment for them; students must go get what they need, setup any apparatus, any put things away. Last semester, I had done some setup for students, especially when it involved rods and clamps. But this semester, I am having students even do those, so students are becoming more familiar with how to quickly set up things with table clamps and right angle clamps.] While playing with the spring, I also briefly introduced some language students were familiar with from last semester, such as equilibrium position and (linear) restoring force. Anyway, with a motion detector, we observed the motion of the mass-spring system and I recorded just the position vs. time graph. Students were then given a laboratory exploration to first predict and then observe the velocity vs time graph and acceleration vs time graph, focusing on how they line up. There were some additional questions for them to discuss about where is it moving fastest, where is the velocity zero, where is the acceleration greatest, etc.
3. Direction Instruction on Sinusoidal Graphs: Back together as a class for some direct instruction, I introduced sinusoidal functions a way of mathematically describing the graphs they’d just seen. I used a Desmos file to show them what effect the parameters had on the shape of the graph (corresponding to Amplitude and Period), pointing out that the mathematics allows us to control Amplitude and period separately. Some of this involved reminding them about frequency and period which they had learned last semester for circular motion, and part of this involved making more connections to their reading assignment.
4.  Modeling and Practicing a Logger Pro Lab Skill-Curve Fits for Sine Graphs: Next, I modeled how to use a curve fit in Logger Pro to model their data, and how to decode the parameters that Logger Pro provides. Students were sent back to practice this new skill, since it’s a technique we will be using a lot over the next 3 weeks.
5. Clicker Question and Mini Exploration-Does Amplitude effect Period:  Back as a whole class, the question was posed to students, how the period would change if the amplitude was increased? [I didn’t actually use the words period or amplitude… we watched the vertical spring again, and I drew attention to how far I pulled it and how much time it seemed to take to go back and forth. I then drew attention to pulling it farther (and without letting go), asked, how will the time to go back and forth compare? We discussed this clicker question–> then instead of resolving the issue with debate, we set out to do more mini-experiments. Students of course found that amplitude didn’t effect period, and we had the clicker question debate to make sense of why that answer seemed plausible. I related it back to mathematics—with the math we saw that the Amplitude and Period could be changed independently. With the physical system, too, you could change the amplitude without changing the period.
6. Direct Instruction on Maximum Speed and Maximum Acceleration: This is probably the weakest part of the day, but with limited time, I talked about how we had seen with the position vs time graph, the “Amplitude of the Graph” corresponded to the maximum distance that the mass got on either side of equilibrium. On the velocity (and acceleration) graphs, the “Amplitude of the Graph” would correspond to the maximum speed (and maximum acceleration) that the object would achieve. I opened up another Desmos file and walked students through what factors seemed to effect the maximum speed— changing the amplitude or changing the frequency made the maximum speed greater. I related this to their reading, where they had an equation that related maximum speed to amplitude and frequency. This section could have been improved with just one or two clicker questions, where students predict what will happen to the maximum speed when I change Amplitude, and then the frequency. We’d could have then observed on the simulation after having collected our thoughts. I didn’t specifically go through the observations and reasoning about acceleration, but pointed out where in the reading they had seen this.
7. Problem-solving: Example and Whiteboard. With some new quantitative relationships in their belts, we needed to do some problem-solving. I don’t always do an example problem, but I did today, because I wanted to model a skill of making good sinusoidal graphs. In the problems, there was not a specific question, but rather information about an oscillation was given, and the task was to draw the position, velocity, and acceleration graphs. This required having to work out period, amplitude, maximum speed, maximum acceleration, whichever wasn’t known. The specific modeling I wanted to do, was identifying “land marks” in the graph. For example, in a position vs time graph, before scribbling a graph, marking some places that you can reason about where the particle will be—half way through the period it will be on the opposite side. Half way through half way, it will be back in equilibrium. Once you’ve identified those landmarks, it’s easier to draw the sinusoidal graph correctly. Anyway, after I modeled solving the problem, students got a problem of their own and worked them out in groups on the whiteboard.