Scaffolding the Uniform Acceleration Paradigm Lab

Since being told about the Hudl Technique slow motion video app (free in Android/iOS), an entire world of new labs opened up for my students. Hand-controlled timers are just too inaccurate for most experiments, but a slow motion video that also shows real time to the hundredth of a second is more than adaquate for almost any experiment in an introductory physics class. Shown below is one group setting up their experiment to be recorded by the app.

Past Failures

My college-prep physics class (non-AP, trig-based, but more conceptual than not) began this lab today. Since the goal of this lab is to have them model the relationship between velocity and time, the follow up to generating the position vs. time graph for the tennis ball is then calculating the velocity of that ball at different times.

I’ve tried having them do this in several different ways over the years, all of which that have been some shade of failure. Most of my attempts have revolved around having them draw tangent lines at different points of the curve (a curve which they hand-drew), calculate the slopes of those lines, and then using that to generate the velocity data. The problem there is that having them hand-draw a curve then hand-draw tangent lines makes the uncertainty in the velocity data skyrocket thus destroying any real linearity with the data.

Additionally, there was just too much for my students to grasp all at once. Even the students that got “good” data still didn’t really understand what they were doing.

I’ve also tried having them use a motion detector to generate velocity data, which they then pick out and graph. But I didn’t like that either as it felt too much like magic hand-waving.

So with this lab approaching, I was dreading how all of it would go again. A lot of confusion, poor data, and blank stares.

Using Desmos

One of the primary issues I’ve discovered with the tangent line method (accuracy issues aside) is that it skips a step. A tangent line is the point at which two points on a curve converge (i.e., calculus), and understanding that the definition for instantaneous velocity comes from squeezing two points infinitely close is essential for understanding how the operational definition of velocity (slope of a position vs. time graph) is generalized to non-linear position functions.

Once I realized this, my goal became to figure out a way to take the above into account without a mountain of cognitive overload. Usually this means turning to Desmos and, as expected, Desmos continues to be one of my favorite tools for the physics classroom. Here’s the tool I made.

Students first enter their position vs. time data in the table to the left…

desmos1

…which then desmos uses to generate the function that best fits the data (all of which is in the WIZARDRY folder):

desmos2

The back-end of this tool generates a fit for any function ax^n.

Understanding Tangent Lines

Once students have entered their data, I then have them talk in their groups about how they can calculate the velocity of the ball. When the topic of “the” slope comes up, I then point out that there is no “the” slope because the line gets steeper and steeper… which makes sense because the ball’s velocity gets greater and greater.

But then how do we calculate the velocity of the ball at any particular point? What points do we choose?

After a few more minutes of them discussing this amongst themselves, I allow groups to throw out suggestions to the class. Often times they won’t come to the “choose two points very close together” conclusion on their own, but one group will say “choose a point before” and “choose a point after.” From there, I lead that into “but how close? should the points be?” Or, an even better question: “But why do they need to be close?”

At this point I put Desmos up on the projector and show them two things:

desmos4

Note: the tangent line is turned off by default as I think it can confuse things when they’re making their calculations later on. To turn it on, go into the Wizardry folder and click the empty circle next to the very last equation

I point out that the orange and black lines deviate quite a bit for almost all of the graph. However, if you zoom in…

desmos5

…they overlap perfectly. This is the justifcation for “pick two points really close together.” To wrap up, I ask them to write a few sentences that explain and justify the process of calculating the velocity at a particular point in time.

This also just-so-miraculously-happens (read: completely and totally on purpose) sets them up for calculus later on should they take it.

Generating Velocity Data

At this point, students are ready to start generating their velocity data. I tell students to drag the blue dot (the red follows along) such that they center each of their data points. See the two screenshots above for what that looks like for data points not on/close to the curve. The slope of the line between the red and blue points is the velocity at he x-coordinate of that data point (1 s for the point above). Students generate around 10 velocity-time data points, and then move to graphing their new data.

Because the velocity data is generated from a perfect curve, the velocity data is almost perfectly linear, thus making the further analysis much easier.

If you have any questions about how all of this works or suggestions on how to improve, please let me know!

 

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Replacing Textbook Problems with Lab Experiences

This was published in the Talkin’ Physics column of The Physics Teacher’s October 2017 issue.

End-of-the-chapter textbook problems are often the bread-and-butter of any traditional physics classroom.  However, research strongly suggests that students be given the opportunity to apply their knowledge in multiple contexts as well as be provided with opportunities to do the process of science through laboratory experiences (Mestre, 2001). Little correlation has been shown linking the number of textbook problems solved with conceptual understanding of topics in mechanics (Kim & Pak, 2002). Furthermore, textbook problems as the primary source of practice for students robs them of the joy and productive struggle of learning how to think like an experimental physicist. Methods such as Modeling Instruction tackle this problem head-on by starting each instructional unit with an inquiry-based lab aimed at establishing the important concepts and equations for the unit, and this article will discuss ideas and experiences for how to carry that philosophy throughout a unit.

Practicums, practicums, practicums!

Designing the right kind of lab experience is more than simply having students make calculations based on making real measurements. What makes a lab experience a practicum[1] is that the calculation students make is verifiable. An experiment should be able to be carried out that undeniably shows, either through visual confirmation or an additional measurement, that their prediction is accurate or not. This allows nature to be the arbiter of the quality and accuracy of their work instead of the teacher, thus placing students in an environment far more representative of how real science actually works.

Through lab practicums, students also get to see the importance of measurement uncertainty and significant figures. If in a multi-step calculation a student rounds their results in between each calculation, their final prediction is likely to greatly deviate from reality regardless of the quality of their solution or initial measurements. The same applies for poor measurement techniques in that an accurate analytic solution is worthless for predictions if the measurements are taken poorly. Practicums also emphasize that equations are not black boxes that take in numbers and magically eject answers, but instead are mathematical models that provide predictions (accurate or otherwise) for how the world works.

Lab practicums can be as simple or as complicated as they need to be for the desired level of rigor or available time. After students have begun tackling the concepts of constant velocity, instead of assigning them problems of the A car drives at a constant velocity of…variety, why not instead give them a constant-velocity tumble buggy and have them make predictions about its motion ? Have them calculate the velocity and then place a piece of tape on the floor where they predict the buggy will be after some amount of time determined by the teacher. If more pressed for time, calculate the velocity beforehand and provide that to students. For an added challenge, provide students two buggies with different velocities and ask them to place tape on the floor where the two will collide.

For calculus-based classes, have students use a large rubber band to launch a wooden block across the floor and predict where on the floor the block will skid to a stop. They will need to model the non-linear force-position function for the rubber band to figure out the initial amount of elastic potential energy that is then transferred to kinetic energy in the block and thermal energy due to the friction between the block and the floor.

pic1

Accurate time measurements for experiments like this are often difficult to achieve. However, a free mobile app called Hudl Technique, on both Android and iOS, takes not only slow-motion video, but also provides accurate time measurements down to the hundredth of a second. This opens the door for experiments for virtually all topics within mechanics at whatever level of challenge desired.

For statics problems hang objects from multiple spring scales all pulling on an object from different angles. Give students the mass of the object and have them predict a reading on a scale or vice-versa.

pic2

For energy, use two spring scales with different k-values that are tied to low-friction carts along a string. Task students with figuring out how far to stretch each spring such that the carts have the same velocity (O’Shea, 2012).

pic3

Once thinking has been shifted towards lab practicums, an entire new world of practicing physics in the literal sense is opened up. But even in the face of time and equipment limitations, there are Direct Measurement Videos[2] (DMV). The Science Education Resource Center (SERC) at Carlton College has a series of high-quality slow motion videos of different experiments relevant for a variety of physics topics, primarily mechanics. What sets these apart is that students can use the DMV web player to advance movies frame-by-frame as well as use screen overlays to take time and position measurements directly from the videos. They work seamlessly on laptops, desktops, and mobile devices.

What ties all of this together is the attempt to provide students with as authentic a scientific experience as possible. Real scientists solve new problems every day without an authority figure, other than nature itself, to appeal to when things get tough. They must deal with all kinds of problems associated with accurate data collection, reliable experimental setups, and finicky equipment. The end product of an experiment is rarely what it started out as, and allowing students a variety of opportunities to experience that process, to do science, will provide them with the kind of problem solving skills that physics teachers claim as the benefits of studying physics.

References

Kim, E., & Pak, S.-J. (2002). Students do not overcome conceptual difficulties after solving 1000 traditional problems. American Journal of Physics, 70(6), 759.

Mestre, J. P. (2001). Implications of research on learning for the education of prospective science and physics teachers. Physics Education, 36(1), 44.

O’Shea, K. (2012, February 5). Building the Energy Transfer Model. Retrieved from Physics! Blog!: https://kellyoshea.blog/2012/02/05/building-the-energy-transfer-model/

 

[1] Practicums for Physics Teachers: http://www.physicsforce.com/class-practicums/

[2] Direct Measurement Videos http://serc.carleton.edu/dmvideos/index.html

 

Lenz’s Law Activity

This is the second year I’ve taught AP Physics C: E&M, and we’re getting to the final leg of the course: electromagnetism. I was preparing the unit’s opening activities when I came up against the struggles of Lenz’s Law. One pesky negative sign within Faraday’s Law creates more trouble than just about anything in this class.

At this point in the year, my students have studied electrostatics, including potential and Gauss’ Law, RC circuits, and magnetostatics, including Ampere’s Law.

I’ve never really understood how to use Lenz’s Law to consistently predict the correct direction of an induced current. I was trying to figure it out again a few days ago, and it ended with some choice 4-letter words and my head on my desk.

But I’ve got it now! And I’ve come up with an activity that will help students understand it as well all thanks to the HTML5 version of PhET’s Faraday’s Law simulation.

The HTML5 version is particularly good because of the option to display the magnetic field lines coming from the magnet. This is the key to the entire activity as it’s what allows students to reason through how the magnetic flux through the wire loop is changing. It also being an HTML5 sim allows it to run on phones and tablets. Also, it’s FREE.

Faraday’s Law First

I had students first explore Faraday’s Law through the old java version of the sim. This one allows you to vary the size of the loops, which the HTML5 version doesn’t. I ask students simply to play with the sim and figure out the conditions required for current to be generated in the wire as well as to list the factors that affect the magnitude and direction of the current. Students don’t have much trouble figuring all of this out.

We then have a quick discussion on how to predict the direction of the induced current, and this is where things get sticky. There are lots of patchwork theories that arise that are basically a list of if-then statements involving the direction of motion of the north or south end of the magnet relative to the loop.

This is ultimately unsatisfying because a list of if-then’s is not a good scientific model, especially situations that aren’t “moving a bar magnet near a wire loop.” I then take a few minutes to review flux, which they’d seen from electrostatics, and then introduce magnetic flux as basically the same thing, but with magnetic fields.

I conclude the discussion by emphasizing that looking at this from a “how is the flux changing” perspective is more generalized than “which end of the magnet is moving how” perspective and, thus, will result in a much more generalized and useful model.

Data Collection

The goal of the activity is to collect “data” by drawing before-and-after pictures of the magnet moving into and out of the loop. Students use each set before-and-after drawings to figure out the change in magnetic flux due to the motion of the magnet, the voltage reading (whether it’s positive or negative) to determine the direction of the current, and then the right-hand rule (which they know from the previous unit) to figure out the field created by the wire.

Here’s one example of moving the north end of the magnet towards the loop. Having the “field lines” box checked is essential.

Before:

lenz1

After:

lenz2

In the “before” snapshot, students draw an arrow* indicating the direction of the flux through the loop. In the “after” snapshot, students draw another arrow showing the direction of the flux, longer if the flux increased and shorter if it decreased, then a dashed arrow showing the change in the flux. They also use the voltage (negative voltage = flow from positive terminal to negative terminal, positive = opposite) to determine the direction of the current, which they draw, and then they draw the direction of the field in the wire due to that current.

To figure out if the flux increases or decreases, they use the density of the field lines in the loop. More density = greater field = greater flux. The direction of the change is figured out by looking at the direction that they had to stretch or compress their initial flux arrow to turn it into the final flux arrow. For example, the first picture has a flux arrow pointing to the left, and the after has one longer, but still pointing to the left. If you wanted to make the first one look like the second, you’d have to stretch it to the left. Therefore, the change is to the left.

Students look at 4 scenarios, which require 2 drawings each:

  • The north side of the magnet moves from to the right of the loop to the front edge being inside the loop (pictured above).
  • The far end of the south side of the magnet moves from the center of the loop to the left of the loop (pictured below).
  • The far end of the south side of the magnet moves from the left of the loop to the center of the loop.
  • The far end of the north side of the magnet moves from the center of the loop to the right of the loop.

Basically they pull the magnet into and out of the loop from the right, then reverse and pull it into and out of the loop from the right.

lenz3

lenz4

*Since flux isn’t a vector, I’ve been careful not to call them vectors. Though, what we’re kind of doing is vector subtraction to figure out the direction of the change. I’m not quite sure why we can treat it like a vector and it still work out that way. Oh well.

Data Summary

Once they’re done, they’ve got 8 drawings with all kinds of arrows, so it’s kind of a mess. The goal is to see a pattern that they can use to predict the direction of the induced current based on the change in flux. When they’re eventually figuring this out on their own, I want them to think about it like this:

  1. Figure out the direction of the change in flux.
  2. Figure out the direction of the field that would oppose that change.
  3. Use the right-hand-rule to figure out the current required to generate that field.

To facilitate that, I had my students make a summary page that filtered out most of the data so that they could easily see the pattern. I put this up on the board and told them to fill in the rest from their data once they finished.

14A571DA-48A7-401F-BC20-F9FDCECF0E96

The conclusion to this was to figure out the pattern between the direction of the change in flux and the direction of the induced magnetic field. If done correctly, all of them should be opposite of one another, which students easily saw.

Here is an example of some student work.

1BBC7FCB-63D5-4694-AF45-EE08AC3DB4F0

19B3A3AB-280C-43CF-8C28-CD9A234EA441

A few other things

The devil is in the details for this activity, so I walked through the first set of drawings with everyone to make sure they were doing everything correctly. I told them that they needed to draw the same number of field lines for each snapshot (I chose 6, the simulation draws 8), and that when drawing them, they needed to make sure and basically draw them as they’re drawn on the simulation. Doing this right is key to being able to use the density of field lines to deduce the magnitude of the field/flux within the loop.

My students also have lots of practice in figuring out the direction of change by looking at before and after vectors. Even if they hadn’t, explaining it as I did above (by looking at the direction required to stretch or compress one vector to the other), I don’t think it would have been too difficult for them to figure out.

I also made sure to check on each group frequently to make sure they were doing everything correctly because, again, getting everything correct is key to seeing the pattern at the end.

If you have any suggestions for improving this, please let me know! If you use this (or something similar) and would like to share your experience with it, please do so as well! And if you need help or clarification with anything here, leave a comment, and I’ll do my best to help out.

Brainstorming a Special Relativity Activity

My AP Physics C: E&M class finished all the content we needed to get through with a few weeks to spare, so I decided (amidst their great excitement) to do a unit on relative motion and special relativity. Thankfully, the physics education twitter community is awesome, and I’ve been rescued from what would otherwise be certain doom with this unit, especially given that I had almost as much to learn about it as my students. Special thanks to @LCTTA and @kellyoshea for equipping me with a life preserver before I dove out into such choppy waters.

Through my various readings and video watchings, I came across this fantastic video by MinutePhysics. Take a moment to watch it before reading further; it’s only a 2 minute video, and it’ll save me a lot of typing. The inspiration for this activity came from what he call’s the switch-a-roo, in particular when “rotates” the slices vs. just sliding them (at about 1:03).

Beginnings of an activity

My idea is really nothing more than making strips of paper that students can slide and rotate much like the “slices” shown in the MinutePhysics video. This will be given to them after a week of basic relative motion, so frames of reference and relative velocity are part of their vocabulary now. This will also go right after an introduction to the Michelson-Morley experiment and their “discovery” that the speed of light is constant in all frames of reference.

There will be two objects in motion relative to one another along with a photon. Both objects and the photon start at the same position. One object remains stationary, the other moves away with a constant velocity as does the photon in the same direction. Here’s how I’m imagining this might go:

Change the frame of reference such that the other object is at rest.

From this:relmod5To this:rel4mod2.pngHow does this violate the rules of special relativity? I would be looking for them to notice what’s in green above. I’m also curious as to what discussion the jaggedness of the photon’s line (look closely at the second picture) might generate.

How can you perform a switch-a-roo that doesn’t violate the rules of special relativity? I’m a little unsure how to guide them towards a condition for …doesn’t violate the rules of special relativity beyond that they have to keep the photon line “unbroken.” That makes it work out right, but I’d like it a little more grounded in physics. However, even if I can’t figure that out, it at least allows them to see the creation of a new time axis.

Rel2mod.pngWhat do you think lines parallel to the bottom of each strip mean? It’s a new time axis! This will rely on their previous discussions about spacetime diagrams (which we’ve done for only non-relativistic scenarios in preparation for this part) about lines parallel to the time axis indicate constant position and vice-versa.

Does time in the new frame tick at the same rate as the first frame? Justify you claim with evidence and reasoning. This one’ll be a zinger! Time dilation!

Some other follow-up questions that I’ve yet to think of. Suggestions? Does the new frame have the same position axis as the old one? If not, what would a new one look like? See the next section on why I might ask this as I thought of it while typing it out. However, upon further reflection, I’m not even sure this can be done? Even if not through this particular example, it’s still a great discussion to have.

Some other follow-up questions that I’ve yet to think of. Suggestions?

Ponderings

What I like: I think the most effective (and coolest) thing about this is that it gives students some kind of visual for how to think about time dilation and why the constancy of the speed of light necessitates such an effect. Linking the effects of special relativity to one of the key postulates in relativity, the speed of light in vacuum is constant in all frames of reference, was something that I never really understood until recently, and I want my students to appreciate that link. I think it also helps them appreciate the genius of the theory in that everything works out just fine if you just abandon the idea of absolute simultaneity. It sounds so simple when you say it like that, but it’s really rather profound!

It also sets them up for utilizing spacetime diagrams for relativistic scenarios, especially if we want to get into drawing the new time and position axes for a moving frame on top of those for a stationary frame, as shown here:

What I don’t like: I begin the activity by them seeing that applying a regular ‘ol switch-a-roo (i.e., the Galilean transformation) makes light travels a shorter distance in the same amount of time, thus violating special relativity. And that’s a great starting place, I think. But there’s not really a way to circle back to that to verify that the new switch-a-roo (i.e., the Lorentz transformation) succeeds where the old one fails. I see how that happens when the new position axis is drawn (like in the SparkNotes picture above), but I’m not sure my students will see that. Perhaps I should just ask them to discuss whether or not a new position axis is needed and, if so, what would it look like? Hmm..

I’ll be giving this to my students in a few days, and I’m hoping for the best!

 

 

E&M Modeling: Inductance

This post is part of a series of posts documenting my efforts to keep with a modeling philosophy in my AP Physics C: E&M class where sticking to the Modeling method can often times be difficult. Getting exact measurements on things like electric field strength, electric and magnetic flux, and current changing over time can be difficult or outright impossible without a bunch of expensive equipment. Here’s another way that I’ve figured out how to adapt and keep as best I can do the method through the amazingly useful and 100% free PhET simulations paired with the Hudl Technique app.

Sequencing and setup

This “lab” was conducted after a 2 week introduction to electromagnetism through Faraday’s Law and Lenz’s, which they constructed a conceptual model of by using the Faraday’s Electromagnetic Lab simulation. The remaining time was spent working the typical run-of-the-mill induced current/EMF problems, including my favorite E&M demonstration of dropping a magnet through a copper tube.

I began by drawing a simple circuit with nothing more than a DC voltage source and a loop of wire, and they discussed how the magnetic field within the loops would change as the current and number of loops increased producing a simple drawing like this:

circuit1

I essentially wanted to get them thinking more about the time-behavior of circuits, which we haven’t done a lot of. Most of their experience has been that either the circuit is on or it’s off, no mention of anything in between.

Students then setup a simple RL circuit using the PhET Circuit Construction Kit (AC+DC) Virtual Lab simulation and answered a series of questions mostly aimed at making sure they noticed all of the things I wanted them to notice.

circuit2.png

question1

question2

To accompany question 5, I asked them to model time time-behavior the potential difference across the bulb and the inductor using a diagram I came up with last year. The idea here was to both solidify their understanding of the time-behavior of the potential difference as well as to set them up to be able to explain the results using Lenz’s Law.

circuit3.png

Recreation of what my students came up with.

They were already familiar with the negative sign due to Lenz’s Law within Faraday’s Law, but they viewed it from the context of determining current direction by looking at the direction of the change in magnetic flux. From here, I added to their drawing to help them understand exactly why the bulb started with no difference in potential across it.

circuit4.png

The opposite emf induced by the inductor cancels out what would otherwise be a 10V potential difference across the bulb.

Armed with the beginnings of a conceptual understanding of what was going on in the circuit, they were now tasked with developing a quantitative model for the current and potential differences within the circuit.

Obtaining and analyzing time-dependent behavior

I have neither the equipment, funding, or even the knowledge of what kind of equipment I would need to procure in order to get the kind of data I want. Instead, I had my students use the free Hudl Technique app, which not only turns any smartphone into a slow motion video camera, but also allows you to advance frame-by-frame through the video with the accompanying time-stamp of each frame. They placed an anmeter and voltmeter in frame of their camera, took a video, and then scrolled through afterwards in order to record the current and potential difference across the inductor at various points in time.

circuit5.png

Armed with a notebook full of data, my students jumped into Plot.ly to determine equations for V(t) across the inductor and I(t) within the circuit. Below are their results.

graph1.png

R-squared of 0.9998. Current function I(t) = 0.998 – 0.992*exp(-0.0544*t)

Students quickly realized that the 0.998 and 0.992 were awfully close to the maximum current of 1 A so were easily able to generalize that part of the equation. The 0.0544, on the other hand, was not so obvious. They’ve yet to begin tackling that yet, though. They’re potential difference graphs produced equally solid results:

graph2.png

R-squared of 0.9999. Potential difference function V(t) = 0.0626 + 9.89*exp(-0.541*t)

Like with the maximum current, students saw the parallel to maximum potential difference. They also noticed that the term within the exponential was almost identical, though they’ve yet to delve into what that means.

Finally, I asked them to plot the potential difference vs. the derivative of the current with respect to time as I knew that the slope of this graph would be the definition of inductance. They, of course, don’t know that yet. Taking the derivative of their current function and plotting it with respect to potential difference values gives the following:

graph3

R-squared of 0.9999. Potential difference function V = 182*dI/dt + 0.0762

Hey! It’s linear! And here’s why my students are for now, which we’ll pick up on the next time we meet. Everything after this is my attempt to explain everything out in the way that I’d want my students to see it. It will essentially be a record of me trying to figure out what all this means because I’m still not yet sure what inductance is beyond being able to qualitatively describe what higher or lower inductance does to a circuit’s behavior. But that’s ok because…

But what is inductance really?

Dimensional analysis can be used to figure out that the units of the slope are volt-seconds per amp, which is a little difficult to explain in the typical For every ___ [unit 1], ____ changes by [unit 2] framework that my students (and me) are used to. It can be shown that the units can also be written in terms of tesla-meters-squared per amp, giving units of magnetic flux per amp, which is much easier to understand. Hey! Inductance basically is a measure of the magnetic field per amp that can be generated in an inductor. Neat.

However, that doesn’t quite help determine how the circuit behaves as the slope, i.e. the inductance, changes.

Solving the equation for dI/dt yields the following:

equation

So, as the inductance increases, the rate of change of current decreases, meaning that it takes a longer amount of time for the circuit to reach the maximum current value. Since the current takes longer to ramp up, this means that the magnetic field generated within the inductor also takes longer to ramp up…

After running off and checking with the simulation, it turns out that I’m correct. Hooray! Though, to be honest, since both I and V are functions of time, I can’t quite shake why my logic feels a bit shakey even though the prediction is accurate. Something to think on.

Actually feeling like I understand inductance is a rather new feeling to me, but please let me know if I’m totally missing something here. You’ll do me and my students a huge service!

 

 

Learning to Appreciate The Choice of a System

A New Found Appreciation

On my seemingly mountainous, yet steadily disintegrating, pile of Really Important Physics Things That I Never Learned in Undergrad (TM) was the importance of choosing a system when analyzing multiple objects in an interaction. Physics is, among other things, a study of how things interact, but when your analysis is only focused on one object, then deliberately choosing a system seems unnecessary. A rope pulling a crate is certainly “multiple objects interacting”, however in problems such as this, nobody actually cares about the rope. For much of the traditional kinematics/dynamics sequence, the choice of a system is largely invisible and often a single object. The car. The crate. The ball.

But this becomes a problem when anything involving a conservation law comes into play. Whether or not some quantity is conserved is completely dependent upon the chosen system. Upon reflection, I imagine the unspoken, implicit choice of “system” becomes something murky like “all the objects mentioned in the problem” or even “the entire universe.” I also suspect that not explicitly defining a system causes issues with students applying conservation laws [citation to be added later, possibly?]. If students internalize “energy conservation” as “energy doesn’t change”, then I can see that causing issues with incorporating work/change in energy into their problem solving process. Same goes with the impulse-momentum theorem and momentum.

This new appreciation for the explicit choice of a system started with a twitter conversation earlier this year in which I expressed my dislike of the phrase “closed system.” Yet until recently, I wasn’t sure how to help my students develop the same appreciation and understanding of what it means to define a system in relation to conservation laws. Until now. I think.

Introducing Momentum

My students were working through the paradigm lab for the momentum unit from the AMTA Modeling Curriculum. Two frictionless carts, one of which is spring-loaded, “explode” away from one another. Students are tasked with placing them on the track at such a location that the carts reach the edge of the track at the same time. Through varying the mass of one of the carts, students discover the inherent ratio nature of momentum and interactions and come up with the following equation (momentum is defined as mass*velocity after they come up with the top one):

momentum

carts1

Afterwards, they worked on using momentum bar charts to represent the experiment. I doctored the numbers a little to keep the focus on the concept of thinking proportionally while not being overshadowed by seemingly random decimals and fractions. I asked them to determine the initial (as defined before the carts exploded) total momentum and the final total momentum (as defined as after the carts stopped touching), to which their surprise was zero. Peculiar!

To be perfectly honest, I wasn’t planning what came next until the night before I did it, which is when I remembered the twitter conversation from earlier this year. As it turns out, asking students to sum up the momenta of the carts came in handy later on.

Taking a Second Look

My general strategy for students discovering foundational laws and principles is to provide them both with an experiment and a way to view the results in such a way that whatever it is I’m wanting them to discover screams at them with a megaphone “HEY! LISTEN! HEY!” until they can’t stand it anymore. In this case, I’m wanting them to discover the Law of Conservation of Momentum while simultaneously approaching the Impulse-Momentum Theorem. Students were prompted at the beginning of the class to organize their notebooks into two columns with three rows each.

do this

I started with analyzing the carts separately because that’s what they’re used to. We’d not formally discussed how to analyze a collection of objects, much less that such a thing was “allowed.” I allowed students to work primarily in their groups without assistance from me, and they required little help as all of the diagrams and analysis was nothing new to them. Once I was certain that all the groups had gotten everything, I put my analysis on the whiteboard.

separate

The diagrams with all the circles on the left are system schemas, which were introduced months before when students were first learning how to draw free-body diagrams. Emphasis was placed on determining whether the momentum of each “system” (i.e., cart) was equal or not equal to zero, along with the final change in momentum from before, to during, to after.

Next up was for students to re-do the analysis, but treating both carts as a single system.

What’s in the booooxxxx?

What I was most worried about was the level of abstraction required for students to truly understand what it meant to analyze both carts as a single system. We hadn’t discussed center-of-mass, nor had we discussed what it would mean for the center of mass of an object or collection of objects to be located outside of those objects. While center-of-mass would be the most accurate way to articulate the effects due to conservation of momentum, it would require more new concepts and abstraction, which I was sure to be too much for my students to assimilate at once. Instead, I placed a box over both carts.

box

I emphasized that this sort of analysis is something that we’d done many times before. The carts themselves are made of individual atoms all interacting with each other, yet we never worried about it because it was unnecessary. Treating the carts as a “single” object was really no different, and the visual of the box really seemed to drive the point home.

During the times in which students were moving through each of the before, during, and after steps, I would raise the box, set the carts accordingly, and put it back down. I emphasized that we were only concerned with what happened to the box from the outside. Just like we didn’t care what individual atoms were doing before, we don’t really care about what happens inside the box. Without much help from me, students produced the following analysis in their notebooks. Again, emphasis is placed on determining the total change in momentum from before, during, to after the interaction.

after

At this point, students are asked to make a claim, and support it with evidence and reasoning. Most groups’ claims were of the following form:

If the forces on a system are balanced, then it’s momentum doesn’t change. If the forces are unbalanced, then it’s momentum does change.

Sounds a lot like a conservation law to me, which is typically stated like this:

In a closed system, the momentum of that system doesn’t change.

I’m with Frank; ditch the phrasing of “open” and “closed” system all together.

Closing Thoughts

There’s no reason to wait until momentum to introduce the idea of analyzing systems of single and multiple objects. Next year, I certainly won’t wait so long. I imagine the same type of demonstration can be performed and analyzed to get the point across. This will be particularly useful once we get into energy, where the entire concepts of work and power are defined by what’s happening by or on the system and, of course, the link between the system definition and the law of conservation of energy.

This has also helped me realize something about the kind of physics teacher that I am. I am continually fascinated by all the connections I’m discovering that I never found on the first, second, or third time around with all of this content. Now, more than ever, I see physics as an intricately constructed puzzle-that’s-also-a-tower, built from the ground up from a few simple principles and definitions. I want to help my students construct this tower for themselves and see all the beauty in the details that I do. More practically, I think that students being aware of these details helps them achieve a more deep understanding of physics.

E&M Modeling: Flux part 2

Setting the Stage

In my last post I wrote about a theoretical approach to having students discover the concept of electric flux for a point charge. While they struggled with explaining what the proportionality constant for their Electric field vs. enclosed area graphs meant (aka, the flux), I was still satisfied with the results. What turned out to be the most useful question was asking them why I might have chosen a spherical surface to draw around the point charge as opposed to some other shape. After all, it’s a completely imaginary surface, so why this one?

They didn’t have much to say until we spent some additional time refining the equation we’d come up with:

fluxequation2

Those of you more familiar with the concept of flux will recognize the missing cos(theta), so my next goal was to have them choose to add that term to the equation.

I started the discussion by holding up a whiteboard and told them to imagine a steady stream of wind flowing with a constant speed directly at me. I asked At what angle should I hold the whiteboard so that I could “harness” the wind stream as best I can?

It only took a few moments for them to realize that angling it completely perpendicular would allow them to maximize their “use” of the wind stream and that holding the board parallel would minimize it. I then asked about the wind then blowing at some in-between angle and put this picture up to help them visualize.

flux2

They first decided that this would yield some level of “usefulness” that was in-between the maximum and minimum values based on the angle between the vectors and the surface. Asking which “part” of the vector contributed to the “usefulness” led them to conclude that the perpendicular component of the vector was the only part that contributed. They were unable to see a difference between using the sin of the angle between the vertical and the vector and the cos of the angle perpendicular to the surface, which is fine because there isn’t one. I said that we should use the angle with the perpendicular only because that’s the convention, which will come in handy later. Their equation was now this:

flux3

Revisiting my original question

My original question was this: Why did I choose to draw a spherical surface around the point charge? Why not any other surface?

I put this picture back up on the board to help with visualization. I’m discovering more and more that the difficulties with flux and Gauss’ Law are primarily with visualizing the various surfaces and charge distributions, so any chance I have to aid in that I take.

fieldlines

They quickly determined that the field lines were perpendicular at each point of the circular surface, which was easily extended to a spherical one. Since each point was perpendicular, cos(theta) is just 1, thus the flux calculation is almost trivial.

Making the flux calculation trivial is one of the primary ways to effectively utilize Gauss’ Law. My purpose here was to continue their development of seeing spatial symmetries and using those symmetries to simplify otherwise complicated calculations.

Finalizing the flux equation

The equation they’d come up with was more generalized, but still not as general as it needs to be for both using and understanding Gauss’ Law at a deeper conceptual level. I also continue to emphasize that the more general a scientific model is, the more useful it becomes. So the issue with their equation is that it assumes a surface in which the field lines are perpendicular at each point of the surface. What if that’s not possible? How can we generalize the equation to work for any surface?

I drew a square around the point charge to help jump start the discussion.

flux4

The question now is this: How can we calculate the flux through the square?

I’m careful to clarify that I’m not looking for them to jump to a mathematical equation, but rather I want them to brainstorm on a conceptual way to do it. We’ll worry about the mathematics later. Just talk me through how you might calculate this.

I should mention that this isn’t the first time I’ve had them do this kind of exercise. I first exposed them to this when calculating electric fields of extended bodies, pre-flux/Gauss’ Law as a way to get them thinking in “calculus-mode.” Even though all of my students are in AP Calc BC, they haven’t gotten to integrals yet, but that doesn’t mean they can’t start “thinking calculus.” Writing this post now has me thinking that I’ll write up my approach to that. So I’ll do that at some point.

Anyway, when they got stuck, I asked them how they could get the contribution to the flux due to an individual field line and then reminded them that the total flux was the contribution due to all the field lines. They came up with this:

We just need to know the E at each point on the square, multiply it by the cos of the angle perpendicular to the square, add all that up, and then multiply by the area of the square.

I then asked them to write that out mathematically yielding this:

flux5

While that may cause a mathematician to cringe (and the Asquare part isn’t fully correct), I wasn’t worried about the correctness of their notation. The essence is there. This also let them see if E was the same at each point of the square and if cos(theta) was always 1, then those things could be pulled out of the sum, which will be useful once they get to Gauss’ Law. I then introduce the more formal definition of flux and made sure to show how each element of their equation translated to this definition:

flux6

I also briefly discuss the dot product, though I honestly haven’t done a great job of helping them build their understanding of it. I’m not sure that I care right now… but we’ll see if that comes back to bite me later one. I’m careful to emphasize that the only real differences between their equation and this one is that of notation. I explain what the circle on the integral means along with the S below the integral.

Next up: Gauss’ Law

My students still weren’t exactly sure why we we were spending so much time on this concept of flux. I’d mentioned several times that this would help make deriving expressions for electric fields much easier than the brute force method they’d done previously, but that doesn’t mean they see why this will be a path to that.

One student had alluded to looking at the relationship between the flux and the charge inside the surface, which is exactly what Gauss’ Law is, so I brought that point back up for discussion. In my next post, I’ll talk about how I had them approach investigating that and how they used it to come up with Gauss’ Law.