Linear data sets (for your enjoyment)

Kicking off the year in my modeling physics course means practicing working with and interpreting linear data. Some students quickly pick up the modeling method of describing slope and intercept, while other students just need more practice for the data to speak to them in the same way.

I use the spaghetti bridge lab to introduce graphing linear relationships and have a pretty good handout1 with a few more linear data sets as practice.

However, my experience has taught me two things:

  1. There will be students that just need a bit more practice to really nail down the skill.
  2. It's deceptively hard coming up with linear data sets.

So, I sat down earlier and surfed the internet, found some real data sets, cleaned them up a bit, and imported them into Excel & Google Sheets (from whence they are easily copyable and paste-able). I might as well share them, because I know you'd prefer to avoid converting .txt data files to .xlsx. In fact, I'll make the Google Sheets version editable, so you can add your own awesome data sets).

Download the data sets

The Excel version includes graphs with the equation for best fit lines. Google Sheets doesn't do best-fit lines yet, so those have the graphs (as interpreted by the Google Sheets converter), but no equations.

Enjoy!

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  1. from the AMTA Modeling Physics curricular materials, which is why I'm not sharing them here. But, seriously, you should just join the AMTA so you can access the huge wealth of resources they provide. Here's the link. Do it. (back)

My first weeks as a Modeling Physics teacher

As I mentioned earlier, I'm teaching Physics at a new-ish high school this year. I've been spending a large chunk of time designing the curriculum and materials for this class. So far, the year has been a bit hectic (thus the lack of posts here), but the school community is really amazing, supportive, and progressive. A few things that are making what can be a difficult first year much better than average:

  1. Experts at my fingertips & time to develop curriculum. The curriculum people at my new school were very proactive in trying to connect me to experienced physics teachers. I was (and continue to be) impressed with the level of support they're providing for teachers developing new curricula. Unfortunately, none of the teachers had used Modeling Instruction. Fortunately, I've curated a twitter feed that includes 15-20 active modelers and I've found countless helpful resources from those very helpful people. We've also had dedicated time to work on curriculum development. Besides a (paid) week in June, we've also been given time during our professional development time to simply work on building the curriculum. As someone new to the school developing the curriculum for a class that has never before been offered at this school, this has been invaluable.
  2. The willingness to help a n00b. Here's a Venn Diagram showing Teachers using Twitter/Blogs and Teachers willing to help out a poor Modeling Instruction rookie who wasn't able to make it to a Modeling Workshop this summer due to his crazy schedule:
    You people are awesome.
    Perhaps this shouldn't be surprising- I mean, if someone is actively spending time writing a blog or sharing via twitter they're more than likely into the whole "sharing" thing. I'm sure I've asked (and will continue to ask) more than my share of dumb questions. Amazingly, despite my frequent questions that surely induce heavy eye-rolling on the other side of the Internet, I've continued to receive an amazing amount of help with zero snark (and zero snark when we're talking about The Tweeter is nothing to shake a stick at!).
  3. The huge resource of online materials. I chose Modeling Instruction as the curriculum for my Physics classes because I believe in the process it supports- not because it's the easiest to design and implement. To be honest, it's a bit scary (especially because I couldn't get to a Modeling Workshop prior to implementation). However, there is no shortage of materials to be found online- and not just general "modeling-instruction-is-great-and-here's-why" materials (there's a lot of that too, though). There are detailed descriptions of labs and their results, handouts, tips for whiteboarding, worksheets, etc., etc., etc.

    Here's a partial list:

    • The American Modeling Teacher's Association. Yes, you need to be a member to access the resources, but the resources are huge. I shelled out the $250 for a lifetime membership. The materials and support I gained access to for that money is easily worth the $250 by itself.
    • Kelly O'Shea's Model Building Posts & Unit Packets. Kelly's an expert modeler. Her posts really helped me first visualize what a modeling classroom looks like. Her materials are also excellent.
    • Mark Schober's Modeling Physics. Contains materials and resources for every modeling unit, along with calendars- which was nice as someone new to modeling to get a rough timeline for each unit.
    • Todd K's DHS Physics Site. Even more modeling materials and calendars.
  4. Paying it forward. It's my plan to make the materials I develop and implement for my Physics classes readily available online in some format, at some point. I've gained so much value from the resources others have posted that it is (perhaps with some hubris) my hope that others in the future might gain something from my experience. Obviously I'm no expert- but my hope is that through sharing both the materials and my reflections on how they were implemented will, if nothing else, help me to become a more purposeful and reflective educator.

Perhaps I'm odd, but I really enjoy designing new curricula- which is a lucky break since I'm responsible for designing the Physics curriculum from the ground up. So far it's been a challenge given the specifics of my particular situation (which will undoubtedly be a topic for future post), but as I come to know my students better and gain more experience implementing modeling instruction, I've found the process more and more enjoyable.

The spring constant of Winston Churchill's belly

This will be the most scientific and precise post regarding Winston Churchill's belly you'll read today. Maybe all week.

Today, we'll be analyzing the following video:

After randomly embedding the preceding video while thinking about Hooke's Law and the spring constant in my last post, what I, and I'm sure you as well, immediately wonder is, of course, "I wonder what type of spring constant Winston Churchill's belly had?" This seems like something worthy of my time.

Here we go!

If we're going to figure this out, we need some data. First, we need some sense of scale. Since I have no idea the how tall the Animaniacs are, let's focus on the historical figures. I'm going to go with Winston Churchill's height to give the video some scale since he's pretty stretched out whilst his belly is being jumped upon1. It's surprisingly hard to find Churchill's height online with any sort of citation. I found what seems like a pretty solid source (via Wikipedia) for the height of Harry S. Truman (1.75 m). Using that information along with the following picture, I can figure out Churchill's height after throwing the image into Tracker:

Churchill and Truman were nearly the same height. I got 1.76 m (5 ft, 9 in) for Churchill. That seems pretty close to most of the unsourced figures for his height I found online.

I think the best way to go about finding the spring constant for Winston Churchill's belly is to use gravitational potential energy and elastic potential energy. If we can find the gravitational potential energy Stalin has at the top of his bounce and the maximum compression of Churchill's belly, we should be able to do the following:

mg\Delta y = \frac{1}{2}kx^2 \\ \\ k = \dfrac{2mg\Delta y}{x^2}

Where m is Stalin's mass, Δy is Stalin's maximum height above Churchill's belly, and x is the maximum compression of Churchill's belly.

I can fairly easily find Δy and x using Tracker to analyze the video.

I used 1.70 m for Churchill's height in the video instead of the 1.76 m figure above since his knees are bent slightly. Using that information to scale the video, Stalin's maximum height (Δy) is 0.65 meters and the maximum compression of Churchill's belly (x) is 0.28 m.

Finding Stalin's mass will require another long and probably fruitless internet search. Instead, I'm going to assume from the above picture Stalin is approximately the same height as Harry S. Truman and then assume Stalin's BMI is slightly above average (he was a dictator- which means he has access to lots of food). I'm going to say Stalin's BMI is 26. According to this BMI calculator, that would give Stalin a weight of 175 lbs, or 79.4 kg.

Now we've precisely (ha.) figured out all our variables, so we can go ahead and solve the equation for the spring constant (k):

k = \dfrac{2mg\Delta y}{x^2} \\ \\ \\ k = \dfrac{2(79.4\text{ kg})(9.8\text{ m/s}^2)(0.65\text{ m})}{(0.28\text{ m})^2} \\ \\ \\ k = 12,900\text{ N/m}

OK, so what's that mean? It means that if you could compress Winston Churchill's belly by a full meter it would require 12,900 Newtons of force. On the surface of the Earth, that would take a mass of 1,315 kg (2,900 lbs) sitting on his belly to compress it by a full meter2. WolframAlpha helpfully notes that this is approximately a mass equivalent to approximately 2 "typical dairy cows."

We can also learn something about the Animaniacs' collective mass now that we know the spring constant. If we rearrange the previous equation to solve for the mass, we get:

m = \dfrac{kx^2}{2g\Delta y}

It looks like the maximum height the Animaniacs attain is 0.77 m with a maximum belly compression of 0.16 m. Now solving for the mass we find:
m = \dfrac{(12900\text{ N/m})(0.16\text{ m})^2}{2(9.8\text{ m/s}^2)(0.77\text{ m})} \\ \\ \\ m = 21.9\text{ kg}

Collectively the three Animaniacs have a mass of 21.9 kg (48.3 lbs). Wow. They're lighter than I anticipated. If you divide that figure evenly by three, the average Animaniac weight is 16.1 lbs. Clearly Dot and Wakko are smaller than Yakko. This may, in fact, prove Dot's hypothesis that in addition to being cute, she's a cat:

Watch animaniacs - what are we? in Animation  |  View More Free Videos Online at Veoh.com

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  1. Also, I came across a few places that speculated that Stalin may have use elevator shoes to make himself seem taller, so it might be harder to get an accurate figure for him. However, this isn't exactly going to be a super-accuracy fest anyway, so maybe I shouldn't let that bother me. (back)
  2. I'm not sure if Churchill actually has a meter of stomach to depress, but you get the idea. (back)

Worksheet labs aren't that great: Hooke's Law

In a recent post, I strongly suggested that a physics class should be a place where students are actively involved in the exploration of the relationships that exist between different variables (force and mass, for example)- not a place where students are simply given a list of equations they are told explain how the world works. Let's continue down this line with an example.

Example: Simple Harmonic Motion and Hooke's Law

This is a lab from a college class I took last semester:

Analysis

This lab isn't terrible. I mean, who doesn't like bouncing springs?


In the first part, we were required to find the spring constant by examining the relationship between the force applied to the spring and the spring's elongation using a graph. That's not too shabby, right? Well...no...but...

What the lab doesn't require is any thinking about the relationship between force and elongation. You make a nice graph, but are told right in the instructions that the slope of the graph is this thing called the "spring constant." We aren't expected to know anything more about how the relationship between force and elongation and the spring constant works.

In part two, we varied the mass on the spring and measured the period of the spring's oscillation, which we then compared to the expected period based upon our calculations and a formula we were given ahead of time:

T = 2 \pi \sqrt{ \dfrac{m}{k}}

I didn't need to know much to write up the lab report:

  1. The period of a spring's oscillation depends on the mass attached to the spring.
  2. The formula we were given to find the period of a spring's oscillation works.

That's it. If I was an astute student I might've realized that the slope of a Force-Elongation graph will give you the spring constant- but we were walked through that step in such a way that it would have been easy to miss that tidbit. Never mind understanding what having a larger or smaller spring constant would mean in real life.

Rethinking the lab

So now you're thinking that I'm just a cranky-pants who likes pointing out the failings of other people's labs. Let me try to improve your perception of myself by explaining how I'd like to run a lab covering the same content.

First, I think it's important to identify what I want students to understand as a result of completing this activity. I'd like them to understand:

  1. The nature of the relationship between the force applied to a spring and the spring's elongation.
  2. The slope of a Force-Elongation plot is the "spring constant."
  3. The nature of the relationship between the mass hanging on a spring and the spring's oscillation period.

Second, I want the students to play be the primary investigators. I'm not going to give them a sheet explaining step by step exactly what they have to do. I want the students to handle that part. Maybe I give each group of students a few springs and a set of masses and simply set them free to play around and make observations for 10 minutes or so- after which we discuss as a class observations they have made and decide upon a path for further investigation. Maybe I give some guidance right away and tell them to investigate the relationship between the mass on the spring and the elongation of the spring.

Third, we draw some Force-Elongation graphs. We discuss the relationship between force and spring elongation (it should be pretty obvious it's a direct linear proportionality- i.e., if you double the force on the spring, you double its elongation). So now we know that F \propto x. Next, we look at the difference in the graphs for each spring. Why are some lines steeper than others? What is the difference between a spring with a steep slope and a spring with a more gradual slope? Then explain the slope on a Force-Elongation graph is called the "spring constant." So now we've figured out that if we know the force acting on a spring and that spring's spring constant, we can figure out how much the spring will stretch: F=kx. Hey...that looks an awful lot like Hooke's Law...

Fourth, I'd play this video clip:

Fifth, I'd tell students to investigate the relationship between the amount of mass on a spring and the period of the spring's oscillation. We'd collect data, make some graphs, and hopefully come to the conclusion that T \propto \sqrt{m}.

If we stop here, we've already done a lot. We've discovered Hooke's Law. We understand a stiffer spring has a bigger spring constant. We know how doubling the mass on a spring will affect the spring's oscillation. At this point I could introduce the equation T = 2 \pi \sqrt{ \frac{m}{k}}. Maybe we could then do the second part of the lab posted above and see how close the observed periods of the springs match the values calculated with that forumla. We'd probably notice all of our observed periods were off by a little bit. This opens up a discussion of why we all have this systematic error. Why are we all off? What could be off? Looking at the formula, there are really only two places we could have error: the spring constant or the mass. Maybe we draw a free-body diagram for the mass on the spring. At this point a student will probably suggest we need to draw a free-body diagram for the spring as well. Hmm...you know...this spring has mass too...could the mass of the spring itself be affecting the spring's period? Now we've independently figured out we need to consider the spring's mass as well. From there we could figure out a test to determine how much of the spring's mass we need to include.

Overcoming the traditional lab format

If you randomly visited physics courses in high schools and colleges across the nation, you'd most likely see a lot of labs similar to the first lab. Traditionally physics labs have been designed so you're given a formula and are asked to make observations that fit with the formula. This is despite the fact that the student-led investigation requires deeper thinking, encourages greater engagement and thinking about the concepts, a better understanding of how the world works, and an understanding of what an equation actually means.

Why should this be so? I believe it's because traditional labs are easy. Print out a sheet with a step by step procedure. Hand out the supplies. Make some measurements. Maybe make a graph. Answer a couple quick questions. Done. The student-led investigation is tricker to share and explain. The entire process I described in the student-led investigation could be preformed without any worksheets whatsoever. It's harder for teachers looking for a new lab to stumble on a description of this type of lab. It's really easy to hit up The Google and find a lab handout, save it, print it, and pass it out. Student-led investigations also lead to potential student errors. Students may struggle. It may take more class time. Sometimes you'll get data that doesn't turn out as well as you'd like. This can be scary and frustrating for teachers. And yet...

Struggling with what this or that graph is telling us, or being forced to think about where errors came from, or having to defend your results and process requires a lot of thinking. Critical thinking. And helping students learn to think critically is worth the extra time and effort. As a bonus, they'll also actually understand the physics better, which is also a good thing in a physics class. 🙂

What is the purpose of Physics class?

I took three physics classes through a local community college last semester. From how the content was presented in each class, it would be fair to say Physics is primarily concerned with learning a set of equations and then figuring out which equation you need to use in order to find the right answer.

This is not a very useful skill. People wiser than I have pointed out similar things. So why, in high school and introductory college physics classes, do they lean so heavily on "learning the formulas?" Here are the two arguments I've heard the most often:

They'll need it in college/their careers

It could be argued, perhaps, that it is good preparation for students who will be pursuing engineering or scientific careers- after all, they'll be taking college classes and graduate classes and probably use a couple equations during their careers. However, there's a big problem with this line of thinking. Are all the students in a high school physics class there because they're planning on becoming scientists and engineers? A few, maybe. Most of them will not- and that's OK, but this realization should cause us to rethink how we present the material.

The equations explain the relationship between variables

I'm sympathetic towards this line of thinking (more on this later)- but not enough to think it's valid. Whenever I hear this argument the first question that comes to mind is "Is this the best way to explore those relationships?" In my experience, students who have struggled understanding physics often did so because they couldn't make sense of what the equations actually describe. Given an equation and all the variables but one and they'd be able to work though a problem, but they weren't understanding why that answer makes sense and any further obfuscation of the problem quickly threw them off track. I agree that the relationship between variables is an important bit. I don't believe that equations clarify that relationship for the vast majority of students.

How I'd like to teach physics

Understanding the relationship between variables, in my mind, is the key to a useful understanding of physics. If I push twice as hard on this shopping cart, what happens to the cart's acceleration? That's a tangible situation that is easier to understand than simply throwing out F = ma and hoping students figure out that relationship on their own. Further, students should discover these relationships. Give students some equipment and tools and have them measure what happens to an object's acceleration as they apply more or less force on the object (some tracking software would be really handy for this). Then have them apply the same force but change up the mass. Chances are pretty good they'll be able to discover F = ma on their own. Chances are they'll have a much better conceptual understanding of what F = ma means at this point than if you simply gave them the equation and had them do some problems. Or if you simply had them prove the formula is correct in a lab.

Why it matters

  1. I believe the focus on relationships promotes a better conceptual understanding of physics- the students can more effectively internalize the way the world around them works. A populace with a healthy baseline of physics knowledge could prevent silly and potential harmful pseudoscience such as magnet therapy from becoming an issue.
  2. There's been a focus on increasing interest in STEM careers- and a special focus on recruiting women and minorities into STEM fields (see this White House press release). An equation-focused physics curriculum can seem intimidating to students. A collaborative, constructivist approach can be perceived to be less intimidating and more welcoming (I'd recommend giving Episode 32 of the Shifted Learning podcast for some interesting bits on gender issues in STEM).

Modeling Instruction

I don't have much experience with Modeling Instruction, but it seems from my reading that the instruction I've been describing is essentially what it is. As a bonus, it's well developed, well researched, and well used instructional method to improve students' ability to construct a better understanding of the physical world around them.

If you're interested, I've found both Kelly O'Shea's series on model building and Frank Noschese's primer to modeling instruction to be great resources. Check their blogrolls for even more good stuff from teachers using modeling.

As I look forward to potentially teaching physics next year I want students who take my classes to come out with a lasting understanding of the topic. I don't want them to half-heartedly memorize equations that they'll forget two weeks after we finish a unit. I'd like to teach for all the students, not just the future scientists and engineers.

Pipe Insulation Roller Coaster Assessment

Welcome back. If you haven't joined us for the last two posts, let me recommend that you first read about determining rolling friction on the coaster and the project overview.

On to the assessment...

Assessment is extremely important. It explicitly informs students what things we value (and thus the things we value). If we assess the wrong things, students will focus on the wrong things. This can turn an otherwise excellent project into a mediocre project. For this post, I'll share two methods of assessment: First, the "old" method I used when I last taught physics (in 2008). Second, my updated assessment scheme that I'd use if I did this project again.

The old assessment strategy

Embedded below is the document I gave to students at the beginning of the pipe insulation roller coaster project. Most noticeably it includes a description of the assessment scheme I used way back in January of 2008.


As you can see, I split the assessment of this project into two equal parts:

An assessment of the finished roller coaster

I wanted students to think carefully about the design, construction, and "marketing" of their coasters. I wanted them to design coasters that not only met the requirements, but coasters that were beautiful and interesting. Individual items being assessed under this rubric were weighted differently. For example, "Appropriate name of the coaster" was only worth 5%, while "Creativity, originality, and aesthetics" was worth 20%. Here's a link to the sheet I used when assessing this aspect of the coaster project.

An assessment of the physics concepts

In the embedded document above, you can see the breakdown of what items were being assessed. In my last post on pipe insulation roller coasters, you can see how students labeled their coasters with information on the marble's energy, velocity, and such along the track. Groups were required to turn in a sheet with the calculations they performed to arrive at these numbers. These sheets were the primary basis for determining whether students understood the physics concepts.

Problems

There are a lot of problems with the assessment scheme as described above. I'm not going to try to address them all, so here are a couple of the biggest issues:

  • Assessing coaster design
    • I'm a fan of elegant design. For this project I'm a fan of finished coasters that look well designed and exciting. That's why I included the first part of the assessment. I wanted to incentivize students to think about the design and construction of their coasters. In retrospect this is probably unnecessary. Students generally came into this project with plenty of intrinsic motivation to make their coaster the best in the history of the class. While I'd still stress the importance of quality design in the future, I'd completely cut this half of the assessment. Students already cared about the design of their coaster. If anything, awarding points for coaster design had an net negative effect. Especially because it doesn't assess anything related to the understanding of physics.
  • Assessing student understanding of physics concepts
    • As a normal part of working in a group while attempting to complete a large project in a limited time, students split up the work. Students are generally pretty smart about this in their own way. While I stressed that everyone in the group should contribute equally towards the calculations. Most groups would have the student who had the best understanding of the physics do most of the calculations. Why? Because it was faster. They needed to finish their coaster and just having the fastest person do the calculations meant more time for construction. While I generally knew when students in a group were adding very little to the calculations (and would assess them accordingly), on the whole this method didn't give me a good picture of each individual students' level of understanding. There were certainly students who skated through the project while minimally demonstrating their understanding of the energy and friction concepts involved.

The new assessment strategy

You've probably already picked up on a few of the improvements I'd make for this project.

  1. Use standards-based assessment. Standards-based assessment is an integral part of the classroom throughout the year- not just for projects. If you're unfamiliar with what this "standards-based" business is all about click the little number at the end of this sentence for plenty of links in the footnotes1. Here are a list of standards that would be assessed through this project:

    Content standards assessed

    • Energy
      • Understand and apply the law of conservation of energy.
      • Explain and calculate the kinetic energy and potential energy of an object.
      • Explain and calculate the amount of work done on and by an object.
      • Solve basic conservation of energy problems involving kinetic energy and potential energy.
      • Solve conservation of energy problems involving work and thermal energy.
    • Circular Motion
      • Solve basic circular motion problems using formulas.
    • Habits of Mind
      • Collaborate and communicate with others to meet specific goals.
      • Handle and overcome hurdles creatively and productively.

    The specific standards used can vary based on your specific implementation.

  2. No points for coaster requirements. As I mentioned earlier, it proved unnecessary to incentivize their coaster designs and meeting the basic requirements of the project. This decision also comes out of standards-based grading, which focuses assessment around, "Do you know physics?" instead of "Can you jump through the right hoops?" That isn't to say we don't talk about what makes a coaster "exciting" or "aesthetically pleasing" or whatever. It just means a student needs to demonstrate their understanding of the physics to earn their grade.
  3. A focus on informal assessment. Rather than heavily relying on a sheet of calculations turned in at the end of the project (and probably done lopsidedly by one or two group members) to determine if the group understands the physics, I'd assess their understanding as I walked around the classroom discussing the coasters and their designs with the students as they work on them. Starting with questions like, "Why did you make that loop smaller?," or "Where are you having trouble staying within the requirements?" can be used to probe into student thinking and understanding. The final calculations would still be a part of the assessment, but no longer the single key piece of information in the assessment.

On the whole I was very happy with this project as I used it in the past. As I've learned and grown as a teacher I've found several ways I can tweak the old project to keep up with the type of student learning I want to support in my classroom. If you have other suggestions for improvement, I'd be happy to hear them.

As a bonus, here's a student produced video of the roller coaster project made for the daily announcements. The video was made by a student who wasn't in the physics class, so there's a little more emphasis on the destruction of the roller coasters at the end of the project than I'd like. Kids. What can ya do?

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  1. Here are posts I've written about my experience implementing standards-based assessment. I'm not an expert, so let me also direct you my bookmarks related to standards-based grading, and some resources written by a couple people who are more expert: Shawn Cornally and Frank Noschese (who offers blog posts, a shared google doc foler, and a collection of bookmarked links). There are certainly other great resources out there, but these are a great starting point. (back)

Pipe Insulation Roller Coasters

The Hazard Zone team with their coaster
The Hazard Zone team with their coaster

I like projects. I really liked this project. The pipe insulation roller coaster project is one of the most enjoyable projects I've ever used in class.

History

It was my second year teaching physics. During the unit on energy, the book we were using frequently used roller coasters in their problems. We even had a little "roller coaster" to use with photo gates. I thought we could do better.

My original idea was to get some flexible Hot Wheels tracks and make some loop-de-loops and hills. Turns out a class set of Hot Wheels track is pretty expensive. On an unrelated yet serendipitous visit to my local big box hardware store, I ran across the perfect (and cheap!) substitute: Pipe Insulation!. For $1.30 or so you can get six feet of pipe insulation- which doubles nicely as a marble track1 when you split the pipe insulation into two equal halves. It's really easy to cut pipe insulation with a sharp pair of scissors. Just be sure you don't buy the "self-sealing" pipe insulation, which has glue pre-applied- it's more expensive and it'd turn into a sticky mess.

At first I planning to simply design a one-period long investigation using the pipe insulation (my original ideas morphed into the pre-activity for this project). As I started to think through the project more and more, I realized we could go way bigger. And thus, the pipe insulation roller coaster project was born.

Building the Coasters

In groups of three, students were given 24 feet of pipe insulation (4 pieces), a roll of duct tape2, and access to a large pile of cardboard boxes3. All groups had to adhere to a few standard requirements:

  • Construction requirements
    1. The entire roller coaster must fit within a 1.0m x 2.0m rectangle4.
    2. There must be at least two inversions (loops, corkscrews, etc.).
    3. All 24 feet of pipe insulation must be used.
    4. The track must end 50 cm above the ground.

    The coaster is under construction

  • Physics requirements
    In addition to meeting the above requirements, students were required to utilize their understanding of the work-energy theorem, circular motion, and friction to do the following:
    1. Determine the average rolling friction, kinetic energy, and potential energy at 8 locations on their roller coaster.
    2. Determine the minimum velocities required for the marble to stay on the track at the top of all the inversions
    3. Determine the g-forces the marble experiences through the inversions and at least five additional corners, hills, or valleys.
    4. The g-forces must be kept at "safe" levels5.
Labeling the Physics Data
The labels included information about the energy, velocity, and acceleration of the marble at specific points (Note: there are some sig fig issues here).
Calculations
  1. Rolling friction, kinetic energy, and potential energy
    • The potential energy (U_g = mgh) is easy enough to find after measuring the height of the track and finding the mass of the marble. The kinetic energy is trickier and can be done by filming the marble and doing some analysis with Tracker, but since the speed of the marble is likely to be a little too fast for most cameras to pick up clearly, it's probably easier (and much faster) to simply measure the time it takes the marble travel a certain length of track. I describe how this can be done in a previous post, so check that out for more info. That post also includes how to calculated the coefficient of friction by finding how much work was done on the marble due to friction- so I'll keep things shorter here by not re-explaining that process.
      • Pro-tip: Have students mark every 10 cm or so on their track before they start putting together their coasters (note the tape marks in this pic). Since d in W=F\cdot d in this case is the length of track the marble has rolled so far, it makes finding the value for d much easier than trying to measure a twisting, looping roller coaster track.
    
    
  2. Minimum marble velocities through the inversions.
    • This is also called the critical velocity. That's fitting. If you're riding a roller coaster it's pretty critical that you make it around each loop. Also, you might be in critical condition if you don't. While falling to our death would be exciting, it also limits the ability to ride roller coasters in the future (and I like roller coasters). Since we're primarily concerned with what is happening to the marble at the top of the loop, here's a diagram of the vertical forces on the marble at the very top of the loop:

      The red curve is the roller coaster track. Just so you know.

      So just normal force (the track pushing on the marble) and gravitational force (the earth pulling on the marble). Since these forces are both acting towards the center of the loop together they're equal to the radial force:

      When the marble is just barely making it around the loop (at the critical velocity), the normal force goes to zero. That is, the track stops pushing on the marble for just an instant at the top of the loop. If the normal force stays zero for any longer than that it means the marble is in free fall, and that's just not safe. So:

      Then when you substitute in masses and accelerations for the forces and do some rearranging:

      There you go. All you need to know is the radius of the loop, and that's easy enough to measure. Of course, you'd want a little cushion above the critical velocity, especially because we're ignoring the friction that is constantly slowing down the marble as it makes its way down the track.

  3. Finding g-forces
    • An exciting roller coaster will make you weightless and in the next instant squish you into your seat. A really bad roller coaster squishes you until you pass out. This is awesomely known as G-LOC (G-force Induced Loss of Consciousness). With the proper training and gear, fighter pilots can make it to about 9g's before G-LOC. Mere mortals like myself usually experience G-LOC between 4 and 6g's.

      As I mentioned, I set the limit for pipe insulation roller coasters at 30g's simply because it allowed more creative and exciting coaster designs. While this would kill most humans, it turns out marbles have a very high tolerance before reaching G-LOC.

      To find the g-forces being pulled on corners, loops, or hills you just need to find the radial acceleration (keeping in mind that 1g = 9.8 m/s^2):

    Pipe Insulation Roller Coaster construction underway

    Raise the stakes

    Students become fiercely proud of their roller coasters. They'll name them. Brag about them. Drag their friends in during lunch to show them off. Seeing this, I had students show off their creations to any teachers, parents, or administrators that I was able to cajole into stopping by for the official testing of the coasters. I even made up a fun little rubric (.doc file) for any observers to fill out for each coaster. This introduces some level of competition into the project, which gives me pause- though from day one students generally start some friendly smack talk about how their coaster is akin to the Millenium Force while all other coasters are more like the Woodstock Express. The students love to show off their coasters, and it seems the people being shown enjoy the experience as well.

    Coaster judging in progress.
    Coaster judging in progress.

    Assessment

    Assessment is massively important. However, this post is already long. The exciting conclusion of this post will feature the assessment piece in: Part 2: Pipe Insulation Roller Coaster Assessment.

    The Pipe Insulation Roller Coaster Series

    1. Pipe Insulation Roller Coasters and Rolling Friction
    2. Pipe Insulation Roller Coasters
    3. Pipe Insulation Roller Coaster Assessment
    
    

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    1. The first day we played with pipe insulation in class I had students use some marble-sized steel balls. Unfortunately because the steel balls are so much heavier and the pipe insulation is spongy and flexible, there was just too much friction. When we switched to marbles the next day everything worked like a charm. (back)
    2. Most groups typically use more than one roll of duct tape. My first couple years I bought the colored duct tape and gave each group a different color. That was a nice touch, but also a bit more expensive than using the standard silver. Whatever you decide, I highly recommend avoiding the cut-rate duct tape. The cheap stuff just didn't stick as well which caused students to waste a lot of time fixing places where the duct tape fell and in the end used a lot more duct tape. (back)
    3. I had an arrangement with our school's kitchen manager to set broken down boxes aside for me for a few weeks before we started the project. If that's not an option, I've also found if you talk to a manager of a local grocery store they're usually more than willing to donate boxes. (back)
    4. I made it a requirement for groups to start by building a cardboard rectangle with the maximum dimensions. This served two functions: (1) It made it easy for the groups to see what space they had to work with, and (2) it allows the roller coasters to be moved around a little by sliding them across the floor. (back)
    5. Originally I wanted students to keep g-forces below 10. Very quickly it became apparent that under 10g's was overly restrictive and I upped it to 30g's. That's not really safe for living creatures, but it would certainly make it more "exciting." (back)

Pipe Insulation Roller Coasters: Rolling Friction

Fair warning: This isn't a description of the pipe insulation roller coaster (a.k.a. PI Coaster) project. It is the activity we did immediately before starting on the roller coasters.

The PI coaster project was one of those quality projects that students enjoyed while still requiring solid content knowledge. I last used this project in 2008- the last year I taught physics. I'd like to think that I've grown as a teacher since then, so I decided I should update it to be what I'd expect of a project from myself today. You know. SBG-it up. Throw in some video analysis. Etc. Suddenly I found myself driving to the local hardware store to pick up some pipe insulation at 9:30 at night.

The Goal

The goal of this activity is to find the coefficient of friction acting between the marble and the track. By the time we get started on this project, we would have already gone over kinematics, F=ma, friction, and uniform circular motion in class, and we'd be right in the middle of the Work & Energy unit.

Specifically, the following concepts are needed for this investigation:

  • Energy may change forms, but is conserved (minus any work done by friction):

    [latex, size=2]\Sigma E_{first} = \Sigma E_{last} - W_{fr}\(

  • The amount of work done on an object depends on the size of the net force acting on the object and the distance the force is applied:

    [latex, size=2]W=F\cdot d\)

  • The size of the frictional force depends on the coefficient of friction between the two surfaces and the weight of the object:

    [latex, size=2]F_{fr}=\mu F_N\(

Here's the setup:

Students set up 12 feet of track as shown in the picture above and measure the height from which the marble is dropped (on the left of this image). In order to find the coefficient of friction, you first need to find the amount of work done by friction on the marble as rolls through the track. To do this students use the following formula:

[latex, size=2]PE_g = E_k - W_{fr}\)

Solving for work done by friction and doing a little substitution for the energies:

[latex, size=2]W_{fr}=mgh - \frac{1}{2}mv^2\(

Looking at the right side of the equation, we need to find the mass of the marble, the height from which the marble is dropped, and the velocity of the marble at the end of the track. The first two are easy enough to measure.

Finding the final velocity of the marble isn't terribly tricky, but the method I used in 2008 had a lot of error. Students would measure out the final 50 cm of the track (as seen below). Then they'd send the marble through the track 10 times- each trial they would use a stopwatch to time how long it took the marble to travel the final 50 cm.

Timing the marble was hard. Depending on the height of the track, the marble takes less than half a second to whip through the final 50 cm. Using a handheld stopwatch often led to large differences between one trial and the next. Not so great for accurate data.

Using Tracker to find velocity

In rethinking this activity, it struck me that Tracker Video Analysis might be great to cut down on these timing errors. Only one way to find out: Break out the tripod.

After fiddling with the setup of the tripod and camera for a bit, I realized two things.

  1. The marbles were too dark to stand out in the video. No easily deterred, I took a few marbles out to the garage and spray painted them orange. I'd have used hunter's orange or neon green, but I didn't have any of that laying around.
  2. My "video camera" (a.k.a. an iPhone) only films at ~24 frames per second. When I started the marbles on the track 1 meter above the ground, they showed up as a long, faint blur when on an individual frame. I lowered the track to 0.75 m. The marbles still showed up as a blur, but they were much more distinct blurs1.

Once I troubleshot my way through those issues, I filmed this amazing & exciting clip for analysis:

I did six trials to get a good set of data I could average. You could easily get away with 3 trials and still get good data. I also measured the velocity of each marble during the final five data points to use as a final velocity.

The average final velocity from the trials above: 1.720 m/s

Calcumalations

Using the same energy-loss method detailed above, I calculated the coefficient of rolling friction (\mu_r) for the marble over the entire length of the track:

[latex, size=2]W_{fr}=mgh - \frac{1}{2}mv^2\)

[latex, size=2]W_{fr}=(0.0045 \text{ kg})(9.8 \text{ m/s}^2)(0.75\text{ m})- \frac{1}{2}(0.0045\text{ kg})(1.720\text{ m/s})^2\(

[latex, size=2]W_{fr}=0.034\text{ J}\)

Then solving for the friction force:

[latex, size=2]W_{fr}=F_{fr}\cdot d\(

[latex, size=2]F_{fr}=\dfrac{W_{fr}}{d}\)

[latex, size=2]F_{fr}=\dfrac{0.034\text{ J}}{3.66\text{ m}}\(

[latex, size=2]F_{fr}=0.0093\text{ N}\)

Solving for the average coefficient of friction:

[latex, size=2]F_{fr}=\mu_rF_N\(

There's no up or down acceleration, so F_N = F_g.

[latex, size=2]\mu_r=\dfrac{0.0092\text{ N}}{(0.0044\text{ kg}\cdot 9.8\text{ m/s}^2)}\)

[latex, size=3]\mu_r=0.21\(

Is that a reasonable figure? According to the EngineersHandbook.com, wet wood on wood's coefficient of friction is 0.2. From my vast experience slipping and falling on a wet decks, I know wet wood is dern slippery, and I would've expected\mu_r for the marble to be pretty low as well.

Alternate method

Using Tracker, I can find the acceleration of the marble as it rolls along at the end of the track. Using someF=ma magic I can find\mu_r using acceleration instead of velocity.

I created velocity-time charts for each marble and added best-fit lines to find the average velocity and acceleration of the marble. I found the average acceleration of the marble to be -0.065\text{ m/s}^2.

[latex, size=2]F_{fr}=ma=(0.0045\text{ kg})(-0.065\text{ m/s}^2)= -0.00029\text{ N}\)

Then finding the coefficient of friction:

[latex, size=2]F_{fr}=\mu_rF_N\(

[latex, size=2]\mu_r=\dfrac{0.00029\text{ N}}{(0.0045\text{ kg}\cdot 9.8\text{ m/s}^2)}\)

[latex, size=3]\mu_r=0.0066$$

"Wait, what? That's two orders of magnitude smaller!" That's what I said when I first got that number. Then I realized I this method was calculating\mu_ronly for a straight and level section of the track. You'd expect the friction to be much less along a straight track than when the marble's being forced to do loops and turns.

Is it worth it?

Using video analysis is more time-consuming, but I also think it helps students see more clearly that the coefficient of friction between the marble and the track is constantly changing. I think I'd have to try this out with students once or twice before deciding whether it's an effective use of class time. The basic concepts are covered sufficiently using my old method, though they're fleshed out in more detail using video analysis.

Additionally, I think I'd have each group of students use a different track configuration- one with two loops, one with S-curves, etc. That'd give us an even better idea of how the track layout will effect the friction between the marble and track.

The Pipe Insulation Roller Coaster Series

  1. Pipe Insulation Roller Coasters: Rolling Friction
  2. Pipe Insulation Roller Coasters
  3. Pipe Insulation Roller Coaster Assessment

 

 

 ______________________________

  1. If anyone would like to chip in for the Buy Ben a High Speed Camera Fund, let me know. 🙂     (back)

Learning Tracker Video Analysis with Napoleon Dynamite

I know I'm late to the game. Rhett Allain, John Burk, Frank Noschese, among many others have been sharing how they use Tracker (or a similar tool) to analyze the physics of videos. Since I'm working on picking up my teaching certification in Physics this year, I figure this would be a nice addition to the teaching toolbox1.

So, what is Tracker? It's a free and open-source video analysis and modeling tool designed to be used in physics education. It works on Macs, PCs, and Linux boxes. Logger Pro is a similar tool, but it's not free or open-source2.

Getting going

To begin, I watched Rhett Allain's video tutorial, but it includes a few more complicated pieces that I wasn't quite ready for. Luckily sitting in the Related Videos sidebar on YouTube was this tutorial, which went over the super-basics for n00bs like myself. Alright. Tracker downloaded & installed. Basic tutorial viewed. Now I need me a video to analyze.

I wanted something pretty easy to break myself in: a fixed camera angle, no panning, with an object moving directly perpendicular to the camera. I figured YouTube must be full of videos of people jumping bikes, and I went out to find my first video analysis victim. Amazingly, one of the first videos I found was both interesting, funny, and had the perfect still camera and perpendicularly-moving object:

Perfect! OK, now I needed to calibrate Tracker so it can accurately determine scale. Hmm...well Napoleon is standing fairly close to the sidewalk. I wonder if Jon Heder's height is online? Well, of course it is. In fact, Google gives me an estimated height right on top of the search results by just typing in height Jon Heder. However, I think I'll use IMDb's data, which lists his height at 185cm (sans 'fro).

Napoleon Dynamite's height
Calibrating size with Napoleon Dynamite

There might be a small error there since he is standing a few feet back from the ramp, but it should be OK.

Did Pedro get, like, 3 feet of air that time?

It took me awhile to realize that I needed to shift-click to track an object...once I figured that out things went smoothly. I tracked the back tire of Pedro's bike. Here's a graph of  the back tire's height vs. time:

There are a couple hitches in the graph. A few times the video would advance a frame without the screen image changing at all. Must be some artifact of the video. I added a best-fit parabola to the points after the back tire left the ramp. Hmm...the acceleration due to gravity is -8.477 m/s^2. That's a bit off the expected -9.8 m/s^2. That could be a result of the hitches in the data, my poor clicking skills, or my use of Napoleon Dynamite's height as my calibration. We'll go with it, since it's not crazy bad.

Coming up to the ramp the back tire sits at 0.038m and reaches a maximum height of 0.472 m. How much air does Pedro get? ~0.43m, or 1.4ft. Napoleon's estimate is a little high.

Maybe Napoleon meant Pedro's bike traveled forward three feet in the air? Let's check the table.

I highlighted the points of interest. We can look at the change in x-values from when the tire left the ramp (at 0 meters) until the tire lands back on the sidewalk (at y = 0). The bike traveled 1.3 meters while airborne; about 4.25 feet. So maybe that's what Napoleon meant.

Who was faster?

Let's check the position-time graphs for Pedro and Napoleon.

I added best fit lines to both sets of data. We can easily compare their velocities by checking the slope of their best fit lines.

  • Pedro's velocity: 5.47 m/s (12.24 mph)
  • Napoleon's: 5.44 m/s (12.16 mph)

If I account for potential errors in measurement, their velocities are basically the same. Though if forced to pick a winner, I'd Vote for Pedro.

How tall is Pedro?

It should be fairly straightforward to find Pedro's height using the data in the video. The first thing I need to do is verify that the camera angle is exactly the same when Pedro is standing behind the sidewalk as it was earlier. After switching back and forth between the two parts, it's pretty clear that the camera angle is a little different. Nuts.

So, I need to find and measure an object that is visible in both parts of the video. I chose the left window on the (Pedro's?) house. Going to the first part of the video where I'm pretty sure the calibration is accurate, I used the measuring tape to measure the height of the window. I got 1.25 meters.

Jumping to the second part, I calibrated the video by setting the height of the window to 1.25 meters. Then I used the measuring tape to determine Pedro's height. I got 1.67 meters, or about 5' 6". Seems like a reasonable result. Let's compare it to what the Internet says about Pedro's height. IMDb gives Efren Ramirez's (a.k.a. Pedro) height as 1.70 meters (5' 7").

Not too shabby for my first time using Tracker.

Bonus

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  1. You might notice this post is pretty similar in style to Rhett Allain's video analyses on Dot Physics. Well, it is. When just learning how to do something, it's always best to start by imitating the masters, right? Oh, if you haven't yet, you should definitely check out his many, many amazing examples using video analysis to learn all sorts of crazy things. The guy's a Tracker ninja.     (back)
  2. To be fair, it's only $189 for a site license of Logger Pro, which ain't too shabby. According to Frank Noschese, Logger Pro is a little more user-friendly. Tracker has a bit of a learning curve.     (back)