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)

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

 

 

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

Rubber Band Cars

There's something powerful about physically making something that works yourself. The tinkering, trial and error testing, and early frustration often lead to some impressive feelings of accomplishment in the end.

This year when covering the types of energy and energy transformations, I realized a project I ran for 6 years at my school in Michigan would fit in quite well: The Rubber Band Car Project.1

You can check out the handout and guidelines I provided to students, though the basic gist of the project consists of:

  • Building a car from found materials;
  • Using no more than two #33 size rubber bands to power the car;
  • Getting said car to move as great a distance as possible (6 meters is the goal);
  • Describing how the energy stored in the rubber bands is transformed and conserved as the car does its thing.

Testing Rubber Band CarsInitially students are generally pretty worried because the guidelines ban items like CDs & DVDs as wheels, and Legos or other such objects from being used. However, as I share some examples of cars from the past (see them here), and as students start tinkering and sharing ideas with each other, the worries start to fade.

Most of the building process takes place at home, but I provide one day in class for students to bring in their cars (or materials that will eventually become their cars) and work on them in class. This is often extremely helpful for students who are struggling to figure out how to put their cars together and get them to work. As they walk around the room, they can see how everyone else is tackling similar problems and get ideas for how to solve their own.

Issues

Standards-Based Grading. I had a pretty solid assessment system that I was quite happy with before I went all-SBG. I'm not sure I'm quite as comfortable with how I'm assessing it using the SBG system. As of right now I'm not too worried by this. The old system had many years of tweaks and adjustments to get it to that sweet spot, and it'll probably take a couple tweaks to get the SBG-assessment for the project there too.

"I didn't do it." In the past there was always a small minority (~2% to 5%) of students who just didn't make anything for the project. This year it seems like the percentage of students with no car will be higher. I'm not sure what to think of that, but it's worrying.

Cool stuff

Non-competitiveness. I try my hardest to make sure the assessment system and the general classroom environment is as non-competitive as possible for this project. I want students to share ideas and collaborate with each other even though they're all making cars individually. For the most part this works out. Students who've figured things out are generally happy to share their knowledge with students who don't. However, there's no getting away from the fact that most students want the bragging rights for having the car that went the furthest.

Engaging the unengaged. Having to physically make something that works is a different sort of project for many students. It's interesting to see how some of the "I-need-an-A-or-I'll-die" students struggle with the project while some who often struggle with traditional projects become the super stars.

Results. I've always recorded every students' results and shared who had some of the most successful cars,2 and this year I'll be using a self-sorting Google Spreadsheet to automatically post the results to the Rubber Band Car Project Page in near real-time.3 I'm not sure if that's really necessary, but it is a fun trick. Perhaps I'll have to do a post on creating self-sorting spreadsheets if anyone is into that sort of thing.

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  1. A big tip o' the hat to Mr. Randy Commeret at Grand Rapids Christian High School; from whom I grabbed this project from nearly wholesale. Rumor has it this project has been around for 20ish years in total.    (back)
  2. Which might feed the competitive nature that I'm trying to avoid, but to date it hasn't gotten too competitive between students.     (back)
  3. Which means you can follow along with the results as we test cars on Thursday, March 24 & Friday, March 25. 🙂      (back)

A lesson in numeracy

I find great irony in the push for offshore drilling. As schools are being held accountable for improving numeracy in our students some political leaders seem to be lacking in that exact same skill when proposing policy decisions.

Offshore drilling sites wouldn't be online until 2017 (according the US Energy Information Administration), then would only fill 1.2% of the U.S.'s expected demand for oil, and 0.6% of the demand for energy.

What I find frustrating is that this isn't simply an environmentalist vs. big oil argument. The proponents of drilling are proposing the hypothesis: "Offshore drilling will reduce costs and the need for foreign oil." Look at the data. Does it seem likely that this hypothesis will be validated?
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Chart and data from Architecture 2030 via Treehugger