## 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: 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 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}\)

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\)

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.

### CalcumalationsUsing the same energy-loss method detailed above, I calculated the coefficient of rolling friction () 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}\)

[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{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}\)

[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)}\)

There's no up or down acceleration, so .

[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 some$F=ma$ 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}\)

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 for the marble to be pretty low as well.

### Alternate methodUsing Tracker, I can find the acceleration of the marble as it rolls along at the end of the track. Using some magic I can find 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 .[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=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$\mu_r$only 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.

## 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).

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

______________________________

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)