## Pipe Insulation Roller Coasters

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 $1.30 or so you can get six feet of pipe insulation- which doubles nicely as a marble track1 when you split the pipe insulation into two equal halves. It's really easy to cut pipe insulation with a sharp pair of scissors. Just be sure you don't buy the "self-sealing" pipe insulation, which has glue pre-applied- it's more expensive and it'd turn into a sticky mess. At first I planning to simply design a one-period long investigation using the pipe insulation (my original ideas morphed into the pre-activity for this project). As I started to think through the project more and more, I realized we could go way bigger. And thus, the pipe insulation roller coaster project was born. ### Building the Coasters In groups of three, students were given 24 feet of pipe insulation (4 pieces), a roll of duct tape2, and access to a large pile of cardboard boxes3. All groups had to adhere to a few standard requirements: • Construction requirements 1. The entire roller coaster must fit within a 1.0m x 2.0m rectangle4. 2. There must be at least two inversions (loops, corkscrews, etc.). 3. All 24 feet of pipe insulation must be used. 4. The track must end 50 cm above the ground. • Physics requirements In addition to meeting the above requirements, students were required to utilize their understanding of the work-energy theorem, circular motion, and friction to do the following: 1. Determine the average rolling friction, kinetic energy, and potential energy at 8 locations on their roller coaster. 2. Determine the minimum velocities required for the marble to stay on the track at the top of all the inversions 3. Determine the g-forces the marble experiences through the inversions and at least five additional corners, hills, or valleys. 4. The g-forces must be kept at "safe" levels5. ##### Calculations 1. Rolling friction, kinetic energy, and potential energy • The potential energy ($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: 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): ### 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. ### 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 ______________________________ 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 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. ### Using Tracker to find velocityIn 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. 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. 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 ### 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.