# 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.