Can You Lift a Car with Two Phone Books?
|Time Required||Short (2-5 days)|
|Material Availability||Readily available|
|Cost||Average ($40 - $80)|
|Safety||Adult supervision required for testing heavier weights.|
AbstractHave you ever heard that two phone books with the pages interleaved are impossible to pull apart? This might seem crazy, right? It is not that hard to slide a sheet of paper off the top of a stack of paper. How much friction can there really be between sheets of paper? In this experiment, you will use pads of sticky notes instead of phone books. How much weight can they support when you interleave the pages? Do you think you will be able to pull them apart by hand? The results might surprise you!
Measure how much weight can be supported by sticky note pads with interleaved pages.
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How much friction—or the force that resists motion between two sliding surfaces—do you think there is between two pieces of paper? Try a very quick test before you even start this project. Take two stacks of sticky notes with 5 pages each, and interleave the pages, or overlap them one by one. Then try to pull them apart, like in Figure 1. Easy, right? Now try it with 20 pages per stack. Can you pull them apart at all? Why is it so much harder?
Figure 1. Try to pull apart two small stacks of sticky notes with interleaved pages.
The pages of the sticky note pads are held together by friction. Even a small number of interleaved pages can result in a surprisingly large amount of friction, making the sticky notes impossible to pull apart. This phenomenon is commonly demonstrated with phone books. People have tried pulling two interleaved phone books apart by playing tug-of-war or even attaching them to cars or tanks driving in opposite directions. See the Bibliography and search the internet for "phone book friction" and you will find plenty of videos showing the experiment. There are even demonstrations that show a car hanging from a crane supported by two phone books.
How can the phone books support so much force? To understand this, first you need to understand a little more about friction. The friction force between two surfaces depends on two factors: the normal force, or the force acting perpendicular to the surfaces pushing them together, and the coefficient of friction, a constant that describes the friction between two materials. For example, your shoes and a carpeted floor have a higher coefficient of friction than your shoes and ice (note: technically there are different coefficients of friction depending on whether an object is sliding or static [not moving], see the Physics Classroom reference in the Bibliography or consult a high school physics textbook for more details). The friction force is related to the normal force and coefficient of friction by Equation 1:
- F is the friction force in newtons (N)
- μ is the coefficient of friction (no units)
- N is the normal force in newtons (N)
So, the higher the coefficient of friction, or the higher the normal force, the higher the resulting friction force. What, then, results in such a high friction force between the interleaved pages of the sticky notes? The coefficient of friction does not change because the materials stay the same (paper), so it must be a high normal force. What causes a high normal force? When you try to pull the sticky notes apart, the interleaved pages are squeezed inward. This increases the normal force on the pages. Conceptually, this works just like a Chinese finger trap—the harder you pull, the harder the pages squeeze together. Take a look a Figure 2 to help you understand.
Figure 2. This diagram shows a side view of two stacks of sticky notes with interleaved pages. The section of interleaved pages is twice as thick as the binding of either individual sticky note pad. This means that the pages in the non-overlapping area of each pad must be at an angle (labeled θ in the figure; note that the angle is measured from the page to a horizontal line, not between pages). Because of this, when you try to pull apart the pads, the tension in each individual page (red arrow) has a horizontal component (blue arrow), but it also has a vertical component (green arrow). This vertical component squeezes the pages together, increasing the normal force acting on each page and thus increasing the friction.
To help you visualize this, take your interleaved stacks of 20 sticky notes again. Hold them very loosely and look at them edge-on. You can even try pushing them toward each other a bit. You should be able to see small gaps between the pages, and maybe some of the pages will bend a bit. Now, pull on the sticky notes as hard as you can. The pages should flatten out and squish together, and the gaps between them will disappear.
In this project, you will measure exactly how much weight different numbers of interleaved sticky notes can hold by hanging weights from the sticky notes, vertically. How do you think the amount of weight they can support will change with the number of pages? Will the change be linear, exponential, or something else? For example, if the change is linear, then if you double the number of pages, you would expect the weight supported to also double. However, if the change is exponential, the weight supported would more than double. What do you think is more likely, based on your background research?
Common misconceptions about the phone book friction experiment
Due to the popularity of this experiment, a few common misconceptions or incorrect explanations of the physics involved can easily be found online. We have listed a few of these misconceptions here, along with explanations. Make sure you avoid them in your science project! For advanced students, a detailed explanation of the physics can be found in the Bibliography reference The enigma of the two interleaved phonebooks.
Weight of the pages
A very common misconception is that the weight of all the pages pressing down on each other increases the normal force, therefore increasing the friction. Technically, it is true that when the sticky note pads are horizontal, the weight of each page presses down on the pages below it. However, this cannot explain the full effect observed in the experiment. This is easy to verify; take your sticky notes and stand them on edge or hold them vertically, as shown in Figure 3. In these orientations, the weight of the pages are not pressing down on each other. However, they are still difficult to pull apart. If the friction was only from the normal force resulting from the weight of the pages pushing down on each other, then the experiment would only work when the pages were horizontal, and they would be easy to pull apart when they were on edge or vertical. This is clearly not the case.
Figure 3. The sticky notes are very difficult to pull apart, regardless of their orientation. This means that "the weight of the pages pressing down on each other" when the pages are horizontal cannot explain why there is so much friction.
Air pressure or suction
Another popular misconception is that air pressure, or suction, holds the pages together, resulting in an increased normal force. This sounds convincing because the gaps between pages, shown in Figure 4, are so small that you can barely see them. However, this explanation could only be true if there was actually a vacuum between the individual pages. Air molecules are very tiny—much, much smaller than the gap between two sticky notes, so they can easily fit into the gap. That means that there are actually air molecules in between the pages, pushing them outward, which cancels out the effect of air molecules pushing inward.
Figure 4. The gaps between individual pages are so small that you can barely see them, but air molecules can still fit into the gaps, exerting air pressure on the pages from the inside.
Still do not believe it? Put a piece of paper on a flat surface like a desk or table. The atmosphere exerts a pressure of 101.3 kilopascals (kPa) or 14.7 pounds per square inch (psi). On an 8.5x11 inch piece of paper, that means a force of over 1,000 pounds (or over 4,000 newtons [N]) on each side. Yet, you can slide the paper around on the table surface with ease. Why doesn't the paper feel like it has a thousand-pound weight sitting on top of it? This is because, even though it looks like the paper is "flat" against the table, there are still air molecules between the paper and the table pushing up on it. This cancels out the effect of air molecules pressing down on the top of the paper. The same goes for the interleaved pages of phone books or sticky notes.
Another tempting explanation is that the increase in friction results from "increased surface area." However, note from Equation 1 that friction force does not depend on surface area. To prove this, consider this example. Interleave a stack of 20 3x3 inch sticky notes with a stack of 21 sticky notes (this way, both sides of each sheet in the smaller stack are in contact with a sheet from the larger stack, instead of leaving one side exposed). This gives a total contact area of 2×3×3×20=360 square inches (20 sheets, 3x3 square inches in size, 2 sides per sheet). As you have experienced already, the sticky notes are very difficult to pull apart. Compare this to two 24x36 inch sheets of posterboard stacked on top of each other, with a contact area of 864 square inches, like in Figure 5. Despite having more than twice the contact area of the sticky notes, the posterboards are very easy to pull apart.
Figure 5. Despite having much more total surface area, the two sheets of posterboard are much easier to pull apart than two relatively small stacks of interleaved sticky notes.
This theory can be even more appealing, because at first glance, it looks like you can back it up with experimental evidence. Try interleaving the pages of some sticky note pads just a little bit, with only a centimeter or so of overlap. It will be much easier to pull them apart than if you interleave the pages with a lot of overlap (and thus more contact surface area). However, changing the overlap distance also changes the angle of the interleaved pages in the non-overlapping section, as shown in Figure 6. As explained in Figure 2 and the reference titled The enigma of the two interleaved phonebooks, it is this angle that results in increased normal force and therefore, increased friction—not the increase in contact area.
Figure 6. (Top) When there is a small overlap between the interleaved pages, the angle each page forms with a horizontal line is smaller, so the increase in the normal force is not as large. (Bottom) When there is a large overlap, the pages are at a much steeper angle, leading to a much larger normal force. It is this angle—not the increase in contact area—that leads to higher friction.
Terms and Concepts
- Normal force
- Coefficient of friction
- What is friction? What does friction depend on?
- Why can two phone books with interleaved pages support so much weight?
- Do you think the relationship between the number of pages and maximum weight supported is linear, exponential, or something else?
- What are some of the incorrect explanations for the "phone book friction" experiment? Do you understand why they are wrong?
- Mythbusters. (2008, September 10). Phone Book Friction. Discovery Channel. Retrieved October 28, 2015, from http://www.discovery.com/tv-shows/mythbusters/videos/phone-book-friction/
- Henderson, T. (n.d.). Types of Forces. The Physics Classroom. Retrieved October 28, 2015, from http://www.physicsclassroom.com/Class/newtlaws/U2L2b.cfm#frict
- Alarcón, H. et. al. (2015, August 13). The enigma of the two interleaved phonebooks. Retrieved October 28, 2015, from http://arxiv.org/pdf/1508.03290.pdf
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Materials and Equipment
The materials you need for this experiment will vary depending on the number of interleaved pages you want to test, and the resulting weight range that the sticky notes will support. Be aware that testing more pages will take longer, since it takes more time to interleave the pages. If you want to test a wide range, you can build more than one support rig for your experiment. See Figures 9, 10, and 11 in the Procedure for example setups, to get an idea of what you want to do before you start your experiment.
- Light-duty testing rig made from common office supplies, recommended for testing up to 15 pages per pad (30 interleaved pages total). See Figure 9.
- Pads of sticky notes (2)
- Mini C-clamps or spring clamps (2)
- Large binder clip
- Paper clips (at least 20)
- Vertical surface to clamp the sticky notes to, like the edge of a table or desk
- Small kitchen or pocket scale
- Medium-duty testing rig, recommended for testing 15–25 pages per pad (30–50 interleaved pages total). See Figure 10. These materials may require a trip to a hardware store.
- Pads of sticky notes (4)
- Medium C-clamps (2)
- Mini clamps or spring clamps (2)
- Small bucket
- Metal chain and clip to hang the bucket by the handle
- Coins, rocks, or other heavy objects to fill the bucket
- Sturdy vertical surface to clamp sticky notes to, like the edge of a table or desk
- Padded surface like a carpet, blanket, or towel to put under the test rig so it does not break or damage the floor when it falls
- Medium kitchen scale
- Heavy-duty testing rig, recommended for testing more than 25 pages per pad (more than 50 interleaved pages total). Ask for help at a hardware store if you do not know what some of these materials are. See Figure 11 for materials and to get a rough idea of the dimensions.
- Pads of sticky notes (6)
- Large C-clamps (2)
- Medium C-clamps (2)
- Spring snap links, also called carabiner hooks or clips (3). Important: Do not use "keychain" clips since they are not actually designed to support a large amount of weight.
- Eye bolts (4)
- Nuts that match the threads of the eye bolts (8)
- Washers for the eye bolts (8)
- S-hooks (4)
- Segments of metal chain (5); length will depend on the size of your plywood
- Square piece of plywood, approximately 16 by 16 inches or larger
- Power drill
- Heavy-duty vertical surface to clamp the sticky notes to, like the underside of a workbench or outdoor deck
- Padded surface like a carpet, blanket, or towel to put under the test rig so it does not break or damage the floor when it falls
- Bathroom scale
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- Decide what page number range you want to test for your experiment and the type of testing rig you want to use (see the Materials section and Figures 9, 10, and 11). For example, if you are using a lightweight testing rig, you could decide to test 2, 4, 6, 8, 10, and 12 pages per stack of sticky notes (so 4, 8, 12, 16, 20, and 24 pages total when they are interleaved). Separate your sticky notes into two stacks for each number, and label them with a pen or pencil, as shown in Figure 7.
Figure 7. Sticky notes divided into stacks with different numbers of pages, two for each number. The number of pages you test is up to you, and does not need to match the numbers in the figure.
- For each matching pair of stacks, interleave the pages by folding them on top of each other one by one, as shown in Figure 8. Important: Make sure you keep the overlap distance the same for each one of your interleaved stacks. The exact distance does not matter, as long as you keep it consistent for each stack.
Figure 8. Two stacks of 30 sticky notes each with interleaved pages.
- Set up your test rig. Figures 9, 10, and 11 show light, medium, and heavy-duty test rigs, respectively.
- What type of rig you build will depend on the materials you have available and the number of pages you decide to test. If you want to test a wide range of number of pages, you may need to build more than one test rig. In general, a test rig consists of clamping one of the sticky note pads to a vertical surface, leaving the other pad hanging below. Then, you clamp something that can hold weights to the bottom sticky note pad. For the experiment, you will gradually add weights until the sticky notes slide apart.
- It is very important to make sure your clamps are tight, otherwise the clamps might break or slide off before the sticky notes slide apart. You should also make sure the hanging weights are balanced and pulling straight down on the sticky pads, not twisting them or causing them to rotate, as this will affect your measurements. In Figures 10 and 11, this is accomplished by putting the two lower C-clamps on, facing in opposite directions.
- You should make sure your test rig is not very far off the ground, and that it will not damage the floor when it falls. This is especially true for the medium- and heavy-duty test rigs. You may want to place blankets or towels under the test rig to soften its landing.
- Important safety note: for the medium- and heavy-duty test rigs, adult supervision is recommended for building the test rig and doing the experiment. The sticky notes or clamps can slide apart very suddenly when you add weights, causing the weights and supports (like chains and clamps) to fall. Keep your hands and other body parts out from under the clamps, so you do not get hit if something falls.
Figure 9. A light-duty test rig. In this setup, mini C-clamps are used to clamp the upper sticky note pad to the edge of a workbench. A larger binder clip is attached to the lower sticky note pad. Paper clips, used as weights, are hung from the binder clip, one at a time, until the sticky notes slide apart.
Figure 10. A medium-duty test rig. In this setup, medium C-clamps are used to attach the upper sticky note pad to the edge of a workbench. Mini C-clamps and a chain are used to hang a plastic bucket from the lower sticky note pad. Coins are added to the bucket as weights until the sticky notes slide apart.
Figure 11. A heavy-duty test rig. In this setup, large C-clamps are used to attach the upper sticky note pad to the edge of a workbench. C-clamps, metal chains, hooks, and eye bolts are used to suspend a square piece of plywood from the lower sticky note pad. Plates from a weightlifting set are added as weights until the sticky notes slide apart. Notice how the chains are sized such that the plywood is only a couple inches off the ground, so it does not fall very far before it hits the floor.
- Prepare a data table like Table 1 in your lab notebook.
- You will have to fill in the number of pages based on what you decide to test. Be consistent as to whether you write down the number of pages per pad or the total number of interleaved pages. For example, two stacks of 20 pages each will result in 40 interleaved pages total.
- Fill in the appropriate units for weight or mass depending on the scale you are using and the expectations of your science fair. For example, a bathroom scale might only display weight in pounds, but your science fair might require that you use metric units. Younger students may not be required to understand the difference, but older students should be careful to distinguish between weight and mass. For example, a jewelry scale might show units in grams, which are a unit of mass, not weight. Check with your science teacher if you have questions about mass versus weight.
|Maximum Weight Supported|
|Number of Pages||1||2||3||Average|
- Hang your first set of interleaved sticky notes using your test rig.
- Carefully start adding weights to the test rig, one at a time.
- Continue adding weights, one at a time, until the interleaved pages slide apart, and the lower stack of sticky notes falls.
- If your clamps slide off or your test rig breaks before the sticky notes slide apart, you will need to build a stronger test rig, or test fewer pages.
- Use a scale to measure the weight that caused the sticky notes to separate. Include the weight of the hanging part of the test rig itself (for example, the binder clip in Figure 9, the C-clamps, chain and bucket in Figure 10, or the C-clamps, chains, hooks, and plywood in Figure 11) and the lower sticky note pad. Record this value in your data table.
- Repeat steps 5–8 for each additional interleaved stack of sticky notes that you want to test.
- If they did not get damaged or ripped during your tests, you can re-interleave each stack of sticky notes for your next trial. If your sticky notes got damaged somehow, you should start over with fresh stacks.
- Repeat steps 5–10 two more times, so you have a total of three trials for each number of pages.
- Analyze your results.
- For each number of pages, calculate an average weight and record the value in your data table.
- Make a graph with number of pages on the x-axis and maximum weight on the y-axis.
- What is the shape of the graph? Is it linear, exponential, or something else?
- How many pages are in a phone book? Can you extrapolate your data to predict how much weight two phone books could support? Does this match up with the results of demonstrations you can find online?
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- Try keeping the number of pages constant but changing the overlap distance. How does the maximum weight change with overlap distance?
- Try the experiment with notebooks that have removable perforated sheets. What happens if you remove every other sheet from two notebooks and interleave the pages? In theory, this means that all the pages would remain "flat" (the angle θ in Figure 2 from the Introduction would be zero). Does this make it easier to separate the notebooks? Do an experiment to compare the force it takes to separate notebooks with all their pages and those with every other page removed.
- For advanced students (high school physics or higher): can you develop a mathematical model that predicts the maximum friction force based on the number of pages? How does this theoretical model compare to your experimental data?
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