Phonebook Friction

I sat down for a game of cards the other night at home, and while shuffling the deck, interweaving the cards one by one, I couldn’t help my brain from wondering back to an old episode of MythBusters. In this particular episode, Adam and Jamie are attempting to pull apart two phone books which have had their pages interwoven one by one.  As is now a popular trivia fact – this task is actually quite difficult.

Now, I haven’t seen the episode since I was a kid, and I remember at the time being much more mesmerized with the fact that the MythBusters were able to find two functioning tanks to attempt (and fail) to pull the phonebooks apart than I was with any of the math behind why its so hard to do. At first glance it seems like there might be a complex relation to do with the pages interacting with each other. The answer, however, is actually quite simple, and comes down to geometry.

Before we dive too deeply, let’s first define how friction actually works. With respect to the phonebook phenomenon, it’s tempting to ascribe the excess force to surface area­ – after all, two 300 pages books with pages carefully interwoven does kind of lend itself to this theory. However, friction acts completely independently of surface area.  Friction, defined as a force that resists motion, is proportional only to what we call the normal force, that is a force perpendicular to that direction of motion. In layman’s terms, a book sliding along a flat, level table experiences friction proportional to its weight. This is because the normal force in this scenario is the force the table exerts on the book according to Newton’s third Law (the equal and opposite reaction). For movement on any flat surface, it holds that friction is proportional to weight. This gets a little more complicated when dealing with ramps or other non-flat surfaces, but the idea is the same. To fully define the force of friction, we assign a coefficient of friction that essentially describes how slippery the two moving surfaces are to each other (the cover of the book and the table in the above example). This coefficient, commonly assigned to Greek letter μ, is then multiplied by the normal force to give the total force of friction, which acts opposite the direction of motion

General Equation for the Force of Friction

As we can see, surface area has no place in the equation. How then, do two phonebooks seemingly amplify the force of friction so much? There are only two mechanisms for doing so. The first is to increase the coefficient of friction, that is to make the pages stickier. If this were the case it would be hard to pull apart even 2 pages of the phonebook. Option 2 is to increase the normal force on the pages, squeezing them against each other. But, pulling the books apart by their spines isn’t doing that, right?

This is where our geometry comes in to play. Interleaving the pages of the two books together causes the pages to spread out. Where interwoven together, the books are twice the thickness of an individual. The pages then bend as they approach the spine of either book, where each is still roughly the original thickness. This is illustrated in the picture, which also demonstrates that the more pages that are interwoven, the more each page has to bend to reach its position in the stack.

A Simple illustration of interwoven pages, showing the angle must increase as more pages are added

What does this mean for normal force? Let’s take a more abstract view, and examine this with simple geometry. In the diagram below, the path of one particular page is shown from the spine of the book to its position in the stack. I have added a few labels to the diagram, as well. F_P is the force being used to pull on the pages. This force results in tension in each page.

A simplified diagram of the path of one page of a interwoven phonebook

This is where the magic starts to become apparent. Because the page travels at an angle – a certain portion of the pulling force in the page acts in the downward direction, squeezing the pages together. This is shown as F_y. Let’s return quickly to equation for friction. Normal force is the force perpendicular to motion – in our scenario of trying to pull the phonebooks apart this is, you guessed it, F_y. Taking a closer look,  it stands to reason that the harder you pull (the larger F_P, the more the pages are squeezed together (the larger F­_y). This is true, but it isn’t the only multiplying factor. Trigonometry tells us that the steeper the angle the page travels at, the larger F_y becomes, even if F_P is the same. For our phonebook example, the more pages that are interwoven, the higher the angle, thus the more force squeezing the pages. All in all – the harder you pull, and the more pages you weave together, the higher the friction between the pages. The act of pulling the phonebooks apart actually increase the frictional force holding them together! More remarkably, as shown by some relatively recent research, a tenfold increase in the number of interwoven pages amplified the force required to separate the pages by 10,000 times!

While seemingly trivial, there are actually some real-world applications to this research. These same principles can be applied to things like muscle fibers in reconstructive surgeries and construction materials.  I love these types of experiments because of how well they demonstrate the importance of human curiosity. What started as a question with no greater meaning, a strongman competition of pulling apart two phonebooks, actually has meaningful impact on the way we understand and quantify real systems – be it a human ankle joint or bundle of cables in your wall. That’s why it’s always important to ask “Why.”