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New Mathematical Model Helps Explain the Strength of Interleaved Phonebooks

Phonebooks made of paper have been going out of style, but they are still of interest to physicists. A few years ago, an episode of Mythbusters explored the strength of interleaved phone books. (Also see the sequel in Mythbusters, or maybe even try it yourself.)

First, some context, in case you are a child of the 21st century, and so perhaps have no personal experience with paper telephone books, which could be hefty. Here’s an old TV advertisement for “the yellow pages”, a telephone directory listing businesses and their telephone numbers:

Now experiments and an accompanying mathematical model have been published in Physical Review Letters by a team of physicists. Frédéric Restagno of the University of Paris-Sud and CNRS in Orsay and his colleagues measured the force needed to separate interleaved pairs of books with between 12 and 100 pages, and they developed a mathematical model based on simple geometric and mechanical ideas to explain the impressive strength of interleaved books.


Figure 1 from the article “Self-Amplification of Solid Friction in Interleaved Assemblies” by Héctor Alarcón, Thomas Salez, Christophe Poulard, Jean-Francis Bloch, Élie Raphaël, Kari Dalnoki-Veress, and Frédéric Restagno


The strength of the interleaved books arises because the book-separating force on each page is applied at a slight angle, and this increases the perpendicular force and hence the friction of each page. Restagno and colleagues also fit the data to a curve of force versus a dimensionless amplification parameter –– following the continuum-mechanics tradition of using cute names for dimensionless parameters, let’s call it the “Hercules number” –– that depends on the number of pages, the page thickness, and the size of the overlap region between the books.

Not very closely related: Another fascinating dimensionless parameter is the “Repunzel number” from research on ponytail physics, which earned the 2012 Ig Nobel Prize in Physics.

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