# Physicists Solve the Mystery of Interleaved Phone Books

## Try pulling apart two interleaved phone books and you’ll inevitably be defeated by huge frictional forces. Just how these forces arise has been a long-standing mystery… until now.

Aug 17, 2015

A famous demonstration of the counterintuitive power of friction involves two telephone directories with their pages alternately interleaved. People are then invited to pull the directories apart, a futile task since the force required to do the job is mind-bogglingly huge.

Indeed, experimenters have variously tried to separate the directories with trucks and military tanks. They’ve even used them to lift a car off the ground.

The force in question is the friction between individual sheets amplified by the huge number of pages in each directory. Of course, friction only occurs when two surfaces are pushed together, and experimenters often explain that gravity is the force that pushes the pages together in these experiments. That turns out to be easily refuted by turning the books on their side or holding them vertically.

Another possibility is air pressure. But this can be refuted by removing every other sheet from the directories before interleaving them, in which case they are easily pulled apart. The pages are still in contact so if air pressure were responsible, it should still work.

So what generates the force normal to the sheets that produces friction? Today, we get an answer thanks to the work of Hector Alarcon at the Universite Paris-Sud in France and a few pals who have investigated the phenomenon and devised a mathematical model that explains what is going on.

Their conclusion is that the pulling itself generates the normal force and this leads to the paradoxical effect that the harder you pull, the more tightly the pages bind together.

Alarcon and co begin with a little background briefing about friction, which was first investigated by Leonardo Da Vinci in the 16th century and later by Guillame Amontons and Charles Augustin de Coloumb in the 17th and  18th centuries respectively.

These guys discovered that friction is more or less independent of the area of the surfaces in contact but proportional to the load during sliding; the constant of proportionality being the coefficient of friction.

Alarcon and co describe their experiment which measured the force required to pull two interleaved books apart and determined how this varied with the number of pages and changes in the contact area.

They found that a relatively small increase in pages dramatically increases the pulling force necessary to separate them. “A tenfold increase in the number of sheets induces a four orders of magnitude increase in the traction force,” they say.

More puzzling was that increasing the area of overlap made the traction force greater too.

Both these effects are straightforward to explain with their new model, say the team. As each page is added to the pile, it is displaced from its original position in the book by the extra pages that have already been added.

So the sheets do not lie entirely flat. Instead, the part of each sheet closest to the spine has to bend at an angle. And this angle increases as more pages are added to the pile.

This angle is hugely important because it converts a fraction of the horizontal pulling force into a normal force that pushes the pages together.

That’s why adding extra pages increases the traction for in a nonlinear way.  The extra pages make an even bigger angle, converting more of the pulling force into a normal force.

It also explains why increasing the contact area magnifies the traction force. The contact area can only increase by making the pages overlap more fully, so that the edges are closer to the opposing directory’s spine. When the pages are closer to the spine like this, the sheets end up making a bigger angle and generating more downforce when they are pulled.

The model additionally explains why removing alternate pages from the directory before interleaving them allows them to be pulled apart easily. In this case, the overlapping pages fit into the spaces left by the missing sheets and so do not bend at all. Without this angle, any pulling force is not converted to a normal force, so there is little or no friction and the directories slide easily apart.

The new model allows all these forces to be calculated for the first time, and Alarcon and co say it should be relevant to a wide range of friction-related phenomenon. They give two everyday examples. First, the ability to moor a ship simply wrapping a rope around a capstan. The second is the Chinese finger trap where a helical braid wrapped around a finger tightens as it is pulled. “The trapping mechanism results from a simple conversion of the traction force to an orthogonal component, which enhances the load and thus the friction,” they say.

But there may be more exotic applications too. “This type of braid is applicable to sutures in surgery and is also thought to play a role in adhesive proteins,” say Alarcon and co, who add that the principle behind mooring a ship might also be relevant to the interaction between DNA and a bacteriophage capsid.

That’s interesting work that clears up the longstanding mystery associated with a common science demonstration.

Ref: arxiv.org/abs/1508.03290 : The Enigma of the Two Interleaved Phonebooks