JUNE 2016

PHYSICS TODAY 75
sheets ranged from 24 to 200. A tensile tester measured the total
traction force as the notebooks were pulled vertically at a con
stant velocity, typically 1 mm per minute. We stopped pulling
intermi"ently so that we could probe the coeﬃcient of friction
at various distances d separating the clamp of one book from
the overlap region. Figure 2a shows the geometry of the inter
leaved books and some of the important parameters used in
our analysis. Figure 2b presents our measurements. It clearly
shows the nonlinear variation of T with both the number of
sheets and the separation distance.
The mechanism leading to ampliﬁed friction can be under
stood with the help of ﬁgure 2a. The index n labels individual
sheets in a booklet, and the nth sheet makes an angle θ
n
as it
goes from the clamp at the end of the notebook to the overlap
region. Therefore, a component of the local traction T
n
exerted
on a sheet by an external operator at point A results in a local
normal force T
n
tanθ
n
exerted on the stack of sheets below it at
point B. The result is a selfampliﬁed friction force that resists
the traction: The harder the operator pulls, the greater the local
loads and frictional resistance.
Summing over all the local tractions yields the total trac
tion T. To proceed analytically, we use a continuum descrip
tion corresponding to a large M. Introducing the quantities
z = n/M, T(z)=T
n
, and the dimensionless ampliﬁcation param
eter α =2µεM
2
/d yields the ordinary diﬀerential equation
T′(z)+2αzT(z)=0, where the prime denotes diﬀerentiation with
respect to z. To integrate and solve that equation, one needs a
boundary condition T
∗
for the traction on the outermost sheet
of the stack. The source of that bounding traction could be fric
tion introduced by the elasticity of the paper as it bends or by
any tiny adhesion. Upon integration, we obtain the total traction
force T =2MT
∗
·(π/4α)
1/2
· e
α
·erf(√
‾
α), where “erf” denotes the
error function.
Note that as α approaches zero, which corresponds to par
allel sheets and small angles, T =2MT
∗
. That makes sense: As
the angles tend to zero, the friction force is due to all the inde
pendent sheets, each experiencing local friction T
∗
. In general,
however, the total traction force depends nearly exponentially
on a single quantity—the dimensionless ampliﬁcation param
eter α, which has been given the lovely name “Hercules num
ber.” And as the inset to ﬁgure 2b shows, when the appropri
ately normalized traction is plo"ed against α, the results of all
our experiments collapse onto a single master curve.
Not just a compelling muse
We began our phonebook study because we wanted to solve
an intriguing popular mystery. But the eﬀect we have studied
can be seen in many other systems in which a pulling force re
sults in an orthogonal load that enhances friction. An example
is a toy known as the Chinese ﬁnger trap. Fingers inserted
into the trap—a braided tube—are diﬃcult to extract because
tension tightens the tube. Surgeons use socalled ﬁngertrap
sutures based on that principle. Likewise, a ship can be moored
to a capstan because the rope can tighten around the capstan
and thereby increase friction. The frictionamplifying mecha
nism also recalls catch bonds, biomolecular links that strengthen
with tensile stress and facilitate cell adhesion. We hope that our
study, in addition to elucidating the longstanding and fun
phonebook puzzle, will help to clarify the mechanical behav
ior of more complex interleaved systems involving textiles,
biological entities, and nanoscale mechanical devices.
Additional resources
‣ ”Mythbusters—Phone Book Friction,” www.youtube.com
/watch?v=AX_lCOjLCTo.
‣ Q. J. Wang, Y.W. Chung, eds., Encyclopedia of Tribology,
Springer (2013).
‣ A. A. Pitenis, D. Dowson, W. G. Sawyer, “Leonardo da Vinci’s
friction experiments: An old story acknowledged and re
peated,” Tribol. Le%. 56, 509 (2014).
‣ H. Alarcón et al., “Selfampliﬁcation of solid friction in
interleaved assemblies,” Phys. Rev. Le%. 116, 015502 (2016).
‣ J. Cumings, A. Ze"l, “Lowfriction nanoscale linear bearing
realized from multiwall carbon nanotubes,” Science 289, 602
(2000).
‣ A. Niguès et al., “Ultrahigh interlayer friction in multiwalled
boron nitride nanotubes,” Nat. Mater. 13, 688 (2014).
PT
FIGURE 2. WHEN SUITABLY SCALED, the friction of interleaved
books obeys a universal law. (a) As this schematic shows, when
interleaved books are separated by a distance d, the nth sheet
makes an angle θ
n
as it goes from the overlap region to a binding
clamp. Here, T
n
denotes the local traction force on the nth sheet.
(b) Total traction force T is measured as a function of distance d.
From bottom to top, the number 2M of sheets in each book is 24,
30, 46, 54, 100 (two data sets), 150, and 200. As the inset shows,
when the total traction is appropriately scaled and plotted as a
function of the Hercules number α =2μεM
2
/d, all the curves from
the unscaledtraction plot coincide. The solid black line corresponds
to the expression for T given in the text.