Trying to understand the simple model in https://arxiv.org/abs/1508.03290, describing the friction between the pages of two interleaved phonebooks. Specifically I don't see how to get Eq. 1.

Here's the geometry (from Fig 1 of the paper):

Fig 1 from paper

The sheets are labeled by $n$, starting with $n=1$ at the center of the phonebook. The thickness of each sheet is $\epsilon$. As $n$ increases the sheets are bent at larger and larger angles $\theta$: $$\tan \theta_n = n \epsilon / d \equiv H_n.$$

I'm looking at sheet $n$ (i.e. the one with points A and B marked). Looking at point B there is a force $T_n$ to the left (the applied traction force). There is an opposing frictional force between the sheet and the dashed one below it. This is caused by the normal force acting at point B (i.e. $T_n \tan \theta_n$). I feel like there should also be another frictional force to the right caused by the dashed sheet above, acting on the far right at the free end of sheet $n$.

I think the magnitude of the two frictional forces is the same: $\mu T_n \tan \theta$.

But then I end up with the equation $$T_n = 2 \mu H_n T_n,$$ which is clearly wrong. The authors get Eq. 1: $$T_n - T_{n+1} = 4\mu H_n T_n.$$

What is the proper way to think about this?

  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/135716/2451 $\endgroup$ – Qmechanic Jan 15 '19 at 18:19
  • $\begingroup$ Yes, that answer references the same paper I am looking at. My question is about a specific derivation within the paper. $\endgroup$ – Alex Jan 15 '19 at 18:58

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