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Firstly, I understand and apologize that this is more of a physics question than a math question.

I noticed that when I pulled a book out from under a pack of gum, the gum stayed largely in place (it did not move with the book). This confused me, so I decided to draw a diagram of the situation, assuming that my answer would be apparent once I could represent all of the forces in play. Of course (as the book was orthogonal to the ground) gravitational force and normal force could be ignored. The only other force that I see that can be in play is the friction between the book and the gum, which should make the gum move along with the book. There must, then, be some sort of force acting opposite and with a greater magnitude than friction. Is this correct, and if so, what is that force and how can it be calculated?

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    $\begingroup$ If there were no friction, there would be no forces acting on the gum, and the gum would not move. Thus, if there is only a small amount of friction acting on the gum, the gum won't move much. You don't need a counteracting force to make a body that stays at rest remain (largely) at rest. $\endgroup$ – Peter Shor Jul 20 '14 at 13:03
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There is, effectively, only gravitation and friction acting on the pack of gum. However, the friction is not that strong (it is mostly independent of the velocity of the book, and dynamic friction is weaker than static friction) and it doesn't have that much time to act. Hence it doesn't affect the momentum of the gum noticeably.

This is very related to the party trick where you pull a table cloth out from under all the tableware. If you pull hard enough (and don't lift) the cloth, there is only a fraction of a second in which the friction acts upon the tableware, and it therefore stays mostly in place.

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