# How does one calculate frictional force for an object sliding down a wall? [closed]

Let's say that there is a book sliding down a vertical wall such that the only fundamental force acting on it is gravity. I want to say that there is a frictional force slowing the book down; however, I can't find a normal force acting on it, so I can't calculate the kinetic friction force.

Is there any frictional force?

For reference, the book has a mass of $3.5kg$ and a coefficient of kinetic friction of $0.13$.

## closed as off-topic by sammy gerbil, Jon Custer, Bill N, stafusa, Daniel GriscomOct 19 '17 at 21:16

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• -1. Why don't you apply the usual formula $F=\mu N$? – sammy gerbil Oct 19 '17 at 10:56

If there is no normal force to the surface there is no friction. The book will just fall by gravity.

In real life if starting pressed on the wall and released it may hit a small extrusion from the wall and get a rotation by scattering off it.

As you correctly point out in your opening paragraph, there is no normal force acting on the book, which means that there is no frictional force.

# Comprehensive

I'm writing this mainly for my own amusement, but you're welcome to read it if you like.

In general, the frictional force $F_f$ is proportional to the normal force $F_N$ because, on the microscopic level, the surface of two objects in contact are uneven, meaning that—like a car tire—the entire surface of the objects are not in contact. If the normal force increases, the amount of surface area in contact will also increase, driving up the amount of resistance between the objects.

Let us first assume an infinitely flat book and wall for your example. Because the two are infinitely flat, we can bring the book close to the wall without worrying about unevenness. Down to a certain minimal separation from the wall, there would be no frictional force between the book and the wall, because there would be no interactions between the particles of the book and the particles of the wall, and therefore the book would fall straight down, parallel to the wall. If, however, we brought the book extremely close to the wall, the fundamental attractive interactions between the particles would cause drag on one side of the book, resulting in some angular acceleration in the book. This would cause it to spin off the wall, due to the center of gravity of the book being so far away from the contact surface, despite there being no normal force.

New let us assume a more realistic scenario, where the book and the wall are uneven and jagged. Firstly, the minimum separation for the book to fall smoothly past the wall is increased to the maximum height of a ridge on the book + the maximum height of a ridge on the wall. If the book is outside that distance, it will fall smoothly beside the wall. If the book is within that range, the result will be similar to that of the infinitely flat surfaces being brought together—only much more chaotic. The difference in this situation is that the uneven ridges in the book and the wall would collide like a falling rock hitting an outcropping on a cliff—suddenly and aggressively. After the first collision, the book would pivot (microscopically)—due to the separation between the center of gravity of the book and the point where the normal force is being applied—with some angular velocity about a point near the edge of the wall's microscopic ridge until a point lower down on the book collided with the wall, forcing the higher point of off the wall. This would continue until the lowest point of contact, normally the bottom edge of the book, collided with the wall, sending the book spinning off the wall.

Side Note:

If the object falling near a vertical surface has liquid properties, it can, through surface tension, adhere to the surface and be slowed down through frictional and adhesive interactions.

## Summary

Given that the two objects are sufficiently close to one another:

If the two surfaces are not infinitely flat, microscopic unevenness in the surfaces will give rise to normal forces, launching the book off of the wall.

If the two surfaces are infinitely flat, fundamental intermolecular forces will cause the book to be spun off of the wall.