# Why does a book leaned up against a wall sometimes fall over after being stable for many hours?

This problem has been vexing me for years.

Sometimes, you lean a book up against a wall and it stays there (the static frictional force balancing gravity, etc), but then after a long time (hours or even days) it suddenly falls down without any obvious external perturbation (e.g. bumping the shelf). What causes this?

• Is the book slowly slipping over time until it reaches an unstable point and falls over?
• Is it just being perturbed by some sudden jolt that I can't detect myself (like a low-frequency vibration)?
• Is it a combination of these two things? Something else entirely?
• Is there a name for this phenomenon? Is there research on it?
• There could also be some tiny deformation happening in the part of the book that’s in contact with the floor that adds up over time to make the state unstable. Apr 10 '20 at 9:34
• In addition to tiny deformations, there can be a micro-movements of the book, which you don't see, until these accumulate in avalanche effect up to the point the book begins to slide rapidly. I.e. in principle if you put book in un-stable position, it instantly starts falling down, but just bit-by-bit in long time period, until gravity rapidly accelerates the process. Apr 10 '20 at 9:56
• Agnius, that is exactly the question: is there slow microscopic/hard to detect movement leading up to a final sudden fall, or is the book mostly steady until it falls? Apr 10 '20 at 9:59

It probably depends on multiple factors but one of the most important is how close the static friction force is to the maximum where you have pending or imminent slipping. At that point the book is unstable and the slightest movement, vibration, etc., will cause it to fall.

For the book the static friction force is a variable reaction force that keeps the book in static horizontal equilibrium, like at the foot of a ladder. But the static friction force has a maximum possible value of $$\mu N$$ where $$N$$ is the force normal to the surface. If the book is initially leaning so that the static friction force is at or near the maximum the book will be unstable and collapse with the slightest vibration or movement.

The reason it might take several hours Is it may require the cumulative effects of imperceptible vibrations to cause the center of gravity to move and the maximum static friction force reached.

As far as I know no ordinary environment is vibration free. From a microscopic view static friction is due to microscopic irregularities at the contacting surfaces causing them to interlock. Slight vertical oscillatory vibrations briefly separate the surface. A brief resulting net horizontal force in the direction away from the wall has the surfaces come back together to interlock microscopically further away from the wall slightly reducing the normal force and thus the static friction force. Over time the maximum static friction force is reached and the book falls

Hope this helps

• Is this the only explanation? Is there any research on this type of phenomenon? I've been unable to find any, but I might just be using the wrong keywords. Apr 10 '20 at 9:05
• As far as I know no ordinary environment is totally free of ambient vibration. Maybe you should research that. Because if it’s true then ambient vibration is the only reason needed for your observation. Depending on how much the book is initially leaning vibration can put it over the threshold of collapse or gradually move it towards the threshold of collapse Apr 10 '20 at 9:17
• From a microscopic view. Static friction is due to microscopic irregularities at the contacting surfaces causing them to interlock. Slight vertical oscillatory vibrations briefly separate the surface. A brief resulting net horizontal force in the direction away from the wall has the surfaces come back to interlock microscopically further away from the wall slightly reducing the normal force and thus the static friction force. Over time the maximum static friction force is reached and the book falls Apr 10 '20 at 9:33
• Would you say that the book falling is the cumulative result of many small perturbations? Or is just waiting for a suitable perturbation to come along? I would lean towards the cumulative thing, since that would explain why the book might not fall over during earlier perturbations/vibrations. Apr 10 '20 at 9:55
• Either/or but I’d be inclined to agree with you it being cumulative, particularly if the book was placed in position and left undisturbed for a long time Apr 10 '20 at 10:07