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You may have this experience, when you rush out, usually can take the door lock suddenly disobedient, hard to take, or not! It's fighting you. You have no choice but to press the latch to close the door.

What is the reason behind this seemingly ordinary thing? Maybe you have no time to think or interest. However, there is another related thing that you may have heard and been interested in, and it is the fact that two books can resist great tension when they are stacked one on top of the other.

A well-known experiment This experiment is easy to do. Fold the two thick books alternately as shown below, and then pull hard on both sides. It turns out: you may pull them apart, but it's hard, but you may not be able to pull them apart at all!

This event has aroused intense interest for hundreds of years, and many explanations have been given one after another. Many people try various experiments and publish special research papers for this purpose.

In 2011, Discovery Channel's MythBusters conducted a validation experiment. They overlap two books, 800 pages thick, page by page, and then try to pull them apart.

They tried to pull the two of them first, but of course they failed.

So they clamped the spine with strong boards and attached a string with strong tensile strength. Then they found some people to pull it together, but they could not pull it open.

Then they fixed the spine with an iron plate, connected it with a thick chain, and tried to pull it with a car, but it still failed!

It wasn't until they used an armored personnel carrier and a light tank to pull the two books apart.

In 2016, researchers from France began a study on this issue. In their experiment, the static friction between two phone books with overlapping pages was enough to lift a car!

Their study showed that the amount of tension the system could withstand was determined by the number of sheets, their thickness, and the width of the overlapping area, and increased sharply as the number of overlapping pages increased. Their results were published in the international physics journal PHYSICAL REVIEW LETTERS [1].

So what is the physics behind this?

Some people say, Bacon said-knowledge is power! It had to be said that this cleverness was indeed quite good!

Seriously, there are several reasons behind this, the most important of which is friction self-locking. Let's talk about what friction self-locking is.

02. What is friction self-locking? According to the Coulomb model, the tangent of the angle between the resultant of the maximum static friction force and the positive pressure and the positive pressure is equal to the static friction coefficient, as shown in the following figure.

This special angle corresponding to is called the friction angle, indicating that it is obvious that when the combined force of gravity and tension is greater than the angle with vertical, the tension will exceed the maximum static friction force and the object will be pulled; otherwise, the object will always remain stationary.

More generally, if gravity and tension are combined together and called active forces, then for a plane with a certain static friction coefficient, when the angle between the active force and the normal of the plane satisfies

The object will always remain stationary. As shown in the following figure, the object will remain stationary as long as the direction of the active force does not exceed the cone surface.

If explained in a more intuitive language, although the active force is increasing, the positive pressure is also increasing, so that the maximum static friction force is always not exceeded, thus maintaining balance.

For example, there is a woman who is a big spender, and her friends think that she may soon lose her labor family. But her ability to earn money grew with her ability to spend, and she kept spending growth rates just below labor's earnings growth rates. As a result, her sisters still saw her family getting richer.

So spending money is like friction, and as long as you stay within the wealth angle, your wealth can be locked in to increase rather than decrease.

There is no proof of this. Here are some more concrete examples from physics.

03, friction self-locking of those things The first is about the friction on the slope. As shown in the figure below, if the static friction coefficient between the slider and the inclined plane is, then how to select the inclination angle of the inclined plane to ensure that the slider does not slide down?

Gravity is the only active force, as long as it does not exceed the angle with the normal of the inclined plane, but exactly equal to the inclination of the inclined plane, so long as it satisfies that no matter how heavy the slider is, it will not slide down. But if this condition is not met, it must slide down.

There is this kind of experience in daily life: others put a stack of books on the slope and stand steadily, but I can't put a piece of paper stably. I can't help but suspect that God is bullying people!

You also have the experience that when you push an object on the horizontal ground obliquely downward, if the angle between the force and the vertical is too small, for example, the object will not move no matter how much brute force you apply.

Another lesson is that wedges can be driven in, but once driven in, there is no fear of being squeezed out. As shown in the figure below, after the yellow wedge is inserted, although it is under pressure from above, it will not be squeezed out.

Because for horizontal thrust, as the main force, its angle with the normal of the inclined plane is greater than the friction angle, so that the wedge can move; for vertical force, its angle with the normal of the inclined plane is less than the friction angle, no matter how large it is, it cannot push the wedge.

Looking at this, you might think: The wedge won't be squeezed out, but it can be pulled out. Yes, nails can be pulled out, but you've never seen nails squeezed out.

The acute angle is called acute because the force it exerts on the obstacle must be outside the angle of friction, so it can push forward. The reason why obtuse angle can not kill in, because of friction self-locking occurred.

No wonder the blade of a sharp knife is always thin and slippery. Africa, however, has a knife that is too dull to be used, because Africans regard it only as a monetary equivalent.

Wedges are common in ancient wooden buildings in China. For example, the Xuankong Temple at the foot of Mount Heng in the north was built by inserting huge wooden wedges into the stone walls.

Those wedges take advantage of the expansion screw principle, which is essentially a physical mechanism of frictional self-locking.

The principle of the expansion wedge (quoted from the Institute of Planetary Research [2]) is similar to the one mentioned at the beginning of this article: friction self-locking.

Let us return to the problem of friction between the pages of the book mentioned in Section 01.

04. Why are overlapping pages so tensile? What makes those overlapping sheets so strong?

Let's first see what characteristics there are in the structure formed by the overlapping pages of the two books. Usually, it looks something like this:

When two pages are interlaced, each page has three horizontal regions, a small section near the spine is flat, the overlapping part is flat, and between the two is an inclined non-overlapping region with a width of.

In order to facilitate analysis and calculation, several parameters of the book are given. Let the thickness of a single page be, the number of pages in both books be, and the thickness of each book be.

Obviously, the force between the pages is due to static friction between the overlapping sheets. Assuming that the friction coefficient between the pages is determined, it is sufficient to study the pressure between the pages. What factors contribute to these pressures?

First, the gravity of the page itself.

A single sheet of paper, flat in its natural state. Put it on the table, the pressure is determined only by its weight, which is of course insignificant. So you might think: It doesn't seem like much friction to be generated by gravity alone.

Is that right?

You're missing a detail. There's friction between every two interlaced pages! For thick enough books, this friction due to page gravity accumulates to a large value.

Let's analyze it briefly.

Suppose that two books are pages, the weight of each page is, and the coefficient of static friction is. After interleaving, from top to bottom, the maximum friction between each two pages is: Therefore, the maximum static friction force caused by gravity is recorded as the mass of a package of 70g A4 copy paper (500 sheets) is 2.2 kg, assuming that the number of pages in a book is also the page, the mass kg, and the friction coefficient is 0.3. Then the maximum friction force is 6461.53 N, which is equivalent to more than 1200 jin of force, and it is more than enough to lift a fat pig.

It can be seen that the friction force caused by the gravity of the page itself is far greater than you think!

However, this value is always finite, and when the book is suspended vertically, the influence of gravity does not exist. So gravity is certainly not the decisive factor.

Second, there is the effect of page bending.

Suppose there is a wire on the ground. Now put one end of it under your foot and lift the other end. You will feel a resistance.

This resistance from deformation is universal, and it also exists for weak paper.

The pages are flat and they are most comfortable. When you cross them over, the pages deform, and it inevitably creates a resilient resistance. These forces build up and also cause pressure to increase between overlapping pages.

However, this part of the calculation needs to use elastic mechanics, which is more complicated. There is no need for careful deduction here. Simply put, the total pressure caused by this part is proportional to the thickness of the paper, the number of pages in the book, and the elastic properties (stiffness) of the paper. Write the maximum static friction caused by this pressure as where is a number related to the number of pages.

Again, this friction is finite, so it is not the determining factor.

The third factor, which is really decisive, is the pressure caused by the pull outward along the spine.

Let the tension on the spine be, and the total number of pages in both books be. Therefore, the horizontal tension distributed from the bottom to the top of the page is

Since the page is in equilibrium, the tension on the oblique section of the page should be the downward component of this force applied to any page by all the pages above it, considering that the same is true for another page, so that the downward force applied to a page is

Sum this pressure over all the pages to get the total pressure. Because the calculation is very complicated, I will not push it carefully here. Simply put, the maximum static friction resulting from this component can ultimately be expressed as

Note: For details of the above calculations, please refer to the analysis procedure provided by user Huxley, i.e. Reference [3].

Therefore, for a flat book, the total maximum friction between pages is

The condition for the book to be pulled is that it is obvious that if the coefficient in the above equation is negative or zero, this condition cannot be satisfied, that is, it cannot be pulled.

Therefore, as long as no matter how much tension is satisfied, the tension will not exceed the maximum static friction force, and the pages will have friction self-locking and cannot be pulled apart.

Therefore, the condition of page friction self-locking is based on the calculation result of user Huxley, when, the above condition is approximately the width of the inclined non-overlapping area, and the thickness of the book. It can be seen that the thicker the book, and the narrower the inclined part, the easier it is to produce friction self-locking.

The above analysis takes into account gravity, if the book is lifted vertically, the influence of gravity is no longer there, but the decisive role is still the coefficient of tension, so the above analysis results still hold true.

Okay, now calculate an example.

If the non-overlapping width of the inclined pages is 10cm, the number of pages per book is 100, and the thickness is 1 cm, then the self-locking condition must be satisfied. The requirement of this friction coefficient is easily satisfied. This explains why, if you take any book and you have the patience to overlap the pages, the experiment will be successful.

Warm reminder: some people are used to entertainment, figure out how to do things like this, the possibility of experimental failure is high!

At this point, he finally understood why the friction generated by the intersection of pages was so great. Because of frictional self-locking. As long as the page is not broken, the greater your pulling force, the greater the friction, and always greater than the pulling force.

One can imagine that each piece of paper is like a long wedge, inserted directly into the depths of each other. A wedge may be weak and break even if it doesn't pull out, but hundreds of wedges, you can't deal with it!

references

https://doi.org/10.1103/PhysRevLett.116.015502

https://baijiahao.baidu.com/s? id=1639930388412359613

https://www.zhihu.com/question/30567010/answer/1310750980

This article comes from Weixin Official Accounts: University Physics (ID: wuliboke), by Xue Debao

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