Mathematics and physics fight: put a perfect iron ball on a flat surface, the contact point is infinitely small?

A few days ago, a friend asked a question: If a perfect round iron ball is placed on a perfect plane, how big is the contact point between the ball and the plane, and will it be infinitely small?

Iron ball on the plane

This is an interesting question. Mathematically speaking, when a sphere touches a plane, it is a point. The point is one-dimensional, which means infinitesimal. But in fact, this is impossible, because infinitesimal contact area means infinite pressure. SoWe still have to analyze it physically.

Friends who have studied plane geometry know that when a circle is tangent to a straight line, the straight line just touches a point on the circumference. Any point on the circumference and the two adjacent points on the left and right are not on the same straight line.

But when we learn this knowledge, you can only "understand" the relationship between the circle and the tangent in your mind. No matter how you draw on the paper, the circle and the straight line will touch "a line segment" instead ofone point.

Mathematically, the tangent point needs to be "understood"

Similarly, in mathematics, if a sphere is in contact with a plane, they must only be in contact with a point. If you think that it is a surface, it means that your imagination is not enough.

Many of the most basic things in mathematics depend on our imagination. One-dimensional "points", "lines" and two-dimensional "surfaces" do not exist in the real world. At least in physics, you cannot find absoluteIdealized points, lines and surfaces, these concepts are just created for the convenience of calculation. Use a pen to poke a point on paper, or draw a line segment, the teacher will teach us that this is a "point and line", but if you use a magnifying glassObserving it, you will find that it is actually a surface left by ink; and if you put it under a microscope, you can see the blackened fiber clusters—they are neither dotted lines nor flat surfaces, but"body".

Line drawn by pencil on paper under microscope

Physically, even the roundest, smoothest and most perfect iron ball, it is still an object composed of a bunch of iron atoms gathered together. At room temperature, the iron atoms form an individual heart through covalent bonds.Cubic crystals, and then combined into iron.

Using the most advanced electron microscope to observe the iron ball, its surface is not smooth and flawless, but a "ball" surrounded by electron clouds around the iron nucleus.

The surface of the iron ball is closely arranged atoms

It can be seen that the iron ball that we think is the "most perfect" is not physically perfect, and its surface is not as smooth as we imagined, but like a neatly arranged egg.

No matter what kind of plane these atomic "eggs" touch, their contact surface must not be a "point", but at least "three points", because the mathematics teacher told us that there are only three pointsTo form a stable surface.

Can an iron ball really be placed on a plane with three atoms as supporting points? Still not. Because when the iron ball contacts the ground, under the action of gravity, the place where the iron ball contacts the ground will deform.

Some friends will say that iron is very hard. If we use the hardest diamond to create a smooth surface, will it also be deformed? Of course it will.

The surface of the diamond is a plane composed of unit cells

Diamond is composed of carbon atoms. In a diamond crystal, every 4 carbon atoms form a stable tetrahedral unit cell. Even if the diamond is polished extremely flat, the physical structure determines that its surface is staggered like an egg.

We assume that there is a 100g iron ball with only three atoms in contact with the "smooth" diamond ground. Calculate the pressure at the contact point.

The covalent radius of iron atoms is only 132pm [picometers, one trillionth of a meter], the plane area formed by three iron atoms is about 3.48×10^-20 square meters, a 100gThe pressure of the iron ball is 0.98 Newton, and the pressure acting on the contact surface reaches 2.8×10^19Pa, which is about 2.78 trillion atmospheres!

Under such a huge pressure, the surface of the iron ball quickly deforms, and the contact surface between the diamond and the iron ball will also dent, so the contact area between the two increases until the pressure is balanced with the elastic modulus of the iron ball 211GPa, This requires the support of billions of iron atoms on the contact surface!

The iron ball will deform elastically when it contacts the ground

Now you should understand: Many common concepts in mathematics do not have corresponding objects in reality; the most perfect iron ball is also made of atoms, and its surface is not absolutely smooth; when the iron ball is placed on a flat surface,The atoms on the contact surface will be displaced due to the huge pressure, the iron ball will be deformed, and the ground will also dent. So the sphere and the ground are no longer a point, but a circular area formed by the interlocking of atoms.


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