# Forces other than the fundamental interactions, e.g. friction

Forgive me for the silly question, but I just don't get it.

I just completed an elementary course in mechanics, and I am curious to know what I am about to ask.

We have, all year, dealt with many forces like gravity, friction, normal forces, tensions etc.

But only one of them is listed as a fundamental force, that is, gravity.

I know that the only forces that exist in nature are the four fundamental force, and all of these are, apparently, non-contact forces.

But then how do you account for, for example, friction? We know that $F_\text{frictional}=\mu N$, But how do we arrive at that? Is this experimental?

I cannot see how contact forces like friction can exist, when none of the fundamental force is a contact force.

Again, forgive me for my ignorance.

• It is not silly question at all. Feynman was also thinking about exact same question -- how to get above formula for the friction from basic principles. Formula for friction is experimental. Aug 2 '14 at 16:19
• Comment to the question (v2): Consider restricting the question to only the friction force to avoid getting too broad. Aug 2 '14 at 16:25

AlanZ2223 has given a nice summary of what's going on. I'll just make a couple of points that are orthogonal to his and that wouldn't fit in comments.

The electrical force is a non-contact force; it falls off with distance like $1/r^2$. But most of the objects we deal with in everyday life are electrically neutral, i.e., they contain equal amounts of positive and negative charge. You would think that this would mean the attractions and repulsions would exactly cancel out, but that's not quite true. When two electrically neutral objects are close together, they can influence each other to rearrange their charges somewhat, so that the cancellation isn't perfect due to the different distances and angles involved in all the force vectors that are being added. This is called a residual interaction. The residual electrical interaction falls off much more quickly than $1/r^2$ at large distances -- more like $1/r^6$. This is the basic reason why bulk-matter forces, which are electrical, appear to be zero-range contact forces.

The other thing to realize is that it is not possible to explain forces such as the frictional and normal forces purely by using classical mechanics and an electrical interaction. If you try to do that, you'll find that bulk matter isn't stable, and that one piece of bulk matter won't prevent another from penetrating into it. In fact, you need two ingredients to explain these forces: (1) electrical interactions, and (2) the Pauli exclusion principle. If you try to explain it using only one of these ingredients without the other, it doesn't work.

• Nice answer. Any suggestions on where I could learn more (at a fairly basic level) about the $r^{-6}$ interaction? (E.g., why is the next term not $r^{-4}$?) Aug 3 '14 at 5:27
• @Charles: they are called van der Waals forces. Aug 3 '14 at 9:19
• Does this imply that any surface, however smooth (If we consider a hypothetical 100% smooth surface) would exhibit a frictional force if an object were to be pushed atop it? Regardless of the nature of the surface these interactions would remain, would they not?
– SNB
May 24 '18 at 6:05

These "non-contact" forces that are ubiquitous in everyday life are mainly attributed to electromagnetism. Basically the four fundamental forces which are the strong force,weak, electromagnetic, and gravitational all have a sort of realm within which they influence most.

The strong/nuclear reigns within the subatomic domain, the weak as well but this kind of force is not nearly as prominent. Then comes the electromagnetic force whose influence falls within our human sized domain and gravity which does have effects in our everyday life but it is mostly prominent in larger bodies such as our solar system or galaxies.

Now all of these forces are able to manifest by the exchange of particles. Trillions and trillions of them are created and annihilated every second. For the electromagnetic force the exchange particle is the photon.

Now for example whenever you touch an object your hand does not just go through the object, the electric forces are acting to repel the object. The electrons that compose your hand repel the electrons of the material keeping your hand from going though, not by directly touching the other electrons but by the exchange of the carriers of the electromagnetic force, the photon, and this is what lies withing the "forces" of friction, normal force, so on and so forth.

Let's examine friction. Whenever you push an object lets say a book across a desk. Friction opposes the direction of motion, you push left, friction pushes right. This is because atomically the molecules of the book are "pushing" or more technically repelling and attracting by effect of the electromagnetic force which is able to manifest by exchanging photons with the atoms of the desk, and this is what is the sort of big picture of what composes these non fundamental "forces".

• Does this imply that any surface, however smooth (If we consider a hypothetical 100% smooth surface) would exhibit a frictional force if an object were to be pushed atop it? Regardless of the surface these interactions would remain, would they not?
– SNB
May 24 '18 at 6:00

I think it's best to just quote Feynman here:

Friction: the force of friction against a dry surface is $-\mu N$, and again you have to know what the symbols mean: when an object is pushed against another surface with a force whose component perpendicular to the surface is $N$, then in order to keep it sliding along the surface, the force required is $\mu$ (friction coefficient) times $N$. You can easily figure out which direction the force is; it's opposite to the direction you slide it.

Does friction force result from a potential energy like gravitational forces do?

The answer is No: friction does not conserve energy, and therefore we have no formula for the potential energy for friction. If you push an object along a surface one way, you do work; then, when you drag it back, you do work again. So after you've gone through a complete cycle, you haven't come out with no energy change; you've done work-and so friction has no potential energy.

Finally, in the example gravitational force (equivalently can also be shown for Coulomb's force and the electric field), it results from a difference of potential energy in the gravitational field $\mathbf{g}$, which can be shown to be conservative (through a complete cycle, 0 work is done, energy is conserved), is equal to the gradient in gravitational potential $\Phi$: $$\mathbf{g} = -\nabla \Phi$$

Friction is indeed a phenomenon that is difficult to harness theoretically. The formula you mention is not derived from first principles, but is justified only by experimental evidence, i.e. it drops from heaven. Moreover, it gives a false impression of simplicity. The friction coefficient is all but constant, as it may depend on temperature, pressure, etc. Just think of a ski on snow. The friction there does depend on snow temperature, the consitency of the snow, how well you prepared the ski, ...

What is next is that there is not just one single friction phenomenon. The friction between ski and snow for example is described by a thin film of liquid water between the ski and the snow, whereas the friction of a shoe on the street is something else entirely. Related to friction is diffusion and dissipation, e.g. the air resistance when you ride a bycicle is essentially explained by the viscosity of air and the turbulent cascade towwards small scales.

However, as the previous answers explain, all of these phenomena are fundamentally explained by electromagnetic forces between molecules; and these never "touch" each other, their electron hull prevents that from happening.

The four fundamental interactions (gravity, weak, strong and electromagnetic) are useful to understand how the basic ingredients (fundamental particles of the standard model) of our world interact with one another. It doesn't imply that all the forces witnessed in our world can be reduced to these forces only. Most physical objects comprises many many many of these fundamental particles and the point consists in realizing that "the properties of the sum" is not the same as "the sum of the properties" and, in fact, "the properties of the sum" is much richer than "the sum of the properties". This is what I meant when I said that physical forces cannot be reduced to the four fundamental interactions only.

Any physical system that displays properties which are not captured by the properties of the ingredients is said to have emergent properties. And basically, all the forces you are having questions about are emergent forces that do not have any equivalent when you just look at the most fundamental ingredients alone.

I preach for my chapel here (and we'll see if people disagree) but the most general strategy one can use to understand how these effective forces at the macroscopic/mesoscopic scale emerge from the microscopic world (that of basic ingredients) is that of statistical thermodynamics (quantum or classical) which is a huge field that one cannot cover in few lines only I am afraid.