Contact forces or interactions are always due to fundamental forces, and therefore we should understand that true contact between objects in nature does not exist. The sensations of holding someone's hand or hitting your elbow on the ground as you fall both seem like real contact. On closer inspection both forces are really electrostatic in origin and arise because the materials making up the two hands, or making up the elbow and concrete, refuse to come too close to one another and therefore repel due to the electromagnetic forces like Coulomb's law as discussed in the last section. If you find this troubling, you will be more troubled in third semester when we discuss the nature of matter in general.[br][br]I am not trying to convince you that interactions are meaningless - just that they are very different than you may think. All of nature and existence likely is. That's the point of this journey we are on together - to get a glimpse of what really underlies our reality, and you'll most likely eventually conclude that reality is stranger than fiction.

While true contact is an illusion, it is still a useful concept to use in describing very short range forces that appear to our eyes and other senses to be true contact.  There are a few such contact forces that we will see often: Friction, the normal force, and air resistance. In addition to these we will have some discussion about tension in ropes or strings, and forces exerted by springs and other elastic systems. Furthermore we will discuss things like rocket propulsion and forces due to humans exerting themselves. All of these forces are really due to the electrostatic force. In our everyday life, besides gravity, the electrostatic force is the only one that is part of our experience. The nuclear forces, while crucial for the behavior of nuclei, do not enter into our experience.[br][br]Which of the two forces that are part of our life experience (gravitation or electrostatic) do you suppose is the stronger one? Spend a moment or two pondering this before reading on. It turns out that the electrostatic force is stronger... and not just a little stronger. In your homework I will have you calculate the ratio. It turns out that using a hydrogen atom as a candidate system, that the electrostatic force holding the electron to the proton (nucleus) is as many times stronger than the gravitational force as the diameter of the physical universe is larger than an atomic nucleus. Yes, you read that right. I was looking for a way to emphasize how large a number the result is. It turns out that within an order of magnitude the analogy holds... and certainly conveys a strong message! [br][br]If you imagine sky diving and the unfortunate event of your chute not deploying, what causes your death? Gravity? Of course not. It's the impact with the electrostatic force when you meet the ground! In fact, think of it this way: If you are sitting right now, quite literally every atom in the entire planet is trying to pull every atom in your body downward via the gravitational force. Opposing gravity is just the outermost layer of your pants pushing against the top layer of molecules in your chair via the electrostatic force, and those relatively few atoms and molecules easily prevail over an entire planet's gravity!

The most common contact force in nature is certainly the normal force. In this context, the word [i]normal[/i] is used in the mathematical sense of perpendicular or orthogonal, and not to mean something like ordinary or common - even though such forces are very much so. We will use the symbol [math]\vec{F}_N[/math] to denote the normal force.[br][br]The normal force is the reaction of a surface when some other object pushes against it. It is the force that the ground exerts on the underside of your shoes to keep you from sinking into the ground, or on the underside of your butt to support you as you sit. While it's true that some surfaces give way and collapse, the details of such situations are beyond the scope of our discussions, so we will assume the surfaces always have enough integrity to support whatever objects we place on them.[br][br]One question that's worth considering is: How does the ground know how hard it needs to push in order to support you? The underlying answer is subtle. All solids in nature are elastic. We will have a later chapter devoted to this topic in detail, but for now that just means they are always to some extent springy. If I placed a large spring on the floor and you sat upon it, it would compress just enough to support you. If a heavier person sat upon it, it would compress farther before reaching equilibrium and supporting them. In both cases the same spring "knew" just how hard to push to support the weight placed upon it. The force of its push increases the farther it is compressed, so regardless of what weight you place upon it (within reason) there will be a level of compression that will support the weight. [br][br]Molecules making up solids like the floor or a chair are like this. Materials are in fact springy or elastic entirely because molecules are also elastic. So when you stand on the ground outside or on the floor in your room, you are compressing little molecular springs just far enough so that they support you.[br][br]Keep in mind that the direction attributed to the normal force is normal to the surface. It is only one of two forces that surfaces commonly exert against objects. The other one is friction, which we will discuss next.

The study of friction is called [b]tribology[/b]. It is actually a very complicated field having roots in both Van der Waal's forces that you may have discussed in chemistry and in strengths of materials studies. Friction characteristics prove to be very difficult to compute or predict accurately. Many engineers and scientists effectively work as tribologists - even if they don't call themselves by that title. Those individuals might work for a tire manufacturer trying to develop a better tire compound for cars that has both lots of friction and also longevity, or they might work for an oil company trying to develop a better synthetic oil for use in engines. They might work for a chemical company trying to develop new glue technologies or a new material for non-stick pans. The US government has been attempting to reproduce the toe pads of geckos at a size scale that could be used by soldiers to give them Spiderman-like abilities to climb walls. All of these are deeply theoretical fields. [br][br]Luckily for us at this level of education, in many cases we can approximate the effect of friction with a very simple expression. You would call it a 0[sup]th[/sup] (zeroth) order approximation - one which has no functional dependence, but is rather treated like a constant. In terms of approximations, a 1[sup]st[/sup] order approximation would be one which depends on some variable in a linear fashion, 2[sup]nd[/sup] order is a second order polynomial, etc. If you have seen Taylor series in calculus, this is one place where they find direct application.[br][br]So our zeroth order approximation is going to be simply this: The force that friction can exert to prevent or fight slippage of two objects relative to one another is simply some constant called the friction coefficient times the force with which the objects are being pressed together. In symbols this is [math]F_f\le\mu F_N.[/math] We will label the friction force [math]F_f.[/math] The symbol [math]\mu[/math] (a Greek 'mu') is the friction coefficient or the coefficient of friction. Notice that the force with which the objects are being pressed together is just the normal force. Also notice the inequality, which goes along with the words "can exert". Friction will only exert a force if something else is trying to cause the object to slide. So the force of friction will be between zero and a maximum value given by the equation above.[br][br]When something in engineering or science is called a coefficient, it is expected to be a constant and is usually unit-less.  While the friction coefficient is a unit-less constant you will find in practice that some engineering coefficients will have associated units.  [br][br]In the present context, what we mean by a friction coefficient is a value that is independent of things like the value of the normal force or the size of the contact surface area. While friction is often much more complex than this, we will nonetheless use this for sake of laying useful foundations on which to later build. What the friction coefficient does depend on are the two materials involved - the one of which an object is made, and the one of which the surface is composed. In this sense it is meaningless to say something like "The friction coefficient of aluminum is 0.34". Rather, you'd have to say "The friction coefficient of aluminum on rubber is 0.34", or something of that sort.

As mentioned above, treating friction as a product of a constant-valued coefficient times the normal force is a simplification of reality. For instance, if you were to study the friction characteristics of a car tire on pavement, you'd find that the tire seems to actually exhibit more grip when it slips just a little bit. Slipping more than a little bit reduces the friction - like a car doing a burn-out and not going much anywhere.[br][br]Lots of objects approximate this behavior, so to improve upon our zeroth order approximation, we allow the friction coefficient to take on two different values - one for static situations (no slippage) and one for kinetic situations (slippage is occurring). We just ignore the case of sliding just a little bit and go straight from no slipping to slipping a lot.  [br][br]Think about trying to move a piece of furniture.  The behavior we're trying to capture is that if you push an object in attempts to make it slide, it usually takes more effort to break the object free and initiate sliding than it takes to keep the object sliding once in motion. Thus, we assume it will always be true that the static coefficient is greater than or equal to the kinetic coefficient, or [math]\mu_s\ge\mu_k.[/math] The alternative would suppose that it's easier to start something sliding than it is to slide something, and that would be a logical contradiction.