Reading the content in the friction equations

The friction equation as often given in high school physics classes often looks quite simple: $f = \mu N$. But friction is about two surfaces sticking together and is quite complex. It's better to think of friction as being described by three equations, with each marker on the equation as carrying information about the complexity of the phenomenon.

$$F^{static friction}_{A \rightarrow B}   \le \mu^S_{AB} N_{A \rightarrow B}$$

$$F^{kinetic friction}_{A \rightarrow B}   = \mu^K_{AB} N_{A \rightarrow B}$$

$$ \mu^S_{AB} \ge \mu^K_{AB}$$

Many things about these equations tell us important conceptual ideas about friction with the things that are missing telling us almost as much as what's there!

1. Two equations — The fact that we have two equations tells us that friction forces when the surfaces are not sliding over each other and sliding over each other are different.

2. $\mathbf{\le}$ —The "≤" in the static friction equation tells us that the static friction equation only gives us a MAXIMUM. Yhe static force of friction may adjust itself, just like a spring, to be whatever it needs to be to prevent motion, and it might possibly even be 0.

3. = — The "=" sign in the kinetic friction equation tells us that when the two surfaces are sliding on each other the friction has a well determined value.

4. No v — The fact that the speed does NOT appear in the kinetic friction equation tells us that the force of kinetic friction is independent of the speed of the two surfaces sliding over each other. 

5. N — The N on the right side tells us that it's how hard the two surfaces are being squeezed together that determines the (maximum) friction force.

Warning! Note that many high school physics problems the only place that friction is used is for isolated objects moving horizontally on a flat surface. In this case the normal force squeezing the two surface together is $mg$. But that's an accident of the special example. Remember to determine what normal force is actually squeezing the two surfaces together in whatever situation you have — and do NOT write f = μmg!

6. AB — The subscripts on the coefficients of friction tells us that they depend on the two surfaces that are touching.

7. $ \mathbf{\mu^S_{AB} \ge \mu^K_{AB}}$ — This equation tells us that it can be harder to overcome the friction to get something moving than the force that's needed to keep it moving at a constant speed. (This is what the height of the triangular peak in the graph of the friction force shows you in the Friction page.)

8. No area! — The fact that there is no area in the equation tells us that the surface area of contact does not affect the friction force! Surprising!

9. No vectors! — Despite the fact that this is an equation relating two force vectors (the friction force and the normal force), this is NOT a vector equation. The friction force is parallel to the surface of contact trying to prevent the two touching objects from sliding over each other. The normal force is perpendicular to the surface of contact trying to squeeze them together. The friction equations only refer to magnitudes.

Friction is a tricky subject so here's a third dangerous bend in our discussion of the topic! Remember! Friction tries to stick two surfaces together, so is one object is being moved, it will exert a force to drag the other object along with it. This means that 

Friction acts to keep two surfaces together. It does NOT act to oppose motion. In fact, it's what produces motion in many situations.

Consider for example holding a pen on your flat, horizontal palm. If you start moving your palm horizontally, the pen speeds up and moves with your palm. What force speeds up the pen so it stays with your palm? The friction between the pen and your palm! 

Joe Redish 1/31/19