# Appendix A: Normal Contact Force and Frictional Force

Normal contact forces (N) arise when two surfaces are pressed against each other.

Frictional forces (f) arise when the two surfaces are sliding, or trying to slide, against each other.

Together, NCF and frictional force make up the contact force between two surfaces. NCF is the perpendicular component (to the surfaces), and frictional force is the parallel component.

Personally, I find it helpful to think that two surfaces repel each other perpendicularly (hence N arises), but attract each other parallelly (hence f arises).

Friction Opposes Relative Motion

A common misconception is that friction always destroys motion. Do realize that friction does not oppose motion per se. What friction opposes is relative motion (between the two surfaces). In fact, we use friction to create motion when we walk, when we run and even when we grip a pen. Without friction, one can neither get a car to start nor stop!

Coefficients of Friction

Through experiments, we have discovered some empirical laws of friction. (“Empirical” means “not theoretical”. Empirical laws are derived from observations without a universally accepted theoretical basis. Empirical laws are usually accurate most of the time, except when they aren’t.)

It turns out that the frictional force f (between two surfaces), is directly proportional to the normal contact force N (between those two surfaces).

$f=\mu N$

In other words, two surfaces can grip each other better if they are pressed against each other harder. This is how fridge magnets work. The magnetic attraction creates a large N, allowing for a larger f.

The constant of proportionality is called the coefficient of friction m. It is dependent on the material of the two surfaces. Some surfaces are by nature “stickier” than others.

Static and Kinetic Friction

I am sure everyone has pushed a very heavy object (e.g. cupboard or table) along the floor before.

Usually, the force F that we apply to the heavy object is initially too weak to overcome the frictional force f. So we increase F gradually, eventually reaching the tipping point whereupon the object jerks forward. Once the object starts moving, we only need to maintain a smaller F to keep it moving (at a constant speed). If we sketch the variation of F with time t, we get a graph that looks something like this.

Since $F=f$ all the time (do you know why?), this graph is also showing us how f varied with t during the motion.

Cleary, friction behaves differently before and after slipping. The friction when the two surfaces are still at rest (relative to each other), and when the two surfaces are already moving (relative to each other), are called the static friction fs and the kinetic friction fk respectively.

The static friction fs can range from zero to some maximum value to match the external force that is trying cause the surfaces to slip. The maximum static friction is given by the formula

${{f}_{s}}\le {{\mu }_{s}}N$

where ms is the coefficient of static friction.

Once the surfaces are already slipping, the kinetic friction fk is a constant value, given by the formula

${{f}_{k}}={{\mu }_{k}}N$

where mk is the coefficient of static friction.

The coefficients of friction between some common materials are tabulated below. The value of m is usually between 0 and 1 (but can be greater than 1). Note also that mk is always smaller than ms.

Contact Area

It has also been observed that friction does not depend on the contact area between the two surfaces. For example, if you are dragging a cuboid along the floor, it does not matter which face of the cuboid is on the floor. The friction is the same.

Speed

It has also been observed that kinetic friction does not depend on the speed. As long as the surfaces are already moving relative to each other (i.e. slipped), it does not matter how fast they are moving.