# Resistive forces

#### Prerequisite

One of the things that the theoretical framework provided by Newton's laws does for us is to let us see "invisible actors" — forces that act in a situation that we might not otherwise notice. One example is the fact that a block sitting on a table actually feels a force from the table that prevents the block from falling through it. This force (a normal force) arises because the table compresses like a spring, exerting more and more force on the object until the object's weight is balanced by the upward force from the table. But the compression is so small that we typically do not see it unless we measure it with special instruments.

Friction (and other resistive forces) are other "invisible actors". So many of the motions we see are dominated by friction that we assume things "just slow down naturally" and we don't notice the friction force and the object causing it.

Figuring out what forces there are and how they behave (what they depend on) creates models of how objects interact and what they do to each other. What models we choose depend on the level we are observing. If we watch macroscopic objects (or even microscopic ones down to the size of cells), we tend to do phenomenology — we look, measure, and model, creating equations that work over some range of phenomena. (Hooke's law for is a good example of a phenomenological law.)

The class of forces we are interested in for this section are the group known as resistive forces. These tend to act to reduce the relative motions of two objects.  We will consider three different models of resistive forces, appropriate for different situations: friction, viscosity, and drag.

• Friction — When two solid objects slide over each other, each exerts a force on the other that is parallel to the interacting surfaces and in a direction to reduce the relative sliding.  Friction depends on what the two surfaces are made of and how hard they are squeezed together. It does NOT depend on the speed of relative motion. The force is a constant independent of the relative velocity of the objects.
$$F^{fric}_{B \rightarrow A} = \mu N$$
• Viscosity — When a solid object moves through a fluid it drags the fluid along with it. The rubbing of each layer of fluid on the next exerts internal fluid forces that act to reduce the relative sliding. This results in a force on the object that acts to reduce the relative motion of the object and fluid.  The magnitude of the viscous forces is proportional to the relative velocity of object and fluid.
$$F^{visc}_{fluid \rightarrow object} = bv$$
This model is particularly relevant for small objects moving slowly in a fluid (bacteria, white blood cells, ...).
• Drag — When a solid object moves to push fluid out of the way in front of it, it has to exert a force on the fluid to speed it up. This results in a force of the fluid back on the object. The magnitude of the drag force is proportional to the square of the relative velocity of the object and fluid.
$$F^{drag}_{fluid \rightarrow object} = Cv^2$$
This model is relevant for macroscopic objects moving through a fluid (swimming dolphins, diving hawks, cars on a highway,...).

Both viscosity and drag typically act on any object moving in a fluid. The ratio of these forces is the Reynolds number. It tells you which of the two forces dominate. For more detail, see the follow-ons.

The convention for separating the part of the resistive force that is proportional to velocity and the part that is proportional to the square of the velocity is not uniform in the scientific literature. In some fields, both are referred to as "drag" with the former called "viscous drag" and the latter called "inertial drag". In others, both are referred to as a "viscosity". We will try to stick to the naming convention introduced here.

Joe Redish 9/26/11

Article 368