#### Prerequisites

In our basic overview of Newton's laws (Formulation of Newton's Laws as foothold principles) we stated that a good mathematical model for forces was the model of vectors that we developed using the idea built on the concept of displacement in space. In this page, we elaborate on some of the more subtle issues of adding forces and discuss the limitations of this mathematical model.

## Why vectors are a good mathematical model for forces

As with any mathematical model of a physical property, we have to rely on a good understanding of how the physical property behaves before we can choose a math to describe it. For forces, we have some pretty good intuitions. We know that if we want to push a stuck car out of the mud, two people pushing does a better job than one, and they both had better push on the same side (not one in front and one in back). This suggests that forces acting in the same direction tend to add, while forces acting in the opposite direction tend to cancel. This is just what vectors -- or the displacements on which they are modeled -- do. If you go two displacements in the same direction, you get farther, if you go one displacement in one direction and then an equal displacement in the opposite direction, you wind up where you started -- at zero displacement.

But forces are not displacements, so to see if they follow the same math as displacements, we have to do lots of careful experiments to check. One such experiment often done in introductory physics is the force table shown in the figure at the right. Known forces are exerted on a ring at the center of the circular table by connecting it to strings attached to known weights. (The ring is free-floating -- not connected to the table.) The angles are adjusted to the ring is stationary and not touching the center rod. Then the forces on it applied by the strings must balance. This experiment, and many others, provide convincing evidence that treating forces as vectors is an appropriate mathematical model and works extremely well. (Image source: Cenco Physics,  Sargent-Welch
http://blog.cencophysics.com/2009/08/
composition-of-concurrent-forces/

## Dangerous bend: Forces are not displacements

Although vectors are an excellent model for adding forces, we have to be a bit careful. We created vectors to match the behavior of displacements, but "vectors" are a mathematical tool -- they are not displacements unless that is what we are modeling. When we treat forces as vectors we have to not accidentally attach to them ideas about displacements that are properties of displacements but are not part of the mathematics of vectors.

For example, displacement vectors connect two points in space. We think about them as telling us something about location. But forces are not things that live in space. Although we draw an arrow that extends over space (since that's the easiest representation of it), we do NOT mean that the force itself extends over space. When we have a force on an object, the only spatial point that is relevant is the point on the object at which the force is (or can be taken to be) applied. The region of space over which the arrow extends has no physical meaning. This can be particularly tricky and confusing when we talk about vector fields, like the gravitational field, g, or the electric field, E. In this case we have arrows that appear to fill space, but the spatial extent of the arrows doesn't have any meaning since force or field vectors do not have units of distance.

## Dangerous bend: Forces only add simply if you have a good model of your objects

A subtler point that is usually not discussed in introductory physics is that simply treating forces as fixed vectors that add is only correct if your model of your objects has not ignored important degrees of freedom. We might, for example, model a water molecule as a rigid uncharged object (it's neutral) if we are talking about how it contributed to the pressure on a wall of its container by bouncing off the wall.

But if we are talking about its interaction with a potassium ion, we have to pay attention to the charge distribution of the neutral water molecule — that the hydrogen ends are more positive and the oxygen end is more negative. Moreover, if there are other charges in the problem, they may polarize the water molecule, changing the distribution of its electron cloud a bit. Then the forces it exerts on the potassium ion can change a bit. So while the forces from the electrons and nuclei still add simply when you have inter-molecular interactions, the forces from the molecules themselves may not.

The moral is that forces add like vectors if you are careful about your model of your objects and interactions. If you are talking about complex objects that might deform, you may have to go down to a finer level of description for your forces to add simply.

Joe Redish 10/6/13

Article 348