A simple model of solid matter

Prerequisites

In our page on Normal forces we had started with a simple model: a box sitting on a table.  And in our model, we ignored any structural properties of either the box or the table.  Applying the Newtonian framework, we realized that the principles of that framework imply that the table has to exert an upward normal force on the box and it has to be exactly equal to the box's weight (unless I put another box on top of it, or if I push down on the box, or if I tilt the table, or ...).   But then we wondered, "How did the table know how much force to exert?"

To answer that question, we have to develop a more detailed model of matter — one that allows us to see how it might respond to a push or a pull on itself. Now real matter is quite complex — especially biological matter, which contains a complex mixture of many different chemicals.  But the idea in physics is to start simple and become increasingly complex. This allows us to understand the pieces of a complex system and to build up our understanding of underlying structure and mechanism a step at a time. So our next step will be to consider a model that is only slightly more complex than an inert chunk with no internal structure that we pay attention to.  Let's consider a uniform piece of matter consisting on one kind of molecule.

A ball and spring model of a solid

We know that matter consists of atoms arranged into molecules. A solid holds together, so those molecules must hold together somehow. In our Newtonian language, this means the molecules must exert attractive forces on each other.  We also know that those attractive forces don't pull the molecules completely on top of each other. If you push on a solid, it pushes back; it doesn't collapse down to nothing.  So even if we know nothing about quantum physics and nothing about the real forces between atoms and molecules, we know if you start at their average distance in a solid,

1. if you try to pull molecules in a solid apart, they attract each other;
2. if you try to push molecules in a solid together, they repel each other.

This looks a lot like a spring — qualitatively at least.  If you have a spring at its rest length,

1. if you try to pull the ends of a spring apart, the ends try to pull back
2. if you try to push the ends of a spring together, the ends push back against you.

This is exactly the kind of forces we need.  So it suggests that as our first model of the structure of matter, we might try bits of matter connected by springs.

[Although this is only an analogy, it is an example of the last paragraph of the section of the page Springs just before the section on Scales.  While the actual force between molecules is a cancelling pattern of positive and negative electric forces stirred in with quantum mechanics, the result pattern of energy vs distance between molecules looks just like the pattern around the resting length of a stretched spring.  This makes a spring an excellent model for inter-molecular forces in many cases.]

We know that a spring knows how much to push back when we push or pull on it.  It changes its length from the rest length.  Since the property of a spring is to push or pull back more the more its length is changed, the force it will exert is determined by how much it's squeezed or stretched and the spring constant:

$$T = kΔL.$$

So let's make a model of matter consistent with small masses (which we will represent for now as spheres) connected by springs. The model is shown in the figure on the left below, and the response of matter to something pushing on it is shown in the figure on the right. (Only a 2D slice of the matter is shown in that figure to make it easier to make sense of.)

We can now understand how the table knows how much to push back. When the box is placed on the table it only feels its weight, so it starts to accelerate downward. This deforms the table, bringing upward spring forces into play. The table deforms until the forces are enough to cancel the weight.

Scales

This still seems a bit weird. We can't see the deformation of the table. How can that have an effect? Let's estimate how big the spring constant of the table has to be in order for it to be invisible to our eye.

Suppose I put a 2 kg lead brick on a table. It has a weight of about

$mg =$ (2 kg)(10 N/kg) or 20 N.

I think I could see a deformation of 2 mm — but probably not much less.  This means that to be invisible to me, the spring constant of the table would have to be big enough to cancel the weight with a deformation of 2 mm or less. So solving for the spring constant,

$k = T / ΔL =$ (20 N)/(2 mm)
= (20 N)/(2 mm) x (1000 mm)/(1 m) = 104 N/m.

So that's about 10,000 N/m. Is that reasonable?  We'll find out when we study Young's modulus and the deformation of solids.

Joe Redish 9/24/11

Article 374