The parameters of matter

Dimensional analysis helps us decide how to most conveniently describe matter and to separate what depends on the kind of matter we are talking about and what depends on the specific material object we are talking about. We might call this the creation of density-like quantities.

For example, if we look at a cup of water and a cubic meter of water, there are ways in which the two objects are very different and ways in which they are very much the same. They are different in that one is MUCH heavier than the other and occupies much more space. They are the same in that if I dip into either one and pull out a cubic centimeter of material they will be identical as far as I can tell.  What's different about the two is the size; what's the same is what they're made of, bit by bit. If we want to identify what it is about the two systems, we have to think a bit more carefully how to express that mathematically — and it depends on the given situation and what we are talking about.

Suppose we are talking about the mass (and indirectly, about the weight, mg).  Since each bit of water is the same as every other bit, what makes the two objects different is how many bits we have.

We could imagine breaking each one up into bits of the same size, finding the mass of the bit, and then deciding how many bits we have in each case.  This is how we define a measurement in general.

We have an arbitrary choice to make. A natural one might be to consider a cubic centimeter (cm³) of water as our bit. Since that has a mass of (about) 1 g, the counting is simple.

For other materials — rock for example — the calculation isn't so simple.  You would have to multiply the number of cm³ by the mass of one cm³.

In general, what we are doing in this case is extracting what it is about the mass that depends on the size -- and perhaps the shape (though not in this example) -- and what depends on the character of the material.  So for the case of the mass this analysis suggests that we write the mass as

mass = volume x (some property of the material, but not the specific sample).

Just looking at dimensions, we see that this "property of the material" has to have dimensions of mass/volume or a dimensionality of M/L³.  This property is called the density.  It only depends on the material and not on the specific sample.

Careful!  As you can see by how we defined it, the property of matter that we have defined as density has a numerical value that is equal to the mass of one volume unit of the substance. But that is NOT what the density is. We want to define density so it is a property of the material, not of the material and the unit we choose to measure volume in. So we define density as "mass per unit volume" NOT as "the mass of a unit volume". The difference is that our density will have dimensions of mass/volume.  This allows us to have the same density defined in any units we want.

To see what this means, consider the density of water. If we pay attention to units and think of it as a "mass per unit volume", we can write it as 1 g/cm³ or as 1000 kg/m³. While these look very different, they are in fact equal to each other! To see how this works, we note that since 1 kg = 1000 g, that dividing both sides of the equation by 1000 g, we get

1 = (1 kg)/(1000 g).

We can always multiply anything by "1" without changing the meaning of our expression. Multiplying our density by various forms of "1" we get

d = 1 g/cm³ = (1 g/cm³) x (1 kg/1000 g) x (100 cm/1 m)³
= 10⁶/10³ x (g-kg/g) x (cm³/m³-cm³) = 10³ kg/m³ = 1000 kg/m³

So by our rules of dimensions and units, these two densities are in fact the same.

For density, we extracted a factor of the volume from the mass and obtained a parameter which only depended on the material, not on the particular sample of it.  For some situations, the shape matters and we would want to extract a length factor and an area differently in order to obtain a parameter that described the material alone and not the particular sample. We'll encounter many examples of this throughout the term. Two good example of this are Young's modulus for the deformation of a block of solid material under a stretch or a squeeze, and the resistivity of a material that can carry a current.

Joe Redish 9/25/11