Dimensions

Natural Scales

In mathematizing a physical problem or situation, we typically have a number of dimensioned parameters involved: a mass, a spring constant, the graviational field, .... What we want from our analysis of the situation is typically some other dimensioned parameter: a distance, a velocity, an energy,.... We expect to do some mathematical manipulations and come up with, hopefully, some analytic expression for the answer. Since the goal of any problem is to express an unknown quantity in terms of known ones, our generic solution will express an unknown dimensioned variable or parameter in terms of a set of dimensioned parameters that are presumed known.

In some cases, we have a limited set of dimensioned parameters. There may turn out to be only one combination of our known parameters that will yield the dimension of our unknown parameter. In this case, we know exactly what our answer is going to turn out to look like -- the appropriately dimensioned combination of our known parameters times some unknown dimensionless constant. Interestingly enough, the dimensionless constant typically turns out to be on the order of 1 (though we sometimes get 2π or 1/2π, depending on how we choose to describe our known parameters). This can give us a plausible estimate for the scale of our unknown up to an order of magnitude. An example is given in our section on Dimensional Analysis.

When we combine the dimensioned parameters of our problem to create a mass (M), length (L), or time (T) we refer to this as a natural scale for the problem. That is, it is a scale that can be created from the problem parameters. These natural scales have a number of useful characteristics to help us think about our problem and how to solve it.

If we can create a mass, M_N, a length L_N, and a time T_N, then we can also create a natural velocity, V_N = L_N/T_N, a natural acceleration, a natural energy, etc. The solutions for these quantities can then be expected to be proportional to our natural quantities times some dimensionless number.

For an example, see the solution to our problem of analyzing the falling of a body in a viscous medium.

Another way of looking at this is to observe that any dimensioned parameter is constructed as a combination of a number of distinct measurements. Thus a velocity may be constructed from a distance measurement and a time measurement. An energy may be constructed from a measurement of a mass, a distance, and two times. When we have a set of parameters, they typically are not simply basic measurements of mass, length, and time. Constructing the "natural scales" gives a set of measurements of fundamental (operationally defined) quantities, which, when combined appropriately, will yield the given parameters of the system. (I am grateful to Adam Franklin for this insight.)