Three Ways to Understand Mathematical Relationships in Physics

by Professor Sam Matteson

Students in introductory physics often express alarm and even panic when they are confronted with the task of thinking quantitatively or when they are called on to make qualitative inferences based on the mathematical relationship of physical quantities. This short note is designed to help identify three ways one can use the mathematical relationships expressed by physical law or "equations."

Rote Computation

When one is given two quantities (say the mass m and the spring constant K of a simple harmonic oscillator) and required to calculate a third quantity (say the frequency f, note this not F, the force) one can use the equation that one has memorized in a rote way, as one might use a recipe or an instruction book:

f =1/(2) (K/m).

This is a mathematical sentence. It reads "the frequency of a simple harmonic oscillator is computed thus (or is equal to) 1 divided by the product of 2 and  pi (pi is approximately 3.1416), then multiplied by the square root of the ratio of the spring constant divided by the mass."

The important issue here is to know what all the symbols stand for, for example f is the frequency of the oscillation of the simple harmonic oscillator, K is the spring constant (in N/m) and m is the mass (in kg); in addition one must be sure to do the arithmetic correctly. For example one must perform the following operations on one’s calculator: K, , m, =, ,  , 2,  , p =.

Example: What is the frequency of oscillation of a simple harmonic oscillator that has a spring constant of K = 250. N/m and a mass m =0.20 kg?

f = 1/(2)(K/m)

f = 1/(2 3.1416) (( 250. N/m)/(0.20 kg))

f = 0.159235.36 hz

f= 5.629 hz

This is often called "plug and chug;" put the numbers in the right places and chug the equation engine along until the answer pops out. This can get one the answer but this is rarely enough to understand the physics. Qualitative Relationships

The equations of physics are succinctly stated relationships between physical parameters. For example, if one increases the mass of a simple harmonic oscillator (and keeps everything else the same) the frequency of oscillation must decrease. This means that the oscillator will move more slowly. On the other hand if one increases the spring constant K the frequency will increase. These relationships are inherent in the equation:

f = 1/(2 (K/m).

We say that the frequency is directly proportional to the square root of the spring constant and inversely proportional to the square root of the mass. Symbolically we write these statements this way:

f  K

f  m.

This means that no matter what other things the frequency may depend upon, it depends directly on the square root of the spring constant and it depends on the inverse of the square root of the mass. Thus, if K increases f must increase; conversely if m increases, f must decrease because one will be dividing by a larger number than before.

This is qualitative mathematical inference and is very powerful in analyzing real phenomena. But one can go even further without having to resort to a calculator.

Semiquantitative Analysis

If one understands the dependence on the important parameters one can predict how much of a change will result if one (or more) of the physical parameters are altered. For example, consider a simple harmonic oscillator whose mass is increased four times (4). Since the frequency is inversely proportional to the square root of the mass, increasing the mass four times (4x) decreases the frequency by the square root of four (4´ ) or by two (2 ). If instead the mass had been decreased by a factor of one half (1/2), let’s say, then the frequency would have increased by a factor of the square root of two ( 2 1.414), since the reciprocal of the square root of one half is 2.

In a similar way one can analyze the effect of increasing or decreasing the spring constant. Doubling the spring constant would increase the frequency by the square root of two (2). Thus, one can make quantitative predictions without actually computing the new frequency from "scratch." This is type of analysis requires one to really understand the equation and can help students gain a sense of how various physical quantities are related.