3.8 Inverse Relations
The final type of fundamental relationship that we will study is the inverse relation. An inverse relation is basically the opposite of a direct relation. In an inverse relation, as the independent variable increases the dependent variable decreases. This can take multiple forms, but the most common type that you will encounter in this physics course looks like the sketch to the right.
Consider an experiment where students make waves on a spring by shaking the spring at a certain frequency, and measuring the resulting wavelength of the waves.
The data from this experiment indicate an inverse relation between Wavelength and Frequency. As Frequency increases, Wavelength decreases. How do we determine an equation for this obviously nonlinear graph? If you said, we need to make a test plot, you are headed in the right direction. But what mathematical manipulation of the data might possibly linearize the graph? How can we make both variables change in the same direction? The answer might be to take the reciprocal of one of the variables. This will cause the variable you have manipulated to decrease if it was increasing or to increase if it was decreasing. The convention that we will follow for our test plots of inverse relations is to take the reciprocal of the independent variable. We will therefore divide one by every frequency value to make a column of one over frequency. Remember that whatever manipulation you make of the physical quantity, you must also make of the units for that physical quantity. The reciprocal of waves/s is s/wave. You will plot a new graph of Wavelength vs. 1/Frequency.
The mathematical analysis of this experiment follows:
In this experiment, since Wavelength is directly proportional to 1/Frequency, we can also say that Wavelength is inversely proportional to Frequency. Such a relation is known as an inverse proportion.
The mathematical model that describes the relationship between Wavelength and Frequency for this experiment is:
Wavelength = 19.8 m/s · 1/Frequency
As in the direct proportion, the constant slope of 19.8 m/s may be replaced with a variable name that describes the nature of the rate of change represented by the slope of the straight-line graph, Wavelength vs. 1/Frequency. The physical meaning of this graph’s slope can be determined by examining two forensic clues. The first clue involves the units of measure. Since the slope is measured in m/s and speed is measured in m/s, the slope must be the speed of something. The other clue is the numerical value of the slope. The speed is related to the value 19.8 m/s. The slope is constant, and so the value of the variable held constant in the lab is either 19.8 or some multiple of that value, say for example 1/2 of 39.6, or 19.8. In evaluating the physical meaning of the slopes of your graphs, remember to refer to (1) the units of measure and (2) the numerical value of the slope.
While the relations described on this page and the previous pages do not describe every possible physical situation that might be encountered, it does serve as the basis of a great many of them and will cover nearly every situation you are likely to encounter in an introductory physics course. Sometimes your test plot will not yield a linear relation initially, but it might suggest yet another test plot. In most situations in this course, a maximum of three graphs will allow you to linearize the data and obtain a mathematical model.