3.4 Linear Relationships with a y-intercept

When the data you collect yields a linear graph with a non-zero y-intercept, you will determine the mathematical equation that describes the relationship between the variables with the slope-intercept form of the equation of a line. Consider the experiment shown here in which the experimenter tests the effect of adding various masses to a spring on the amount that the spring stretches.

A proper mathematical analysis of this graph would include the following:

The result of this experiment, then, is a mathematical equation that models the behavior of the spring:

Stretch = 0.30 cm/g · mass + 3.2 cm

With this mathematical model we know many characteristics of the spring and can predict its behavior without actually further testing the spring. In models of this type, there is physical significance associated with each value in the equation. For instance, the slope of this graph, 0.30 cm/g, tells us that the spring will stretch 0.30 centimeters for each gram of mass that is added to it. We might call this slope the "wimpiness" of the spring, since if the slope is high it means that the spring stretches a lot when a relatively small mass is placed on it and a low value for the slope means that it takes a lot of mass to get a little stretch. The “wimpiness” is the rate of change of stretch per unit of mass and remains constant for a given spring. The rate can be represented with a variable that describes it. We used “S” to represent Stretch, “m” to represent mass. The rate of changing stretch per mass is constant, so we will call the constant slope “w” for “wimpiness”. Now the mathematical equation may be expressed in more general terms as

Stretch = wimpiness · mass + 3.2 cm

The y-intercept of 3.2 cm tells us that the spring was already stretched 3.2 cm when the experimenter started adding mass to the spring. The intercept represents the initial stretch of the spring and can be represented in more general terms as the initial stretch by subscripting the variable “S” with a “0”, “S0”, to indicate the stretch at the beginning of the experiment. Now the equation may be expressed in more general terms as

S = w · m + S₀

With this mathematical model, we can determine the stretch of the spring for any value of mass by simply substituting the mass value and initial stretch into the equation. How far would the spring be stretched if 57.2 g of mass were added to a spring initially stretched 1.0 cm? Mathematical models are powerful tools in the study of science and we will use those that we develop experimentally as the basis of many of our studies in physics.