3.5 Direct Proportions

When the data you collect yields a line that passes through the origin of the axes (0,0) we call the relationship a direct proportion. This is because for such relationships, changes in one variable result in proportional changes in the other variable. Consider another spring experiment such as the one described earlier. Suppose in this case that the spring initially is unstretched and begins to stretch as soon as any mass is added to it. The data for such an experiment might look like the following:

The mathematical analysis follows:

The mathematical model for this experiment states:

Stretch = 0.30 cm/g · mass

What is the physical significance of this statement? Notice that the mathematical analysis of a direct proportion begins with a statement that it is in fact a proportion. Any direct proportion of the form, y∝x, can be written as an equation of the form y = kx, where k represents the constant of proportionality between the variables. This is also the slope of the graph. The same equation could have been arrived at using the slope-intercept form of the equation of a line. Since the y-intercept term is essentially zero, b drops out of the equation y = kx + b, and the equation becomes y = kx. To express the model in more general terms, we replace the slope, 0.30 cm/g, with the same variable name for rate of change of stretch per mass that we used before, “w”.

S = w · m

When you are evaluating real data, you will need to decide whether or not the graph should go through the origin. Given the limitations of the experimental process, real data will rarely yield a line that goes perfectly through the origin. In the example above, the computer calculated a y-intercept of 0.01 cm ± 0.09 cm. Since the uncertainty (±0.09 cm) in determining the y-intercept exceeds the value of the y-intercept (0.01 cm) it is obviously reasonable to call the y-intercept zero. Other cases may not be so clear-cut. The first rule of order when trying to determine whether or not a direct linear relationship is indeed a direct proportion is to ask what would happen to the dependent variable if the independent variable were zero. In many cases you can reason from the physical situation being investigated whether or not the graph should logically go through the origin.