Guide 1-3c. Graphical Analysis for a Linear Relationship
Circumference vs. diameter of cylinders
One common type of laboratory investigation is to determine the relationship between physical variables. As an example, we'll use a familiar situation for which we expect a linear relationship to exist between the variables. Suppose we have a collection of cylinders of different diameters ranging from 1 to 4 centimeters. Our goal is to experimentally determine the relationship between the circumference and the diameter and compare the results to what we expect from mathematical theory. We'll take as the independent variable the diameter and as the dependent variable the circumference.
Here's how the measurements are made. In order to find the circumference of each cylinder, a pencil mark is placed on one end of the cylinder. Then the cylinder is rolled without slipping across a piece of paper until two marks appear on the paper. The distance between the marks is measured with a ruler. The series of sketches below illustrate the process.
In order to find the diameter of each cylinder, the ruler is placed on a table. Then the cylinder is placed on the ruler in such a position as to give the greatest distance from one side of the circular cross section to the other as shown in the sketch to the right.
Note that in order to take accurate readings from the ruler, the ends of the ruler are never used as endpoints for the distances measured. Also, lines of sight to the ruler scale are as nearly perpendicular to the scale as possible.
Data and Graph
Example data and a graph are shown below. Note the following about the format of the table and graph. You should format your graphs and tables similarly.
Theoretically, we expect the circumference to be proportional to the diameter. So it's no surprise that the graph above appears linear. Therefore, we use a straightedge and draw a best fit straight line through the data points. Then we find the slope and intercept. This is shown below. Note the following about this method of finding the slope:
slope = (C2 - C1)/(D2 - D1)
= (0.121 m - 0.006 m)/(0.0380 m - 0.0020 m)
= (0.115 m)/(0.0360 m)
= 3.19 (units divide out)
intercept = -0.001 m
The next step in determining the function that describes the relationship between the variables is to write the equation of the line. In math class, you learned that the equation of a straight line is y = mx + b. You also learned how to determine the values of the slope, m, and intercept, b. In physics class, you have to translate the variables into physically meaningful terms and use corresponding symbols to represent them. You also have to determine values of the slope and intercept and relate these to the physical situation. A matching table provides an aid to writing the equation. The matching table has 6 columns. These are described first.
The matching table for the experiment at hand is the following.
Now we can write the equation of the relationship: C = 3.19D - 0.001 m
This is an experimentally-determined relationship between circumference and diameter. However, we also know from theory that C = (pi)D, where pi is 3.14 to 3 significant figures. Therefore, we can say that we expect the slope of the experimental relationship to be pi and the intercept to be 0. In this case, experiment and theory agree on the value of the slope to within 2%. This and the small size of the intercept are easily explained by the errors inherent in the methods of measuring circumference and diameter.
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