The results of many experiments are best displayed in graphical form. The independent variable is usually plotted on the abscissa (x-axis or horizontal axis), and the dependent variable is usually plotted on the ordinate (y-axis or vertical axis). You should follow a few simple rules when constructing a graph. 1. Give the graph a title so that the reader knows what the graph is about. 2. Label each axis, state the units for the variable, and be sure the scale is clear. The scale need not be the same for each axis. 3. Choose the scale so that the graph occupies most of the graph paper, but choose one that is not difficult to work with. At times, you may want the origin to be away from the corner of the page to allow space for negative values. 4. Draw a smooth curve through the plotted points. The curve need not pass through the points, but make the curve come as close to as many of the points as possible. Generally, this means that you try to miss the most points by the least amount. Do not force the graph through the end points. You will work mostly with straight-line graphs, so the process is relatively easy to do. The straight-line graph is so important for this laboratory that two examples are given; one is relatively straightforward, and the second is more challenging. Suppose you did a very simple experiment to determine the ratio of the circumference of a circle to its diameter. You probably recall from elementary mathematics that the connection between the diameter d and the circumference C is C = π d. This equation shows that a graph of C versus d is a straight line whose slope is π and whose intercept on the vertical axis is zero. This statement can be understood by comparing the equation C = π d to the standard slope-intercept form of the equation for a straight line, y = m x + b. Here m is the slope (Δy / Δx) and b is the value of y when x = 0, the y-intercept. The connection is easy to see by writing the two equations directly on top of one another as follows: y = m x + b and (1) C = π d + 0. (2) It is clear that C plays the role of y, π is m, d is x, and b = 0 in this case. Suppose that the following data are obtained when the diameter and circumference of several circular objects are measured.

Diameter: Circumference:

1.00 cm 3.20 cm

1.95 cm 6.05 cm

3.04 cm 9.60 cm

3.93 cm 12.22 cm

5.01 cm 15.80 cm

A graph of these points is shown below. Note that the straight line misses the most number of points by the least amount. The slope calculation is based on the two points on the straight line marked with open squares. Since none of the data points lies on the line, you should not use the difference between any of the data points for the slope calculations. Also, notice that for the slope calculation, two widely separated points on the line are used. This minimizes the error in reading the graph.

For this case, ΔC = (15.0 – 5.0) cm = 10.0 cm, Δd = (4.8 – 1.55) cm = 3.25 cm, and m = ΔC/Δd = 10.0 cm/3.25 cm = 3.08.

Since the accepted value of π is 3.14, the experimental value has a percent error of [(3.08 – 3.14)/3.14] × 100% = –1.9 %. The value determined experimentally is about 1.9 % less than the accepted value.

In some cases, it is not so easy to identify the appropriate variables to plot to obtain a straight line. Consider the equation v2 = vo2 + 2aΔx. (3)

This equation relates the speed v of an object to its initial speed vo, its acceleration a, and its displacement Δx. Suppose data were available relating v and Δx, and our task was to determine vo and a by plotting a straight-line graph. It is clear that a graph of v versus Δx is not a straight line. If, however, you were to graph v2 versus Δx, you would obtain a straight line with vo2 representing

the vertical axis intercept and 2a representing the slope. Again, write the two equations on top of each other to make the identifications, as shown below: y=mx+b v2 = 2a Δx + vo2 (4) (5)

In this form (note that the order of Equation (3) has been changed), the connections are clear; v2 plays the role of y, 2a is m, Δx is x, and vo2 is b. You will gain experience in this process throughout the semester; it is important that you become comfortable with the process of identifying the variables and plotting the straight-line graph.