Finding and Analyzing a Sinusoidal Function

The sinusoid is a very common function and can be gathered in a number of ways. Two very simple ways are to use the program GENERIC to measure sound generated from a tuning fork with a microphone, a ranger to measure the distance from a pendulum swinging back and forth in front of the motion detector, or using a motion detector to measure the distance to a mass hanging from a spring. Any graph of harmonic motion will yield good results. While students may use the sinusoidal regression function on the calculators, manually finding these values promotes a great deal of learning about transformations and what the actual values mean. The programs used in this lab are available as generic.83p

Collecting the Data

For this lab, we will hang a simple pendulum so that at it's closest point, the pendulum is at least 0.5 meters from the motion detector. Smaller pendulums work better than larger ones as the object may move up and down and effect the motion detectors ability to measure the distance. As you will note in the image, it is quite simple to cobble up something which will work. In this case, tape is used to hang a ball from a cabinet and allowed to swing back and forth.

Using the GENERIC, 99 points were collected every 0.05 seconds. The distance is recorded in meters. The graph of the collected data is shown below.

Analyzing the Data

To manually determine the equation of the sinusoid, we will attempt to fit the data to the model y = k + a sin(d(x-h)). To find each of the values a, d, h & k, one needs to identify one complete period of the graph. In this case, the period between the two red lines in the image below will be used.

Graph 1

Since the sine function normally has a midline at the x-axis, we need to determine the midline of the current data. Using the trace key, identify the minimum point of the data and a maximum point of the data. By averaging the y-value of these two points, the vertical shift (k) value can be determined. The two values are shown in Graph 2 and Graph 3.

Graph 2

Graph 3

The vertical shift k = (.502909 + .881431)/2 = .69217. This value may be stored into K. By entering the graph Y=K or Y=.69217 into the function grapher, the midline is shown on Graph 4 (without the red line).

Graph 4

Next, we will find the amplitude or vertical stretch (a) value. The vertical stretch is the distance from the midline to the peak as show in red on Graph 4 above. Graph 3 demonstrates that the y-value of the peak is .881431 and we calculated the midline to be at y = .69217. Subtracting to find the amplitude "a", we get a = .881431 - .69217 = .189261. Store this value to "A" on the calculator.

The horizontal stretch factor "d" is the number of periods of the graph which occur in the natural period of the function. Normally, 1 period occurs every 2pi radians. Setting up a proportion, one can show that d = 2pi divided by the period of the function. Since we decided to use the period running from the first minimum value to the second minimum value, we must determine the horizontal distance between these two points. By tracing to the second minimum value (Graph 5), one can find the x-value at this point, subtract it from the x-value of the previous minimum (shown in Graph 2) to determine the period of this function.

Graph 5

For this data, period = 2.44855 - .349797 = 2.098753. Since "d" equals 2pi divided by the period, find d and store it in location D on the calculator. D = 2pi/2.098753 = 2.99377

Finally, determine the horizontal shift. The graph of sine normally passes through the origin and increases as one moves to the right. Hence, find a point where the graph crosses the midline and is increasing as one moves from left to right. Trace to this point and determine the approximate x-value. One may need to average or "guess" if two values are close. Store this value in "H". Note Graph 6 below.

Graph 6

The midline occurred at K = .69. The point shown in Graph 6 is the closest value of y to .69 but it is slightly larger. Since H is the horizontal shift we must choose an x-value to shift the graph horizontally. After some discussion, we decided to use a value slightly less than x=.949. Set H = .93 and store it into the calculator.

Now that all of the transformations have been determined and stored, enter the function Y = K+Asin(D(X-H)) into the calculators function grapher. The resulting graph of both the stat plot and the function graph is shown in Graph 7.

Graph 7

Note that the portion of the curve which we chose fits quite well. To help students understand what each of the transformations does, it is possible to try and adjust the function to fit the entire curve better. The red arrow points to the function graph during the second period. Note that the graph is horizontally too far to the right. In other words, we are not going to get enough periods in the original period which means we need to increase the value of D. Allow students to guess and check to find a better value for "D". Note that some students may decide to shift the equation horizontally by changing H. In this case that may improve the match in the second period, but it deteriorate the match in the first period. After a little guessing and checking, we settled on a value of 3.15 for D. The image in Graph 8 is the stat plot of the data with the function y = .69 + .19 sin (3.15(X-.93)) graphed in the same view. Note that it is very difficult to distinguish the two graphs.

Graph 8

Interpreting the Model

Interpreting the data is as important as finding the model. Here, the value of k is at the midline of the motion. Since distance was measured in meters, the pendulum would be 0.69 meters from the motion detector if hanging straight down. The amplitude is the distance the pendulum moved to the left or right. Hence, the parabola moved 0.19 meters to either side of the vertical position. Finally, it took 2pi / D = 2pi / 3.15 = 1.99 seconds to travel through one complete cycle of the pendulum. The horizontal shift is only significant in that the individual triggering the collection of data ultimately determines the horizontal shift of the graph.

For another application of this technique, check out graphing the reciprocal trig functions of secant and cosecant.

If you are an educator or interested in this activity and wish to have more information, send an e-mail to jwelker@lps.org.