Appendix III Graphs in the Introductory Physics Laboratory

Appendix III Graphs in the Introductory Physics Laboratory 1. Introduction One of the purposes of the introductory physics laboratory is to train the student in the presentation and analysis of experimental data. The two forms commonly used to present data are the tabular form and the graphical form. In general, the tabular form is used when experimental observations are being recorded. To summarize observations, either the tabular form or the graphical form might be employed, depending upon the purpose. If the data are presented graphically, only three significant figures can be shown on most of the graph paper commercially available. This limitation does not apply to data presented in tabular form. Thus, in some cases one may wish to both tabulate and graph his/her data. The tabular form is used when presenting several observations, which are independent of one another. For example, if one measures the dimensions of several parallelepipeds, he/she may wish to tabulate the length, height, and depth of each object. The tabular form is also useful when presenting repeated measurements of the same quantity; for example, if one measures the length of an object ten times he/she may wish to record each individual measurement. In the previous examples, no useful purpose could be served by presenting the data graphically. However, the graphical presentation of data is quite useful if one is trying to find a relationship between two physical quantities. For example, suppose one releases a ball from some height and then by some means measures its height at regular intervals of time after the instant it is released. Supposed the above experiment is repeated three times and the observations were shown in Table I. Table I Time after Height above the Floor (meters) Average Height Release (sec) Trial 1 Trial 2 Trial 3 (meters) 0.12 7.75 8.00 7.90 7.88 0.24 7.55 7.65 7.90 7.70 0.36 7.35 7.50 7.65 7.50 0.48 6.95 6.70 6.75 6.80 0.60 6.15 6.00 5.90 6.02 0.72 5.10 5.50 4.85 5.15 0.84 4.85 4.60 4.65 4.70 0.96 3.55 3.50 3.70 3.58 1.08 2.7 2.77 2.90 2.79 1.2 1.13 0.90 1.25 1.09

Height above floor (meters) Height of ball vs. Time of fall 9 8 Uncertainties are observerd deviations 7 6 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time of Fall (seconds) Figure 1

Height Height vs Time 2 experimental theoretical 9 8 7 6 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Time 2 Figure 2

It is quite difficult to deduce any quantitative relationship between the time after the ball was released and the height above the floor by considering the numbers in the table. If the average height is plotted versus time after release, as in Figure 1, one can more easily deduce that there is some relationship between the height and time. One can also obtain other information from the graph that is not easily obtained from the tabulated data. One may wish to estimate the position of the ball at some time other than one of those at which measurements were made (for example, at 0.65 seconds). The process of estimating between measured points is called interpolation. It may be desirable to estimate the time that the ball strikes the floor. Note that this is outside of the time range for which measurements were made. The process of extending the apparent relationship between two quantities to a range outside of the range in which measurements were made is called extrapolation. Graphical presentation of the data would aid in extrapolation of the measurements. The graphical presentation of data can also be used to show how the observed dependence of open physical quantity upon another compares with that predicted by theory as shown in Figure 2. 2. Preliminary Considerations A. Type of Graph Paper In order to present observations graphically, one must first decide what type of graph paper is suitable for the data. There is a wide variety of graph paper commercially available, most of which fall within one of the following general types: rectangular (also called linear and Cartesian), semi-log, log-log, polar, and projective (sometimes called triangular). The two types of paper most commonly used in the introductory physics laboratory are rectangular and semi-log paper. For rectangular graph paper, parallel lines are drawn at regular intervals in both the horizontal and vertical directions. The number of lines drawn and the spacing of the lines determine the number of significant figures to which data can be displayed; the finer the grid of lines, the greater the numbers of significant figures that can be displayed. One therefore must choose the graph paper on which the grid lines are fine enough to display the data to sufficient accuracy. For semi-log graph paper, vertical lines are drawn at regular intervals, but the horizontal lines are not spaced at regular intervals. Instead, the horizontal lines are spaced according to a logarithmic scale. Suppose, for example, that only ten horizontal lines are to be drawn. The spacing of these lines would be as follows: the first line would appear at the bottom of the page corresponding to log 1 = 0, the second line about one third of the way up the paper since log 2 = 0.3, the third line about halfway up the page since log 3 = 0.477, etc. It is also possible to place horizontal lines between the ten major lines; for example, a horizontal line can be drawn about four-tenths of the way up the page to represent 2.5 since log 5 = 0.4. Semi-log graph paper upon which there are ten major horizontal lines is called one-cycle semi-log paper. It is possible, of course, that ten major lines (i.e. one cycle) might fill only

half of the page. Another cycle might fill the other half of the page, and, in this case, one obtains two-cycle semi semi-log paper. Three-, four-, and five-cycle semi-log paper is obtained in a similar manner. Semi-log paper is useful if one wishes to display data in which one of the quantities varies by several orders of magnitude (factors of ten) and the other varies by an order of magnitude or less. It is especially useful in displaying data in which one of the quantities varies exponentially, for a straight-line result for such a relationship. For log-log graph paper, both the horizontal and vertical lines are spaced according to a logarithmic scale. For polar graph paper, the grid lines are formed by concentric circles of regularly increasing radii and by straight lines which are spaced at even angular increments and which diverge radially from the center of the circles. The value of some physical quantities depends upon the values of two other different physical quantities. Such a quantity might be displayed, not as a single curve, but as a surface. Projective (or triangular) graph paper is useful for displaying such quantities. B. The Independent and Dependent Quantity To show graphically the dependence of one physical quantity upon another, one must first decide which of the two quantities is the independent quantity and which is the dependent quantity. The independent physical quantity is the one that is controlled, that is, the one at values of which the experimenter chooses (usually at convenient intervals) to make measurements of the other quantity. In Table I the height of the ball was measured every 0.12 seconds. In that case, the experimenter chose the times 0.12, 0.24, 0.36, etc. to measure the height of the ball. The time, in this example, was controlled and was, therefore, the independent quantity. Of course, one could devise another experiment in which he/she would measure the timed elapsed when the ball passed fixed positions (for example, 7.00, 6.00 and 5.00 meters, etc.). In this case, the height would be the independent quantity and the time the dependent quantity. In general, controllable quantities depend upon the type of equipment available and the experimenter s personal choice of how he/she wishes to perform the experiment. It is the custom in scientific work to let the independent quantity be represented by the horizontal coordinate axis (the abscissa) and the dependent quantity is represented by the vertical coordinate axis (the ordinate). The axes may be interchanged if it is more convenient or if the results are to be compared with some specific algebraic form. C. The Coordinate Scale Once the type of graph paper has been chosen and the dependent and independent quantities identified, one must choose a coordinate scale, i.e., one must decide what each major line of the graph paper is to represent. For example, suppose one wishes to plot the data given in Table I on rectangular paper which has 7 major horizontal divisions and 9 major vertical divisions (see Figure 1). Height will be the dependent quantity and should be represented along the vertical coordinate axis. The independent quantity is the time of fall

and should be represented by the horizontal coordinate axis. The time of fall extends from 0 sec to 1.2 sec. One must decide what fraction of a second each of the 7 major horizontal divisions will represent. The heights observed vary from about 8 meters to about 0 meters. The next decision to be made is what fraction of a meter each of the 9 major vertical divisions will represent. In making these decisions, two things should be kept in mind. The first consideration is that the graph should occupy as much of the paper as is practical. The reason for this is simply that one can present data more accurately if more divisions are available for the plot. Suppose that only 4 major division are used to represent the heights tabulated in Table I. The accuracy to which a height could be read from the graph would be only one-half of the accuracy, which could be read if 8 divisions had been used. The second consideration is that the major divisions should be assigned values that allow easy interpolation between them. If, in the example of the freely falling ball, the major horizontal divisions were assigned the values 0.12 sec, 0.24 sec, 0.36 sec, etc., one would have difficulty in reading a value of 0.15 sec from the graph. If, on the other hand, each major horizontal division is assigned the values 0.1, 0.2, 0.3, etc., interpolation is greatly facilitated. Let us now return to the problem of how to choose the coordinate scale if we wish to present the data in Table I on a piece of graph paper with 7 major horizontal divisions and 9 major vertical divisions. Since the heights in Table I vary from 0 meters to 8 meters, one might let 1 major vertical divisions represent one meter in order to fill as much of the page as possible. Note that 8 of the 9 major vertical divisions would be used. By assigning 0.2 seconds to each horizontal major division, 6 of the 7 major horizontal divisions are used. The graph will fill most of the page and, interpolations between major divisions are easily accomplished. (See Figure 1.) In assigning values to each major division, scientific notation should be used. For example, suppose a set of data extends over the time range from 0.001 sec. to 0.01 sec. in 0.001 sec intervals (i.e. 0.001, 0.002, 0.003, etc.). One might wish to assign the value 0.001, 0.002, 0.003, etc., to the major divisions of the coordinate axis representing time. Instead of labeling the major divisions 0.001, 0.002, 0.003, etc., one should label the divisions 1, 2, 3, etc., and in labeling the coordinate axis express the units as 10-3 seconds. (See later section on labeling the graph.) Once the coordinate scales have been chosen and values assigned to each major horizontal and vertical division, one is prepared to plot the graph.

3. Plotting the Graph A. Use of Circles of Other Identifying Figures The data that we plot usually consist of two sets of numbers: one set for the independent quantity and one set for the dependent quantity. For each value of the independent quantity there is (usually) only one value of the dependent quantity. Corresponding to each pair of numbers, one places a dot on the graph paper at the appropriate grid point. It is desirable to draw a small circle about each point in order to make the point easily distinguishable. Sometime more than one set of data is presented on the same graph. In this case the data points from each set are enclosed by different geometric figures. Circles might enclose the points of one set of data, squares might enclose the points of second set, and triangles might enclose the points of a third set. By this means, the different sets can be easily identified. B. Experimental Uncertainty in Each Point All experimental measurements involve some uncertainty. It is desirable to indicate the uncertainty at each point on a graph. This can be done by drawing a solid vertical line at each point, such that the top of the line represents the positive uncertainty and the bottom represents the negative uncertainty. There are three kinds of uncertainties commonly displayed graphically. These are, (1) the standard deviation of the measurements used to compute the average (see section on accuracy of measurement), (2) the standard deviation of the average value, and (3) the maximum and minimum deviations experimentally observed. Whenever uncertainties are displayed on a graph, it is necessary that the kind of uncertainty be clearly identified (Figure 1). Of the three kinds of uncertainties mentioned above, only the experimentally observed deviations are considered in the introductory physics laboratory. Three trails were made in the freely falling ball example (Table I). At any of the times that the observations of height were made, the difference between the maximum and minimum heights observed represents a known experimental uncertainty. Hence, if the average height is plotted versus the time of fall, the observed experimental deviation at each point can be represented by a vertical line, the top of which is terminated at the minimum height observed at that time (Figure 1). C. Smooth Curves Once the points have been plotted on the graph, one must draw a curve that best represents the relationship between the dependent quantity and independent quantity. This is not a simple task and requires considerable judgment on the part of the experimenter. First, he/she must decide whether the data are better represented by a straight line, a smooth curve or a very irregular curve connecting each point. One s judgment in this matter is usually guided by his/her knowledge of the theory underlying the experiment and the uncertainties in the measurements. An irregular curve connecting each point indicates that the mathematical relationship between the two quantities is very complicated and further suggests that the known error in each measurement is small. In the introductory physics laboratory, such

relationships are seldom encountered. Thus, the irregular curve connecting each point is generally unacceptable in that laboratory, and one must decide only whether a straight line or a smooth curve should be used. Several different criteria may be used to obtain the best fit. The best curve may be the one for which the sum of the differences between the measured points and the curve is a minimum. Another criterion commonly used is to minimize the sum of the squares of the difference between the curve and the measured value. Considerable care should be taken in drawing the curve through the data. The measurements are but samples of the truth of the relationship between the variables. The curve you draw through the data is your professional judgment of what you believe this truth to be. 4. Labeling the Graph A. Labeling the Coordinate Axes Each graph should be labeled neatly and properly. Printed parallel to each of the coordinate axes is the name of the physical quantity and in parentheses the units of the quantity represented by the axis. In the freely falling ball example, the words - - Time (seconds) -- are printed parallel to the horizontal axis and the words -- Height above Floor (meters) -- are printed parallel to the vertical axis. (See Figure 1.) If the scientific notation has been used in assigning values to the major divisions of the coordinate axes, this should be indicated in the coordinate axis label. For example, if the time is measured in milliseconds, the coordinate axis label should read as follows: Time (10-3 seconds), or Time (ms). B. Legends If more than one set of data is plotted on a graph, or if the theoretical relationship between the quantities is shown on a graph, each set of data must be clearly labeled. This is done in one of the following ways: If the sets of data are spread apart, the curve for each set of data is labeled wherever space permits. If the sets of data are very close together and identifying geometrical figures are used to indicate the set to which data points belong, each identifying figure is defined somewhere on the graph (usually in the upper right-hand corner). Both of these methods are shown in Figure 2 for illustrative purposes only. Only one of the methods should be used for any particular graph. In placing legends on a graph there is no generally accepted convention that is followed. However, it is general rule that if more than one kind of information is placed on the graph, each kind must be clearly identified.

C. Title of Graph Each graph should have a title printer at the top of the page that describes in as few words as possible what is being presented. 5. Graphical Analysis of Data In the introductory physics laboratory, the student is not usually required to determine the formal mathematical functional relationship between the dependent and the independent quantities. The procedure for doing this requires an understanding of some advance mathematics and is beyond the scope of this course. Instead, the student is usually given the relation between the quantities and then is required to perform one or all of the following operations: (1) to determine certain constants in the relationship from his/her data; (2) to interpolate his/her data; or (3) to extrapolate his/her data. The data taken in the experiments performed in the introductory physics laboratory will usually plat as (1) a straight line, (2) a parabola (or half a parabola) symmetric about an axis, (3) an exponential function, or (4) a trigonometric function. In the discussion to follow, these functions are described. A. The Straight Line A set of data will fit a straight line on a rectangular graph if the two quantities (for example x and y) are related by an equation of the form y = mx + b [1] In this equation m represents the slope of the line and b represents the y-intercept of the line when x = 0. Both m and b are constants for a given straight line. To obtain the slope of a straight line drawn from the experimental data, one divides the change in the ordinate y by the corresponding change in the abscissa (x). In order to minimize the error in determining the slope, one should utilize as much of the line as possible. Thus, in figure below, the uncertainty in the slop determined from the change in the ordinate and abscissa shown in part (a) will be greater than the uncertainty in the slope determined from the corresponding changes shown in part (b). Often, we must extrapolate the straight line back to the point that it intercepts the ordinate axis to obtain the intercept of the line. (See, for example, Figure 2.) It is important to note that if the quantities x and y have units, the intercept and the slope will in general also have units. y y y y x x x x a. poor b. bett er It is also possible to determine the equation of the straight line, which best represents, a

set of data by analytical methods. Suppose that one had n pairs of data points (x i, y i ). The slope and the intercept of the straight line which best represents this data is found by the method of least squares to be given by the equations: and b = i i m = n x y x y i i i i [2] 2 2 n x x 2 n xi xi y x x x y i i i i i 2 2 [3] B. The parabola Symmetric About the Ordinate Axis Experimental data will fit a parabola symmetric about the ordinate axis on rectangular graph paper if the quantities are related by an equation of the form y = ax 2 + b [4] In equation [4] a, and b are constants. The constant b is easily obtained if the plot intersects the ordinate axis (b is the value of y at x = 0). The constant a, is not easily obtained if one plots y versus x on rectangular paper. Actually, if measurements are made for positive values of x only, it is quite difficult to decide whether the quantities are even related by equation [4]. (See, for example, Figure 1.) One method of deciding whether the quantities are related in accordance with equation [4] is to plot y versus x 2 on a rectangular graph paper. If the quantities are truly related as indicated by equation [4], a straight line will result when y is plotted versus x 2. (See Figure 2.) Also, the determination of the constant a, and b are greatly facilitated when quantities related by equation [4] are plotted in this manner. The constant b will be the intercept of the resulting straight line. C. The Exponential Function A set of data will fit an exponential curve on rectangular paper if the variables are related by an equation of the form y = y o e x [5] where y o and are constants and the symbol e stands for the base of the natural logarithm (e = 2.718...). y 0 is the value of the y when x = 0. To determine one must determine by how many units the quantity x must increase for y to increase by a factor of e. Let x be the number of units of x it takes to increase y by the factor e. Then the constant is the reciprocal of x, i.e., 1 x [6]

In general, may be more accurately determined if the data are plotted on semi-log paper. The following equation is obtained by taking the natural logarithm of both sides of equation [5]. ln y = ln yo + x = constant + x [7] Thus, data that fits an exponential function on rectangular paper will appear as a straight line on semi-log paper. Since it is easier to draw a straight line through a set of data points than to draw an exponential curve through a set of data points, generally better accuracy can be obtained by determining from a plot on semi-log paper. It should be noted that the method used to determine from a semi-log plot is exactly the same as that used to determine from a plot on Cartesian paper, namely by taking the reciprocal of x. D. Trigonometric Functions There are six trigonometric functions -- the sine, the cosine, the tangent, the cotangent, the secant, and the cosecant. The sine and the cosine are generally the only ones usually encountered in the introductory physics laboratory. The following discussion will concern the sine function. Experimental data will fit a sine function on rectangular paper if the quantities plotted are related by an equation of the form y = yo sin 2 x - a [8] where y o,, and a are all constants. The constant y o is called the amplitude of the sine function. If it is assumed that, the quantities plotted are related by equation [8], y o is, then, the maximum value attained by y. The constant is the wavelength of the sine function. This constant is determined by finding the number of units of x it takes for the quantity y to go through one complete cycle. The constant a, is called the phase difference of the sine function. If y is zero when x is zero, then a, is also zero. However, if y does not go to zero until x has reached some value, say, then a =. Stated differently, a, is the number of units of x between the origin (x = 0) and the point where y first goes to zero.