Guide 1-3a. Design and Execution of a Relationship Experiment

Variables and experimental design

One common type of laboratory investigation is to determine the relationship between physical variables. For example, consider a simple pendulum which is composed of a compact weight (bob) that is hung from a string attached at its upper end to a fixed support. Suppose the goal of the investigation is to determine which variables influence the period of the pendulum, that is, the time for the bob to execute one complete cycle over and back. Some variables whose influence one could investigate include the length of the string, the mass of the bob, and the angle from the vertical at which the string is released. The latter three variables are termed independent variables, because one selects their values in carrying out the experiment. The period is termed the dependent variable, because its value may depend on the values of the independent variables.

In order to determine how each of the independent variables may influence the period, one needs an experimental design in which only one of the independent variables is changed at a time while the others are held constant. In this way, if an influence (or lack thereof) is found on the period, one can be fairly confident that the independent variable that was changed is the variable that influenced (or didn't influence) the period. Such an experiment is called a controlled experiment. By the way, the reason we said fairly confident is because there's always the possibility in dealing with the natural world that there are variables that the experimenter has overlooked and has not controlled. For the case of the pendulum, for example, suppose you carried out the experiment in an elevator that was moving up and down between floors. You would discover some strange results and find it difficult to reach conclusions about how the independent variables affected the period. That's because the elevator accelerates and decelerates and, as it turns out, such motion influences the period of a pendulum. Of course, it's not likely that you would do this experiment in an elevator and, if you did, you would probably guess that the motion of the elevator influenced the results. That's part of being a competent scientist. However, even if you're competent, you can still overlook variables. An example might be your location on the surface of the Earth. Location, in fact, influences the period, although the effect is so small that one typically doesn't notice it. But if you were making very precise measurements, you would have to take the effect into account.

The typical way to carry out an experiment to investigate the possible influence of an independent variable on the dependent variable is to measure the value of the dependent variable for several values of the independent variable. A graph is then made of the dependent variable vs. the independent variable in order to visually represent the relationship between the variables. We illustrate next for the pendulum.

Tabulating and graphing data

Let the independent variable be the mass of the bob. Suppose we use a stopwatch to measure the period of the pendulum for different masses ranging from 0.200 to 1.000 kg while keeping the length of the pendulum constant at 0.500 m and the angle of release constant at 10.0°.  Example results are shown in the table below. A graph of Period vs. Mass is shown to the right of the data table.

Table of Data	Period vs. Mass for a Simple Pendulum
Mass (kg) Period (s) 0.200 1.41 0.400 1.47 0.600 1.44 0.800 1.51 1.000 1.49 Length = 0.500 m Angle = 10.0°

Note the following about the format of the graph above.

• The table includes all the data, including the values that are held constant.
• The independent variable is listed before the dependent variable in the table.
• The units for the table columns appear once in each column heading. This takes the place of writing units beside each number.
• The dependent variable is plotted on the vertical axis. This is conventional practice and should be followed unless there's reason to do otherwise. In this course, we'll usually follow this convention but there will be occasional exceptions.
• The axes are labeled with the names of the variables and their associated units. This is also standard practice. Never use generic x and y axis labels for a scientific graph, and always include units.
• The axes are numbered in equal increments.
• The graph is titled in the form "Dependent Variable vs. Independent Variable for Name of System or Object". This is a standard form in physics.
• The data points are clearly indicated. (The symbols indicating the data points are called point protectors.) The points are not connected with lines.

Note also that the numbering of the vertical axis does not start at 0. This isn't necessarily the best practice, and one has to be aware of this in order to avoid making hasty conclusions.  A quick glance at the graph seems to indicate a general increasing trend of period with mass. However, consider the graph of the same data below for which the vertical axis starts at 0.

Period vs. Mass for a Simple Pendulum

From this second graph, one could conclude that the mass doesn't influence the period. The variations in the values of period might be deemed small enough to be explained by error in deciding when to start and stop the stopwatch. Of course, one could be more confident of any conclusion by refining the method of measuring the period, by taking more measurement trials under the same conditions in order to see how reproducible the results are, and by extending the range of masses. We'll consider such refinements later. They're not a subject of this particular guide.

By the way, if you're surprised at the result above, you can easily test it for yourself. Hang a set of keys from a string about half a meter long and measure the period for different numbers of keys in the key ring. The independence of mass and period actually says something very fundamental about the nature of mass and corresponds with the observation that objects of different mass fall from the same location with the same acceleration.

Linearizing the data

Now let's consider the influence of the length of the string on the period. We measure the period of the pendulum for lengths ranging from 0.100 to 1.900 m while keeping the mass of the bob constant at 0.400 kg and the angle of release constant at 10.0°.  Example results are shown in the table below. A graph of Period vs. Length is shown to the right of the data table.

Table of Data	Period vs. Length for a Simple Pendulum
Length (m) Period (s) 0.100 0.70 0.500 1.47 0.900 1.97 1.400 2.50 1.900 2.88 Mass = 0.400 kg Angle = 10.0°

This time the period shows an unmistakable increasing trend with length. What is the functional form of this trend? The latter is a question that must be asked whenever one does a quantitative experiment to determine the relationship between physical variables. One can approach an answer by hypothesizing about what mathematical function the trend follows. Then one could test the hypothesis. In the above case, noting that the period would be expected to decrease toward 0 as the length did the same, a curve similar to a square root function might fit the trend of the data. A way to test this would be to re-express the independent variable as the square root of the length. If the hypothesis were correct, then a graph of period vs. length would be linear. Here are the results for the data above.

Table of Data	Period vs. Square Root of Length for a Simple Pendulum
Square Root of Length (m½) Period (s) 0.316 0.61 0.707 1.40 0.949 1.88 1.183 2.38 1.378 2.75 Mass = 0.400 kg Angle = 10.0°

Note the following about the re-expressed data and the graph:

• The name of the independent variable and its units have been changed to be consistent with the re-expression of the variable of Length to the Square Root of Length.
• The relationship between Period and the Square Root of Length appears to be linear. This is consistent with the hypothesis that the functional form of the graph of Period vs. Length is a square root function.
• The data points are no longer spaced by equal increments along the horizontal axis.

From an experimental point of view, it would make sense to collect additional data to fill in the gaps for values of Square Root of Length from 0.0 to 0.7 m^½. If one suspected from the outset that the length would increase faster than the period, the experiment could have designed so that the spacing between consecutive values increased with increasing length. This would improve the experimental design. Of course, even if one didn't suspect this, one could always repeat the experiment with a better design. This is something that a good experimental scientist does.

Let's work with the data that we have to complete this example. The next step is to draw a straight line to represent the trend of the data shown in the last graph. See the graph below.  If the line is drawn by hand rather than with a computer, one uses a straightedge to ensure linearity. The straightedge is positioned in such a way as to "split the differences" between the points. That is, the deviations of points above the line are about the same as the deviations below the line. The line isn't drawn through the origin, since that isn't a data point. Even though we expect that the period is 0 when the length is 0, we don't let that influence the way that we draw the line. We call the resulting line the line of best fit.

The fact that the points don't all lie on the same straight line is presumably due to experimental errors such as the error in starting the stopwatch mentioned previously. Another possible error is the measurement of the length. By drawing a straight line with a straightedge, however, we are expressing our conviction that the relationship between the Period and the Square Root of Length is linear.

Period vs. Square Root of Length for a Simple Pendulum

Interpreting the Results

The next step in determining the function that describes the relationship between the variables is to prepare a matching table for the fit and 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. Here's what we mean.

Step 1. Translating the variables

The variable on the vertical axis is the Period. Let's represent that with the symbol T. Then y translates into T. The variable on the horizontal axis is the Square Root of Length. Let's represent the length with the symbol L. Then x translates into L^½.

Step 2. Physical significance of the slope

The slope can be determined from the line on the graph by application of the definition of slope. Click here to see how the value is determined. The units of slope are always the units of the vertical variable divided by the units of the horizontal variable. In this case, that would be s/m^½.  Here's how we relate the slope to characteristics of the system. We have to know some theory for this. We know that a pendulum approximates simple harmonic motion. For a spring undergoing SHM, the period of oscillation is given by:

T = 2π(m/k)^0.5.

The quantity under the square root is the ratio of an inertial property (mass) to an elastic property (spring constant or springyness). For the simple pendulum, we can hypothesize that the corresponding inertial property is length (the longer the length, the greater the period) and that the corresponding elastic property is the gravitational field (the stronger the field, the smaller the period of oscillation). Thus, we can hypothesize the following form for the period of the pendulum.

T = C(L/g)^0.5,

where C represents a constant. Perhaps this constant is the same 2π as for the spring. We will hypothesize that and see if the graphical analysis bears it out. One thing for certain is that C has no units since the units of (L/g)^0.5 are seconds. Let's see how to relate C to the slope of the fit. First, let's re-express the equation above.

T = (C/g^0.5)L^0.5.

Since T was plotted vs. L^0.5 in the graph above, we can see that the slope of the fit is C/g^0.5. We call this the physical significance of the slope.

Step 3. Physical significance of the intercept

The intercept read from the graph is a little above 0. While physically we expect that the period is 0 when the length is 0, we don't force the intercept to be 0. There may be something about the experimental design or the measurements that introduces a systematic error which causes all values of the period to be a bit high in this case. One possibility is that the experimenter always started the stopwatch early and stopped it late. Another possibility is that all the measured lengths were low, because the experimenter didn't include the diameter of the bob in the length of the pendulum.

Step 4. Summarizing results in a matching table

We summarize the results above in what is called a matching table. The Math and corresponding Physics quantities are listed in two columns. The values of the fit coefficients are given beside the corresponding values expected from theory. Finally, the units are listed. The grayed-out cells don't have entries, because these are in rows for variables.

Step 5. Writing the equation of the fit

We're now ready to write the equation of the relationship between the variables. We replace y and x with the symbols representing the corresponding physical quantities, and we replace m and b with the values and units of those constants. The final result for the relationship between the period and length of a pendulum is the following.

T = (2.07 s/m^½)L^½ + 0.05 s

Note that the above is an empirical (experimental) result rather than something derived from theory.

Step 6. Calculating the experimental value of the constant

The final step is determine the value of C from the experimental results. We expect the slope of the fit to be C/g^0.5. Solving for C and substituting values gives the following.

C = (slope)g^0.5 = (2.07 s/m^0.5)(9.80 m/s²)^0.5 = 6.48

This result is within 3% of 2π.

Inductive and deductive reasoning

The process described above in which data is collected and then used to determine a relationship is called inductive reasoning or induction. The nature of this process is that it begins with specific information (in this case, data) and concludes with a general relationship that can be used to predict values that were not part of the original data set. For example, we could use the general relationship, T = (2.07 s/m^½)L^½ + 0.05 s, to design a clock that has a period of 1.00 s. We solve the equation for L and substitute 1.00 s for T.

L  = [(T - 0.05 s)/(2.07 s/m^½)]² = [1.00 s - 0.05 s)/(2.07 s/m^½)]² = 0.211 m

The units reduce to meters as expected. This process of making a specific prediction based on a general relationship is termed deductive reasoning or deduction and is the opposite of induction. Scientists use induction to discover relationships, and they use deduction to make predictions based on known relationships. These processes are part and parcel of the scientific method.

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