CHAPTER 3

Analysis

After completing the measurements develop the habit of taking a few moments to examine your Data Table to see if you notice any patterns, trends or correlations in the data. What do you notice in the Table above? Note that both the variable "pull" and the variable "stretch" varied in this experiment. Which variable did you control? Which variable resulted from the effect of changing the other variable? What trends do you notice in the two variables when compared? The stronger the pull on the spring the longer it stretches. Sometimes results are intuitively obvious as in this case, but sometimes you will notice results that seem to be counter your intuition. This happens frequently in science. A better way to display correlations or trends in experimental data is to graph two variables against each other. What do we find when this is done? Recall that the independent variable in this experiment (the one you controlled) was the load on the spring that caused the spring to extend. Thus the dependent variable is the amount that the spring stretched. Mathematicians generally plot the independent variable on the horizontal axis and the dependent variable on the vertical axis of graphs. However, scientists relax this convention, and in the case of this experiment, usually "pull" is plotted on the vertical axis and "stretch" on the horizontal axis. You should also always try to select scales

for your graphs sufficiently large that you can read the axes to the most significant figure you measured. In this case you could read your meter stick to the nearest cm and the Spring Scale to the nearest 0.05 Newton. Make sure that you can read your graph scales to these accuracies. It is also a good habit to select scales for your graphs so that any correlated data fall on a line which makes roughly a 45° angle with the graph axes. Why do you think this is a good way to plot graphs?

The horizontal and vertical bars through the data points on the graph represent the errors in both coordinates for each data point. The lengths of the error bars are determined by the measurement errors (discussed in Appendix B) given in the sample Data Table above. The error bars are plotted at the scale of the graph. It is important to plot error bars on your graphs, so that you know at a glance the inherent accuracy of your experimental data. Note that these are random and unavoidable errors in your data. (Be sure to read Appendix B so that you understand the difference between an error of a single measurement and the error of the mean of several measurements.) Why is it important to know your measuring errors when you are trying to understand the meaning of your graphs?

Now that you have summarized your measurements in a graph and determined your measurement errors, you can analyze the data. Examine your graph. Do the data points show any trends or patterns? What shape line might best be drawn through the data? In assessing your graph, be sure to take account of the error bars. You must use your judgment to decide whether a straight line through the points will come sufficiently close to most of the data points that a straight line fit is justified. If a straight line through the points misses the error bars of most of the points by distances on the graph of three or more times the size of an error bar, then a straight line fit may not be appropriate. Instead you may need to consider some other mathematical function such as a quadratic curve or an exponential curve. Because there are many different kinds of mathematical functions that give rise to curved lines, you should think carefully about what kind of function you wish to draw through your data. Sometimes known physical laws provide a guide to the kind of function you might try to fit to your experimental data.

What mathematical function do you think would best fit the data in this experiment? In science we usually seek the simplest mathematical relation that is consistent with the experimental data.

After examination of the graph, we decide that a straight line will fit the data quite well. But how should we draw the line through the data? We judge from the linear trend of the graphed points that there must be a linear relation between the pull exerted on the Slinky and the amount it extends in response to that pull. But what is the best method to draw a straight line through the points in the graph?

Recall the general expression for a straight line, y=a+bx. Here let us the straight line expression for our graph, F = a + kx where x is the amount that the spring extends, a and k are constants and F is the "strength of the pull" or force exerted by the coins hanging from the Slinky suspended from the ceiling. Since you have no a priori knowledge about the Slinky's characteristics, you do not know the values of a and k. How can you determine the values of a and k from your graph?

What is the best way to draw or to fit an equation of the form, F = a +kx, to your data in the graph? Finding the best line to fit your data is one of the ways to "model your data" because both the straight line you will draw on your graph and the equation for that line represent a model of how much the Slinky stretches when you suspend different weights from it. The model is thus a generalized representation of the experimental data. Sometimes the model incorporates known physical laws or theory, and sometimes the model does not. In this case have we used any known physics laws or theories yet?

The importance of fitting a mathematical expression to the experimental data, is that you will have an expression that will allow you to predict how much the Slinky will stretch when you apply different "pulls" to it. Thus modeling the data with a mathematical expression provides a powerful tool that allows us to predict what the Slinky will do under many kinds of different circumstances. This ability to predict is a powerful tool in science.

Modeling your data is a very important step in the process of doing science, because you are interpreting your data. Someone else might not agree, for example, that a straight line is best to fit to the data. Perhaps a curved line would fit just as well. In this case, is there any doubt in your mind about what kind of mathematical function should be fit to the data? Note that in science we never just connect the points in a graph. Why do we not do this?

If you decided that a straight line of the form F=a +kx would be the best mathematical function to fit to your data, then you next should draw the best straight line possible through the data, and determine a and b from the line drawn on your graph. When you draw the line, use a clear plastic ruler so that you can minimize the deviations of the data points from the straight line. Although if care is taken in drawing the line on your graph, you can achieve a very accurate straight line fit to your data points, there is another way to determine the best straight line that fits your data points.

The graph on the left above represents an idealized experiment where the data have no errors. The experimenter made readings of the ruler and the Spring Scale with perfect accuracy, or no errors. This is of course impossible, because all experimental data contain errors. But if such an ideal experiment were ever performed, and the two variables, stretch and pull, were related in a linear way, that is by a straight line relationship, then the graph of pull vs. stretch would look like the graph on the left. The real experimental data are plotted in the graph on the right. Note the error bars associated with each data point in this case. Also note the way the points scatter about the line that has been drawn through the data. Is there a mathematical way to determine the slope, k, and the intercept, a, of the line in the graph at the right, F=a+kx?

For every coin that you loaded on the end of the Slinky, there is a Slinky extension value which would fall on an ideal straight line represented by the graph on the left above. The mathematical way to determine the values of a and k from your real experimental data values is to minimize the sums of the squares of deviations of the experimentally measured spring extensions from the value determined by that ideal straight line. If your experimental data had no errors. That is, if you could have made your measurements with perfect accuracy, then the measurements would have fallen exactly on the straight line in the left graph, and the values of a and x would be determined by any combination of two of your "errorless data" points. By minimizing the sums of the squares of the deviations of your real data points in the right hand graph from the ideal straight line, we can calculate the best estimates for the constants a and k. You will not be asked to do that here because, if done with care, drawing a line through the points on your graph produces essentially the same results. However, when you use the computer program Graphical Analysis in the laboratory to fit a straight line to your data, the software uses this very useful statistical technique which scientists frequently use to minimize the sums of the squares of the deviations from the "best fit" line or curve to the data. Hence we call this graph fitting method the Least Squares Method. Also when you fit a curve to your data on the TI-83 graphing calculators, or using the computer software program called, Graphical Analysis, you are using the Least Squares Method to fit a curve to your data. Fitting curves to your experimental data is one to model your data.

Let us proceed with a graphical fit of a straight line to your data, by carefully drawing a straight line through your data points. Can you then estimate the y intercept and the slope of your line? The slope should be determined from the line you drew through the data points. Why should you not use the actual data points in your graph to calculate the slope and the intercept? Recall that the slope is the ratio of the difference between any two points on the vertical axis divided by the corresponding difference between two points on the horizontal axis. The y-intercept is the value of P when x = 0 (see figure below).

The diagram illustrates how the values a and k in the expression F= a + kx can be determined from the line you draw in your graph. After drawing a line through your data points, your graph may look like the example below. What value did you find for the slope, k?. Be sure to give the units for the slope. How can you determine them? In science it is imperative to include units in your results, because pure numbers have no scientific meaning. Likewise a measurement without units or an associated error, has no meaning in science because one does not know how to interpret a measurement unless the error and the units of measurement are known. What value did you find for the y-intercept, a?.

Now that you have a result for your experiment, you should pause a few moments to reflect on the model you have just fit to your data. What does the slope mean? What if the slope were larger? smaller? What would this tell you about the Slinky? What do you predict the slope of the pull vs. stretch graph for the spring in the Spring Scale would be, greater or smaller, than the Slinky slope? What is the meaning of the y-intercept value? Is this number among those you measured in your Data Table? What does the straight line model for the relation between the extension and the Slinky and the size of the "load" tell you? How does is the pull on the Slinky related to the amount it extends? Is the pull inversely or directly proportional to the amount the Slinky stretches? What if the trend in the data points had indicated a curved relation, like a parabola? After contemplating answers to the above questions, you are now ready to make a statement or two about the model you fit to your data, and conclusions you can draw about the characteristics of the Slinky.

What kind of spring is the Slinky Jr.? What kind of spring is in the Spring Scale? Both of these springs are called extension springs, and you have just discovered a very important property universal to all extension springs. If you had performed the above experiment on a statistical sample of springs, say 50-60 of them, you would find the same result within your experimental errors. If your result were to be generalized as true for all springs, how would you state your conclusions?________.

Finally, you should take time when writing your Lab Reports to make a comment on how the experiment might be improved. After performing an experiment, a scientist usually can think of ways that the experiment could have been done slightly better. For example, the weights available may not have stretched the Slinky over its entire range. Therefore you really did not measure the overall characteristics of the Slinky. Instead your data and therefore your results apply only to the extension range you observed. Did repeated stretching of the Slinky introduce a permanent change in its "unloaded" length? In other words did the measurements affect your results? Frequently they do in scientific experiments. How could you determine whether your results are representative of all Slinky Jr.'s?

Summary: In doing experiments remember that the time available is limited. You should plan your experiment carefully and work efficiently, but not so fast as to make careless errors. Keep a careful record of what you do in your Lab Book. A scientist may actually spend months or years planning and doing a single experiment.

Be sure to record all experimental data and relevant details in your Lab Book. Date all Lab Book entries. Include informal comments about the experiment as you go. Make careful measurements. Repeat each measurement at least twice to guard against "human" errors. Avoid doing arithmetic in your head which inevitably does lead to "human" (i.e. avoidable) error. When measurements are made, always create a summary Data Table which lists the averages of your measurements, including errors associated with each measurement, and units for each column in your table. As you make your measurements be sure to try to minimize errors as much as possible. Also never discard a measurement just because it is different from the others. It might be the only correct point! Or the discordant point might be a clue that something you had not anticipated in happening in your experiment. Be alert for such events.

Unlike a student laboratory, generally, no scientific experiment is only performed once. Usually the experiment is repeated several times, and sometimes various variables are held constant while others are varied to see how the experiment is affected. Repeating the experiment is important, as you can judge the reproducibility of your results. Any well-designed experiment should produce reproducible results. If you do not get the same the results when repeating an experiment, what might this mean?

Basic Method of Doing Science
The basic procedure for doing a scientific experiment can be summarized by the following diagram. Note the upward arrow indicates that you may have to perform several related experiments to test alternative hypotheses or to refine your model. When do you decide that you have a satisfactory explanation or model of a phenomenon?

Newton's Three Laws of Motion
Galileo performed many experiments on motion. In fact he discovered the First Law of Motion before Newton rephrased this important law. Galileo called his version the "law of inertia", and stated that an object remains at rest or in motion in a straight line until acted upon by a force.

Newton formulated the Second Law relating force (F), acceleration (a) and the mass (m) of any object. The law can be stated most simply as F = ma. This simple expression has a profound meaning in that it relates three fundamental concepts, force, mass and acceleration related, and defines the "inertial mass" of any object as m = F/a. The fundamental significance of the Second Law is that a force is required to change the motion of an object. Any object with changing motion is undergoing an acceleration (or deceleration). Newton discovered that a force is required to accelerate any object.

Newton's Third Law of Motion is also related to forces and motion. We experience effects of Newton's Third Law all the time, every day whenever we push, pull, lift, move or otherwise alter the position of anything. Every time a force is applied to anything, there is always an equal force in the opposite direction applied in response by the object. Sometimes Newton's Third Law is stated: For every action there is an equal and opposite reaction. What happens when you engage in a game of pool or billiards? Can you identify the "active" and "reactive" forces?

Personal Lab 3: What Factors Determine the Period of a Pendulum?

[Lab 3]

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