Graph Plotting

Since you will be required to express many of your results in the form of graphs, the following suggestions will be of use.

1. The general idea is the same as that used in graphical methods in algebra, with the same conventions in regard to signs and location of points. If the values of the two quantities to be plotted are not already arranged in columns, time will be saved by so arranging them before attempting to plot points on the graph.

2. A scale of units for each axis should in general be chosen so that the final curve will practically (but not necessarily) fill the page, though the unit chosen must be one which can readily be applied. Neglect of this caution causes much waste of time and also defeats the purpose of the graph.

   A common difficulty in this respect arises from an attempt to use a scale such as ``one square for each three units" on decimal co-ordinate graph paper. A convenient rule for decimal co-ordinate paper is to let one square equal one, two or five units or any of these three numbers multiplied by a power of ten (e.g. 0.1, 0.2, 0.5, or 10, 20, 50).

3. The prominent divisions should be plainly marked on the two axes. Do not mark every square -- too many numbers on the axes generally make the graph confusing. The values of the individual readings should not appear on the axes.

4. Each axis should bear a label which includes the unit used.

5. The separate points should be accurately located and marked in such a way as to be distinguishable after the curve is drawn. An encircled dot is especially convenient for locating points. When it is important to record the uncertainties of each plotted point, horizontal and/or vertical bars are passed through the point, the lengths of which represent the uncertainty of the point with respect to the x and/or y co-ordinates.
6. If several curves are drawn on the same sheet, it is best to represent them in different ways, e.g. broken and solid lines, and to mark the points of each curve by different symbols.

7. If there is a theory concerning the functional relationship of the two variables plotted, then a curve corresponding to that hypothesis should be drawn through the experimental points. In many experiments the theory is not well known in advance and the only hypothesis that can be made is that it will yield a smooth curve. The curve should be drawn through the mean of the various points. The curve need not pass through the first and last points. Instead, each point should be considered as accurate as any other point (unless there are experimental reasons for some points being less accurate than others), so that the curve is drawn with about as many points above the curve as are below it with the `aboves' and `belows' distributed at random along the curve (i.e. not all points above the curve at one end and below at the other end).

8. Each graph should bear a title telling briefly what the curve represents.

As an example, suppose that we wish to plot the curve showing the relation between the observed position of an indicator on the end of a spring and the force pulling the spring, and suppose that we have made the following observations:

Which shall we plot along the ``horizontal" x-axis and which along the ``vertical" y-axis?

[ The terms ``horizontal" and ``vertical" are of course not literal, but based on yet another perceptual convention about the orientation of the two-dimensional surface on which the graph is being drawn; all this may become considerably more confusing when 3D ``virtual reality" computer interfaces become more widely used. ]

A general rule is to plot the independent variable along the x-axis (also known as the ordinate) and the dependent variable along the y-axis (also known as the abcissa). In other words, if the position of the indicator on the spring is to be expressed in terms of the force on the spring, then plot the force on the x-axis and the indicator position on the y-axis. The scale units are chosen with consideration both for convenience and for the size of the diagram.

We now make the assumption that the relation between the force F applied to the spring and the extension s of the spring is given by Hooke's Law,

written to include , the position of the indicator when F = 0.

We therefore expect a linear relation between F and s, so we draw the best straight line through the points.

[ Of course, if we stretch the spring too far we will discover deviations from this ideal behavior. Most physical ``laws" are applicable only within certain limits with which the user is expected to be familiar. This expectation is frequently frustrated.... ]

If we now wish to determine the spring constant, k, we can do so easily from the slope of the line. The slope is obtained by taking two points on the straight line (NOT two experimental points, because the straight line represents the best fit to the experimental data), say and , and forming the quotient:

As a rule, it is convenient to choose points such that the difference comes out to be a simple number; in this way, the process of division is simplified. In this example, since F is plotted on the x-axis and s on the y-axis,

where and are any two points on the best fit line and so

Subtracting the first of the above equations from the second,

so that

The slope in the example shown on the following page is 0.233 cm/N. Thus

Note that since we used the slope of the best-fit line rather than a single point to calculate k, we did not need to find . However, this calculation is only as good as our ability to draw a straight line through the mean of the points. There are criteria, such as the ``least squares fit," which make it possible to find the curve which best fits a set of points. If such a method had been applied, the use of the slope would certainly yield a better estimate than a single point or pair of points. Usually we will not use such techniques and will fit curves to data visually. Such a fit is easily made with the aid of a clear flat plastic ruler.

 

Figure: Hooke's Law: Position of Indicator versus Spring Force