PHYSICS
LABORATORY
MANUAL
ENG TECH 1PH3
GRAPHING TECHNIQUES, ERROR ANALYSIS and REPORT FORMAT
1) INTRODUCTION
When obtaining a result from experimental data, a numerical calculation may be appropriate. At other times, a graph of the data should be used as a visual presentation of the results. Graphical presentation of data often allows us to assess more satisfactorily a series of results than the examination of the same results in a table, because the graph presents a visual display of the continuous relationship between the variables studied. It also allows interpolation and extrapolation as to the behavior. between and beyond the data points taken in the experiment and, it may also be a valuable tool in the analysis of errors.
2) GRAPH PREPARATION
The following recommendations should be considered when preparing a graph:
1. Choose the scales and type of scale to make the most effective use of the graph paper.
2. Graphs should normally fill the paper.
3. Plot all the data points clearly.
4. Different symbols should be used when more than one curve is plotted on the same piece of paper.
5. Each graph should have a title.
6. Axes should be labelled with both their scale and their units.
7. It is conventional to choose the x-axis (abscissa) to represent the independent variable (the one which you change) and the y-axis (ordinate) to represent the dependent variable (the one which is found to change as a result). This would be referred to as plotting the dependent variable versus (or against) the independent variable.
8. Remember you must DECIDE whether to use all the points or omit some. Or, use a straight line or a curve or a series of straight lines joining the points.
9. Above all, plot a graph as the experiment proceeds. Alternatively, the graph should be plotted before the equipment is dismantled and put away. Remember that points cannot be checked afterwards except for arithmetical and plotting errors.
10. All information must be on the lined part of the graph paper. Not on the blank margins.
EXAMPLE:
1. The mile markers were noted at 15-minute intervals from a car moving along the highway. The data obtained was tabulated as follows:
Time (hours) (± 5 sec) 0.25 0.5 0.75 1.0 1.25 (independent)
Distance (km) (± 1 km) 317 328 337 348 358 (dependent)
From this data a graph similar to Figure I.1 shown below, may be prepared. It is good practice to number tables and graphs (labelled as Figures). A brief description in the Figure title is also useful to someone reading your report.
Figure I.1: Graph of distance versus time.
3) GRAPH INTERPRETATION
1. The graph is a straight line within the limits of error, i.e. the line passes through the error bars about each point.
2. The slope of the line represents the way in which the distance (y) changes with respect to time (x).
In this case the slope of the graph is:
As distance divided by time is speed, the slope of this graph represents the speed of the car.
Further, as the slope is constant (the line is straight), the speed of the car is constant within the experimental errors assessed. Figure I.2 illustrates the slope as the rise over the run.
Figure I.2: Graph showing how to determine the slope of a line.
Numerically, using as long a segment of the line as possible, the slope from Figure I.2 is:
3. The intercept of this line with the y-axis occurs at a y-value of 307, which may be interpreted as the starting point of the journey (provided that the speed in the first period was the same as the average).
4) GRAPH ANALYSIS
In this example the basic theory is that speed is distance divided by time, and as data concerning distance and time is available, speed may be deduced.
Alternatively, a graphical treatment may be used to investigate the form. of the functional relationship between the variables considered — that is, to help develop the theory.
For most purposes the straight-line relationship is most desirable as it is easy to analyze and extrapolate.
When an experiment is being designed, the attempt is usually made to achieve this relationship either by rearranging the formula or using different graph paper (logarithmic or semi-log, etc...). Mathematically, a straight line is represented by the general formula:
Graphically, this formula will produce a straight line of intercept "b" on "y" axis and slope m, which is equal to:
In order to plot the graph, two suitable variables (x and y) must be selected, which means that all other possible parameters of the experiment must be kept constant.
One of the variables, say "x", is then deliberately put through a range of set values and displayed on horizontal axis (abscissa) as INDEPENDENT VARIABLE.
The value of variable "y" is then measured for each value of (independent variable) "x" and displayed on vertical axis (ordinate) as DEPENDENT VARIABLE.