Plotting

PLOTTING DATA FROM A CHART When we work in a lab, especially in a Physics lab, we measure magnitudes. A magnitude is everything that can be measured: lengths, time intervals, forces, angles, speeds, are magnitudes. Happiness, love, disgust are not magnitudes. Many times magnitudes are related to each other in such a way that changing one of them, the other one changes in a very specific way. The faster the speed of a car, the sooner it will drive you home. If you walk twice as fast you will move twice as far from the starting point. We say that one of the magnitudes is a function of the other one: the distance you move is a function of (depends on) the time you were moving. It also depends on how fast you walked. The cost of heating some water to the boil, depends on the mass of water you want to heat, its starting temperature, the power of the gas flame, the time it took, and of course on the money you have to pay for each m3 of gas. When scientists want to investigate about how a magnitude changes when another one varies, they must be very careful to keep all other possible variables unchanged. That makes our experiment valid. The results of an experiment are reported in a chart or in a graph or plot. RESULTS CAN BE SHOWN IN CHARTS A chart will show the results of a set of experiments in an ordered sequence. Most times each row (horizontal line) corresponds to the first, the second, the third, etc. experiment and each column to the different variables or conditions that are been studied, the last column showing the results. Columns must have a heading (“title”): the name or symbol of the variable and the unit used for measuring it. A chart can also have columns that show further information or some calculations made with the data you have obtained for that particular experiment. Suppose you are investigating about the time it takes to dissolve completely 10 g of salt in 100 g of water at different temperatures. Your chart should look like this one: Experiment Nr 1 2 3 4 5

Temperature (°C) 20 30 40 50 60

Time (s) 145 120 85 68 44

The chart shows that the time taken depends on the temperature of the solvent (water). Moreover, it decreases as the temperature grows higher. We say that this time is a function of the temperature. You will study about functions in maths. CHANGING A CHART INTO A GRAPH To plot or draw a graph is to represent the information above visually. In physics and chemistry, line graphs are by far the most vastly used. In the previous experiment you have studied how to variables are related. you changed the water temperature and the

time changed “automatically”. The temperature in the case was the independent variable (the one you arbitrarily changed) and the time the dependent variable. The independent variable is represented along a horizontal line or axis and the different times are represented along a vertical axis at right angles to the first one. These are frequently called a pair of “Cartesian axes” or “Cartesian coordinates” or “orthogonal (right angles) axes”. Both axes are properly labelled with the name of the magnitude you are representing and the unit that is being used. Next, a scale must be chosen to represent your information. To do this, the axes are divided into regular intervals from a starting value (say zero) at the intercept between them. The size of the intervals must be set, so that all of your values can be included in the graph occupying at least an 80% of the total length of the axis. The marks are assigned increasing values of the magnitude they represent: Equal segments represent equal changes in the magnitude. Remember you are drawing the axes and not yet the values you have obtained. Graph paper (marked in mm) or squared paper are most convenient for plotting purposes. Always use pencil for drawing graphs and never a pen or ball pen. To plot the data you have recorded in the chart, you should follow the following instructions

1- Draw the pair of Cartesian axes (two perpendicular lines). The axes should be sized about 10 to 30 squares long on each direction. In some cases longer axes can be used but this is an adequate size for most purposes. Smaller graphs are confusing and bigger are usually anti aesthetic. The point at which both axes meet is called the “origin” of the axes.

2- Write on the horizontal axis the symbol or name and the unit of the independent variable (the one you change freely) in this case “temperature / °C” and on the vertical axis the name or symbol and the unit for the other variable (the one that changes as a consequence of your changing the first one) in this case “time / seconds”

3- Draw the scales on the X axis. This is the most difficult thing about plotting. Your data have to fix in the plot and don’t look too crowded at the same time. Follow the following instructions: 

The origin is labelled “0”

 The first square should be labelled 1, 2 or 5 or any multiple or sub multiple of these numbers. (0.01 or 0.5 or 20 or 500) This is to make your scale easy to work with.

 All other squares are equivalent: they represent the same interval or change in the variable.  The scale should be chosen so that you can draw the highest data you have. 

Do the same for the Y axis.

Remember that the X and Y scales can be absolutely different. In our example, the maximum temperature found is 60º. If the scale had been chosen so that 1 sq = 1ª we should have used 60 sq! Too many! If it had been chosen as 10º the graph will look too narrow (we would be using 6 sq. in a scale with more than 20 sq.) An appropriate choice would be 2º or 5º. Squares will be marked: 2 4 6 8 10 12 and so on, or 5 10 15 20 25 and so on up to 60 or 70. Follow the same procedure for the Y axis. In this example we have labelled the Y axes from 40 onwards. You can do this if your points start at a value far from the “0” point and never go down. Start at a point slightly below your lowest measurement.

4- Now draw a small cross at each of the points defined by the chart with the data you have previously recorded, e. g. for the third experiment the cross (x) must be drawn where a horizontal line drawn from time = 93 seconds intercepts a vertical line drawn from the point at 40 °C in the temperature axis.

5- Join the points with a smooth line (no angles). If there is a point absolutely out of the general trend don’t take it in account. It must be a mistake.

6- If a sequence of points show like this: 85 – 85.5 – 85 – 85.5 – 86 -85.5 don’t draw a wavy line but draw a smooth line crossing an “average” position.

See how clearly the decrease of the time with increasing temperature is shown! You can immediately infer that a point was not properly determined and the experiment can be repeated for checking that nothing odd is happening. In case that the mischievous point insists on being “there”, you should repeat the whole experiment. If the anomaly persists, well … Start risking hypotheses about what’s going on and try them with new experiments!!

Exercises. The chart shows the solubility in water of two different substances (A and B ) Plot the data in two separate graphs. Temperature 0°C 10°C 20°C 30°C 40°C 50°C 60°C 70°C 80°C 90° 100°C

Solubility 0.56 0.56 0.67 0.78 0.91 (g/l) A

Solubility (g/l) B

60

1.1

66.7 73.9 81.8 88.7 96.0 106

120

1.32

1.72

132

153 160

The following chart shows the stretching of a spring because of a pulling force Plot them. Stretch. Force (dm) (N) 0.093 0.24 0.225 0.49 0.415 0,98 0.588 1.23 0.710 1,47