Graphing

Most people at one time or another during their careers will have to interpret data presented in graphical form. This means of presenting data allows one to discover trends, make predictions, etc. To take seemingly unrelated sets of numbers (data) and make sense out of them is important to a host of disciplines. An example of graphing techniques used in physics follows.

When weight is added to a spring hanging from the ceiling, the spring stretches. How much it stretches depends on how much weight is added. The following slide depicts this experiment.

Starting level Add a mass We control the mass that is added. It is the independent variable. The stretch is dependent on what mass is added. It is the dependent variable. Stretch is now here Add another mass Stretch is now here

The following data were obtained by adding several different amounts of weight to a spring and measuring the corresponding stretch.

Stretch (meters) Weight (Newtons) The Newton is a unit of force or weight

There are two variables or parameters that can change during the experiment, weight and stretch. As mentioned earlier the experimenter controls the amount of weight to be added. The weight is therefore called the independent variable. Again as mentioned before the amount that the spring stretches depends on how much weight is added. Hence the stretch is called the dependent variable. The dependent variable is the quantity that depends on the independent variable.

A graph of this experimental data is shown on the next slide. The independent variable is always plotted on the horizontal axis, the abscissa. The dependent variable is plotted on the vertical axis, the ordinate. Notice that each axis is not only labeled as to what is plotted on it, but also, the units in which the variable is displayed. Units are important.

Each graph should be identified with a title and the experimenter's name. Weight, the independent variable, will be plotted along the horizontal axis (the abscissa). Weight (Newtons) Stretch, the dependent variable, will be plotted along the vertical axis (the ordinate). Stretch (meters) Stretch Versus Weight - Susie Que The graph should be made so that the data fills as much of the page as possible. To do this, sometimes it is better not to start numbering an axis at zero, but rather a value near the first data point.

Each data point should be circled, so that it can be easily found and distinguished from other dots on the paper. Weight (Newtons) Stretch (meters) Stretch Versus Weight - Susie Que This is not a connect-the-dot exercise. The data appears to fit a straight line somewhat like this one. Let’s plot the data. (6.0, ) (14.0, ) (22.0, ) (38.0, ) (40.0, ) (47.0, ) (54.0, ) (58.0, ) (30.0, )

Weight (Newtons) Stretch (meters) Stretch Versus Weight - Susie Que If there is a general trend to the data, then a best-fit curve describing this trend can be drawn. In this example the data points approximately fall along a straight line. This implies a linear relationship between the stretch and the weight. A wealth of information can be obtained if the equation that describes the data is known. With an equation one is able to predict what values the variables will have well beyond the scope or boundaries of the graph. A timid mathematician should not be scared away, since finding the equation is not hard and requires very little knowledge of math.

Weight (Newtons) Stretch (meters) Stretch Versus Weight - Susie Que If data points follow a linear relationship (straight line), the equation describing this line is of the form y = mx + b where y represents the dependent variable (in this case, stretch), and x represents the independent variable (weight). A very important equation.

Weight (Newtons) Stretch (meters) Stretch Versus Weight - Susie Que y = mx + b The value of the dependent variable when x = 0 is given by b and is known as the y-intercept. The y-intercept is found graphically by finding the intersection of the y-axis (x = 0) and the smooth curve through the data points. (From the equation y = mx + b, if we set x = 0 then y = b.) In this case b = 0.11 meters.

Weight (Newtons) Stretch (meters) Stretch Versus Weight - Susie Que The quantity m is the slope of the best-fit line. It is found by taking any two points, for instance (x 2, y 2 ) and (x 1, y 1 ), on the straight line and subtracting their respective x and y values. Note y 2 = mx 2 + b y 1 = mx 1 + b Subtracting one equation from the other yields y 2 - y 1 = mx 2 - mx 1 y 2 - y 1 = m(x 2 -x 1 ) We’ll call this y = mx + b Therefore

Weight (Newtons) Stretch (meters) y-intercept = 0.11 meters Stretch Versus Weight - Susie Que run = (34.0 – 17.0) Newtons = 17.0 Newtons rise = (0.21 – 0.16) meters = 0.05 meters To find the slope of this line pick a couple of points on the line that are somewhat separated from each other Y 1 = Y 2 = X 2 = X 1 = Point 2 Point 1 rise = (0.21 – 0.16) meters = 0.05 meters

y = mx + b b+ Weight (Newtons) Stretch (meters) y-intercept = 0.11 meters Stretch Versus Weight - Susie Que At this point everything needed to write the equation describing the data has been found. Recall that this equation is of the form y=mx 0.11 meters

b+ Weight (Newtons) Stretch (meters) Stretch Versus Weight - Susie Que Here are two ways we can gain useful information from the graph and from the equation of the line. If we wanted to know how much weight would give us a 0.14 m stretch, we could read it from the plot thusly. This would be about 10.2 Newtons. Solving the equation for x when y=0.14 m gives x=10.2 Newtons. y=mx

A curve through this data is not straight and x and y are not linearly related. Their relationship could be complicated. y2y2 y x Suppose you have some x and y data related to each other in the following way. A replot of this data might straighten this line some to give a linear relationship. Let’s try y 2 versus x. This relationship would be

In the earlier example of stretch vs. weight one over the slope of this curve is called the spring constant of the spring. This method of determining the spring constant of a spring is better than alternate methods such as calculating the spring constants of individual measurements and taking an average or taking an average of weights and dividing by an average of the stretches. Graphing is a powerful analytical tool.