GraphingGraphing

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 levelAdd a weight

We control the weight that is added.It is the independent variable.The stretch is dependent on what weight is added.It is the dependent variable.

Stretch is now hereAdd another weightStretch 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)

0.1240 6.00.1475 14.00.1775 22.00.1950 30.00.2195 38.00.2300 40.00.2525 47.00.2675 54.00.2875 58.0

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)

0 10 20 30 40 50 60

Stretch, the dependent variable, will be plotted along the vertical axis (the ordinate).

Str

etc

h (

mete

rs)

0.10

0.14

0.18

0.22

0.26

0.30Stretch 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)

0 10 20 30 40 50 60

Str

etc

h (

mete

rs)

0.10

0.14

0.18

0.22

0.26

0.30Stretch 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, 0.1240)

(14.0, 0.1475)

(22.0, 0.1775)

(38.0, 0.2195)(40.0, 0.2300)

(47.0, 0.2525)

(54.0, 0.2675)

(58.0, 0.2875)

(30.0, 0.1950)

Weight (Newtons)

0 10 20 30 40 50 60

Str

etc

h (

mete

rs)

0.10

0.14

0.18

0.22

0.26

0.30Stretch 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)

0 10 20 30 40 50 60

Str

etc

h (

mete

rs)

0.10

0.14

0.18

0.22

0.26

0.30Stretch 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 importantequation.

Weight (Newtons)

0 10 20 30 40 50 60

Str

etc

h (

mete

rs)

0.10

0.14

0.18

0.22

0.26

0.30Stretch 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)

0 10 20 30 40 50 60

Str

etc

h (

mete

rs)

0.10

0.14

0.18

0.22

0.26

0.30Stretch 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 (x2, y2) and (x1, y1), on the straight

line and subtracting their respective x and y values.

Note

y2 = mx2 + b

y1 = mx1 + b

Subtracting one equation from the other yields

y2 - y1 = mx2 - mx1

y2 - y1 = m(x2 -x1) 12

12

xxyy

m

runrise

We’ll call this

y = mx + b

Therefore

slope

Weight (Newtons)

0 10 20 30 40 50 60

Str

etc

h (

mete

rs)

0.10

0.14

0.18

0.22

0.26

0.30

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

Newtons0.17meters05.0 Newton/meters00294.0

runrise

To find the slope of this line pick a couple of points on the line that are somewhat separated from each other.

17.0

0.16

34.0

0.21

Y1 =

Y2 =

X2 =X1 =

Point 2

Point 1

y = mx + bb+

)meters11.0(

slope

Weight (Newtons)

0 10 20 30 40 50 60

Str

etc

h (

mete

rs)

0.10

0.14

0.18

0.22

0.26

0.30

y-intercept = 0.11 meters

Stretch Versus Weight - Susie Que

Newtons0.17meters05.0 Newton/meters00294.0

runrise

At this point everything needed to write the

equation describing the data has been found.

Recall that this equation is of the formy = mx

Stretch )Newton/meters00294.0( Weight

0.11 meters

b+

)meters11.0(

Weight (Newtons)

0 10 20 30 40 50 60

Str

etc

h (

mete

rs)

0.10

0.14

0.18

0.22

0.26

0.30Stretch 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 = mxStretch )Newton/meters00294.0( Weight

A curve through this data is not straight and x and y are not linearly related. Their relationship could be complicated.

y2y

x0 10 20 30 40 50 60

0

1

2

3

4

5

6

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 y2 versus x.

This relationship would be

bmxy2

In the earlier example of stretch vs. weightIn the earlier example of stretch vs. weight

one over the slope of this curve is called one over the slope of this curve is called

the spring constant of the spring. the spring constant of the spring.

This method of determining the spring This method of determining the spring constant of a spring is better than alternate constant of a spring is better than alternate methods such as methods such as

calculating the spring constants of individual calculating the spring constants of individual measurements and taking an average or measurements and taking an average or

taking an average of weights and dividing by taking an average of weights and dividing by an average of the stretches. an average of the stretches.

Graphing is a powerful analytical tool. Graphing is a powerful analytical tool.

The information on this slide will most likely The information on this slide will most likely be on the lab final exam. be on the lab final exam.