Variables, Graphing, and Functions
Variables, Graphing, and Functions
Variables and experimental design
One common type of laboratory investigation is to determine the relationship between physical variables. For example,
consider a simple pendulum which is composed of a compact weight (bob) that is hung from a string attached at its
upper end to a fixed support. Suppose the goal of the investigation is to determine which variables influence the
period of the pendulum, that is, the time for the bob to execute one complete cycle over and back. Some variables
whose influence one could investigate include the length of the string, the mass of the bob, and the angle from the
vertical at which the string is released. The latter three variables are termed independent variables, because one
selects their values in carrying out the experiment. The period is termed the dependent variable, because its value
may depend on the values of the independent variables.
In order to determine how each of the independent variables may influence the period, one needs an experimental
design in which only one of the independent variables is changed at a time while the others are held constant. In this
way, if an influence (or lack thereof) is found on the period, one can be fairly confident that the independent variable
that was changed is the variable that influenced (or didn't influence) the period. Such an experiment is called a
controlled experiment. By the way, the reason we said fairly confident is because there's always the possibility in
dealing with the natural world that there are variables that the experimenter has overlooked and has not controlled. For
the case of the pendulum, for example, suppose you carried out the experiment in an elevator that was moving up and
down between floors. You would discover some strange results and find it difficult to reach conclusions about how the
independent variables affected the period. That's because the elevator accelerates and decelerates and, as it turns
out, such motion influences the period of a pendulum. Of course, it's not likely that you would do this experiment in an
elevator and, if you did, you would probably guess that the motion of the elevator influenced the results. That's part of
being a competent scientist. However, even if you're competent, you can still overlook variables. An example might be
your location on the surface of the Earth. Location, in fact, influences the period, although the effect is so small that
one typically doesn't notice it. But if you were making very precise measurements, you would have to take the effect
into account.
The typical way to carry out an experiment to investigate the possible influence of an independent variable on the
dependent variable is to measure the value of the dependent variable for several values of the independent variable. A
graph is then made of the dependent variable vs. the independent variable in order to visually represent the
relationship between the variables. We illustrate next for the pendulum.
Tabulating and graphing data
Let the independent variable be the mass of the bob. Suppose we use a stopwatch to measure the period of the
pendulum for different masses ranging from 0.200 to 1.000 kg while keeping the length of the pendulum constant at
0.500 m and the angle of release constant at 10.0°. Example results are shown in the table below. A graph of Period
vs. Mass is shown to the right of the data table.
Table of Data
Period vs. Mass for a Simple Pendulum
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Variables, Graphing, and Functions
Mass
(kg)
Period
(s)
0.200
1.41
0.400
1.47
0.600
1.44
0.800
1.51
1.000
1.49
Length = 0.500 m
Angle = 10.0°
Note the following about the format of the graph above.
The table includes all the data, including the values that are held constant.
The independent variable is listed before the dependent variable in the table.
The units for the table columns appear once in each column heading. This takes the place of writing units
beside each number.
The dependent variable is plotted on the vertical axis. This is conventional practice and should be followed
unless there's reason to do otherwise. In this course, we'll usually follow this convention but there will be
occasional exceptions.
The axes are labeled with the names of the variables and their associated units. This is also standard practice.
Never use generic x and y axis labels for a scientific graph, and always include units.
The axes are numbered in equal increments.
The graph is titled in the form "Dependent Variable vs. Independent Variable for Name of System or Object".
This is a standard form in physics.
The data points are clearly indicated. (The symbols indicating the data points are called point protectors.) The
points are not connected with lines.
Note also that the numbering of the vertical axis does not start at 0. This isn't necessarily the best practice, and one
has to be aware of this in order to avoid making hasty conclusions. A quick glance at the graph seems to indicate a
general increasing trend of period with mass. However, consider the graph of the same data below for which the
vertical axis starts at 0.
Period vs. Mass for a Simple Pendulum
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Variables, Graphing, and Functions
From this second graph, one could conclude that the mass doesn't influence the period. The variations in the values
of period might be deemed small enough to be explained by error in deciding when to start and stop the stopwatch. Of
course, one could be more confident of any conclusion by refining the method of measuring the period, by taking
more measurement trials under the same conditions in order to see how reproducible the results are, and by
extending the range of masses. We'll consider such refinements later. They're not a subject of this particular guide.
By the way, if you're surprised at the result above, you can easily test it for yourself. Hang a set of keys from a string
about half a meter long and measure the period for different numbers of keys in the key ring. The independence of
mass and period actually says something very fundamental about the nature of mass and corresponds with the
observation that objects of different mass fall from the same location with the same acceleration.
Linearizing the data
Now let's consider the influence of the length of the string on the period. We measure the period of the pendulum for
lengths ranging from 0.100 to 1.900 m while keeping the mass of the bob constant at 0.400 kg and the angle of
release constant at 10.0°. Example results are shown in the table below. A graph of Period vs. Length is shown to
the right of the data table.
Table of Data
Period vs. Length for a Simple Pendulum
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Variables, Graphing, and Functions
Length Period
(m)
(s)
0.100
0.70
0.500
1.47
0.900
1.97
1.400
2.50
1.900
2.88
Mass = 0.400 kg
Angle = 10.0°
This time the period shows an unmistakable increasing trend with length. What is the functional form of this trend?
The latter is a question that must be asked whenever one does a quantitative experiment to determine the relationship
between physical variables. One can approach an answer by hypothesizing about what mathematical function the
trend follows. Then one could test the hypothesis. In the above case, noting that the period would be expected to
decrease toward 0 as the length did the same, a curve similar to a square root function might fit the trend of the data.
A way to test this would be to re-express the independent variable as the square root of the length. If the hypothesis
were correct, then a graph of period vs. length would be linear. Here are the results for the data above.
Table of Data
Period vs. Square Root of Length for a Simple Pendulum
Square Root
Period
of Length
(s)
(m ½)
0.316
0.61
0.707
1.40
0.949
1.88
1.183
2.38
1.378
2.75
Mass = 0.400 kg
Angle = 10.0°
Note the following about the re-expressed data and the graph:
The name of the independent variable and its units have been changed to be consistent with the re-expression
of the variable of Length to the Square Root of Length.
The relationship between Period and the Square Root of Length appears to be linear. This is consistent with
the hypothesis that the functional form of the graph of Period vs. Length is a square root function.
The data points are no longer spaced by equal increments along the horizontal axis.
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Variables, Graphing, and Functions
From an experimental point of view, it would make sense to collect additional data to fill in the gaps for values of
Square Root of Length from 0.0 to 0.7 m½. If one suspected from the outset that the length would increase faster
than the period, the experiment could have designed so that the spacing between consecutive values increased with
increasing length. This would improve the experimental design. Of course, even if one didn't suspect this, one could
always repeat the experiment with a better design. This is something that a good experimental scientist does.
Let's work with the data that we have to complete this example. The next step is to draw a straight line to represent
the trend of the data shown in the last graph. See the graph below. If the line is drawn by hand rather than with a
computer, one uses a straightedge to ensure linearity. The straightedge is positioned in such a way as to "split the
differences" between the points. That is, the deviations of points above the line are about the same as the deviations
below the line. The line isn't drawn through the origin, since that isn't a data point. Even though we expect that the
period is 0 when the length is 0, we don't let that influence the way that we draw the line. We call the resulting line the
line of best fit.
The fact that the points don't all lie on the same straight line is presumably due to experimental errors such as the
error in starting the stopwatch mentioned previously. Another possible error is the measurement of the length. By
drawing a straight line with a straightedge, however, we are expressing our conviction that the relationship between
the Period and the Square Root of Length is linear.
Period vs. Square Root of Length for a Simple Pendulum
Writing the equation of the relationship between the variables
The next step in determining the function that describes the relationship between the variables is to write the equation
of the line. In math class, you learned that the equation of a straight line is y = mx + b. You also learned how to
determine the values of the slope, m, and intercept, b. In physics class, you have to translate the variables into
physically meaningful terms and use corresponding symbols to represent them. You also have to determine values of
the slope and intercept and relate these to the physical situation. Here's what we mean.
1.
The variable on the vertical axis is the Period. Let's represent that with the symbol P. Then y translates into P.
2.
The variable on the horizontal axis is the Square Root of Length. Let's represent the length with the symbol L.
Then x translates into L½.
3.
The slope can be determined from the line on the graph by application of the definition of slope. This method is
shown next.
The enlarged graph below shows data on the period of a simple pendulum vs. the square root of the
length of the pendulum. Using the coordinates of the two points indicated on the graph, the slope is:
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Variables, Graphing, and Functions
slope = (P2 - P1 )/(L2 ½ - L1 ½) = (2.76 s - 0.28 s)/(1.32 m½ - 0.12 m½) = 2.07 s/m½
Note the following about this method of finding the slope:
Two widely-separated points are selected for use in calculating the slope. The wider the
separation, the better the accuracy in the final result will be. With the points shown, the slope will
have 3 significant figures. If points were selected for which ∆P was less than 1.00 s or ∆(L½)
was less than 1.00 m½, only 2 significant figures could be achieved.
Data points are not selected as the two points for calculating slope. That's because we want the
slope of the line itself, and the line doesn't necessarily pass through the data points.
The origin isn't selected as a point for calculating slope, because the origin isn't a data point.
The locations of the two points are indicated with crosses. One could use other symbols as long
as the locations were clearly indicated.
The values of the coordinates are expressed to the greatest precision with which the scales can
be read. This is generally one-tenth of the smallest division.
Units are always expressed with values. The units of slope are always the units of the vertical
variable divided by the horizontal variable. In this case, that would be s/m½.
Without knowing more about the pendulum, we can't say for sure what the slope represents physically
about the simple pendulum. However, a good guess would be that it has something to do with gravity.
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Variables, Graphing, and Functions
And, since the slope is a constant, then whatever it represents must also be a constant of the system.
You'll learn more about this in your physics course.
4.
The intercept read from the graph is a little above 0. While physically we expect that the period is 0 when the
length is 0, we don't force the intercept to be 0. There may be something about the experimental design or the
measurements that introduces a systematic error which causes all values of the period to be a bit high in this
case. One possibility is that the experimenter always started the stopwatch early and stopped it late. Another
possibility is that all the measured lengths were low, because the experimenter didn't include the diameter of
the bob in the length of the pendulum.
We're now ready to write the equation of the relationship between the variables. We replace y and x with the symbols
representing the corresponding physical quantities, and we replace m and b with the values and units of those
constants. The final result for the relationship between the period and length of a pendulum is the following.
P = (2.07 s/m½)L½ + 0.03 s
It's important to note that the above is an empirical (experimental) result rather than something derived from theory.
The next step is typically to compare the empirical relationship to theory (when a theory exists). We'll skip this step for
now, since it's too early in the course to discuss the theory of the pendulum. However, we'll show another example in
which a comparison to theory is made.
Inductive and deductive reasoning
The process described above in which data is collected and then used to determine a relationship is called inductive
reasoning or induction. The nature of this process is that it begins with specific information (in this case, data) and
concludes with a general relationship that can be used to predict values that were not part of the original data set. For
example, we could use the general relationship, P = (2.07 s/m½)L½ + 0.03 s, to design a clock that has a period of
1.00 s. We solve the equation for L and substitute 1.00 s for P.
L = [(P - 0.03 s)/(2.07 s/m½)]² = [1.00 s - 0.03 s)/(2.07 s/m½)]² = 0.220 m
Note that the units reduce to meters as expected. This process of making a specific prediction based on a general
relationship is termed deductive reasoning or deduction and is the opposite of induction. Scientists use induction to
discover relationships, and they use deduction to make predictions based on known relationships. These processes
are part and parcel of the scientific method.
A note about curve fitting with software
The process described above for finding the line of best fit and the equation of the line is a process that can be
carried out by hand using graph paper and a straightedge. This process can, of course, be automated using graphical
analysis software. We will be using such software frequently throughout your physics course. You'll use this software
most of the time.
Circumference vs. diameter of cylinders
We'll use a familiar situation as another example of the graphical analysis procedure. Suppose we have a collection of
cylinders of different diameters ranging from 1 to 4 centimeters. Our goal is to experimentally determine the
relationship between the circumference and the diameter and compare the results to what we expect from
mathematical theory.
Here's how the measurements are made. In order to find the circumference of each cylinder, a pencil mark is placed
on one end of the cylinder. Then the cylinder is rolled without slipping across a piece of paper until two marks appear
on the paper. The distance between the marks is measured with a ruler. The series of sketches below illustrate the
process.
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Variables, Graphing, and Functions
In order to find the diameter of each cylinder, the ruler is placed
on a table. Then the cylinder is placed on the ruler in such a
position as to give the greatest distance from one side of the
circular cross section to the other as shown in the sketch to the
right.
Note that in order to take accurate readings from the ruler, the
ends of the ruler are never used as endpoints for the distances
measured. Also, lines of sight to the ruler scale are as nearly
perpendicular to the scale as possible.
Example data and a graph are shown below.
Cylinder ID
Diameter Circumference
(m)
(m)
1
0.0120
0.0398
2
0.0163
0.0505
3
0.0195
0.0612
4
0.0253
0.0797
5
0.0390
0.1250
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Variables, Graphing, and Functions
Theoretically, we expect the circumference to be proportional to the diameter. So it's no surprise that the graph above
appears linear. Therefore, we draw a best fit straight line through the data points, find the slope and intercept, and
write the equation of the line. This is shown below.
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Variables, Graphing, and Functions
slope = (C2 - C1 )/(D2 - D1 )
= (0.111 m - 0.015 m)/(0.0350 m - 0.0050 m)
= 3.20 (units divide out)
intercept = -0.001 m
Equation of best fit line: C = 3.20D - 0.001 m
The above equation is an experimentally-determined relationship between circumference and diameter. However, we
also know from theory that C = (pi)D, where pi is 3.14 to 3 significant figures. Therefore, we can say that we expect
the slope of the experimental relationship to be pi and the intercept to be 0. In this case, experiment and theory agree
on the value of the slope to within 2%. This and the small size of the intercept are easily explained by the errors
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Variables, Graphing, and Functions
inherent in the methods of measuring circumference and diameter.
These results are summarized in the following table, which we call a matching table.
Math maps to Physics
y
-->
C
m
-->
pi
x
-->
D
b
-->
none
Value
Value
Units
(graph) (expected)
m
3.20
3.14
none
m
-0.001
0
m
The Physics column shows the names of the physical variables. (While you may think this is math, the act of
measuring the circumference and diameter of real cylinders makes it physics.) The values obtained from the line of
best fit are given in the Value (graph) column and the values expected from theory are given in the Value (expected)
column.
Cross-sectional area vs. diameter of cylinders
Let's take this a step further and experimentally determine the relationship between the
cross-sectional area and the diameter of the 5 cylinders. Our method of measuring the
diameter is to place a base of the cylinder onto a sheet of graph paper that is ruled in
millimeters and then to trace around the circular base. We then remove the cylinder and
count the number of enclosed squares, each with an area of 1.0 mm². We also count the
partial squares as best we can and add them in to the total. The process is illustrated to the
right. For this diagram, we estimate the area of the circle to be 9.0 mm².
Perhaps you're asking yourself why we don't find the area simply by squaring the diameter and multiplying by pi/4. If
we did that, we would be assuming what we're trying to show by experiment. We measure the area without reference
to the formula for the area of a circle. After that, we'll plot a graph of area vs. diameter and determine the relationship
experimentally.
Example data and a graph are shown below. Note that the area is given in units of m² in order to be consistent with
the units of diameter.
Cylinder ID
Diameter Area of cross section
(m)
(m²)
1
0.0120
0.00013
2
0.0163
0.00021
3
0.0195
0.00030
4
0.0253
0.00049
5
0.0390
0.00118
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Variables, Graphing, and Functions
We can see that the relationship between area and diameter is non-linear. So we'll need to re-express a variable.
Here's where we can use what we know about the theoretical relationship between area and diameter of circles. Since
we expect that the area should be proportional to the square of the diameter, we should square the values of diameter
and plot area vs. diameter². Of course, this also corresponds with the fact that the curve above has the appearance of
a parabola.
The table of data below includes the values of diameter². Below that is the graph of area vs. diameter². A best fit line
has been drawn.
Cylinder ID
Diameter Diameter² Area of cross section
(m)
(m²)
(m²)
1
0.0120
0.000144
0.00013
2
0.0163
0.000266
0.00021
3
0.0195
0.000380
0.00030
4
0.0253
0.000640
0.00049
5
0.0390
0.001521
0.00118
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Variables, Graphing, and Functions
We won't show the calculation of the slope this time but will simply give the equation of the best fit line. Note that the
intercept is less than 0.00001 m²; hence, we don't write it below.
A = (0.77)D²
Now we compare the experimental result to the theory. We expect from theory that A = (pi/4)D². Thus, we expect that
the slope will be pi/4 = 0.785. The experimental value is within 2% of the expected value. The matching table is given
below.
Math maps to Physics
y
-->
A
m
-->
pi/4
x
-->
D²
b
-->
none
Value
Value
Units
(graph) (expected)
m²
0.77
0.785
none
m²
0
0
m²
Conclusion
You may find it strange that we went to all this trouble to determine relationships that we already knew. Of course, we
were just illustrating the process by which a relationship between two physical variables is determined experimentally
and then compared to theory. This is called a verification experiment. The method of determining relationships
experimentally can be used to support or refute an existing theory or to provide a predictive relationship when a theory
is not known. When the experimental relationship is used in conjunction with an accepted theory, characteristics of a
physical system can be determined. You'll use this process frequently in your physics course.
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