Graphing Assignment
Due July 6, 2012
OVERVIEW
Since the late 1990’s, the AP Physics exam has placed an increased emphasis on physics laboratory experience,
interpreting laboratory investigations, and interpreting graphs that accompany a laboratory experience. Questions
every year appear on the exam, usually centering on a laboratory experience like determining gravity or making sense
of data on a position v. time. Occasionally, an AP exam has included a question asking students to create or devise an
experiment that incorporates certain equipment. Those students serious about earning a “5” will read the lab
procedures handed out to you at the beginning of the year as well as understanding the meaning of the graphs you
obtained and correctly answering all post-lab questions. In this review, we are going to look at graphically
interpretation.
GRAPHING DATA CORRECTLY
The goal of a physics experiment is to determine how two sets of measurements are related to each other. In some
physics experiments, you do not have clearly defined independent and dependent variables. Instead, your focus is in
understanding the two things you are measuring and why you need to measure them. Once you have two sets of
measurements, your next step is to create the graph. NEVER free-hand draw a graph. NEVER use data table points to
determine a slope (best fit line only). Use either graph paper or graphing software. Use the x-axis for the independent
variable (that which is experimentally varied; also known as the manipulated variable) and the y-axis for the dependent
variable (that which is a function of the independent variable; also known as the responding variable). For example, if
your graph is position v. time, position is on the y-axis and time is on the x-axis. You are attempting to see how the
position varies with time, or stated in another way, you are studying time dependent position. You should always use a
straight-edge if possible. Also, always label the axes and include units.
A GENERAL APPROACH TO LINEAR GRAPHING
Let’s say we need to determine the spring constant for a linear spring. Recall that a spring constant is a measure of the
stiffness or strength of the spring. To find the spring constant, we need to set up an experiment with a spring loaded
with various masses. The spring stretches different amounts when we added different sized masses. During this
experiment, we measure two quantities, the weight (force) in newtons and the stretch in meters. The weights represent
both the force exerted on the spring and the force exerted by the spring. Here is a table of results. Graph the data.
PROBLEM 1
Force (N)
Stretch (m)
0.2
0.015
0.4
0.035
0.6
0.065
0.8
0.075
1.1
0.130
1.5
0.160
2.0
0.210
SLOPE = ________________
EQUATION = __________________
This is a typical graph of a physics experiment. The data should clearly show a relationship between the two sets of
measurements. In this case, it is a linear relationship. The line is a trend line. All data does not fall on the line, but that
is to be expected when dealing with real data. However, this best-fit-line should include always include equal
numbers of points above and below the line. Also, consider if the point (0, 0) has physical meaning.
The equation for the spring is the equation of the graph’s trend line. This simply follows the form y = mx + b or for
our example, F = mx + b. We will get into the specifics of this in the fall, but for now, determine the slope (with
units) and write the equation for this data (omit the y-intercept, b.)
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Graphing Assignment
Due July 6, 2012
WHAT IF THE GRAPH IS CURVED?
Here is where the fun begins. If we get a nonlinear relationship between our physical quantities, we will need to
investigate further to figure out the exact model for this relationship. This is where your graphing calculator comes in
very handy. However, while you may be able to obtain a graph on the calculator, you will need to be able to properly
transfer it to the graph paper on the AP exam.
LINEARIZING DATA
The process described in the previous example is called linearizing data. It is a simple five-step process:
1) Graph the original data.
2) Look at the shape of the type of relationship it represents (quadratic, cubic, square root, and inverse are the
most common).
3) Create a second set of data for the type of relationship you determined your graph represents. These data will
have different x values. In the previous example, x changed from time to time squared. If the graphed shape
would have been a square root curve, the altered x would have been the square root of x.
4) Create a graph of the new data.
5) Use the slope-intercept form of the line to create the exact relationship for your data. You can see why this
process is called linearizing the data.
This is the most important data analysis skill you can learn in an AP Physics course. You will be practicing it by
working with graphs that present you with interesting relationships we have seen throughout the course. The following
are the most common relationships:
1) y = kx
2) y = kx2
3) y = k/x
4) y = k/x2
5) y = k√x
PROBLEM 2
Here is a sample set of position v. time data. Graph position as a function of time. Then create a second graph to
linearize the data by manipulating time (t2, 1/t, 1/t2, t½). For the linear graph, determine the slope (with units)
and the equation for the linear plot.
Position (m)
Time (t)
5
0.0
10
1.0
16
1.5
26
2.0
36
2.5
51
3.0
SLOPE = _________ EQUATION = _______________
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Graphing Assignment
Due July 6, 2012
From our data, we were able to figure out the exact relationship between position and time and obtained the
acceleration value from our analysis. The point of graphing is to find the average for the data. You should avoid
simply picking a set of points in the table and calculating an average value. This is not proper lab technique and
will get you in trouble on the AP exam. The purpose of graphing is to understand the relationship between two
physical quantities. Therefore, use your graphs to determine other quantities that directly come from the graphs, such
as acceleration.
PROBLEM 3
We are considering an experiment in which we put a series of masses on a spring and then set the spring/mass system
into motion. The resulting motion is simple harmonic motion. We measure each mass we place on the spring and the
period of the SHM. The period is always constant for any one mass. Here is a set of data that was gathered. Graph
the data of Period vs. Mass. Then, linearize the data to determine the proper relationship and apply meaning to
the relationship. Fill in the data table. Plot Period vs. Manipulated Mass and Manipulated Period vs. Mass.
Both plots should be linear. Determine the slope (with units) and equation for each linear graph.
Mass (kg)
Period (s)
0.1
0.60
0.2
0.84
0.3
1.02
0.4
1.20
0.5
1.38
0.8
1.68
1.0
1.98
SLOPE = _________
EQUATION = _______________
SLOPE = _________
EQUATION = _______________
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Graphing Assignment
Due July 6, 2012
GRAPHING AND THE TI-89 CALCULATOR
In this next section, rather than graphing by hand and drawing best-fit lines to determine slope, you will learn to use the
“Data-Matrix” App on your TI-89 calculator to enter data, to create a graph of the data, and to use linear regression
function to determine the slope. Follow the steps attached to the back of this packet to help you answer the
questions.
PROBLEM 4
Below is data collected in an experiment to determine the equation for kinetic energy. The kinetic energy values were
obtained be applying the conservation of energy to a cart launched by a spring with a known spring constant.
x (m)
0
0.0100
0.0150
0.0200
0.0250
0.0300
0.0350
0.0400
v1 (m/s)
0
0.0621
0.0921
0.135
0.170
0.208
0.246
0.279
a.
b.
v2 (m/s)
0
0.0538
0.0953
0.134
0.174
0.212
0.243
0.277
v3 (m/s)
0
0.0568
0.0977
0.134
0.176
0.206
0.243
0.279
vave (m/s)
vave2 (m/s)2
Ek (J)
0
0.00129
0.00289
0.00514
0.00803
0.0116
0.0157
0.0206
Fill in the Blank Columns and enter Average Velocity, Average Velocity Squared, and Kinetic Energy
into the Data Matrix App in the TI-89.
Plot Kinetic Energy on the y-axis and Average Velocity on the x-axis.
* What is the shape of the graph? Write the answer below.
c.
Plot Kinetic Energy on the y-axis and Average Velocity Squared on the x-axis.
* What is the shape of the graph? Write the answer below.
d.
Based on your observations, determine the slope of the linear plot using the linear regression capabilities
of your TI-89. Write the value below.
e.
Based on your observations, what is the KE equation (circle one)?
f.
Using your slope for the linear plot, what is the mass of the object used in this study? Write the value
below.
g.
If the elastic energy associated with a spring is ½ kx2, then what was the average value for the spring
constant, k? Write the value below.
½ mv2
½ mv
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Graphing Assignment
Due July 6, 2012
In a second energy study, a cart is launched up an incline of length (L) and inclined a height (H). The Potential Energy
(mgh) is determined by applying the law of conservation of energy to a spring ( ½ kx2 and with a known spring
constant) that is compressed varying displacements. The cart traveled different distances (d) on the incline based on
the degree of compressions. The cart height (h) is determined by using proportionality. The results are below.
d (m)
0.00
0.044
0.081
0.134
0.193
0.274
0.349
0.449
x (m)
0.000
0.0100
0.0150
0.0200
0.0250
0.0300
0.0350
0.0400
h (m)
0
0.00123
0.00227
0.00375
0.00540
0.00767
0.00977
0.0126
Eg (J)
0
0.0037
0.0084
0.0150
0.0235
0.0338
0.0460
0.0601
a.
Plot Potential energy (Eg) vs. Height (h) on your TI-89.
b.
Using the linear regression capabilities of the TI-89, determine the slope. Write the value below.
c.
If the slope represents mg, determine the mass of the cart. Write the value below.
As a follow-up experiment, a photogate is used to determine the velocity at various heights. The data is below.
d (m)
0
0.20
0.30
0.40
0.50
0.60
0.70
height (m)
0.000
0.006
0.008
0.011
0.014
0.017
0.020
a.
velocity (m/s)
0.000
0.344
0.413
0.474
0.529
0.578
0.625
Eg (J)
Ek (J)
c.
d.
Using the calculated mass from part (c) above and the height in this table, determine the potential energy
values.
Using the calculated mass from part (c) above and the velocity from this table, determine the kinetic
energy values.
Make a plot of Kinetic Energy vs. Potential Energy on your TI-89.
Determine the slope using your TI-89. Write the value below.
e.
What is the significance of the value of the slope?
b.
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Graphing Assignment
Due July 6, 2012
PROBLEM 5
As a final problem, you will analyze data for simple harmonic motion involving a simple pendulum. Answer the
questions based on the results you obtain by using your graphing calculator.
Data Table
Part I Amplitude
Amplitude (°)
Average period (s)
2
1.972
5
1.975
10
1.979
20
1.984
30
2.015
Length (cm)
50
Average period (s)
1.451
60
1.524
70
1.653
80
1.785
90
1.895
100
1.993
Mass (g)
100
Average period (s)
1.973
200
1.997
300
1.993
Part II Length
Part III Mass
Analysis
1. Using TI-89 Calculator, plot a graph of pendulum period vs. amplitude in degrees.
a. Using “PowerReg” instead of “LinReg”, rounding to 1 significant figure, determine the exponent
“b” to which “x” is raised. For example, 0.004 rounds to zero; therefore, y = ax0. Likewise, 2.101
rounds to 2 (meaning squared) and 0.510 rounds to 0.5 (meaning square root).
b.
What is the equation for Period vs. Amplitude according to the Power Regression?
c.
Does the period depend on amplitude? Explain.
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Graphing Assignment
Due July 6, 2012
2.
3.
4.
5.
Using TI-89 Calculator, plot a graph of pendulum period T vs. length .
a. Using “PowerReg” instead of “LinReg”, rounding to 1 significant figure, determine the exponent
“b” to which “x” is raised.
b.
What is the equation for Period vs. Length according to the Power Regression?
c.
Does the period appear to depend on length?
Using TI-89 Calculator, plot the pendulum period vs. mass.
a. Using “PowerReg” instead of “LinReg”, rounding to 1 significant figure, determine the exponent
“b” to which “x” is raised.
b.
What is the equation for Period vs. Mass according to the Power Regression?
c.
Does the period appear to depend on mass? Do you have enough data to answer conclusively?
To examine more carefully how the period T depends on the pendulum length , create the following two
additional graphs of the same data: T2 vs.  and T vs. 2.
a. T2 vs.  : Using “PowerReg” instead of “LinReg”, rounding to 1 significant figure, determine the
exponent “b” to which “x” is raised.
b.
T2 vs.  : What is the equation according to the Power Regression?
c.
T vs. 2: Using “PowerReg” instead of “LinReg”, rounding to 1 significant figure, determine the
exponent “b” to which “x” is raised.
d.
T vs. 2: What is the equation according to the Power Regression?
Using Newton’s laws, we could show that for some pendulums, the period T is related to the length  and
free-fall acceleration g by
T
2

g
T2
2
4
g

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Graphing Assignment
Due July 6, 2012
or
Does one of your graphs support this relationship? Explain. (Hint: Can the term in parentheses be treated as a
constant of proportionality?)
8