Graphing Assignment
Due July 8, 2011
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.)
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 = ______
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 = ______SLOPE = ______
EQUATION = ______EQUATION = ______
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) / v1 (m/s) / v2 (m/s) / v3 (m/s) / vave (m/s) / vave2 (m/s)2 / Ek (J)0 / 0 / 0 / 0 / 0
0.0100 / 0.0621 / 0.0538 / 0.0568 / 0.00129
0.0150 / 0.0921 / 0.0953 / 0.0977 / 0.00289
0.0200 / 0.135 / 0.134 / 0.134 / 0.00514
0.0250 / 0.170 / 0.174 / 0.176 / 0.00803
0.0300 / 0.208 / 0.212 / 0.206 / 0.0116
0.0350 / 0.246 / 0.243 / 0.243 / 0.0157
0.0400 / 0.279 / 0.277 / 0.279 / 0.0206
a. Fill in the Blank Columns and enter Average Velocity, Average Velocity Squared, and Kinetic Energy into the Data Matrix App in the TI-89.
b. 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)? ½ mv2 ½ mv
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.
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.
Dx (m) / d (m) / h (m) / Eg (J)0.000 / 0.00 / 0 / 0
0.0100 / 0.044 / 0.00123 / 0.0037
0.0150 / 0.081 / 0.00227 / 0.0084
0.0200 / 0.134 / 0.00375 / 0.0150
0.0250 / 0.193 / 0.00540 / 0.0235
0.0300 / 0.274 / 0.00767 / 0.0338
0.0350 / 0.349 / 0.00977 / 0.0460
0.0400 / 0.449 / 0.0126 / 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) / height (m) / velocity (m/s) / Eg (J) / Ek (J)0 / 0.000 / 0.000
0.20 / 0.006 / 0.344
0.30 / 0.008 / 0.413
0.40 / 0.011 / 0.474
0.50 / 0.014 / 0.529
0.60 / 0.017 / 0.578
0.70 / 0.020 / 0.625
a. Using the calculated mass from part (c) above and the height in this table, determine the potential energy values.
b. Using the calculated mass from part (c) above and the velocity from this table, determine the kinetic energy values.
c. Make a plot of Kinetic Energy vs. Potential Energy on your TI-89.
d. Determine the slope using your TI-89. Write the value below.
e. What is the significance of the value of the slope?
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
Part II Length
Length (cm) / Average period (s)50 / 1.451
60 / 1.524
70 / 1.653
80 / 1.785
90 / 1.895
100 / 1.993
Part III Mass
Mass (g) / Average period (s)100 / 1.973
200 / 1.997
300 / 1.993
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.
2. 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?
3. 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?
4. 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?
5. 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
or
Does one of your graphs support this relationship? Explain. (Hint: Can the term in parentheses be treated as a
constant of proportionality?)
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