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Pendulum and Chaos
A swinging pendulum keeps a very regular beat. It is so regular, in fact, that for many years the pendulum was the heart of clocks used in astronomical measurements at the Greenwich Observatory.
There are at least three things you could change about a pendulum that might affect the period (the time for one complete cycle):
- the amplitude of the pendulum swing
- the length of the pendulum, measured from the center of the pendulum bob to the point of support
- the mass of the pendulum bob
To investigate the pendulum, you need to do a controlled experiment; that is, you need to make measurements, changing only one variable at a time. Conducting controlled experiments is a basic principle of scientific investigation.
In this experiment, you will use a Photogate capable of microsecond precision to measure the period of one complete swing of a pendulum. By conducting a series of controlled experiments with the pendulum, you can determine how each of these quantities affects the period.
STUDENT OUTCOMES
Through this experiment, students will be able to:
-Analysis simple harmonic motion.
-Verify a pendulum period’s dependencies.
-Verify the acceleration due to gravity value.
MATERIALS
Tablet PC Computer Laptop / Vernier computer interfaceLogger Pro / Vernier photogate
Protractor, Meterstick / String and hanger
PRELIMINARY QUESTIONS
1. Define clearly what the period of a pendulum is.
2. Three factors might affect the period of a pendulum: the length of the string, the mass of the hanger and the initial angle. Which one do you think has the most effect?
PROCEDURE
1. Attach the string to the hanger at one end and to the horizontal bar at the other on end. Ensure that you have the longest possible length available (around 1.4 m). The photogate detector should be set up vertically on the floor, so that it looks like a U shape. Make several trials so that you can make the pendulum oscillate without touching the detector.
2. Open the file “14 Pendulum Periods” in the Physics with Computers folder. A graph of period vs. time is displayed.
3. Temporarily move the mass out of the center of the Photogate. Notice the reading in the status bar of Logger Pro at the bottom of the screen, which shows when the Photogate is blocked. Block the Photogate with your hand; note that the Photogate is shown as blocked. Remove your hand, and the display should change to unblocked. Click and move your hand through the Photogate repeatedly. After the first blocking, Logger Pro reports the time interval between every other block as the period. Verify that this is so.
4. Now you can perform a trial measurement of the period of your pendulum. Pull the mass to the side about 10º from vertical and release. Click and measure the period for five complete swings. Click . Click the Statistics button, , to calculate the average period. You will use this technique to measure the period under a variety of conditions.
4. Part I Amplitude
Determine how the period depends on amplitude for a fixed length (60 cm) and mass (50 g). Measure the period for different amplitudes. Each time, measure the amplitude using the protractor so that the mass with the string is released at a known angle. Repeat the procedure 5 times for each amplitude. Record the data in your data table, and calculate the average period for each amplitude.
5. Part II Mass
Usedifferent masses to determine if the period is affected by changing the mass for a fixed length (100 cm) and amplitude. Measure the period of the pendulum constructed with each mass, taking care to keep the distance from the ring stand rod to the center of the mass the same each time, as well as keeping the amplitude the same. Repeat the procedure 5 times for each mass, using an amplitude of about 10°. Record the data in your data table, and calculate the average period for each amplitude.
6. Part III Length
Use the method you learned above to investigate the effect of changing pendulum length on the period for a fixed mass and amplitude. Use the 50 g mass and a consistent amplitude of 10º for each trial. Vary the pendulum length in steps of 10cm, from 1.4 m to 0.40 m. Make sure to measure your length from the position where the string is attached down to the estimated center of mass of your hanger. Repeat Step 5 for each length. Record the data in the second data table below. Measure the pendulum length from the rod to the middle of the mass.
Data Table
- Part I Amplitude
Amplitude / Trial 1 / Trial 2 / Trial 3 / Trial 4 / Trial 5 / Average period
(°) / (s)
1
2
3
4
5
6
7
8
9
10
15
20
25
30
35
40
- Part II Mass
Mass / Trial 1 / Trial 2 / Trial 3 / Trial 4 / Trial 5 / Average period
(g) / (s)
50
55
60
65
70
75
- Part III Length
Length / Trial 1 / Trial 2 / Trial 3 / Trial 4 / Trial 5 / Average period
(cm) / (s)
140
130
120
110
100
90
80
70
60
50
40
- Part IVdouble pendulum
There is a demo setup in the lab. Play with it, give the inner ball an angle of 30 degree and the outer ball an angle of 90 with respect to the pole, and let them go. You will observe very weird behavior, while trying to measure the period you might even go crazy… to avoid this little problem, use:
and use the stop watch to estimate the period of the inner pendulum using the following parameters:
Gather 5 measurements of the inner pendulum’s period and calculate the standard deviation. Go back to your table of part I (amplitude vs period) and pick any amplitude row. You have calculated the average for 5 values, now calculate their standard deviation.
Compare these 2 standard deviations. Which one is the largest?
The standard deviation of the double pendulum is expected to be higher. The reason is that a double pendulum is a chaotic system, which varies a lot and very susceptible to initial conditions. In order to play with MORE chaos, let’s investigate the logistics equation defined as:
Now, at first sight this looks like a quite harmless equation… ;)
Use excel, and produce 100 data points using the following parameters. This means that you plug the initial values of and in this equation and obtain (second data point); then you reuse this value which you will plug back for in the above equation to get another (third data point), etc… do this 100 times using EXCEL!!!! Otherwise you’ll be still be doing this next month.... if you do not know how to use formulae in excel, now is a good time to ask.
Case1: , and
Case2: , and
Case3: , and
Case4: , and
Case5: , and
Obtain a graph for each case, and answer the following questions:
Case1: Towards which value does X converge?
Case2: Towards which value does X converge?
Case3: Towards which value does X converge?
What is the difference in the convergence pattern between case2, and case3?
Case4: Towards which values does X converge?
Case5: Towards which values does X converge?
If we were to increase r to 3.55, can you predict the number of values X converges to?
Analysis
1. Using your first table, use excel and plot your average periods vs amplitude. Make sure to includethe best fit line and its equation
You need to make two separate graphs and analysis for this question.
First, plot your values for the amplitudes between: 1 and 10
1)Make sure to includethe best fit line and its equation (do not force the y intercept to be zero! Let Excel choose it for you)
2)Choose a y scale from 0 to 3 seconds.
INCLUDE YOUR 2 GRAPHS HERE
Based on the value of the slope, what is the effect of amplitude on the period of a pendulum?
Second, plot your values for the amplitudes between: 10 and 40
3)Do not include a best fit line – just look at the trend.
4)Choose a y scale from 0 to 3 seconds.
INCLUDE YOUR 2 GRAPHS HERE
Based on the value of the slope, what is the effect of amplitude on the period of a pendulum?
2. Using your second table, use excel and plot your average periods vs mass.
1)Make sure to includethe best fit line and its equation (do not force the y intercept to be zero! Let Excel choose it for you)
2)Choose a y scale from 0 to 3 seconds.
INCLUDE YOUR GRAPH HERE
Based on the value of the slope, what is the effect of mass on the period of a pendulum?
- Using your third table, use excel and plot your average periods vs length.
1)Choose a y scale from 0 to 3 seconds.
INCLUDE YOUR GRAPH HERE
I’m sure you would looooove to include a linear best fit as well, wouldn’t you? Do you think you should? Explain (hint: see the functional of the period on the next question).
4. Using Newton’s laws and the small angle approximation, we could show that for some pendulums, the period T is related to the length l and the acceleration due to gravity g by
c.
Linearize such that this equation looks just like the line equation. Then identify clearlyyour new y value (x is still the length) such that the slope of your new line is proportional to the acceleration due to gravity.
Include a new table that shows the calculation of your new y value; include your values as well.
INCLUDE YOUR TABLE HERE
Using Excel, plot your values of new values of y vs . Make sure to include the best fit line and its equation. Pay special care concerning the y intercept that you must choose ( value).Extract the slope of your line and include the units. Solve for the acceleration due to gravity using your slope.
INCLUDE YOUR GRAPH HERE
Give your experimental value of including units.
You know that the accepted value for is . Calculate the percent error, and based on the value you obtain, make a clear statement concerning the accuracy of your measurement.
5. If the period of the pendulum is determined without using the small-angle approximation, the period is
Using this equation, calculate the periods for the amplitudes you used in your table: Create a new table and a graph of your convenience with these new values. Based on your results, when does the period start to depend on the amplitude? Explain.
All the above procedure and analysis questions were present to HELP you figure out what is going on. Now, you are expected to write a full scientific report for your experiment that includes:
Title
Introduction
Method
Results
Conclusion
Make sure that you include this structure clearly (that is, when you start your introduction, start your paragraph with the word: Introduction; when you start your method section, make sure to include the word: Method, etc..)
Explanation for each part
Introduction.
Introduce the device you will be working with, and describe it (a holy pendulum). Complete your introduction by stating your hypothesis: you have several.
1)Correlation between small angles and period
2)Correlation between large angles and period
3)Correlation between period and mass
4)Correlation between length and period
5)Using 4) you want to show that a pendulum can be used to predict the acceleration due to gravity.
6)Suggest that a double pendulum has more variation in its period than a single pendulum using its standard deviation (suggesting chaotic behavior).
7)Verify how initial conditions in the logistic equation produce chaotic behavior.
Method
Describe the conditions and steps to obtain your values (make it short, concise, but complete enough so that anyone who doesn’t know anything about physics is about to reproduce your experiments just by reading your method section).
Results
Your results section should have as many sections as the number of hypothesis: 5 Make sure to separate them clearly.
PartI – short amplitudes vs period
Include a short statement introducing the following results
Include your table of values (border, units, labels, etc..)
Include your graph of average period vs amplitude (label, units, etc…)
Based on the information you are able to extract from your graph, make a clear statement whether your hypothesis has been verified experimentally or not.
Make another clear statement whether your hypothesis can be verified theoretically as well: in order to do this, use:
PartII – large amplitudes vs period
Include a short statement introducing the following results
Include your table of values (border, units, labels, etc..)
Include your graph of average period vs amplitude (label, units, etc…)
Based on the information you are able to extract from your graph, make a clear statement whether your hypothesis has been verified experimentally or not.
Make another clear statement whether your hypothesis can be verified theoretically as well: in order to do this, use: and the highest value from your table (largest amplitude value) to support your experimental value. Make sure to include a calculation of the percent error to demonstrate your accuracy.
PartIII – mass vs period
Include a short statement introducing the following results
Include your table of values (border, units, labels, etc..)
Include your graph of average period vs mass (label, units, etc…)
Based on the information you are able to extract from your graph, make a clear statement whether your hypothesis has been verified experimentally or not.
Make another clear statement whether your hypothesis can be verified theoretically as well: in order to do this, use:
PartIV – length vs period
Include a short statement introducing the following results
Include your table of values (border, units, labels, etc..)
Include your graph of average period vs length (label, units, etc…)
Based on the information you are able to extract from your graph, make a clear statement whether your hypothesis has been verified experimentally or not.
Make another clear statement whether your hypothesis can be verified theoretically as well: in order to do this, use:
Based on both experimental and theoretical basis, is the relationship between length and period linear? Justify.
PartV – Use length2 vs period to predict the value of g.
Include a short statement introducing the following results and what you are doing to obtain them (linearization).
Include your table of values (border, units, labels, etc..)
Include your graph of average period vs mass (label, units, etc…)
Based on the information you are able to extract from your graph, show your theoretical work needed to predict the value of g.
Calculate the percent error between your experiment value and the acceptable value of g at the surface of the earth. Discuss the accuracy of your result (if your accuracy is larger than 5%, check your graph, calculation and/or redo your data acquisition… yeap… coz it can only mean you have been sloppy, sorry…;).
PartVI – Estimation of the period variation between single and double pendulum
Include a short statement introducing the following results.
Include your table of values (border, units, labels, etc..)
Include both your standard deviations for single and double pendulum. Compare them using the . Depending on which one has the largest standard deviation and the most variability in the data, state which system is more likely to be chaotic.
PartVII – Show how initial condition affect the outcome of a chaotic system
Do not include your data point, but introduce the logistic equation, and a short statement of how you are going to use it.
Include all the graphs for each case, and answer the questions asked in procedure 4.
Conclusion
Restate all the hypothesis you have verified and based on the accuracy of your partV, what can you say about the utility of a simple pendulum to predict g?