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course Phy 201
introductory pendulum experiment#$&*
Phy 201
Your 'introductory pendulum experiment' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Introductory Pendulum Experiment_labelMessages **
** **
In this experiment you will make a simple pendulum and observe how its frequency of oscillation varies with its length.
The goals of this experiment include the following:
•Practice sketching graphs.
•Practice interpreting graphs.
•Obtain a concrete example which will be useful in understanding the essential concept of rate of change.
•Enhance your ability to make accurate observations.
•Quantify limits on the accuracy of data and the effect of these limits.
•Obtain information necessary to use a pendulum as an accurate timing device.
To make the pendulum:
•Tie a light string or thread, about as long as you are tall, around a relatively small dense object.
In the absence of anything more convenient you could use a couple of CDs or DVDs with a string or thread tied through the middle.
A ring might also be a good choice (but avoid using a ring with a lot of value).
A smaller and denser object would be preferable, but don't take a lot of time trying to locate the smallest densest object you can find. Just about anything will do if it is smaller than an average fist and dense enough to sink if it is placed in water (you won't actually be placing the object in water).
•The string or thread should be about 4 feet long.
If you have a ruler or a measuring tape marked in centimeters, you may use it. If you don't, open and print out the file at the link ruler, which can be used to obtain sufficiently accurate measurements.
You will count and time oscillations of a series of simple pendulum, of 9 different lengths.
Determine the lengths of the pendulums to be used in your experiment:
Take your height in inches, subtract 30 and divide your result by 5. Round your result off to the next whole number. This will be the length in centimeters of your first pendulum. (For example if you are 91 inches tall (unlikely but not impossible), you would subtract 30 to get 61, then divide 61 by 5 to get 12.2. This rounds off to 12, so your pendulum would be 12 cm long).
Double the length of your first pendulum. This is the length of your third pendulum.
Double the length of your third pendulum. This is the length of your fifth pendulum.
Double the length of your fifth pendulum. This is the length of your seventh pendulum.
Double the length of your seventh pendulum. This is the length of your ninth pendulum.
Write down the numbers 1 through 9 in the first column of a table, and the lengths you have obtained in the second column, each length opposite the appropriate number. For example if your first pendulum was 12 cm long your table would look something like the following:
number
length
1
12
2
3
24
4
5
48
6
7
96
8
9
192
Note that you aren't asked here to enter your table.
Sketch a rough graph of your lengths
You are going to make a graph of this information.
Sketch the graph by hand (perhaps using the template below). Don't use the computer to construct your graphs in this experiment.
If you want to print and use the template below you may do so, or you may make a rough sketch something like the figure below. Don't bother to use a ruler and make a meticulous graph. Any reasonable freehand sketch is fine (or again, if you wish, you can print the figure below and use it).
Now make a graph of pendulum length vs. number, based on your table. This graph will have length on the vertical axis and the pendulum number on the horizontal. The graph will look something like the one below, which however is based on the unlikely 12 cm initial length for the 91-inch-tall student.
Sketch a smooth curve through the points on your graph. This will give you a graph much like the one below:
Estimate the pendulum lengths which would correspond to numbers 2, 4, 6, and 8.
Complete the table by filling in the lengths corresponding to pendulums number 2, 4, 6 and 8, as estimated from your graph. Try to make reasonable estimates, but don't take a lot of time to make your estimates exact. If you are 91 inches tall your table might be as indicated below:
number
length
1
12
2
15
3
24
4
35
5
48
6
70
7
96
8
140
9
192
Time and count oscillations for the nine different pendulums
Hold the string so that the length from the point at which you are holding it to the center of the object is equal to the first distance on your table (e.g., for the table given above that distance would be 12 cm).
•Start the pendulum oscillating, but don't make it swing too far--keep the distance from one end of the swing to the other less than half the length of the pendulum. To keep it swinging you might need to use the fingers in which you are holding it to slightly nudge the pendulum.
•Using a clock with a second hand, determine how many times this pendulum oscillates in 60 seconds. A complete oscillation, or a complete cycle, is from one extreme point to the other and back. If you count every time your pendulum changes direction, you are counting half-cycles (it's fine to count half-cycles; if you do just be sure to divide by 2 to get the number of complete oscillations). Repeat your count at least a few times, and continue until you are sure you know to the nearest whole cycle (or if you prefer to the nearest half-cycle) how many times it oscillates back and forth in a minute. It shouldn't take more than a few trials to get in the habit of counting accurately, but if you don't count carefully it might. A surprising number of students have difficulty making an accurate 60-second count, and some need the practice.
Repeat this for each of the nine lengths on your table.
Construct a table of the number of oscillations observed vs. the length of the pendulum. For the example given here, corresponding to a person 91 inches tall, the table might look something like the following (the numbers in this example shouldn't be regarded as particularly accurate or useful; your counts should be more accurate than these):
length
number of cycles
12
85
15
75
24
60
35
50
48
40
70
35
96
30
140
25
192
20
Graph your data
You will graph the data in your table.
You may if you wish use the template below to help you construct your graph; you may if you wish print out the figure, or simply make a hand sketch--again be reasonably accurate but don't take the time to be be overly meticulous.
A graph of the sample data for the very tall student would look like the following:
Figure out the time required per oscillation at each length
You know how many times each pendulum oscillated in 60 seconds. From this information you can figure out for each length how many seconds, and/or what fraction of a second, was required for a single oscillation. You should be able to figure out how to do this. Make a table showing the number of seconds required per oscillation vs. the length of the pendulum in cm. The figure below corresponds to the example of the 91-inch-tall individual, and these results are not to be regarded as particularly accurate.
length
time for one cycle
12
0.7
15
0.8
24
1
35
1.2
48
1.5
70
1.7
96
2
140
2.4
192
3
Construct a graph of the time required for a single oscillation vs. the length of the pendulum. You may use the graph below as a template:
For the present example the graph might look like this:
Sketch a smooth curve to represent the data. A good smooth curve for the sample data represented above might look like this:
The curve should come as close as possible, on the average, to the data points, but it should not 'wobble around' in an attempt to actually go through any of the data points. The figure below represents a smooth curve that does wobble around a lot, and this isn't what you want here:
After the 'Your answer' prompt about half a page below you will report your data, using the precise format described below. All reported data will be collected and reported back to the rest of the class (your data will be reported anonymously--only I will know whose data is which), and if anyone's data is not in the prescribed format, it might be necessary to ask everyone to report their data again.
Be sure to save your information as you type it into the copied document. Keep your copies; if the data is required later you will then be easily able to access it.
After the prompt on the first line below the 'Your Answer' prompt, report your vccs email address, the length of the shortest pendulum in centimeters, the number of cycles counted for this pendulum in 60 seconds, and the time required for one complete cycle. Put a comma between each pair of entries. So for the example of the 91-inch-tall student, the first line would read
abc123@email.vccs.edu, 12, 85, .7
Each subsequent line will appear in the same format. So the next two lines for this example student would be
abc123@email.vccs.edu, 15, 75, .8
abc123@email.vccs.edu, 24, 60, 1
It should take you only a couple of minutes to enter this information. You should as usual use copy-and-paste to insert your email address (this will save you time and will ensure that you have given the correct address).
Lines have been provided for up to 8 lengths; however if you followed the instructions you should have observed pendulums of only 6 or 7 different lengths. Extra lines can be left blank.
Your answer (start in the next line):
6, 116, .51
9, 102, .58
12, 90, .67
18, 74, .81
24, 62, .97
36, 50, 1.2
48, 42, 1.4
72, 34, 1.8
your brief discussion/description/explanation: A pendulum was made using fishing line and a deep sea sinker. Because my height in inches is 62, the first pendulum was 6 centimeters. As the length of the pendulum increased, the amount of time for each cycle also increased.
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1. According to your graphs, complete the following tables
length in cm
number of cycles
time for one cycle
10
30
50
70
90
110
130
Enter the numbers from your table in the space below with one line for each length. Each line should contain the length, number of cycles and time for one cycle, separated by tabs.
Your answer (start in the next line):
6 116 .51
9 102 .58
12 90 .667
18 74 .81
24 62 .97
36 50 1.2
48 42 1.43
72 34 1.8
96 30 2
&&&&&& 10 95 .63&&&&&&&&&&&&&&
30 55 1.09
50 40 1.5
70 48 1.25
90 32 1.88
110 27 2.22
&&&&&&130 22 2.73&&&&&&&&&&&
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You were asked to use your graph to predict the number of cycles and time per cycle for lengths of 10, 30, 50, 70, 90, 110 and 130 cm. The results you have reported appear to repeat the original data, rather than making the requested predictions based on your graph.
In the revision I request at the end, you can to insert the requested predictions at this point.
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#$&* length, count, period for given lengths
length in cm
number of cycles
time for one cycle
10
30
50
70
90
Enter the numbers from your table in the space below with one line for each length. Each line should contain the length, number of cycles and time for one cycle, separated by tabs.
Your answer (start in the next line):
6 116 .51
9 102 .58
12 90 .667
18 74 .81
24 62 .97
36 50 1.2
48 42 1.43
72 34 1.8
96 30 2
&&&&&& 10 95 .63&&&&&&&&&&&&&&
30 55 1.09
50 40 1.5
70 48 1.25
90 32 1.88
110 27 2.22
&&&&&&130 22 2.73&&&&&&&&&&&
#$&* length, count, period for given counts
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On this question you were asked to predict the length, according to your graph, and give the time per cycle that would correspond to one-minute counts of 10, 30, 50, 70 and 90.
Please revise this accordingly, as requested at the end of the document.
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length in cm
number of cycles
time for one cycle
0.5
0.9
1.3
1.7
2.1
2.5
Enter the numbers from your table in the space below with one line for each length. Each line should contain the length, number of cycles and time for one cycle, separated by tabs.
Your answer (start in the next line):
6 116 .51
9 102 .58
12 90 .667
18 74 .81
24 62 .97
36 50 1.2
48 42 1.43
72 34 1.8
96 30 2
&&&&&&&& 5 120 .5&&&&&&&&
20 67 .9
40 46 1.3
70 35 1.7
95 29 2.1
&&&&&&&&110 24 2.5&&&&&&&&&
#$&* length, count, period for given periods
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On this question you were asked to give the number of cycles counted and predict the length, according to your graph, for periods of 0.5, 0.9, 1.3, 1.7, 2.1 and 2.5 seconds.
Please revise this accordingly, as requested at the end of the document.
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2. Is the graph of # of cycles vs. length in cm constant, increasing or decreasing? Is it doing so at an increasing, constant or decreasing rate?
On this and on all questions, insert your answer after the 'Answer:' prompt, and include a brief explanation of how you arrived at your answer.
Your answer (start in the next line):
As the length of the pendulum increases, the number of cycles decreases at an increasing rate. I arrived at this answer because for each length in inches, the amount of cycles per minute decreased. This can be attributed to the increase in the amount of space between each endpoint on the cycle. As the length of the cord increased, it allowed the pendulum to swing further to each endpoint but was swinging at a fairly constant speed, therefore, each cycle took longer as the length of the cord increased.
#$&* graph of count vs length
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You've stated this well and you're doing some good thinking about how the graph behaves, but the graph decreases at a decreasing rate. In the 'describing graphs' exercise, the graphs of 2^-x and the graph of x^2 from -3 to 0 were the ones that decreased at decreasing rates.
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3. Is the graph of time required for one cycle vs. length in cm constant, increasing or decreasing? Is it doing so at an increasing, constant or decreasing rate?
Your answer (start in the next line):
As the length was increased, the time for each cycle also increased at a increasing rate. As we discussed in the previous question, when the cord length is increased but the speed of the pendulum movement stays fairly constant, the amount of time for the pendulum to reach in endpoint increases because the distance from these points has increased.
#$&* graph of period vs length
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If graphed with the length on the horizontal axis, this graph will decrease at a decreasing rate, much like the graph of sqrt(x).
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4. How much difference is there between your first two lengths, and how much difference between the number of cycles counted in 60 seconds?
Your answer (start in the next line):
There is a three centimeter difference in my first and second length and 14 cycle difference between these two lengths. This can be seen because the first length was 6 cm and cycled 116 times in 60 seconds, while the second length was 9 cm and cycled 102 times in 60 seconds.
#$&* diff between lengths, between counts
5. How much difference is there between your first two lengths, and how much difference between the corresponding times required to complete a cycle?
Your answer (start in the next line):
There is a 3 centimeter difference between my first two lengths and there is .07 seconds difference between each cycle. This can be seen because the first length was 6 cm and required .51 seconds per cycle while the second length was 9 cm and required .58 seconds per cycle.
#$&* diff between lengths, periods
6. How much difference is there between your last two lengths, and how much difference between the number of cycles counted in 60 seconds?
Your answer (start in the next line):
There is a 24 centimeter difference for the last two lengths and there was a difference of 4 cycles in 60 seconds time. This can be seen because the lengths were 72 and 96 and the cycles per minute were 34 and 30.
#$&* diff between last two lengths, counts
7. How much difference is there between your last two lengths, and how much difference between the corresponding times required to complete a cycle?
Your answer (start in the next line):
The difference between the last two lengths was 24 cm and the time for each cycle was .2 seconds. This can be seen because the length of 72 cm had a cycle time of 1.8 while the length of 96 cm had a cycle time of 2 seconds.
#$&* diff between last two lengths, periods
8. Is your graph of number of cycles counted vs. length in cm steeper, on the average, between the first two lengths or between the last two lengths?
Your answer (start in the next line):
It is steaper for the first two lengths than the last two because there was a 14 cylce difference between the first two and only a 4 cycle difference between the last two.
#$&* count vs. length steeper between 1st two or last two pts
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Good.
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9. Is your graph of time required to complete a cycle vs. length in cm steeper, on the average, between the first two lengths or between the last two lengths?
Your answer (start in the next line):
It is steeper for the first two than it is for the last two because the amount of time for the first two had a larger jump than the last two.
#$&* period vs. length steeper between 1st two or last two pts
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Good.
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10. The curve you sketched for your graph of (time required to complete a cycle) vs. (length) cannot possibly pass through the center of each of your points.
•What is the greatest vertical distance between a point of your graph and the curve?
•What do you think is the least vertical distance?
You will answer these questions at the 'your answer' prompt a little ways below.
(For example, in the figure below a curve has been constructed based on three data points. The first and third data point lie slightly above the curve, the second point slightly below. The second point is probably the one which lies furthest from the curve, at a distance of approximately .03 vertical units below. This distance is roughly estimated based on the scale of the graph. The first point is perhaps .01 vertical units above the curve, and the third is perhaps .02 units above.)
Your answer (start in the next line):
The point for the length of 12 cm is the only point below the curve and is only slightly (.01 below) and the point for 48 cm was the highes above the line at about .08 above the curve.
#$&* greatest, least vert dist between datapt and curve
*#&!
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Very good overall, but the tables you gave did not answer the questions, and will need to be revised.
Your thinking about the graphs was excellent, but be sure to check my notes. No need to revise that part unless you have questions.
&#Please see my notes and submit a copy of this document with revisions, comments and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end).
Be sure to include the entire document, including my notes.
&#
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Good revision. Thanks.
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