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21:48:28 `q001. The frequency of a pendulum is how frequently it oscillates back and forth. A very short pendulum oscillates much more quickly than a very long pendulum.
A cycle is a complete oscillation, from one extreme point to the other and back. Frequency is usually measured in cycles/second. However, it could be measured in cycles/minute, cycles/millisecond, cycles/year, or cycles/(time unit), where (time unit) is any unit of time. In this experiment you will observe and measure frequency in cycles/minute. If you hang your arm loosely at your side and nudge it a bit, it should oscillate back and forth a few times. Try it. Estimate how many complete oscillations it will undergo in a minute if you keep nudging it slightly to keep it moving.......!!!!!!!!...................................
RESPONSE --> 54 oscillations/min would be an estimate of my arm oscillating with a nudging constantly being applied.
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21:49:06 If you managed to keep your arm relaxed it probably took a bit over a second to complete one back-and-forth oscillation. For most people a relaxed arm will oscillate about 40 to 50 times in a minute.
If your arm is tense, then you probably had to force the oscillations in you might have ended up with anything from about 30 to 100 times a minute.......!!!!!!!!...................................
RESPONSE --> I ended up with 54 so I might have been a little tense.
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21:52:18 `q002. In this activity you will:
1. Observe the frequency f of a pendulum vs. its length L. 2. Make a table of f vs. L. 3. Make a graph of f vs. L. Note the following conventions for determining which variable is independent and which dependent, and for placement of the dependent and independent variables on tables and graphs. f depends on L. We control L by holding the pendulum string at different lengths and observe its effect on f, so 1. f is the dependent variable, L the independent. 2. When we make a table, the independent variable goes in the first column, the dependent in the second. 3. When we make a graph, the independent variable goes horizontally across the page, and the dependent variable up and down (dependent is vertical vs. independent, which is horizontal). Note how this relates to the traditional way of graphing y vs. x, with y vertical and x horizontal. Summarize these conventions in your own words.......!!!!!!!!...................................
RESPONSE --> The length of the pendulum is the independent factor which in a graph will take the place of the x-axis. This means that the time will be represented by the y-axis.
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21:54:16 Predictions
Imagine a rock or some other mass hanging by a string over the rail of a high deck. As the rock swings back and forth, you gradually let more string over the rail, lowering the mass. Will the swings take longer and longer, or will they become more and more frequent? If it requires 2 seconds for the rock to swing back and forth from a certain length of string, how long do you think it would take if the string was twice as long? What if the string was half as long?......!!!!!!!!...................................
RESPONSE --> As the length of the rope is increased the time for each oscillation will increase. If the length of the string is doubled it would stand to reason that the time for each oscillation would double as well. The same standard would be used if it were cut in half, it would take only half the time.
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21:55:37 `q003. Sketch a graph of the time required for a swing vs. pendulum length, as you would predict it. Describe your graph in some detail, using the conventions outlined in the 'Describing Graphs' exercise.
As with all graphs, be careful to use consistent units on each axis. If the marks on an axis are equally spaced, then they should represent the same change in the quantity represented by that axis. A common error would be, if for example the numbers 5, 10 and 20 were to be represented, would be to make three equally spaced marks representing the numbers 5, 10 and 20. The spacing between 5 and 10 should be half the spacing between 5 and 20. If you're not careful about your spacing, you will distort the shape of your graph.......!!!!!!!!...................................
RESPONSE --> As the x coordinate increase so would the y coordinate. In other words the length would cause the time interval of the oscillation to increase as well.
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21:56:38 If you said, as many people do, that a string twice as long to would imply a 4 second period (the period is the time per swing) and that a string half as long would imply a 1 second period, then you probably are assuming a linear relationship between period and length. In this case your graph should have been a straight line, a line which is decreasing at a constant rate.
If you said that a doubled length implies less than a doubled period, then to be consistent you probably said that half the length implies more than half the period. In this case your graph would be increasing but at a decreasing rate. Other responses are possible. The experiment will tell you whether your prediction was correct.......!!!!!!!!...................................
RESPONSE --> I understand I was reversed in my thinking.
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22:01:10 `q004. If a pendulum of a certain length swings back and forth 35 times per minute, how many times per minute do you think it would take if the string was twice as long? What if the string was half as long?
Sketch a graph of the number of swings per minute vs. pendulum length, as you would predict it. Describe your graph. Be sure to describe how your graph behaves as pendulum length approaches zero.......!!!!!!!!...................................
RESPONSE --> If the string was twice as long the oscillation period would decrease to 17.5 times per minute. On the other hand if the string was cut in half then the period of oscillation would increase to 70 times per minute.
The graph would iindicate a straight line that would be decreasing at a constant rate. As the line approaches x = zero then the osicillation period would continue to increase..................................................
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22:01:41 The most common responses that the pendulum would swing back and forth 70 times per minute if the string was half is long, and 17.5 times per minute if the pendulum was twice as long. This would be consistent with the most common answer to the preceding question, which resulted in an increasing linear graph. The present graph would be decreasing, as we would expect, and would still be linear.
A prediction that half the length would imply less than 70 cycles per minute would be consistent with the idea that doubling the length will imply more than 17.5 cycles per minute. This would result in a graph that decreases but at a decreasing rate.......!!!!!!!!...................................
RESPONSE --> understood
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22:02:34 `q005. How are your answers to the first two questions related to your answers to the last two? Are they consistent?
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RESPONSE --> The thought pattern was consistent for the most part I misunderstood one of the first question.
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22:06:23 The Experiment
Frequency is measured in cycles/time units. Here we will measure frequency in cycles/minute, because it's easy to count the cycles in a minute. First make the pendulum: Tie a light string around a rock, a potato, or any relatively small, dense object. The length of the pendulum is measured from the fixed point of the swing to the center of gravity of the object. Describe how you constructed your pendulum.......!!!!!!!!...................................
RESPONSE --> I tied a piece of thread to a nut.
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22:08:52 `q006. Get the 'feel' of the pendulum.
Feel how quickly a pendulum 1 foot long swings back and forth. Don't time anything yet; just get the feel of the thing. Then get the feel of the rhythm of a two foot pendulum, then a three foot pendulum, then a4 foot pendulum. Try walking to the rhythm of each, stepping (either one or two steps to the cycle) the same distance on each 'beat' of the pendulum. Does the speed of your walk change by the same amount each time the length of the pendulum increases by a foot?......!!!!!!!!...................................
RESPONSE --> As the length of the pendulum increase my rate of speed decreases.
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22:13:04 What would a graph of your speed vs. time look like? Would it be a straight line, would it curve upward, or would it go upward and tend to level off?
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RESPONSE --> The graph of my speed versus time would be decreasing however I am not sure if it would be decreasing at a constant rate or at a decreasing rate.
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22:26:33 `q007. Observe the number of cycles in a minute for different lengths:
Measure different lengths, from about a foot to the longest pendululum you can easily manage. Increase the length by the same amount each time, and time the pendulum for a minute at each length. Choose your length increment so you will end up with about six observations (i.e., so you will have six different lengths). Don't let the pendulum swing too far (no more than about 10 degrees from vertical). The frequency f, in cycles/minute, is the number of complete cycles in the minute. Write down in a table the frequency f and length L for each length. Let the first column be the length, the second the frequency. Label this table Data Set 1, and include the table in your response here.......!!!!!!!!...................................
RESPONSE --> Data Set 1 Length (cm) Frequency (cycles/min) 5 128 10 90 15 72 20 60 25 54 30 48
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22:28:16 Analyzing The Experiment
Look at your data for Data Table 1 and see how changes in frequency are related to changes in length: Look at the numbers on the table. Do the frequencies change regularly with the lengths, or to they change faster and faster, or slower and slower?......!!!!!!!!...................................
RESPONSE --> The frequencies tend to change at a regular rate. The tend to decrease by 1/3 whenever the length is doubled.
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22:30:32 `q008. As you look at the numbers, try to visualize the graph.
Sketch a graph of the general shape of your f vs. L data. Don't mark off a scale, don't plot points, just sketch the basic shape, from your examination of the numbers.......!!!!!!!!...................................
RESPONSE --> completed
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22:31:17 Describe the graph.
How does the graph reflect the behavior of the numbers on the table? What is it about the graph that is related to the 'feel' of the walking exercise?......!!!!!!!!...................................
RESPONSE --> The graph is a curved line that is decreasing at a decreasing rate. As the length of the pendulum increased my speed decreased.
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22:32:25 As the length increases, your table will show that the frequency decreases. That is, for a longer pendulum you will count fewer complete cycles in a minute.
Your table will also show that equal increases in the length of the pendulum result in smaller and smaller decreases in the frequency. For example, between a 1-foot pendulum and a 2-foot pendulum the frequency changes from about 55 cycles per minute to about 40 cycles per minute, the decrease of about 15 cycles per minute, while changing from a four-foot pendulum to a five-foot pendulum the decrease would be from about 28 cycles per minute to about 25 cycles per minute, a decrease of only about 3 cycles per minute. Note that the numbers given here are very approximate, so if your results differ by a few cycles per minute it is no cause for alarm. This behavior will cause the graph to decrease at a decreasing rate. With the walking exercise you should have noticed that each increase in length resulted in a decrease in walking speed, but that the decrease was less for the longer pendulums than for the shorter.......!!!!!!!!...................................
RESPONSE --> understood
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22:37:18 `q009. Sketch the graph by marking a scale and plotting points and compare with your rough sketch. How does the general shape of your graph compare with the shape of the rough sketch?
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RESPONSE --> The graphs are similar.
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22:38:56 Consider the period of the pendulum vs. length:
The period of a pendulum is the time required for 1 complete cycle. That is, if the pendulum requires 2 seconds to complete a cycle, the period is 2 seconds. Before doing any calculations to find the period, recall your direct experience of the pendulum. Does the period increase or decrease with length? What do you think a graph of the period vs. length would look like? Sketch a rough graph of period vs. length, and describe it in words.......!!!!!!!!...................................
RESPONSE --> The period would increase with length.
The graph would be increasing at an increasing rate. The graph is increasing at an increasing rate..................................................
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22:42:00 `q010. From your data, figure out the period associated with each length. Make a table of period vs. length, label it clearly, and include the table in your response here.
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RESPONSE --> Length Period 5 0.008 10 0.011 15 0.014 20 0.017 25 0.019 30 0.021
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22:44:16 The period is the number of seconds required for cycle. For example, if there are 30 cycles in a minute, then it takes 2 seconds for each cycle and the period is 2 seconds. If there are 40 cycles in a minute, 40 cycles require 60 seconds and a single cycle takes 60/40 seconds = 1.5 seconds.
If you miscalculated your periods, recalculate them and show the corrected table. Look at the numbers, asking yourself the same sort of questions as before. Visualize a graph of the numbers in this table. Then draw the graph. Can you look at the graph in such a way as to invoke the 'feel' of the things you have observed? Describe how the graph you made from the table is like, and how it is different than the graph you sketched before you made the table.......!!!!!!!!...................................
RESPONSE --> The graph seems to be increasing at a constant rate.
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犒‰Äíù «¯©ªÚ̪Íç{éõ± Student Name: assignment #001
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22:46:02 `q001. The frequency of a pendulum is how frequently it oscillates back and forth. A very short pendulum oscillates much more quickly than a very long pendulum.
A cycle is a complete oscillation, from one extreme point to the other and back. Frequency is usually measured in cycles/second. However, it could be measured in cycles/minute, cycles/millisecond, cycles/year, or cycles/(time unit), where (time unit) is any unit of time. In this experiment you will observe and measure frequency in cycles/minute. If you hang your arm loosely at your side and nudge it a bit, it should oscillate back and forth a few times. Try it. Estimate how many complete oscillations it will undergo in a minute if you keep nudging it slightly to keep it moving.......!!!!!!!!...................................
RESPONSE --> ok
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22:46:06 If you managed to keep your arm relaxed it probably took a bit over a second to complete one back-and-forth oscillation. For most people a relaxed arm will oscillate about 40 to 50 times in a minute.
If your arm is tense, then you probably had to force the oscillations in you might have ended up with anything from about 30 to 100 times a minute.......!!!!!!!!...................................
RESPONSE --> ok
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22:46:13 `q002. In this activity you will:
1. Observe the frequency f of a pendulum vs. its length L. 2. Make a table of f vs. L. 3. Make a graph of f vs. L. Note the following conventions for determining which variable is independent and which dependent, and for placement of the dependent and independent variables on tables and graphs. f depends on L. We control L by holding the pendulum string at different lengths and observe its effect on f, so 1. f is the dependent variable, L the independent. 2. When we make a table, the independent variable goes in the first column, the dependent in the second. 3. When we make a graph, the independent variable goes horizontally across the page, and the dependent variable up and down (dependent is vertical vs. independent, which is horizontal). Note how this relates to the traditional way of graphing y vs. x, with y vertical and x horizontal. Summarize these conventions in your own words.......!!!!!!!!...................................
RESPONSE --> ok
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22:46:16 Predictions
Imagine a rock or some other mass hanging by a string over the rail of a high deck. As the rock swings back and forth, you gradually let more string over the rail, lowering the mass. Will the swings take longer and longer, or will they become more and more frequent? If it requires 2 seconds for the rock to swing back and forth from a certain length of string, how long do you think it would take if the string was twice as long? What if the string was half as long?......!!!!!!!!...................................
RESPONSE --> ok
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22:46:18 `q003. Sketch a graph of the time required for a swing vs. pendulum length, as you would predict it. Describe your graph in some detail, using the conventions outlined in the 'Describing Graphs' exercise.
As with all graphs, be careful to use consistent units on each axis. If the marks on an axis are equally spaced, then they should represent the same change in the quantity represented by that axis. A common error would be, if for example the numbers 5, 10 and 20 were to be represented, would be to make three equally spaced marks representing the numbers 5, 10 and 20. The spacing between 5 and 10 should be half the spacing between 5 and 20. If you're not careful about your spacing, you will distort the shape of your graph.......!!!!!!!!...................................
RESPONSE --> ok
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22:46:21 If you said, as many people do, that a string twice as long to would imply a 4 second period (the period is the time per swing) and that a string half as long would imply a 1 second period, then you probably are assuming a linear relationship between period and length. In this case your graph should have been a straight line, a line which is decreasing at a constant rate.
If you said that a doubled length implies less than a doubled period, then to be consistent you probably said that half the length implies more than half the period. In this case your graph would be increasing but at a decreasing rate. Other responses are possible. The experiment will tell you whether your prediction was correct.......!!!!!!!!...................................
RESPONSE --> ok
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22:46:24 `q004. If a pendulum of a certain length swings back and forth 35 times per minute, how many times per minute do you think it would take if the string was twice as long? What if the string was half as long?
Sketch a graph of the number of swings per minute vs. pendulum length, as you would predict it. Describe your graph. Be sure to describe how your graph behaves as pendulum length approaches zero.......!!!!!!!!...................................
RESPONSE --> ok
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22:46:26 The most common responses that the pendulum would swing back and forth 70 times per minute if the string was half is long, and 17.5 times per minute if the pendulum was twice as long. This would be consistent with the most common answer to the preceding question, which resulted in an increasing linear graph. The present graph would be decreasing, as we would expect, and would still be linear.
A prediction that half the length would imply less than 70 cycles per minute would be consistent with the idea that doubling the length will imply more than 17.5 cycles per minute. This would result in a graph that decreases but at a decreasing rate.......!!!!!!!!...................................
RESPONSE --> ok
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22:46:29 `q005. How are your answers to the first two questions related to your answers to the last two? Are they consistent?
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RESPONSE --> ok
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22:46:32 The Experiment
Frequency is measured in cycles/time units. Here we will measure frequency in cycles/minute, because it's easy to count the cycles in a minute. First make the pendulum: Tie a light string around a rock, a potato, or any relatively small, dense object. The length of the pendulum is measured from the fixed point of the swing to the center of gravity of the object. Describe how you constructed your pendulum.......!!!!!!!!...................................
RESPONSE --> ok
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22:46:34 `q006. Get the 'feel' of the pendulum.
Feel how quickly a pendulum 1 foot long swings back and forth. Don't time anything yet; just get the feel of the thing. Then get the feel of the rhythm of a two foot pendulum, then a three foot pendulum, then a4 foot pendulum. Try walking to the rhythm of each, stepping (either one or two steps to the cycle) the same distance on each 'beat' of the pendulum. Does the speed of your walk change by the same amount each time the length of the pendulum increases by a foot?......!!!!!!!!...................................
RESPONSE --> ok
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22:46:37 What would a graph of your speed vs. time look like? Would it be a straight line, would it curve upward, or would it go upward and tend to level off?
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RESPONSE --> ok
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22:46:39 `q007. Observe the number of cycles in a minute for different lengths:
Measure different lengths, from about a foot to the longest pendululum you can easily manage. Increase the length by the same amount each time, and time the pendulum for a minute at each length. Choose your length increment so you will end up with about six observations (i.e., so you will have six different lengths). Don't let the pendulum swing too far (no more than about 10 degrees from vertical). The frequency f, in cycles/minute, is the number of complete cycles in the minute. Write down in a table the frequency f and length L for each length. Let the first column be the length, the second the frequency. Label this table Data Set 1, and include the table in your response here.......!!!!!!!!...................................
RESPONSE --> ok
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22:46:42 Analyzing The Experiment
Look at your data for Data Table 1 and see how changes in frequency are related to changes in length: Look at the numbers on the table. Do the frequencies change regularly with the lengths, or to they change faster and faster, or slower and slower?......!!!!!!!!...................................
RESPONSE --> ok
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22:46:44 `q008. As you look at the numbers, try to visualize the graph.
Sketch a graph of the general shape of your f vs. L data. Don't mark off a scale, don't plot points, just sketch the basic shape, from your examination of the numbers.......!!!!!!!!...................................
RESPONSE --> ok
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22:46:47 Describe the graph.
How does the graph reflect the behavior of the numbers on the table? What is it about the graph that is related to the 'feel' of the walking exercise?......!!!!!!!!...................................
RESPONSE --> ok
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22:46:50 As the length increases, your table will show that the frequency decreases. That is, for a longer pendulum you will count fewer complete cycles in a minute.
Your table will also show that equal increases in the length of the pendulum result in smaller and smaller decreases in the frequency. For example, between a 1-foot pendulum and a 2-foot pendulum the frequency changes from about 55 cycles per minute to about 40 cycles per minute, the decrease of about 15 cycles per minute, while changing from a four-foot pendulum to a five-foot pendulum the decrease would be from about 28 cycles per minute to about 25 cycles per minute, a decrease of only about 3 cycles per minute. Note that the numbers given here are very approximate, so if your results differ by a few cycles per minute it is no cause for alarm. This behavior will cause the graph to decrease at a decreasing rate. With the walking exercise you should have noticed that each increase in length resulted in a decrease in walking speed, but that the decrease was less for the longer pendulums than for the shorter.......!!!!!!!!...................................
RESPONSE --> ok
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22:46:54 `q009. Sketch the graph by marking a scale and plotting points and compare with your rough sketch. How does the general shape of your graph compare with the shape of the rough sketch?
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RESPONSE --> ok
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22:46:56 Consider the period of the pendulum vs. length:
The period of a pendulum is the time required for 1 complete cycle. That is, if the pendulum requires 2 seconds to complete a cycle, the period is 2 seconds. Before doing any calculations to find the period, recall your direct experience of the pendulum. Does the period increase or decrease with length? What do you think a graph of the period vs. length would look like? Sketch a rough graph of period vs. length, and describe it in words.......!!!!!!!!...................................
RESPONSE --> ok
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22:46:59 `q010. From your data, figure out the period associated with each length. Make a table of period vs. length, label it clearly, and include the table in your response here.
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RESPONSE --> ok
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22:47:01 The period is the number of seconds required for cycle. For example, if there are 30 cycles in a minute, then it takes 2 seconds for each cycle and the period is 2 seconds. If there are 40 cycles in a minute, 40 cycles require 60 seconds and a single cycle takes 60/40 seconds = 1.5 seconds.
If you miscalculated your periods, recalculate them and show the corrected table. Look at the numbers, asking yourself the same sort of questions as before. Visualize a graph of the numbers in this table. Then draw the graph. Can you look at the graph in such a way as to invoke the 'feel' of the things you have observed? Describe how the graph you made from the table is like, and how it is different than the graph you sketched before you made the table.......!!!!!!!!...................................
RESPONSE --> ok
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22:50:36 `q011. Let T stand for the period of the pendulum (the time required for a single cycle). You have a table of T vs. L. Now create a table for log(T) vs. log(L):
Use the log key on your calculator to obtain the required values. For example, if the period T is 3.1, log(T) will be approximately .5. For every T and every L on your table, find log(T) and log(L) and place these values in your new table. Show your table here.......!!!!!!!!...................................
RESPONSE --> Length Period 0.69 -2.10 1 -1.96 1.18 -1.85 1.30 -1.77 1.40 -1.72 1.48 -1.68
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22:53:46 Next plot your new table, obtaining a graph of y = log(T) vs. x = log(L).
If you have taken your data carefully, and if you have used your calculator and plotted your points correctly, your graph should form a straight line. If not, go back and fix anything that needs to be fixed and replot it. Now using a straightedge, draw a straight line through your graph points and extend this line to the vertical (y) axis; also extend it a little ways past your last data point. If your line doesn't exactly fit every point, make it come as close as possible, on the average, to your graph points. Describe your graph and how well the straight line fits it.......!!!!!!!!...................................
RESPONSE --> It is a straight line.
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22:57:05 `q012. By measuring the rise and the run between the left and right endpoints of your line (not between two points from your table), determine the slope of the line. Call this value p.
Determine the y-intercept of your straight line (i.e., the value of log(T) where your line meets the vertical axis). Call this value y0. Give your values of p and y0.......!!!!!!!!...................................
RESPONSE --> p = 0.50/0.20
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22:58:07 Your pendulum model will be the function T = A * L^p, where p is the slope you determined in a recent step and A = 10^y0. Use your calculator to determine 10^y0 for your value of y0. What do you get for your value of A?
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RESPONSE --> A = 1.26
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22:58:40 q013. Your function model is T = A * L^p. Substitute the values of A and p into this form. What is your function?
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RESPONSE --> T = 1.26 * L^(0.50/0.20)
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23:03:27 Compare your function T = A * L^p with your data, using the values you have found for A and p, as follows:
For every pendulum length L, calculate the period Tpred predicted by this function. That is, plug in each length L and get your predicted value of T, then call this value Tpred. What values do you get for Tpred?......!!!!!!!!...................................
RESPONSE --> 5 70.44 10 398.45 15 1097.99 20 2253.96 25 3937.50 30 6211.17
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23:03:57 `q014. For each pendulum length, calculate the difference T - Tpred.
Express each of these differences as a percent of Tpred. Give a table with columns for L, T, Tpred, T-Tpred, and percent difference. What is your maximum percent difference, and how nearly do you conclude that your function predicts the actual behavior of the pendulum? Does the function do a good job of predicting the period of your pendulum?......!!!!!!!!...................................
RESPONSE --> I do not understand the difference between T and Tpred.
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23:05:24 The 'correct' function for T vs. L is one of the following two equivalant functions:
If L was measured in feet and T in seconds, the ideal function would be about T = 1.1 L^.5. If L was measured in centimeters and T in seconds, then the ideal function would be about T = .2 L^.5. How close was your function to the ideal function? Figure out the periods corresponding to the lengths you observed. How close are these 'ideal' periods to the periods you observed?......!!!!!!!!...................................
RESPONSE --> My function used cm as length and minutes as time so it differs greatly from the ideal function.