PHSX 104 Michigan State University The Simple Pendulum Lab Report

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PHSX 104

Michigan State University

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Exercise #6: The Simple Pendulum Purpose: In this exercise you will study the behavior of a simple pendulum, a single small mass swinging back and forth on a string under its own weight. You will measure the period of the pendulum, which is the amount of time it takes for the pendulum to move back and forth in one complete motion. You will determine what effect changing one property of the pendulum (the mass at the end of the string, the angle at which it swings, and the length of the pendulum respectively) has on the period while holding all other factors constant. This will give you some experience with the experimental technique of “variables vs. controls”. You will compare the results of each part of the lab against what the physical theory explaining the motion of pendulums predicted. You will also use the results of the last part of the exercise to experimentally derive the value of free-fall acceleration .

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Data and Results Sheets Description of pendulum: Part 1: Period vs. Angle Trial Amplitude (degrees) 1 10o 2 20o 3 30o 4 45o Time for 20 swings (s) Click or tap here to enter text. Click or tap here to enter text. Click or tap here to enter text. Click or tap here to enter text. Period T (s) Click or tap here to enter text. Click or tap here to enter text. Click or tap here to enter text. Click or tap here to enter text. Are the periods all about the same, or are you seeing that amplitude of swing does affect period? Explain. Click or tap here to enter text. Part 2: Period vs. Mass Period of pendulum with heavier weight: Click or tap here to enter text. How does this period compare to the period you measured in Trial 1 of Part 1? Are they comparable, or did the added mass change the period significantly? Click or tap here to enter text. Part 3: Period vs. Length Trial Length L (cm) 0 1 20 2 40 3 60 Period T (s) 0 Click or tap here to enter text. Click or tap here to enter text. Period squared 𝑇 2 (s2) 0 Click or tap here to enter text. Click or tap here to enter text. Click or tap here to Click or tap here to 4 80 enter text. Click or tap here to enter text. enter text. Click or tap here to enter text. Sample calculation for determining slope m of graph: Click or tap here to enter text. Sample calculation for determine value of g from slope: Click or tap here to enter text. % error on 𝑔 (show calculation): Click or tap here to enter text. Do your data points make a good fit to the best-fit line? Was your experimental value of 𝑔 close to the accepted value? Justify your conclusion. Click or tap here to enter text. Discuss a couple of sources of error in this lab that may have affected your results. Click or tap here to enter text. Based on your answers to the last two questions, how confident are you in the accuracy of your results for Part 3? Justify your conclusion. Click or tap here to enter text. Exercise #6: The Simple Pendulum Purpose: In this exercise you will study the behavior of a simple pendulum, a single small mass swinging back and forth on a string under its own weight. You will measure the period of the pendulum, which is the amount of time it takes for the pendulum to move back and forth in one complete motion. You will determine what effect changing one property of the pendulum (the mass at the end of the string, the angle at which it swings, and the length of the pendulum respectively) has on the period while holding all other factors constant. This will give you some experience with the experimental technique of “variables vs. controls”. You will compare the results of each part of the lab against what the physical theory explaining the motion of pendulums predicted. You will also use the results of the last part of the exercise to experimentally derive the value of free-fall acceleration . Introduction: It has long been known that if one allows a weight to swing at the end of a long thin rope or rod, it will swing in a very regular fashion. The amount of time it takes the weight to complete one full swing (from maximum height on one side of its swing to maximum height on the other side, and then back) is called the period of the pendulum. What factors should affect the period of a pendulum? A pendulum is an example of a simple harmonic oscillator. A “harmonic oscillator” is any physical system that repeats the same motion in a regular fashion over and over again with the same period. Any object experiencing a restoring force, a force that always pulls the object back towards some fixed point (called its point of equilibrium), will exhibit harmonic motion. “Simple” means that the size of the motion does not affect the period of the motion. Generally speaking, a pendulum’s period is not affected by the size of its swing as long as the maximum angle of swing (the amplitude) is not too large. This is why pendulums were used as the working mechanism of clocks for many centuries; as the pendulum lost energy over time and swung at smaller and smaller angles, its period stayed constant. The restoring force that is causing the pendulum to swing is gravity, which always tries to pull the pendulum back to a vertical hanging position. The weight is trying to free-fall, but the string constrains it to follow the path of a circle as it falls, leading to its regular swinging motion. All objects, regardless of weight/mass, are supposed to free-fall at the same rate of acceleration (according to Galileo). In principle, the actual total mass at the end of your pendulum shouldn’t affect the rate at which it swings. The length L of a pendulum (measured from its support point at one end of the string to the middle of its weight on the other end) has a very pronounced effect on its period. The longer the pendulum, the farther it has to swing during one period. Therefore, longer pendulums have longer periods. The relationship between the period of a pendulum and the three factors previously discussed (mass, amplitude, and length) is exactly expressed by the following equation: √ where is the gravitational free-fall acceleration. Note that neither mass nor amplitude appears in this equation, i.e. the mass of the pendulum and its swing amplitude should not affect its period. From the theory behind pendulum motion, the statement about mass is always true. However, the theory also predicts that eq. (1) is only valid as long as the amplitude is relatively small, somewhat less than 15-20o. At larger amplitudes, one expects to see increasingly large differences between a pendulum’s actual period and that predicted by eq. (1). Procedure: You will be creating a graph for this lab exercise. You should review graph-making and the concepts of “dependent variable”, “independent variable”, and “controls” in the handout “Introduction to Graphing”. To build a home-made pendulum, you will need the following materials:       A weight that you can securely fasten to the end of a string. The weight should be relatively small (no bigger than your fist) and dense, but otherwise you can use pretty much anything. Some possible suggestions are a bag of marbles, some large metal washers, a small round rock, a pocketwatch you aren’t too attached to, a baseball/racquetball/tennis ball, etc. It should weigh at least a few ounces but no more than a pound. A second weight, about twice the mass of your first weight. A piece of string to swing the weight from. Use whatever means are at hand to securely fasten the weight to one end of the string. Then, make sure that you have at least 1 meter of string (~3 feet) extending from the weight to swing it from. I would not suggest using sewing thread, as it is too breakable. Bailing cord, high-test fishing line, strong sewing thread, etc. work well and are strong enough. Whatever you use, make sure that it is more than sufficiently strong so that there is no danger of it snapping as your pendulum swings. Do not use anything that is too stretchy, like yarn or elastic cord. It will stretch as its swings and your results will be bad. A place from which to suspend the pendulum so that it may freely swing. A plant hanger in the ceiling, a nail in the beams in your garage, a very sturdy tree branch, etc. can be used for this. Verify that the string is firmly attached and won’t come flying off as the pendulum swings. Make sure that the support is firm enough that it doesn’t move significantly as the pendulum swings. Too much movement will throw off your measurements. A time piece that will allow you to measure the period of the pendulum. The timepiece needs to be able to measure at least to the seconds. A stopwatch or digital watch should have this level of precision. A protractor, if you have one in the house. It is helpful, but not necessary. CAUTION: Before you start swinging your pendulum, make absolutely sure that nothing breakable is in harm’s way, just in case your pendulum breaks lose or its string snaps. 1. On the Data Sheet, describe in detail how you constructed your pendulum: What are you using for a weight, how big is it, what did you use for a string, what did you hang the pendulum from, etc. Include enough detail that your reader could reconstruct your pendulum exactly for themselves. Part 1 – Period vs. Angle In this part of the exercise you will test the relationship between a pendulum’s period and its swing amplitude while keeping its length and mass fixed (“controlled”). 2. Adjust the length of your pendulum, measured from the point of support at one end of the string to the middle of the weight at the other end, so that it is 60 centimeters long. 3. Pull the string out to an angle of about 10o from vertical. If you have a protractor, you can use this to conveniently check your angle. If not, a 60 cm string pulled about 4 inches (~10 cm) to the side from its vertical hanging position is at an angle of ~10o. 4. Release the weight and allow it to swing. Time 20 full swings (back and forth completely) and record in your Data Table. The pendulum should continue to swing along the same path (mostly) for all 20 swings. If its swing pattern changes too much, your data won’t be good. A little practice may be needed to get good, steady swings. 5. Divide the total swing time by 20 to get the period T for one swing of the pendulum. 6. Repeat this measurement for ever larger initial angles of swing at 20o (pulled out ~20 cm or 8 inches from vertical), 30o (pulled out ~30 cm or 12 inches from vertical), and 45o (pulled out ~42 cm or 16 inches from vertical). The last swing is quite a large path. Make sure that you’ve given the pendulum enough room to swing this much. Part 2 – Period vs. Mass In this part of the exercise, you will vary the mass on the pendulum and see what effect it has on the period. Length and amplitude will be “controlled”. 7. Using the second weight, double the mass on the end of your pendulum. If you used a bag of marbles, add more marbles. If you use a rock, tie another rock or get a rock that’s twice as big, etc. 8. Pull the pendulum to an angle of 10o. Time 20 swings of this pendulum. Divide the result by 20 to get the period. 9. Remove the extra mass from the pendulum. Part 3 – Period vs. Length In this final part of the exercise, you are going to test eq. (1) and also use it to verify the value of . You will change the length of your pendulum several times and measure the resulting changes in the period. You will make a graph of length (as “independent variable”) vs. period (as “dependent variable”). If we graphed just T vs. L, however, eq. (1) suggest that it would give us a square root curve (like graphing √ ), which would tell us very little. If we square both sides of eq. (1), though, we get the following expression: If eq. (1) is valid, a plot of period squared on the y-axis of a graph vs. length should yield a linear graph whose best-fit slope equals on the x-axis Once you determine the slope of your line, you can calculate g. 10. Adjust the length of your pendulum to 20 cm. Pull the pendulum out to an angle of 10o and let it swing 20 times. Divide by 20 and record this time to determine the period of the pendulum. 11. Repeat this measurement for pendulum lengths of 40 cm and 80 cm. Record the period of your pendulum at these lengths. 12. You have already determined the period at a length of 60 cm in Part 1. Simply copy your results from Part 1 into the table. 13. Square the period for each trial and record in the table. 14. Make a plot of L (on the x-axis) vs. T2 (on the y-axis). Length should be plotted in units of centimeters, T2 in units of seconds squared. Make a best fit line to your data points. Eq. (2) implies that at zero length a pendulum should have zero period, so your best fit line should go through the origin. The point (0,0) has been included on the Data Table as a reminder. 15. Determine the slope m of your best fit line (review “Exercise 3 - Graphing” for a reminder of how to calculate slopes). 16. Algebraically rearrange eq. (3) to solve for g in terms of the slope m. Using your value of the slope from Step 15, find the experimental value of g. 17. Compare your experimentally determined value of by calculating a percent error. to the accepted value of Do not forget to submit your graph when you submit your lab report. Exercise #6: The Simple Pendulum Purpose: In this exercise you will study the behavior of a simple pendulum, a single small mass swinging back and forth on a string under its own weight. You will measure the period of the pendulum, which is the amount of time it takes for the pendulum to move back and forth in one complete motion. You will determine what effect changing one property of the pendulum (the mass at the end of the string, the angle at which it swings, and the length of the pendulum respectively) has on the period while holding all other factors constant. This will give you some experience with the experimental technique of “variables vs. controls”. You will compare the results of each part of the lab against what the physical theory explaining the motion of pendulums predicted. You will also use the results of the last part of the exercise to experimentally derive the value of free-fall acceleration . Introduction: It has long been known that if one allows a weight to swing at the end of a long thin rope or rod, it will swing in a very regular fashion. The amount of time it takes the weight to complete one full swing (from maximum height on one side of its swing to maximum height on the other side, and then back) is called the period of the pendulum. What factors should affect the period of a pendulum? A pendulum is an example of a simple harmonic oscillator. A “harmonic oscillator” is any physical system that repeats the same motion in a regular fashion over and over again with the same period. Any object experiencing a restoring force, a force that always pulls the object back towards some fixed point (called its point of equilibrium), will exhibit harmonic motion. “Simple” means that the size of the motion does not affect the period of the motion. Generally speaking, a pendulum’s period is not affected by the size of its swing as long as the maximum angle of swing (the amplitude) is not too large. This is why pendulums were used as the working mechanism of clocks for many centuries; as the pendulum lost energy over time and swung at smaller and smaller angles, its period stayed constant. The restoring force that is causing the pendulum to swing is gravity, which always tries to pull the pendulum back to a vertical hanging position. The weight is trying to free-fall, but the string constrains it to follow the path of a circle as it falls, leading to its regular swinging motion. All objects, regardless of weight/mass, are supposed to free-fall at the same rate of acceleration (according to Galileo). In principle, the actual total mass at the end of your pendulum shouldn’t affect the rate at which it swings. The length L of a pendulum (measured from its support point at one end of the string to the middle of its weight on the other end) has a very pronounced effect on its period. The longer the pendulum, the farther it has to swing during one period. Therefore, longer pendulums have longer periods. The relationship between the period of a pendulum and the three factors previously discussed (mass, amplitude, and length) is exactly expressed by the following equation: √ where is the gravitational free-fall acceleration. Note that neither mass nor amplitude appears in this equation, i.e. the mass of the pendulum and its swing amplitude should not affect its period. From the theory behind pendulum motion, the statement about mass is always true. However, the theory also predicts that eq. (1) is only valid as long as the amplitude is relatively small, somewhat less than 15-20o. At larger amplitudes, one expects to see increasingly large differences between a pendulum’s actual period and that predicted by eq. (1). Procedure: You will be creating a graph for this lab exercise. You should review graph-making and the concepts of “dependent variable”, “independent variable”, and “controls” in the handout “Introduction to Graphing”. To build a home-made pendulum, you will need the following materials:       A weight that you can securely fasten to the end of a string. The weight should be relatively small (no bigger than your fist) and dense, but otherwise you can use pretty much anything. Some possible suggestions are a bag of marbles, some large metal washers, a small round rock, a pocketwatch you aren’t too attached to, a baseball/racquetball/tennis ball, etc. It should weigh at least a few ounces but no more than a pound. A second weight, about twice the mass of your first weight. A piece of string to swing the weight from. Use whatever means are at hand to securely fasten the weight to one end of the string. Then, make sure that you have at least 1 meter of string (~3 feet) extending from the weight to swing it from. I would not suggest using sewing thread, as it is too breakable. Bailing cord, high-test fishing line, strong sewing thread, etc. work well and are strong enough. Whatever you use, make sure that it is more than sufficiently strong so that there is no danger of it snapping as your pendulum swings. Do not use anything that is too stretchy, like yarn or elastic cord. It will stretch as its swings and your results will be bad. A place from which to suspend the pendulum so that it may freely swing. A plant hanger in the ceiling, a nail in the beams in your garage, a very sturdy tree branch, etc. can be used for this. Verify that the string is firmly attached and won’t come flying off as the pendulum swings. Make sure that the support is firm enough that it doesn’t move significantly as the pendulum swings. Too much movement will throw off your measurements. A time piece that will allow you to measure the period of the pendulum. The timepiece needs to be able to measure at least to the seconds. A stopwatch or digital watch should have this level of precision. A protractor, if you have one in the house. It is helpful, but not necessary. CAUTION: Before you start swinging your pendulum, make absolutely sure that nothing breakable is in harm’s way, just in case your pendulum breaks lose or its string snaps. 1. On the Data Sheet, describe in detail how you constructed your pendulum: What are you using for a weight, how big is it, what did you use for a string, what did you hang the pendulum from, etc. Include enough detail that your reader could reconstruct your pendulum exactly for themselves. Part 1 – Period vs. Angle In this part of the exercise you will test the relationship between a pendulum’s period and its swing amplitude while keeping its length and mass fixed (“controlled”). 2. Adjust the length of your pendulum, measured from the point of support at one end of the string to the middle of the weight at the other end, so that it is 60 centimeters long. 3. Pull the string out to an angle of about 10o from vertical. If you have a protractor, you can use this to conveniently check your angle. If not, a 60 cm string pulled about 4 inches (~10 cm) to the side from its vertical hanging position is at an angle of ~10o. 4. Release the weight and allow it to swing. Time 20 full swings (back and forth completely) and record in your Data Table. The pendulum should continue to swing along the same path (mostly) for all 20 swings. If its swing pattern changes too much, your data won’t be good. A little practice may be needed to get good, steady swings. 5. Divide the total swing time by 20 to get the period T for one swing of the pendulum. 6. Repeat this measurement for ever larger initial angles of swing at 20o (pulled out ~20 cm or 8 inches from vertical), 30o (pulled out ~30 cm or 12 inches from vertical), and 45o (pulled out ~42 cm or 16 inches from vertical). The last swing is quite a large path. Make sure that you’ve given the pendulum enough room to swing this much. Part 2 – Period vs. Mass In this part of the exercise, you will vary the mass on the pendulum and see what effect it has on the period. Length and amplitude will be “controlled”. 7. Using the second weight, double the mass on the end of your pendulum. If you used a bag of marbles, add more marbles. If you use a rock, tie another rock or get a rock that’s twice as big, etc. 8. Pull the pendulum to an angle of 10o. Time 20 swings of this pendulum. Divide the result by 20 to get the period. 9. Remove the extra mass from the pendulum. Part 3 – Period vs. Length In this final part of the exercise, you are going to test eq. (1) and also use it to verify the value of . You will change the length of your pendulum several times and measure the resulting changes in the period. You will make a graph of length (as “independent variable”) vs. period (as “dependent variable”). If we graphed just T vs. L, however, eq. (1) suggest that it would give us a square root curve (like graphing √ ), which would tell us very little. If we square both sides of eq. (1), though, we get the following expression: If eq. (1) is valid, a plot of period squared on the y-axis of a graph vs. length should yield a linear graph whose best-fit slope equals on the x-axis Once you determine the slope of your line, you can calculate g. 10. Adjust the length of your pendulum to 20 cm. Pull the pendulum out to an angle of 10o and let it swing 20 times. Divide by 20 and record this time to determine the period of the pendulum. 11. Repeat this measurement for pendulum lengths of 40 cm and 80 cm. Record the period of your pendulum at these lengths. 12. You have already determined the period at a length of 60 cm in Part 1. Simply copy your results from Part 1 into the table. 13. Square the period for each trial and record in the table. 14. Make a plot of L (on the x-axis) vs. T2 (on the y-axis). Length should be plotted in units of centimeters, T2 in units of seconds squared. Make a best fit line to your data points. Eq. (2) implies that at zero length a pendulum should have zero period, so your best fit line should go through the origin. The point (0,0) has been included on the Data Table as a reminder. 15. Determine the slope m of your best fit line (review “Exercise 3 - Graphing” for a reminder of how to calculate slopes). 16. Algebraically rearrange eq. (3) to solve for g in terms of the slope m. Using your value of the slope from Step 15, find the experimental value of g. 17. Compare your experimentally determined value of by calculating a percent error. to the accepted value of Do not forget to submit your graph when you submit your lab report.
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Explanation & Answer

lab is completed. see attached file. please ask questions if you have any. .

Data and Results Sheets
Description of pendulum:
Pendulum consisted of fishing line attached to a hook in the ceiling at one end and a small plastic
box filled with weights and cotton balls at the other end. For part (I) the weights were washers.
The plastic box was the original packaging the washers were purchased in. The length of the
fishing line was 60cm from the hook to the center of the washers. In part (II), the weights were
BB’s instead of washers.
Part 1: Period vs. Angle

Trial

Amplitude (degrees)

1
2
3
4

10o
20o
30o
45o

Time for
20 swings (s)
31.52
30.58
30.27
31.83

Period T (s)
1.576
1.529
1.514
1.592

Are the periods all about the same, or are you seeing that amplitude of swing does affect period?
Explain.
The periods are about the same regardless of amplitude. The average period was 1.552 seconds

Part 2: Period vs. Mass
Per...


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