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
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Click or tap here to
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Period T (s)
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enter text.
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Are the periods all about the same, or are you seeing that amplitude of swing does affect period?
Explain.
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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?
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Part 3: Period vs. Length
Trial
Length
L (cm)
0
1
20
2
40
3
60
Period
T (s)
0
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enter text.
Click or tap here to
enter text.
Period squared
𝑇 2 (s2)
0
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enter text.
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Click or tap here to
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4
80
enter text.
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enter text.
enter text.
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enter text.
Sample calculation for determining slope m of graph:
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Sample calculation for determine value of g from slope:
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% error on 𝑔 (show calculation):
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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.
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Discuss a couple of sources of error in this lab that may have affected your results.
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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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