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The Pendulum Experiment
The purpose
of this Den Meeting is to teach the Scientific Method. This is a requirement
of the Cub Scout Academics Belt Loop. This meeting was a huge hit with the
Scouts and they learned a lot in the process.
The
Scientific Method
In its most
basic form, the thought process in the scientific method can consists of the
following tasks:
·
Identifying a Problem
·
Forming a hypothesis
·
Designing and Performing Experiments
·
Collecting and Analyzing Data
·
Formulating Conclusions about the Hypothesis
Simple
Science Project
Scouts will
use the scientific method to solve a problem.
A pendulum is
any mass which swings back and forth on a rope, string, or chain. Pendulums
can be found in old clocks and other machinery. A playground swing is a
pendulum.
If you pull
the mass away from its rest position, so that the string is at an angle, and
then let go, the mass will begin to swing back and forth. The length of time
it takes the mass to swing all the way over and back, once, is called the
period of the pendulum.
All three
experiments will examine things we can do to the pendulum that will change
the period. Here are the three questions we are asking:
1.
Does the amount of mass on the end of the string affect the
period?
2.
Does the angle you pull pack the string to affect the period?
3.
Does the length of the string affect the period?
In these experiments,
the dependent variable will always be the time for one full swing, or the
period.
The three
tested independent variables will be the mass, the angle, and the length of
string.
The
controlled variables will be the attachment point of the string, the string
itself, the method used to time the pendulum, and the variables we are not
currently testing. These will remain the same for each test, so that we know
they won't affect the results.
The
experiments are easy to do, and don't require any special equipment. We did
them ourselves using some string, a few large nuts, a pen, and a watch, and
got good results for all three tests in about 20 minutes.
Here'a list
of what you'll need for each group doing the experiment:
 - a piece of
string at least 3 feet long- 3 or 4 weights, all the same - a pen and tape, to attach the pendulum to a shelf - a watch that counts seconds - pencil and paper to record the results
It's also
easier if you have several people doing the experiments, so that one person
is free to time the swings.
Instructions
The
instructions for this experiment are outlined in each of the worksheets.This
is the format we used for our meeting:
1.
Show a pendulum to the Scouts (simply a weight attached to a
string). Swing the pendulum.
2.
Ask Scouts the following: Does the number of times the
pendulum swing depend on the weight attached to it, or the length of the
string or the angle at which it is pulled back?
3.
Worksheets are provided below for three experiments: (i) vary
the length; (ii) vary the mass and (iii) vary the angle.
4.
Organize the Scouts into teams to develop a way to answer the
questions in the worksheets. It is ideal to have a parent or leader to run
and organize each workstation. Make certain the Scouts keep their team
results to themselves until the end.
5.
Scouts test their hypothesis to answer the questions.
6.
Bring the Scouts together and discuss the results.
·
When you change the mass on the end of a pendulum, the period
of the pendulum does not change
·
When you change the length of the pendulum, the period of the
pendulum does change. The shorter the string, the
shorter the period.
·
When you change the angle of the pendulum, the period of the
pendulum does not change.
Interesting History
Here some
interesting history that you may want to include as part of the Den Meeting.
The Scouts found this very interesting. Make certain you do not tell them
about Galileo's conclusion prior to them have determined the results are on
their own.
The Pendulum
Have you ever
been bored in Church? Kids fidget; adults sometimes gently doze. Most of us have
done this at some time or another, but few have put such boredom to as good a
use as Galileo, who made a fundamental scientific discovery that changed the
world.
While
watching a chandelier swing back and forth at the Cathedral of Pisa in 1583,
Galileo noticed something curious. We might expect a chandelier swinging
farther to take longer, but not so. Galileo noticed that the time period to
swing through one complete cycle is independent of the amplitude through
which it swings. One can duplicate Galileo's pendulum experiments by timing a
weight swinging on the end of a string. For not too large amplitudes, the
time period for one complete cycle will be the same regardless of amplitude.
The period does however depend on the length of the string. A longer pendulum
will take longer to complete one cycle. For these experiments we would use a
stopwatch, perhaps one built into most digital watches these days. But in
Galileo's time, wristwatches were not yet available. He timed the swings with
his pulse, the only timing device at hand.
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Galileo's Pendulum Experiments
Galileo used pendulums
extensively in his experiments. Early in his career, he researched the
characteristics of their motion. After investigating their behavior, he was
able to use them as time measurement devices in later experiments.
Pendulums are mentioned
in both Galileo's Dialogue Concerning the Two Chief World Systems and
his Dialogues Concerning Two New Sciences. In these two works,
Galileo discusses some of the major points he discovered about pendulums.
Follow the links to jump to an experimental evaluation of the claim.
- Pendulums nearly return to their release heights.
- All pendulums eventually come to rest with the lighter ones coming to rest faster.
- The period is independent of the bob weight.
- The period is independent of the amplitude.
- The square of the period varies directly with the length.
Galileo also performed
experiments to examine the nature of collisions in which he used pendulums, but
these experiments appear to have provided less insight and to have been less
conclusive than the other experiments. These collision experiments were not
repeated or evaluated.
We attempted to
reproduce Galileo's findings on these main points and verify his claims.
Galileo's techniques had to be modified in several ways to be practical for our
resources. For one experiment in Two New Sciences, string lengths
of four or five yards are suggested. For these experiments, string lengths of
24.0 cm to 99.4 cm were used. The experiments also used lead and cork balls.
For these experiments, egg-shaped fishing weights and a cork fishing float were
used.
Time measurement was a
major issue in many of Galileo's experiments. For his pendulum experiments,
Galileo seems to have compared the pendulums in pairs over the same time. For
example, a person would be assigned to each pendulum of the pair and between
the words "start" and "stop" each person would count the
number of oscillations. This method was used for comparison in these
experiments.
Galileo observed that
the bobs of pendulums nearly return to their release height. Today this fact
demonstrates conservation of energy, a principle not yet discovered in
Galileo's time. As a recreation, pendulums were released from different
heights. The height the pendulum returned to was noted and compared to the
release height. No quantitative measurements were made, but in every trial, the
pendulum's return height was very close to its release height. The estimated
difference between the heights was no more that 3 mm for the range of string
lengths used.
Galileo noted that
lighter pendulums come to rest faster. As a test of this observation, two
pendulums, nearly identical except for their bobs of different weights, were
released at the same time and height. A bob of lead was hung with a string
length of 28.9 cm. A bob of cork was hung to hang at 29.0 cm. The two were
released at the same time after being pulled back about 5 degrees. After
waiting for several minutes, the cork bob came to rest while the lead bob was
still moving. More trials revealed the same result in agreement with Galileo.
Cork and lead pendulums of the same length
Galileo claimed to have
hung pendulums of cork and lead from his ceiling and measured their periods to
be the same. As a test, a pendulum 29.0 cm long of cork and a pendulum 28.9 cm
long of lead were used. Both were suspended and released simultaneously from
the same height. For five trials, the cork was allowed to travel through ten
oscillations and compared to the number of oscillations of the lead during that
time. Then the process was reversed for five additional trials. The lead
pendulum was allowed to travel through ten oscillations and the oscillations of
the cork were counted. The results are below.
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Number
of cork oscillations
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10.0
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10.0
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10.0
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10.0
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10.0
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9.9
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10.0
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10.0
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9.9
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10.0
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Number
of lead oscillations
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10.0
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10.0
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9.9
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10.1
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10.1
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10.0
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10.0
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10.0
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10.0
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10.0
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The average number of
oscillations for the cork bob was 9.98. The average number of oscillations for
the lead bob was 10.01. The percent difference between these averages is
0.300%. For any one measurement, the highest discrepancy was 0.1 oscillation or
1%. Galileo's discovery holds up very well in this test.
Galileo claimed that the
pendulum period was independent of the amplitude in Two New Sciences.
Scholars debate whether he meant that the periods are exactly the same of that
they differ very little. As a test of whether they are exactly the same, two
pendulums with identical lead bobs were suspended 28.9 cm. They were released
at the same time from different angles. One was pulled back about 5 degrees
while the other was released from about 45 degrees. The pendulum pulled back
five degrees was allowed to travel through thirty cycles, and the numbers of
oscillations of the other pendulum during this time were counted. The data is
below.
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Oscillations
of 5 degree release
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30.0
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30.0
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30.0
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30.0
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30.0
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Oscillations
of 45 degree release
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29.5
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29.6
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29.5
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29.5
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29.0
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The pendulum which
traveled through the larger angle had a longer period. It averaged 29.42
oscillations during 30 swings of the other, and had fewer oscillations in every
trial. Clearly, pendulums with different amplitudes do not have the same
period. In fact, it appears that pendulums with larger amplitudes have longer
periods. The difference is quite small, though. Whether Galileo's claim is true
depends on interpretation of the claim, but the interpretation that identical
pendulums of different amplitudes have periods independent of amplitude is
false.
Lead pendulums with one string about four times as long as
the other
Galileo found that the
period squared is proportional to the length for a pendulum. As a test, lead
pendulums differing in length by factors of two and four were compared.
Pendulums of lengths 24.0 cm and 50.5 cm were released simultaneously. The
shorter pendulum was allowed to pass through 28 cycles as the oscillations of
the longer one were counted. The data is below.
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24.0
cm string
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28.0
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28.0
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28.0
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28.0
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28.0
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50.5
cm string
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20.0
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19.9
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19.8
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20.0
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19.9
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Then pendulums of
lengths 24.0 cm and 99.4 cm were compared. They were released simulatneously.
The shorter pendulum was allowed to pass through 20 cycles as the oscillations
of the longer pendulum were counted. The data for these trials is below.
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24.0
cm string
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20.0
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20.0
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20.0
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20.0
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20.0
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99.4
cm string
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9.75
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9.25
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9.7
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10.0
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9.75
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For the first data set,
the longer pendulum averaged 19.9 cycles during the shorter ones 28. 19.9/28 is
0.711. The square root of the ratio of their lengths is 0.689. The percent
different between these ratios is 3.14%. For the second data set, the longer
pendulum averaged 9.69 cycles during the shorter pendulum's 20. The ratio
between these two numbers is 0.485. The square root of the ratio of their
lengths is 0.491. The percent difference between these ratios is 1.23%. For
both experiments, the relationship discovered by Galileo holds well.
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