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.
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.
Pendulums nearly return to their release heights.
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.
All pendulums eventually come to rest with the lighter ones coming to rest
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.
The period is independent of the bob weight.
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
|Number of cork oscillations||10.0||10.0||10.0||10.0||10.0||9.9||10.0||10.0||9.9||10.0|
|Number of lead
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.
The period is independent of the amplitude.
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.
|Oscillations of 5 degree release||30.0||30.0||30.0||30.0||30.0|
|Oscillations of 45 degree release||29.5||29.6||29.5||29.5||29.0
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.
The square of the period varies directly with the length.
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
|24.0 cm string||28.0||28.0||28.0||28.0||28.0|
|50.5 cm string||20.0||19.9||19.8||20.0||19.9
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.
|24.0 cm string||20.0||20.0||20.0||20.0||20.0|
|99.4 cm string||9.75||9.25||9.7||10.0||9.75
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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Last revised April 17, 1995