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 faster.

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 below.

Number of cork oscillations10.010.010.010.010.09.910.010.09.910.0
Number of lead oscillations10.010.09.910.110.110.010.010.010.010.0

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 release30.030.030.030.030.0
Oscillations of 45 degree release29.529.629.529.529.0

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 below.

24.0 cm string28.028.028.028.028.0
50.5 cm string20.019.919.820.019.9

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 string20.020.020.020.020.0
99.4 cm string9.759.259.710.09.75

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

Michael Morgan
mlmorgan@rice.edu