Galileo’s Pendulum:
An exercise in gravitation and simple harmonic motion
Zosia A. C. Krusberg
Yerkes Winter Institute
December 2007
Abstract
In this lab, you will investigate the mathematical relationships governing the motion
of a simple pendulum — a bob swinging back and forth on a string due to the
presence of gravity — and use your deepened understanding of this motion to first
determine the acceleration due to gravity at Earth’s surface, and second, to create
a precise time-keeping device. Until as recently as 1930, pendulum clocks were the
most accurate time-keeping devices around!
Objectives
Upon completion of this lab, you will understand:
• the components of a pendulum.
• that the motion of a simple gravity pendulum can be approximated as simple
harmonic motion for small angles.
• that gravity acts as a restoring force that causes the mass to oscillate about its
equilibrium position.
• the relationship between period and frequency.
• how the pendulum’s period depends on mass, oscillation amplitude, length, and
gravitational acceleration.
Vocabulary
Upon completion of this lab, you will be familiar with the following terms:
pendulum; pivot; bob; simple harmonic motion; period; frequency; restoring force; equilibrium position
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Materials
rope/string; weights of different masses; meter sticks; stands; stop watches; calculators;
post-it boards; markers
A bit of history
Figure 1: It is this chandelier in the Pisa Cathedral that is rumored to have inspired
Galileo to investigate the properties of pendulums.
In the 17th century, Galileo — the same guy who was forced into house arrest by the
Roman Inquisition for defending heliocentrism (the theory that the Sun is at the center
of the solar system) — became one of the first people to study the physical properties of
pendulums. According to legend, Galileo’s interest in pendulums was sparked when he
was watching the swinging motion of a chandelier in a cathedral in Pisa, Italy, and timed
its period with his pulse. His subsequent studies of pendulums led him to discover the
relationship between a pendulum’s period and its mass, amplitude, and length. These
investigations encouraged Galileo to suggest that pendulums could be used as timekeeping devices, and the first patent for a pendulum clock was filed in 1656 by another
famous physicist, Christian Huygens.
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Figure 2: Here, in the Panthéon in Paris, hangs Foucaults original pendulum that proved
Earth’s rotation about its own axis.
In 1851, the French physicist Léon Foucault presented a public display of a large
pendulum consisting of a 28-kg bob and a 67-m wire in the Panthéon in Paris as an
experiment to demonstrate the rotation of the Earth. This type of pendulum — now
named a Foucault pendulum — works because the pendulum is free to oscillate in any
vertical plane, and because of the rotation of the Earth “underneath” the pendulum,
the direction of oscillation rotates with time. At the North or South poles, the plane
of oscillation of the Foucault pendulum undergoes one full rotation in one day (you can
picture the pendulum’s plane of oscillation staying fixed with respect to the stars while
Earth rotates underneath). At the equator, on the other hand, the plane of oscillation
stays constant. At intermediate latitudes, the time for the plane of oscillation of the
pendulum is longer than at the poles, but depends on the exact latitudes.
Procedure
Part I: The relationship between period and mass
Goal: Students will determine the relationship between a pendulum’s period and its mass.
1. Draw a diagram of a generalized pendulum in your lab notebook. Make sure that
you identify the pivot, the bob, and the equilibrium point.
2. Draw a diagram of the experimental setup in your lab notebook, and label the
components appropriately.
3. Create a data table in your lab notebook. The first column should read “Mass,”
the second should read “Time for 10 oscillations.”
4. Attach each weight in your weight collection to the string in turn, and for each
weight, time how long the pendulum takes to make 10 oscillations. Write this
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number down in the table in your lab notebook. Remember to include units in
your data table: to make it easy for yourself, include it in brackets along with the
title of each column, e.g., “Mass (g).” Make sure that you let go of the pendulum
at the same height each time, and that the length of string is the same each time:
we want to keep all variables constant except mass.
5. Add a third column to your data table, and title it “Period.” The period is the
time it takes for the pendulum to perform one oscillation, i.e., the time it takes
for the pendulum to return to its starting position. To obtain this number, divide
the number in the previous column by 10. This number is actually the average
period over 10 oscillations.
6. Add a fourth column to your data table, and title it “Frequency.” The frequency
is the number of oscillations the pendulum performs per second; it is the inverse
of the period:
f=
1
P
where f is the frequency and P is the period. The unit of frequency is the hertz
(Hz). Calculate the frequencies for the periods you obtained in the previous column.
7. Answer the following questions in your lab notebook: Does the period change
significantly as mass changes? What are the advantages and disadvantages of
timing 10 oscillations rather than 1?
Part II: The relationship between period and amplitude
Goal: Students will determine the relationship between a pendulum’s period and the
amplitude of its oscillations.
1. Create a second data table in your lab notebook. The variable we’re testing now
is amplitude, so title the columns appropriately.
2. Using a meter stick, let go of the pendulum at different initial heights (amplitudes),
and time how long the pendulum takes to make 10 oscillations. Write this number
down in the table in your lab notebook. You should collect at least 7 data points,
i.e., release the pendulum at 7 different initial heights. Make sure to use the same
mass and length of string each time, since we only want to vary one variable,
amplitude.
3. Determine the period of the pendulum for each height using the procedure from
Part I.
4. Determine the frequency of the pendulum for each height using the procedure from
Part I.
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5. Answer the following question in your lab notebook: Does the period change
significantly as amplitude changes?
Part III: The relationship between period and length
Goal: Students will determine the relationship between a pendulum’s period and its
length.
1. Create a third data table in your lab notebook. The variable we’re testing now is
length, so title the columns appropriately.
2. Using different lengths of string, time how long the pendulum takes to make 10
oscillations. As before, you should collect at least 7 data points, i.e., time the
pendulum’s period for 7 different string lengths. Make sure to get a wide range
of lengths, from really short to as long as you can manage with the setup. Make
sure to use the same mass and initial height each time, since we only want to vary
one variable, length.
3. Determine the period of the pendulum for each length using the procedure from
Part I.
4. Determine the frequency of the pendulum for each length using the procedure from
Part I.
5. Answer the following question in your lab notebook: Does the period change
significantly as amplitude changes?
Part IV: Determine the acceleration due to gravity
Goal: Students will use their data from Part III to determine the acceleration due to
gravity, g.
1. Using the data table you created in Part III of this activity, make a plot of period
versus length in your lab notebook. Make sure you title your plot, label your axes,
and include units.
2. Describe the general appearance of this plot. Does it resemble any function you’ve
seen before?
3. The equation for the period of a simple gravity pendulum is the following:
s
P = 2π
l
g
where l is the length of the pendulum and g is the acceleration due to gravity. If
this looks confusing, don’t worry too much about it. If you’ve done some algebra,
you can see that the period only depends on the square root of the length (both
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π and g are constants). We’re now going to use this equation to determine the
acceleration due to gravity, g. Remember that we’ve done this once already this
year, so this will be a way to test the result we obtained in a different experiment.
Using the data that you collected in Part III of this activity, make
√ a second plot,
but this time, plot period versus the square root of length ( l). To do this,
√
add another column to your data table and title it “ Length,” and fill it in by
calculating these values. By plotting the data this way, the plot will show a linear
relationship (straight line). Again, make sure you title your plot, label your aces,
and include units.
Find the slope of this line. The slope, let’s call it m, is equal to
2π
m= √
g
So, to determine the acceleration due to gravity, we solve this equation for g:
g=
4π 2
m2
Use this equation to calculate the value of the acceleration due to gravity.
4. Answer the following question in your lab notebook: If the experiment were carried
out on the Moon, how would the period of the pendulum change? What if it were
carried out on Jupiter (though we all know Jupiter has no solid surface to stand
on, so this would be a difficult experiment to carry out in practice!)?
Part V: Challenge
Using your data, design and construct a pendulum that you can use to accurately
measure a time interval of 30 seconds. Test your pendulum clock against a watch or
clock.
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