# LAB #10 THE PENDULUM INTRODUCTION In nature there are

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```					                                            The Pendulum
Introduction: In nature there are many physical phenomena that are repetitive, for instance the motion of
the planets with respect to the sun and the oscillations of a pendulum. If a motion repeats itself after a time
interval T, it is called periodic motion and is characterized by its period, T, and/or its frequency, f = 1/T.
When a physical body in periodic motion moves back and forth over the same trajectory, it is called
oscillatory motion. A simple pendulum consists of a mass m, suspended by a light string. If the suspended
mass is pulled a little to one side and released, it will experience a restoring force (a force trying to push or
pull it back to its equilibrium position) that causes it to swing back and forth, executing a periodic motion.
The time that it takes the mass m to complete one full oscillation is called the period of the pendulum, T.

Our objective in this experiment is to
1. Measure the period of pendulums of varying lengths, masses and amplitudes (displacements from
equilibrium).
2. Determine the effect pendulum length, mass, and amplitude have on the pendulum’s period.

Figure 1: Left the pendulum setup showing the
set of three masses, the support rods, photogate
and meter stick. Right shows the proper
alignment of the pendulum's path and the
photogate.

IMPORTANT: It is essential that as the pendulum swings it does not hit and damage the photogate. See
Fig.1. Proper alignment and release of the pendulum will eliminate this possible problem.

Equipment: Pasco® photogate, ULI, Software Pendulum, (2) two string, pendulum lengths 50-35 and
20cm, (3) three pendulum masses of 50-20-200g, stand, meter stick.
Theory: A simple pendulum is a mass attached to a string, of negligible mass, and displaced from its
equilibrium position by an angle θ, see Figure 2. When this is done the pendulum experiences a net
restoring force (the component of the gravitational force tangent to the pendulum’s trajectory supplies this
force). This force attempts to push or pull the pendulum back to its equilibrium position (hence the term
restoring force). The pendulum’s inertia causes it to overshoot this position, this results in the pendulum
executing periodic motion about its equilibrium position.

Experiment:

We are going to look at the dependence of the pendulum’s period on its length, mass and amplitude. The
length of the pendulum can easily be changed by slightly loosening the upper support rod and turning it so
as to wind, or unwind, the string to the desired length. Your laboratory instructor can help you with this if
necessary. NOTE: The length of the pendulum is measured from the bottom of the horizontal support rod
to the center of the pendulum mass (its center of mass). The amplitude of the pendulum is the horizontal
distance the pendulum mass is displaced from its equilibrium position as shown in the figure. Be sure the
ULI is on and start the software by clicking on the Pendulum-2 software icon on the “At-Ease” panel.
Measure the masses of each of your pendulums if they are not marked with their mass. You will need to
confirm for each trial that the pendulum is properly breaking the photogate beam. If not the height of the
photogate will need to be adjusted.

Part 1: Mass and Period
o   For the 200g pendulum set the pendulum length to 50.0cm.
o   Displace the pendulum bob 10.0cm from its equilibrium position and carefully release the bob. Make sure
it swings as straight as possible.
o   Allow the pendulum to execute about 5 complete oscillations, than click on the START button on the
software.
o   Allow the pendulum to complete 5 oscillations of its motion, than click on the STOP button on the
software.
o   On the Pendulum-2:Control Window click on the statistics icon (∑) to open the statistics panel.
o   The statistics panel shows the maximum, minimum and average of the 5 period measurements, including
the standard deviation for the measurements. Round the mean value to the hundredths place (0.01s) and
record this in your data table.
o   Choose the “Timing” option along the main menu of the software, and select “clear data” so that the
software will immediately begin taking data the next time you click on the START button.
o   Repeat your measurements for the other two masses. Remember that you will have to adjust the string to
get the correct pendulum length and may need to adjust the height of the photogate.

Pendulum                      Mass(g)                      Period(s)
1
2
3

Part 2: Amplitude and Period
o   For the 50g pendulum set the pendulum length to 50.0cm (this will be kept constant throughout this part of
the experiment).
o   Displace the pendulum bob 5.0cm from its equilibrium position and carefully release the bob. Make sure it
swings as straight as possible.
o   Allow the pendulum to execute about 5 complete oscillations, than click on the START button on the
software.
o   Allow the pendulum to complete 5 oscillations of its motion, than click on the STOP button on the
software.
o   The statistics panel shows the maximum, minimum and average of the 5 period measurements, including
the standard deviation for the measurements. Round the mean value to the hundredths place (0.01s) and
record this in your data table.
o   Choose the “Timing” option along the main menu of the software, and select “clear data” so that the
software will immediately begin taking data the next time you click on the START button.
o   Repeat your measurements for amplitudes of 10.0 and 15.0cm. Remember to continue to use the 50.0g
pendulum and the 50.0cm length.

Amplitude(cm)                   Period(s)
5
10
15

Part 3: Pendulum Length and Period
o   For the 50g pendulum set the pendulum length to 50.0cm (The pendulum mass and amplitude will be kept
constant throughout this part of the experiment).
o   Displace the pendulum bob 10.0cm from its equilibrium position and carefully release the bob. Make sure
it swings as straight as possible.
o   Allow the pendulum to execute about 5 complete oscillations, than click on the START button on the
software.
o   Allow the pendulum to complete 5 oscillations of its motion, than click on the STOP button on the
software.
o   The statistics panel shows the maximum, minimum and average of the 5 period measurements, including
the standard deviation for the measurements. Round the mean value to the hundredths place (0.01s) and
record this in your data table.
o   Choose the “Timing” option along the main menu of the software, and select “clear data” so that the
software will immediately begin taking data the next time you click on the START button.
o   Repeat your measurements for pendulum lengths of 35.0cm and 20.0cm. Remember to continue to use the
50.0g pendulum and the 10.0cm amplitude. Recall that it is best to change the pendulum length by
loosening the horizontal support bar and winding, or unwinding, string until the desired length is achieved.

Length(cm)                    Period(s)
20
35
50

Analysis:
1. Plot the Period of the pendulum as a function of its mass using your data from Part 1 of the
experiment. Fit this data to a best straight line1. Comment on the slope of this line in your
conclusions, and what this slope tells you about the dependence of the pendulum’s period on the mass
of the pendulum. (Note that if two physical quantities are truly independent, plotting one as a function
of the other will show no pattern or trend and the slop of the plot will average to zero.) Also comment
on whether your results are consistent with our theoretical understanding of the pendulum (i.e. does the
theory tell us the period should or should not depend on mass and does that agree with your results?).
2.   Plot the Period of the pendulum as a function of its amplitude using your data from Part 2 of the
experiment. Fit this data to a best straight line. Comment on the slope of this line in your conclusions,
and what this slope tells you about the dependence of the pendulum’s period on the amplitude of its
swing. Again, comment on whether your results are consistent with our theoretical understanding of
the pendulum (i.e. does theory tell us the period should or should not depend on the amplitude of the
pendulum’s swing and does that agree with your results?).
3.   Plot the Period of the pendulum as a function of its length using your data from Part 3 of the
experiment.
4.   Use the Graphic Analysis package to graph the period of the pendulum as a function of its length.
After the graph is shown, use the Automatic curve fit feature in the Menu to curve fit this graph. Note
the Mean Square Error for this. Also comment on whether your results are consistent with our
theoretical understanding of the pendulum. That is, does theory tell us the period should or should not
depend on the length of the pendulum? If so what is that dependence? Do your results agree with the
theory, and how good or poor is that agreement?

Be certain to attach a copy of each of your plots to your report.

1
The computer plotting application will likely choose some initial default scale for your plot. Look at this
scale carefully so that you are not mislead. That is, a nearly horizontal line shown on a very fine scale
(over a very small range) may appear to have a non-zero slope. Study your plot, the slop and the Mean
Square Error of the fit carefully.

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