Laboratory 7, Problem 3: How does the Period of a Pendulum change when the Length of String is varied?
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How does the Period of a Pendulum Change
when the Length of String is varied?
Abstract—Periodic motion involving a swinging pendulum was
studied. As the pendulum oscillated, its motion was captured
using video and computer technology. When analyzed, the period of the motion was found to be given by two multiplied by Pi
multiplied by the square root of the quantity of the pendulum’s
length divided by the acceleration due to gravity. Therefore, the
period, or time to complete a full oscillation, of a pendulum was
found to be dependent on its length.
Index Terms—periodic motion, pendulum
I.
INTRODUCTION
T
he examination of a pendulum involves the study of its
periodic motion. Objects that exhibit this type of motion
follow sinusoidal paths and experience oscillations between
their maximum values of position [1]. The period of a pendulum is defined as the time it takes to complete one full oscillation. This relationship is given as follows [2]:
Figure 1. The pendulum is shown from a front view.
T = 2*Π L / g
In rotational motion, it is often questioned how the period
of the pendulum differs when various factors concerning the
pendulum are changed. For example, a manufacturer of
grandfather clocks wonders how to construct a clock, consisting of a pendulum, which will keep the correct time. A proposed hypothesis for this relationship is that the period will
vary greatly when the length of the pendulum string is
changed.
In this experiment, the motion of a swinging bob of a pendulum will be measured using computer technology to prove
or disprove the stated hypothesis. Equations relating the motion of each mass must be used to determine and compare
their periods.
II. EXPERIMENTAL MATERIALS AND METHODS
A. Arrangement and Equipment
An accurate way to represent and study periodic motion is
to configure an oscillating pendulum and analyze its motion in
relation to different lengths of attached string. To achieve
this, a frictionless string was attached to a mass of one kilogram at its center, creating a freely-oscillating pendulum. The
string was secured to a stationary pole to allow for the measurements to reflect only the periodic motion of the pendulum.
The mass was initially stationary and was released from rest to
begin its motion. The mass then oscillated, reflecting principles of periodic motion. The construction of the laboratory
equipment is demonstrated in Figures 1 and 2.
Figure 2. The pendulum is shown from a side view.
B. Utilization of Data
The motion of the mass was recorded using a video camera and digitally reproduced on a Dell computer using LabVIEW® and AviCapture [3]. The mass was released from
rest to begin the motion of the system. As the mass traveled
along its path of periodic motion, its successive quantities of
position and time were recorded using AviAnalysis. Data
was compiled for string lengths of 71 cm and 52.5 cm.
C. Calibration
The distances recorded in AviAnalysis were then related
to the actual physical distances traveled by the oscillating
mass. Using a known distance of (0.10 m) on a meter stick
captured in clear view by the video camera, the calibration
of the ratio of meters to pixels was obtained.
The calibration affects the experimental results due to the
direct correlation made between distance in the laboratory
and on the computer. If the points of calibration are not input with precision, the data will be skewed and accurate results will not be obtained.
Laboratory 7, Problem 3: How does the Period of a Pendulum change when the Length of String is varied?
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x(t ) = A cos(t * 2Π / T )
y (t ) = A sin(t * 2Π / T )
Position of Pendulum of Length 52.5 cm
0.4
0.35
Position (m)
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
Tim e (s)
Figure 3. Relationship between displacement and time for a pendulum
length of 52.5 cm.
Position of Pendulum of Length 71.0 cm
0.5
To ascertain the period from the given graphs, one must
measure the distance between two points of equal height on
the graph. These two points must include a complete sinusoidal curve between them. These values represent the
measured periods of the data in relation to the length of the
pendulum.
C. Analysis of data-based calculations
Overall, the observed data was found to produce values
for the period that were related to the length of the string by
the following equation:
T = 2*Π L / g
0.4
Position (m)
2
0.3
0.2
0.1
0
0
1
2
3
4
Tim e (s)
Figure 4. Relationship between displacement and time for a pendulum
length of 71.0 cm.
III. RESULTS AND DISCUSSION
A. Analysis of experimental error
The possible errors that affected this experiment were
manual errors performed by humans, those contributed by
the technological equipment and computer and the slight rotation of the pendulum during oscillation. First, the manual
errors occur through the calibration and plotting of points on
AviCapture. The human eye cannot distinguish with exact
preciseness the points with which the cart traveled, but it can
achieve a very close estimate. This percentage of error is
approximated to be ± .005 m in relation to points representing the position of the cart. The second type of error occurs
due to the slight distortion from the physical situation when
translated to the computer by the video technology. This error accounts for approximately ±1.2% of measured data.
Thirdly, the pendulum bob experienced minimal amounts of
rotation along its path of oscillation. The error generated by
this rotation is approximately ±.75% of obtained data. Errors in comparison of measured or calculated values in the
laboratory and actual values are calculated as follows:
(Uncertainty)/(Best value)*100=Error
B. Data Obtained Through Experimentation
The position of each point along the path of the bob’s oscillation in each case was plotted against equal time intervals
during the entire period of motion using lengths of 52.5 cm
and 71.0 cm, given by Figures 3 and 4. The curves were
generated by Excel® and used to calculate the period of the
pendulum’s motion [4]. The period is defined as the time an
object takes to complete one full oscillation. In a sinusoidal
equation, the period T is given as follows:
When the length is changed, the pendulum will take more
or less time to oscillate, depending on its length and acceleration due to gravity. Therefore, the period may be varied
by changing either of these two factors. Since acceleration
due to gravity is constant on Earth, the only dependent factor is the length of the pendulum. This finding is in accord
with the aforementioned hypothesis.
The discrepancies in measured and calculated values for
the period are displayed in Table 1. Reasons for obtaining
slightly uncertain values are accounted for by error and
slight rotation in the swinging bob of the pendulum.
Table 1. Comparison between measured and calculated values of period.
Mass of
Pendulum
(kg)
1.0 kg
1.0 kg
Length
of
String
(m)
0.710 m
0.525 m
Measured
Period (s)
Calculated
Period (s)
Error
(%)
1.67 s
1.43 s
1.69 s
1.45 s
1.2%
1.4%
IV. CONCLUSION
A pendulum will exhibit a period that varies depending
on its length, according to the given equation:
T = 2*Π L / g
This finding agrees with the previously stated hypothesis
within accepted ranges of error.
This result means that the manufacturer of the grandfather
clocks should vary the length of the pendulum string to obtain the desired period. Therefore, he or she may ascertain
the exact length of the string to achieve a period of exactly
one second to ensure the clock’s accuracy and usefulness.
The motion of a pendulum can also be studied without the
use of digital technology to alleviate distortion in the data.
However, manually timing and recording measurements will
produce results affected by error as well.
ACKNOWLEDGMENT
The lab participants wish to acknowledge our TA for assistance in conducting the lab and for the utilization of the
Laboratory 7, Problem 3: How does the Period of a Pendulum change when the Length of String is varied?
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lab equipment.
REFERENCES
[1]Osmond, I. Thornton. (1905). Treatment of Simple
Harmonic Motion. [Online]. Available: www.jstor.org
[2]Patterson, Louise Diehl. (1952) Pendulums of Wren
and Hooke. [Online]. Available: www.jstor.org
[3]Sony model DWF-VL500 video camera, LabVIEW®
by National Instruments, Co., v6.1
[4]Excel® 2002
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