Laboratory 7, Problem 3: How does the Period of a Pendulum change when the Length of String is varied? #20346

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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 pe-riod 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 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 pendu-lum is defined as the time it takes to complete one full oscilla-tion. This relationship is given as follows [2]:

gLT /*2 Π= 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, consist-ing of a pendulum, which will keep the correct time. A pro-posed 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 pen-dulum will be measured using computer technology to prove or disprove the stated hypothesis. Equations relating the mo-tion 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 kilo-gram at its center, creating a freely-oscillating pendulum. The string was secured to a stationary pole to allow for the meas-urements 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 princi-ples of periodic motion. The construction of the laboratory equipment is demonstrated in Figures 1 and 2.

Figure 1. The pendulum is shown from a front view.

Figure 2. The pendulum is shown from a side view.

B. Utilization of Data The motion of the mass was recorded using a video cam-

era and digitally reproduced on a Dell computer using Lab-VIEW® 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 in-put with precision, the data will be skewed and accurate re-sults will not be obtained.

T

Laboratory 7, Problem 3: How does the Period of a Pendulum change when the Length of String is varied? #20346

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Position of Pendulum of Length 52.5 cm

00.050.1

0.150.2

0.250.3

0.350.4

0 1 2 3

Time (s)

Posi

tion

(m)

4

Figure 3. Relationship between displacement and time for a pendulum length of 52.5 cm.

Position of Pendulum of Length 71.0 cm

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3

Time (s)

Posi

tion

(m)

4

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 ro-tation 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 represent-ing 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 er-ror 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. Er-rors 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 os-cillation 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:

)/2*sin()()/2*cos()(

TtAtyTtAtx

Π=Π=

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 sinu-soidal 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:

gLT /*2 Π= When the length is changed, the pendulum will take more

or less time to oscillate, depending on its length and accel-eration 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 fac-tor 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)

Length of

String (m)

Measured Period (s)

Calculated Period (s)

Error (%)

1.0 kg 0.710 m 1.67 s 1.69 s 1.2% 1.0 kg 0.525 m 1.43 s 1.45 s 1.4%

IV. CONCLUSION A pendulum will exhibit a period that varies depending

on its length, according to the given equation:

gLT /*2 Π= 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 ob-tain 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 as-

sistance 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? #20346

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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