Simple Pendulum and the determination of the acceleration of gravity

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The pendulum is one of the simplest oscillation systems used in the determination of the acceleration of gravity, besides being used as an instrument for the measurement of time.

In this publication I will show how to determine in a practical and simple way the value of acceleration of gravity based upon the simple pendulum. In principle I'll make a small reference to the laws that govern the movement of a simple pendulum.

The simple pendulum is constituted by an inextensible thread of negligible mass and a punctual mass "m" joined to one of its ends as shown in figure 1.

Figure 1 - Materials and equipment used
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Figure 2 - Simple Pendulum Forces Diagram
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Obtainment of the movement equation of the Simple Pendulum

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The obtained Equation shows the dependence of the period with the pendulum length, being also independent of the angle of oscillation (amplitude) and of the mass. It is important to note that the equation obtained is valid for small oscillations.

This dependence had already been observed by Galileo Galilei in the year 1581, when observing in the cathedral of Pisa the pendular movement of the bronze lamps, using as a measure of time his pulsations, given the absence in his time of instruments of measurement of time. Galileo was able at an early age to observe that the oscillation period of the pendulum for small oscillations was independent of the amplitude to what was called pendulous isochronism.

The equation that related the period to the pendulum's length for small oscillations is obtained later by Christian Huygens in 1673.

The parameters involved in the equation (period of oscillation and length) can be obtained with ease, allowing obtaining the value of the acceleration of gravity.

The oscillation times measured for different lengths are shown in the table in figure 3. To minimize the error in the time readings, four oscillation times for each length were averaged.

Figure 3 - Oscillation times measured for different lengths

Figure 3 - Graphic representation of the Average Period Vs. Length

From the expression

According to the experimental relationship obtained in the graphic representation

Percentage error:

theoretical value of gravity = 9.81m/s2

Error% = 100x (theoretical value - experimental value) / theoretical value

Error% = 100x ❘(9,81 – 9,8307) ❘ / 9,81

Error% = 0,21

The obtained result shows that the simple pendulum allows determining with high precision the value of the acceleration of gravity.

Other applications of the pendulum, such as the one used by Léon Foucault for the verification of the rotation of the earth, are indicative of its scope in different areas of science and engineering.

The successful determination of the acceleration of gravity using this method makes it an easy application instrument.

REFERENCES

Physics for Scientists and Engineers, 3rd Edition, Fishbane Gasiorowicz.
Physics for Scientists and Engineers, 6th Edition 6th Edition, Paul A, Tipler.
http://fisicagalileogalilei.blogspot.com/