# Condition that the Isochronous Nature of the Simple Pendulum Does Not Hold

## Abstract

In this experiment, a simple pendulum device with a length of 1.3m is made and used to measure the period of the simple pendulum with the accuracy of several milliseconds through the Arduino. A light sensor and a laser are used for the measurement.
As a result of the experiment, the period increases as the amplitude increases. The period increased by less than 1% when the amplitude was less than 30 degrees. The period increased by over 10% when the amplitude reached 80 degrees.

## Motivation and Purpose of Research

The simple pendulum is an idealized mathematical model of a pendulum. It is a weight on the end of a massless string pivoted at a point. When given an initial push, it will swing back and forth at a constant amplitude. The period of the simple pendulum does not depend on the weight or the amplitude of the simple pendulum. This is the definition of the isochronism of a simple pendulum. However, it is said that the isochronism of a simple pendulum is valid only when the amplitude is small. Then what amplitude is considered ‘small’?
There was no data that explains the exact amplitude which is the benchmark between isochronism being valid or not. The only data that could be found was that the isochronism of a simple pendulum is valid only at a small amplitude. The previous reports and experiments all ended up on the line that proves the isochronism of a simple pendulum is valid.
Therefore, this experiment aims to find out the maximum amplitude of a simple pendulum that the isochronism of a simple pendulum holds.

## Theoretical Background

1. The history of the study about a simple pendulum
A simple pendulum was first used in a seismograph that directs the direction of the place where the earthquake occurred by using the vibration of the simple pendulum designed by a Chinese scientist Zhang Hyung. ​In the Renaissance era, pendulums were used as power sources for mechanical devices such as large pumps. The first scientist who studied the properties of the pendulum was an Italian scientist, Galileo Galilei. Galileo discovered an important property called the isochronism of the pendulum. He found that the period of the pendulum was independent of the mass and amplitude of the pendulum and proportional to the square root of the length of the pendulum. Descartes discovered that a pendulum is not isochronous in 1636. The period of the pendulum increased by the amplitude. 37 years later in 1673, Huygens published the pendulum theory and confirmed that Galileo\’s isochronism is correct only for small-amplitude vibrations.
2. Period of an ideal simple pendulum when the amplitude is small Figure 1
A simple pendulum can swing by its weight. An object of mass is suspended on the string. With the gravitational acceleration , the gravitational force of the pendulum can be divided into two, as shown in Figure 1. The force that contributes to the motion of the simple pendulum is . Equation 1
Equation 1 is the general equation of motion. is the force acting on the weight, is the acceleration of the weight. The equation of motion of the simple pendulum is as shown in Equation 2. Equation 2 is the length of the pendulum. If the amplitude is very small in Equation 2, can be approximated by . Using this approximation, Equation 3 appears. Equation 3
The known solution to Equation 3 is Equation 4. Equation 4
Where is a constant value depending on the initial condition. The time when the cosine function of Equation 4 is completed, that is, the period of the simple pendulum is given by Equation 5. Equation 5
It should be noted that this result is based on the approximation that the amplitude is small.
3. Ideal period of the simple pendulum when the amplitude is not small
Equation 5 assumes the amplitude is small. However, the period of a simple pendulum can be expressed as shown in Equation 6 if the amplitude is not small. Equation 6
When the amplitude increases, the period should increase according to Equation 6.

## Experimental Methods and Results  Figure 2 (Left) Photo of the apparatus (Right) Design of the apparatus
Figure 2 is a photo of the period-measuring-device. The parts of the apparatus designed by 3D CAD software Fusion 360 are manufactured into materials of wood, acrylic and brass. The brass corresponding to the weight of the simple pendulum was made to weigh about 600g. The weight was intended to be heavier than the weight holder. The weight was hanging on the shaft by the weight holder with a rolling bearing. The distance from the center of the axis to the center of the simple pendulum was designed to be 1.3m. The longer the simple pendulum is the easier it is to observe the change in the period but it is more difficult to carry.

## The Principle of Experimental Apparatus

The light sensor detects the light source as the simple pendulum swings. The light is temporarily blocked when the pendulum passes the light sensor. At this time, the value of the light sensor rapidly decreases and then increases rapidly again. Arduino measures the value of the light sensor about every 1ms and calculates the period. We have experimented with various types of light sources and concluded that we should use lasers to increase accuracy. When a laser is used as the light source, because the intensity of the light is significantly higher than that of the room illumination, the period of the simple pendulum can be measured without being influenced by the surrounding brightness.

## Experimental Process

1. Connect the Arduino board to the computer and run the Arduino program on the computer to activate the serial monitor.
2. Shine the laser beam on the light sensor.
3. Adjust the amplitude to 10° while looking at the protractor attached to the apparatus.
4. After releasing the simple pendulum from the hand, allow the pendulum to oscillate until the period value appears on the serial monitor three times.
5. Repeat the procedure from 10° to 90° by 5° intervals.

## Results

In Equation 5, the period of the simple pendulum is constant regardless of the amplitude. This fact is presented by Galileo Galileo and is widely known as the isochronous nature of a simple pendulum. On the other hand, in equation 6, the period of a simple pendulum increases as the amplitude increases. It coincides with what Descartes and Huygens found. We investigated the change of the period by changing the amplitude. Figure 3 ​Change of period according to increasing amplitude
Figure 3 shows the dependence of the period on the amplitude. Linearly increasing tendency can be observed. However, it did not fit well with the tendency to increase in the parabolic form in Equation 6. It was difficult to observe the tendency when the amplitude is within 16°. Therefore, the amplitude was increased to 90° and the experiment was carried out again. Figure 4. Appearance of the apparatus support Figure 5. Protractor attached to the apparatus
As shown in Fig. 4 and Fig. 5, the wood board was removed and replaced with Teflon and brass parts to improve the durability of the equipment and to change the amplitude freely. In addition, a ​protractor was attached to allow more precise amplitude manipulation. The amplitude is increased from 10° to 90° in 5° intervals. The measured period is shown in Fig. 6 Fig. 6 ​Dependence of the period of the simple pendulum on the amplitude
I​n Fig. 6, the dotted line represents the estimated value including the second-order term for the amplitude in Equation 6. The value of the gravitational acceleration assigned to Equation 6 was 9.8m/s^2 and the length of the pendulum was 1.23 m. The length of the simple pendulum is designed to 1.3 m, but considering the weight of the weight holder, the length of the centre of gravity is shortened. The period of the simple pendulum was recorded while the simple pendulum swung three times, and is indicated by circles, squares, and triangles in Fig. 6. The average of these three values ​​is shown in red circles. The period measured by the experiment increases in the parabolic curve as amplitude increases. It was well agreed with our prediction. The maximum error was 2.5% at 75 degrees of amplitude. At 75 degrees of amplitude, theoretically, 2479ms of the period is expected.
Also, we found that the period decreases as the number of the swing increases. It is interpreted that the amplitude of the third vibration is smaller than that of the first vibration due to the friction, and thus the period decreases. This was an unexpected result contrary to the general expectation that the period of the third oscillation would be larger than the first oscillation by friction. The period with an amplitude of 10° was 2.238s. The amplitude of 2.260s, which is a 1% increase over this period, can be seen as about 30°. Therefore, it can be seen that the isochronism of the simple pendulum is correct within the 30° of amplitude. Figure 7 ​The rate of change of the period as the amplitude increases
Figure 7 shows the rate of change in the period as the amplitude increases. The rate of change of the period is based on the period when the amplitude is 10 °. Up to an amplitude of 30°, it can be verified that the rate of change is less than 1%. If the period increases by more than 10%, it can be confirmed that the amplitude becomes about 80 ° or more. This corresponds roughly to the theoretical calculation of Equation 6. The calculation shows that the period difference between 0° amplitude and 23° amplitude is 1%.

## Conclusion and Discussion

The isochronism of the simple pendulum is correct within 1% under the 30° of amplitude. A device capable of measuring the period of the simple pendulum was made and the isochronism of the simple pendulum was confirmed to be broken. It is confirmed that the period changes less than 1% until the amplitude is 30°. On the other hand, the period increased by more than 10% when the amplitude became larger than 80°.
The misconception that the period of the pendulum will not change was generated by approximation in the derivation of the equation. Equation including the quadratic terms for amplitude is in agreement with our experimental result within 2.5%.
If we hadn\’t used Arduino and laser, these accurate measurements would have been impossible. The period difference between swinging with the 85 degrees and 10 degrees was only 310ms. It\’s a good example of experimentation that uses Arduino\’s precision to uncover what was wrong.

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