On Equivalent Amplitude Motion of the Foucault Pendulum

1. On Equivalent Amplitude Motion of the Foucault Pendulum Shuai Yuan1, 2,a , Hongjie Li1,2, * 1. LMIB, School of Mathematics and System Science, Beihang University,…
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  • 1. On Equivalent Amplitude Motion of the Foucault Pendulum Shuai Yuan1, 2,a , Hongjie Li1,2, * 1. LMIB, School of Mathematics and System Science, Beihang University, Beijing, China 2. School of Economics and Management, Beihang University, Beijing, China a. Email Address: * Corresponding author: Hongjie Li (Email address: Keywords: Foucault pendulum; Gravitational potential energy; Motion compensation; Idle compensator; Arduino controller. Abstract. Foucault pendulum is one of the earliest scientific experimental devices used to accurately verify the rotation of the Earth. However, for many countries, the attenuation of the maximum deflection has made it an insurmountable problem, which makes the Foucault pendulum vibrate only for a short time. In this paper, we propose a novel design method to address this problem, by using MATLAB and Arduino controller during the project simulation process. By means of using real-time control systems, the maximum deflection is reduced effectively, which achieves the goal of extending the swing time without changing its inherent physical properties. Introduction In 1851, in order to prove the self-rotation of the earth, the French physicist Foucault executed a successful experiment in 1851. During the experiment, the swing plane was gradually moving along the clockwise direction, and the direction of the swing was changing continuously. In explaining this phenomenon, the Foucault pendulum was not affected by external force. Therefore, according to the law of inertia, the changing of the direction of the swing is caused by the earth rotating along the counterclockwise direction. Therefore, the Foucault pendulum, one of the most famous experimental devices, has confidentially validated the self-rotation of the Earth The Foucault pendulum has the significant meaning in both the field of scientific research and education. There are many well-known museums and public places all over the world decorating the Foucault pendulum. For example, as shown in Fig. 1 and Fig. 2, the Foucault pendulums placed in Beijing Planetarium and the Headquarter of U.N. in New York have both shown great instructive meaning, and attracted tremendous students and visitors [1]. Fig.1 Foucault pendulum in Beijing Fig.2 Foucault pendulum in U.N. Headquarters There are many different kinds of design and devices of Foucault pendulum all over the world. For example, However, most of them are affected by the gradually amplitude decrease, which could finally force the device to stop swinging. To solve this problem, some researchers tried to set an electromagnet under the pendulum in order to counteract the friction [2] as shown in Fig. 3.
  • 2. Fig.3 The electromagnet device under the Foucault pendulum in U.N. Headquarters However, this kind of design has also brought new problems towards the motion of the Foucault pendulum. For instance, the swing plane is changing constantly, the electromagnetic force can only be effected when the Foucault pendulum passing through the center of the equipment. Therefore, the accuracy of the swing of the Foucault pendulum is doubted through this solution. Meanwhile, the previous methods are not only very complicated, but also changing the fixed physical property of the Foucault pendulum, which makes the Foucault pendulum loses its value. In this paper, we proposed a novel solution that can make the Foucault pendulum operate under the condition of the equivalent amplitude, while its intrinsic physical properties will not be changed at the same time. The vital part of the solution is that it can compensate energy for the equipment and remain its physical property unchanged. Thus, by using our design, t the Foucault pendulum can not only be used in the museum as a better way to show its principles for people, but also could be used to make teaching aid for schools all over the world [3]. The remainder of this paper is organized as follows: in the second section, we will describe our design in great detail, and provide experimental results. Subsequently, we will draw the conclusion of this paper in the third section. Design of the Proposed Foucault Pendulum and Experimental Analysis In order to solve the amplitude decaying problem, it is necessary to think about how to eliminate the friction and the resistance produced by air. Considering the fact that it is almost impossible for us to place the equipment into a vacuum, we turn to find a way to make the Foucault pendulum working well through the compensating energy. Therefore, finding an appropriate system and the moment for energy compensation is the key issue of our research. In our experiment, we will use a smaller pendulum with the same physical property. Experiment of the Amplitude Decaying and Analysis. In our research, we will discuss the problem through experiment directly. During our experiment, the mechanical model of the Foucault pendulum is shown in Fig.4. Fig.4 The mechanical model of the Foucault pendulum The length of the pendulum isl , the starting point is A, the initial swing angle is θ , the point O is the lowest point that the Foucault pendulum could reach when in moving. The point B is the theoretical highest point the Foucault pendulum can reach when we do not consider the friction or the resistance produced by air. The point C is the highest point the Foucault pendulum can reach in the reality. The difference in height between point A and point O is H , the difference in height between point B and point C is s , the friction and the resistance produced by air in the experiment is denoted by f [3, 4]. Therefore, we have could derive the following ( fW is the total of the friction work) g l k dSfW AOf π θ θ 20 sin⋅ −=⋅= ∫ (1) According to the analysis, we can safely draw the conclusion that when the amplitude of the Foucault pendulum is decaying. We can solve this problem by giving the proper amount of energy to
  • 3. the system. And by doing this, we can make the amplitude returning to the original level. We can draw the diagram which shows the compensation of the energy to the system by using MATLAB, as shown in Fig. 5. Fig.5 The diagram of the energy compensation The Scheme of the Energy Compensation and the Experimental Data [5, 6]. According to the analysis, it is clearly that the motion of the Foucault pendulum is a process involving the transformation between the kinetic energy and the gravity potential energy. When the Foucault pendulum achieves the highest point, we can observe that the amplitude is changing obviously. In other words, the energy is decaying. Therefore, when it reach the highest point, we can make the Foucault pendulum move upwards a lot bit by giving the external force from outside to ensure the total energy of the system remains unchanged. The amount of the energy we need to compensate and the magnitude of moving upwards are derived by the following analysis. Assuming that the amount of moving upwards caused by the external force is h . If the ball starts to swinging at point A, and the time used until the ball finally stopped at the point O (When o 3≤θ , we think that the ball has stopped to operate), we can measure the time the ball used as t(s) as follows: Therefore, we have h∆ = t l g l )cos1( θπ −⋅⋅ (2) In order to to get the range of h , we select a iron ball as the swing ball and choose 0.9m、1.2m、 1.5m as the length of the pendulum respectively in our experiment. When deciding the length of the pendulum of a specific value, we select 5 degree, 10 degree and 15 degree as the initial angles of the pendulum, respectively to do repeated times of experiments and report the the average as the final results. By doing these experiments we can derive the amount the pendulum needed to be compensated. According to the data we got from the experiment, we can easily draw the conclusion that when the length of the pendulum is 0.9m, the amount we need to raise is about 5 10×34.5 - m. For the length of 1.2m, the amount is about 5 10×60.7 - m. For the length of 1.5m, the amount is about 5 10×44.9 - m. Therefore, we can safely draw the conclusion that the scope of h is ( the unit is meter): 3 100534.0 − × ≤∆≤ h .10×0944.0 3 The conclusion we drawn can provide us a reasonable scope of h , when considering the practical solution. Fig. 6 The appearance of ISM Fig. 7 The structure principle of ISM The Implemental Details of the Proposed Design. After calculating the approximate scope of h through the analysis above,we finally put forward a solution of Foucault Pendulum system by using the idle speed motor which could provide upward lift power. The idle speed motor (or, ISM for short)
  • 4. is a kind of electric devices that can control linear flex movement precisely. Its appearance and the strructure of are shown as in Fig. 6 and Fig. 7, respectively. In our design, we take advantage of the flexibility of the idle speed motor to achieve the compensation of Foucault Pendulum, in order to avoid complicated mechanical structures and the experimental device simpler. In our solution, the idle speed motor is lift up by h in a half Foucault pendulum cycle to supply the energy of the system. In order to receive the signal of photosensitive sensor (as shon in Fig. 8) and control the idle speed motor simultaliously, an Arduino controller should be fixed right under the Foucault pendulum, as shown in Fig.8. When the pendulum ball swings to its highest point, the idle speed motor will be pulled up at h . When the ball is moving from the highest to the lowest, the idle speed motor will fall a height of h . Then, when it is moving from the lowest to the highest, the idle speed motor will be lifted up by h . In the whole process, the signal of photosensitive sensor will be received by Arduino controller to control the operation of the idle speed motor. Fig.8 The photosensitive sensor Fig. 9 The Arduino controller Considering that the Foucault pendulum solution of this paper would finally be used in the China Science and Technology Museum, we also provide the corresponding design solution for the actual exhibition. Design schemes are shown in Fig. 10 and Fig. 11. The height of the exhibits is 1.4 m, and the square for the base is set to 0.4m. Pendulum length of the exhibits is 1.2 meters long, and the diameter of the pendulum ball is 0.04m. The upper space of the exhibition uses the idling motor previously described in this paper. Photosensitive component will be installed in the hole, right under the Foucault pendulum to collect the swing information of the pendulum ball. The set of scale disc around the hole is to help the observer understand the pendulum trajectory clearly. The set of this device will enable the Foucault pendulum run continuously, which could help audiences better understand the Foucault pendulum and the Earth rotation bias forces. Fig. 10 The model of the Foucault pendulum Fig. 11 The model of the idle speed motor Conclusion This paper reviewed the theory, related topics and research methods of the famous Foucault pendulum, based on which this paper experimentally study the amplitude attenuation problem. We propose a novel design, of which the key is to compensate the energy loss in the process of the movement through gravitational potential energy. In order to guide the practical implementation, this paper also experimentally found the specific solution for the Foucault pendulum energy compensation and the range of important parameters, which could provide some reference advices for the engineering design in the aspect of perpetual motion of Foucault pendulum. Finally, two sets of Foucault pendulum energy compensation equipment are designed to support Planetarium Foucault pendulum. On the other hand, our method still has several problems to be fixed. For instance, the energy attenuation is not calculated accurately, which caused by the experiment condition. Thus, errors
  • 5. cannot be avoided. In our future work, we will improve the accuracy of computing the energy attenuation. Acknowledgement This work is supported by the Natural and Science Foundation of China (Grant No. 61379001). References [1] The website of the U.N., Available at: pendulum.html [2] G.C. Yang, The Foucault Pendulum based on the photoelectric control, College Physics, vol.3, pp.28-36, 1998. [3]Y.S. Chai, The Foucault Pendulum with Micro Size, Patent No.CN02216572.X, 2004. [4]K.S. Kruszelnicki, The Foucault Pendulum, Available at: /pendulum.html [5] H.Richard Crane, Foucault Pendulum ‘Wall Clock’ , American Association of Physics Teachers, vol.1, pp.63, 1995. [6] Joe Wolfe, “The Foucault Pendulum”, Available at: //jw/foucault_pendulum.html
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