This paper deals with the non-linear oscillation of a simple pendulum and presents an approach for solving the non-linear differential equation that governs its movement by using the harmonic balance method. With this technique it is possible to easily obtain analytical approximate formulas for the period of the pendulum. As we shall see, these formulas show excellent agreement with the exact period calculated with the use of elliptical integrals, and they are valid for both small and large amplitudes of oscillation.The most significant feature of the treatment presented is its simplicity because for the level of approximation considered in this paper the required work can be done "by hand". KEY WORDS: Simple pendulum, large-angle period, harmonic balance method 3
This paper deals with the nonlinear oscillation of a simple pendulum and presents not only the exact formula for the period but also the exact expression of the angular displacement as a function of the time, the amplitude of oscillations and the angular frequency for small oscillations. This angular displacement is written in terms of the Jacobi elliptic function sn(u;m) using the following initial conditions: the initial angular displacement is different from zero while the initial angular velocity is zero. The angular displacements are plotted using Mathematica, an available symbolic computer program that allows us to plot easily the function obtained. As we will see, even for amplitudes as high as 0.75π (135• ) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency ω in terms of the complete elliptic integral of the first kind. We can conclude that for amplitudes lower than 135 o the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the initial amplitude). We believe that present study may be a suitable and fruitful exercise for teaching and better understanding the behavior of the nonlinear pendulum in advanced undergraduate courses on classical mechanics. Keywords: simple pendulum, large-angle period, angular displacement.Este artigo aborda a oscilação não-linear de um pêndulo simples e apresenta não apenas a fórmula exata do período mas também a dependencia temporal do deslocamento angular para amplitudes das oscilações e a freqüência angular para pequenas oscilações. O deslocamento angularé escrito em termos da função elíptica de Jacobi sn(u;m) usando as seguintes condições iniciais: o deslocamento angular inicialé diferente de zero enquanto que a velocidade angular inicialé zero. Os deslocamentos angulares são plotados usando Mathematica, um disponível programa simbólico de computador que nos permite plotar facilmente a função obtida. Como veremos, mesmo para amplitudes tão grandes quanto 0,75π (135 o )é possível usar a expressão para o deslocamento angular mas considerando a expressão exata para a freqüência angular w em termos da integral elíptica completa de primeira espécie. Concluímos que, para amplitudes menores que 135 o , o movimento periódico exibido por um pêndulo simplesé praticamente harmônico, mas suas oscilações não são isócronas (o períodoé uma função da amplitude inicial). Acreditamos que o presente estudo possa tornar-se um exercício conveniente e frutífero para o ensino e para uma melhor compreensão do pêndulo não-linear em cursos avançados de mecânica clássica na graduação. Palavras-chave: pêndulo simples, período a grandesângulos, deslocamento angular.Perhaps one of the nonlinear systems most studied and analyzed is the simple pendulum [1][2][3][4][5][6][7][8][9][10][11][12], which is the most popular textbook example of a nonlinear system and is studied not only in advanced but also in introductory university courses of...
ORTUÑO, Manuel, et al. "Optimization of a 1 mm thick PVA/acrylamide recording material to obtain holographic memories: method of preparation and holographic properties". Applied Physics B: Lasers and Optics. Vol. 76, No. 8 (July 2003 AbstractInformation holographic storage is a very promising technique due to its high theoretical capacity. One of the key factors in developing holographic memories is the need for a suitable recording material which must have certain specific characteristics. In particular, in order to achieve a high storage density it is necessary to work with great thicknesses. One of the essential requirements for holographic memories to be competitive is that the material must have a thickness of 500 µm or more, but it is not easy to find such thicknesses with the photopolymers currently available. In this study, we develop a method of preparing layers of a polyvinyl alcohol/acrylamide based photopolymer approximately 1 mm thick. Optimization of this material makes it possible to obtain good results for the main holographic parameters; diffraction efficiency 70% and energetic sensitivity 50 mJ/cm 2 .PACS numbers: 42.70.Ln; 42.40.Pa; 42.40.Ht ORTUÑO, Manuel, et al. "Optimization of a 1 mm thick PVA/acrylamide recording material to obtain holographic memories: method of preparation and holographic properties".
In this work we propose an inexpensive laboratory practice for the laboratory of an introductory course of Physics for any grade of Sciences and Engineering, which was very well received by our students, where a smartphone (iOS, Android or Windows Phone is irrelevant) is used together with some mini magnets, similar to those that can be found in the door of our refrigerators, a 20 cm long school rule, a paper and a free application (app) for measuring the magnetic field using the magnetic fields sensor or magnetometer of the smartphone, which needs to be downloaded and installed. The apps we have used are: Magnetometer (iOS), Magnetometer Metal Detector and Physics Toolbox Magnetometer (Android). Nothing else is needed. Cost of this practice: 0 coins. The main purpose of the practice is that students determine the dependence of the component x of the magnetic field produced by different magnets (from the typical magnets that are decorated in refrigerators even with a ring magnet and spherical magnet). We have obtained that the dependency of the magnetic field with the distance is of the form x -3 , in total agreement with the theoretical analysis. The secondary objective is to apply the technique of least squares fit to obtain this exponent and the magnetic moment of the magnets, with theirs corresponding absolute error.
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