It is well known that hypothetical tunnels net within the Earth is usually considered as an illustration in a gravitation course for undergraduate students. In this context, an uniform mass distribution is usually considered and, consequently, a vehicle inside the tunnel oscillates in a simple harmonic motion (SHM). In this work, we investigate this problem and we argue that, instead to illustrate this problem using Earth, it is better to consider the Moon. In order to support this proposition, we present a twofold argumentation: first, we show that there is a quite discrepancy between the results obtained from Earth uniform mass distribution model with those obtained considering realistic Earth models (PREM model), whose mass is nonuniformly distributed; second, when the Moon is considered, we show that there is not a discrepancy between the results computed from uniform mass distribution model with those computed from nonuniform mass distribution. Indeed, we revealed that a vehicle inside the tunnel within Moon oscillates approximately in a SHM.

<p>Apresentamos uma abordagem essencialmente geométrica do problema da braquistócrona. Utilizamos conceitos de trigonometria básica, acessíveis a estudantes de Física Geral e Experimental I e II, de modo a permitir que o problema já seja apresentado num primeiro momento ainda no primeiro ano de um curso de física ou engenharia. Mostramos que a aceleração resultante sobre uma partícula descrevendo uma braquistócrona tem módulo <italic>g</italic> da aceleração gravitacional e aponta na direção do centro do círculo que gera a cicloide. Apresentamos um método geométrico que permite deduzir a equação diferencial da cicloide sem lançar mão do cálculo variacional. Mostramos que o tempo de descida ao longo desta trajetória é sempre o mesmo, independentemente da posição inicial.</p>

<p>We present an essentially geometric approach to the brachistochrone problem. We apply basic knowledge of trigonometry, accessible to the students in a one or two semester basic physics program, in such a way to allow the problem to be introduced at the beginning of an undergraduate course of Physics or Engineering. We show that the resulting acceleration on a particle describing a brachistochrone has gravitational acceleration of magnitude <italic>g</italic>and points towards the center of the circle that generates the cycloid. We present a geometrical method to deduce the differential equation for the cycloid, without using variational calculus. We show that the descending time along this trajectory is always the same, independently of the starting point.</p>

We present a simple and general method to solve higher-power D'Alembert equation in (1+1) dimensions for a set of completely general initial conditions. Explicit solutions are written down for the cases of []² and []³ and the singularity structure is manifest through the derivative terms of some of the initial-condition functions. The explicit solutions found out may be useful if one wishes to device external sources which may suppress the non-physical modes introduced by the higher derivatives.

Propomos a introdução do método de diferenças finitas como tópico a ser abordado na disciplina de mecânica quântica de um curso de graduação em física. O método é suficientemente simples para ser introduzido e exemplificado em seis horas de aula, permitindo a obten cão de resultados tanto qualitativamente corretos como de alta precisão quantitativa. Devido a sua grande aplicabilidade, seu entendimento por parte dos alunos de graduação em física, futuros pesquisadores, é essencial para a formação acadêmica destes. No presente trabalho, apresentamos o método em detalhes. Sua precisão é verificada calculando o espectro de energia para o oscilador harmônico e comparando-os com os resultados analí ticos conhecidos. Em seguida aplicamos o método a dois outros sistemas, o oscilador anarmônico quártico e o potencial linear. Para cada um destes sistemas, calculamos os dez ní veis de energia mais baixos, bem como seus respectivos auto-estados.

We propose the introduction of the finite differences method as one topic to be inserted in the discipline of quantum mechanics in a physics undergraduate course. The method is simple enough to be introduced and exemplified in about six hours of class allowing both qualitatively correct and high-precision quantitative results. Due to the great applicability of the method, it is essential that the undergraduate students, future researchers, learn it. In the present work, we show the method in detail and verify its precision initially by calculating the energy spectrum of the harmonic oscillator and comparing it to its well-known analytical results. Then we apply it to two other systems, the anharmonic oscillator and the one with linear potential. For each of those systems We compute, for both systems, the ten lowest energy eigenvalues are calculated, as well as their corresponding eigenfunctions.