Abstract
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.
Highlights
The Moon colonization is still a ludic topic that drives the human spirit to embrace nobles causes and some efforts that lift the status quo of the human thought
We can compare the different models treated in the text in Table 4, namely: Note that this table encloses the values that some quantities assume in the center of the Moon and the time interval that a free falling particle takes to cross the entire tunnel drilled along a diameter of the Moon
With respect to Earth, with a more accurate analysis of the dashed curve given in the Fig. 4b, we find that the motion in the first 13.5 minutes is characterized by an approximately constant acceleration, around 10 m/s2, and during the remaining 5.6 minutes, the acceleration drops gradually until it reaches zero in the center of the Earth [20]. This result confirms what is advocated in the reference [4], which states that an Earth model based on constant acceleration is more realistic than that which is addressed in didactic literature in that the authors treat a planet as if it had constant density (SHM)
Summary
The Moon colonization is still a ludic topic that drives the human spirit to embrace nobles causes and some efforts that lift the status quo of the human thought. In order to get this, the Moon colonization can play a very important role, since it can be used as a first camp base to start the exploration of the outer space At this context, the construction of an efficient mobility & transportation net is mandatory to support such an enterprise. [4] the author proposed a more realistic model, in which the Earth density is considered non uniform This geological model is constructed on the data base denominated Preliminary reference Earth model (PREM) [5]. In order to clarify this point, we departure from those Moon models and, through some numerical computation and experimental data, numerical results for some physical quantities are calculated: time, position, velocity, acceleration and pressure on the tunnel walls.
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