Pub Date : 2023-11-08DOI: 10.1088/1361-6404/ad0aa3
Benito Vázquez-Dorrı́o, Angel Paredes, Miguel Angel Queiruga-Dios
Abstract Inertial motion looks like accelerated motion when observed from the point of view of a non-inertial reference frame. A non-inertial observer can then describe the trajectories by introducing fictitious forces, like the centrifugal and Coriolis forces, that arise from the coordinate change between reference frames. This well-known fact is part of the typical Physics syllabus for undergraduate scientists and engineers, and a number of interesting classroom demonstrations have been discussed in the literature. We present a complementary possibility for the visualization of the effect of fictitious forces by shining a blue laser beam on a rotating platform covered with a phosphorescent vinyl sheet. The laser can be moved in order to simulate inertial motion in the laboratory frame for the trajectory of the laser spot. This gets immediately imprinted in the rotating phosphorescent material resulting in non-inertial trajectories that can be readily observed and compared to the dynamics governed by fictitious forces. Since friction is not considered, this hands-on activity can be considered as a direct demonstration of the effect of pure fictitious forces in and out the classroom. The approach is simple, inexpensive, fast and non-destructive, and can therefore be very convenient for active lecture demonstrations or individual activities of students. We also describe some educational possibilities of how to use the procedure in the classroom or in the laboratory.
{"title":"Hands-on visualization of the effect of fictitious forces with a laser pointer","authors":"Benito Vázquez-Dorrı́o, Angel Paredes, Miguel Angel Queiruga-Dios","doi":"10.1088/1361-6404/ad0aa3","DOIUrl":"https://doi.org/10.1088/1361-6404/ad0aa3","url":null,"abstract":"Abstract Inertial motion looks like accelerated motion when observed from the point of view of a non-inertial reference frame. A non-inertial observer can then describe the trajectories by introducing fictitious forces, like the centrifugal and Coriolis forces, that arise from the coordinate change between reference frames. This well-known fact is part of the typical Physics syllabus for undergraduate scientists and engineers, and a number of interesting classroom demonstrations have been discussed in the literature. We present a complementary possibility for the visualization of the effect of fictitious forces by shining a blue laser beam on a rotating platform covered with a phosphorescent vinyl sheet. The laser can be moved in order to simulate inertial motion in the laboratory frame for the trajectory of the laser spot. This gets immediately imprinted in the rotating phosphorescent material resulting in non-inertial trajectories that can be readily observed and compared to the dynamics governed by fictitious forces. Since friction is not considered, this hands-on activity can be considered as a direct demonstration of the effect of pure fictitious forces in and out the classroom. The approach is simple, inexpensive, fast and non-destructive, and can therefore be very convenient for active lecture demonstrations or individual activities of students. We also describe some educational possibilities of how to use the procedure in the classroom or in the laboratory.
","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":"10 s4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135341012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper we extend some recent works about Doppler effect in surface waves on water. We improve the experimental set up exploring several situations: source in motion with constant velocity and receiver at rest, source at rest and receiver in motion with constant velocity, as well as both source and receiver in motion. Thereby we produce fractional frequency changes of the order of 40-50%, far larger than those obtained by more traditional sound experiments. The experimental set-up, the data collection and the data analysis also allow to highlight some aspects relevant from a didactic point of view, in particular the experimental results clearly show the nonlinearity of the Doppler shift with the moving source velocity.
{"title":"Doppler effect in the ripple tank: further experiments with a moving source","authors":"Michele D'Anna, Tommaso Corridoni, Stefano Sposetti, Federico Corni","doi":"10.1088/1361-6404/ad0aa2","DOIUrl":"https://doi.org/10.1088/1361-6404/ad0aa2","url":null,"abstract":"Abstract In this paper we extend some recent works about Doppler effect in surface waves on water. We improve the experimental set up exploring several situations: source in motion with constant velocity and receiver at rest, source at rest and receiver in motion with constant velocity, as well as both source and receiver in motion. Thereby we produce fractional frequency changes of the order of 40-50%, far larger than those obtained by more traditional sound experiments. The experimental set-up, the data collection and the data analysis also allow to highlight some aspects relevant from a didactic point of view, in particular the experimental results clearly show the nonlinearity of the Doppler shift with the moving source velocity.
","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":" 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135340962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-08DOI: 10.1088/1361-6404/ad0a9f
Luis Medrano Navarro, Luis Martin-Moreno, Sergio G Rodrigo
Abstract The research in Artificial Intelligence methods with potential applications in science has become an essential task in the scientific community in recent years. Physics Informed Neural Networks (PINNs) is one of these methods and represents a contemporary technique based on neural network fundamentals to solve differential equations. These networks can potentially improve or complement classical numerical methods in computational physics, making them an exciting area of study. In this paper, we introduce PINNs at an elementary level, mainly oriented to physics education, making them suitable for educational purposes at both undergraduate and graduate levels. PINNs can be used to create virtual simulations and educational tools that aid in understating complex physical concepts and processes involving differential equations. By combining the power of neural networks with physics principles, PINNs can provide an interactive and engaging learning experience that can improve students' understanding and retention of physics concepts in higher education.
{"title":"Solving differential equations with Deep Learning: a beginner's guide","authors":"Luis Medrano Navarro, Luis Martin-Moreno, Sergio G Rodrigo","doi":"10.1088/1361-6404/ad0a9f","DOIUrl":"https://doi.org/10.1088/1361-6404/ad0a9f","url":null,"abstract":"Abstract The research in Artificial Intelligence methods with potential applications in science has become an essential task in the scientific community in recent years. Physics Informed Neural Networks (PINNs) is one of these methods and represents a contemporary technique based on neural network fundamentals to solve differential equations. These networks can potentially improve or complement classical numerical methods in computational physics, making them an exciting area of study. In this paper, we introduce PINNs at an elementary level, mainly oriented to physics education, making them suitable for educational purposes at both undergraduate and graduate levels. PINNs can be used to create virtual simulations and educational tools that aid in understating complex physical concepts and processes involving differential equations. By combining the power of neural networks with physics principles, PINNs can provide an interactive and engaging learning experience that can improve students' understanding and retention of physics concepts in higher education.","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":" 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135340954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-27DOI: 10.1088/1361-6404/ad0790
Pekka Pirinen, Pascal Klein, Simon Zacharias Lahme, Antti Lehtinen, Lucija Rončević, Ana Susac
Abstract Digital signal processing is a valuable practical skill for the contemporary
physicist, yet in physics curricula, its central concepts are often introduced either
in method courses in a highly abstract and mathematics-oriented manner or in lab
work with little explicit attention. In this paper, we present an experimental task in
which we focus on a practical implementation of the discrete Fourier transform (DFT)
in an everyday context of vibration analysis using data collected by a smartphone
accelerometer. Students are accompanied in the experiment by a Jupyter notebook
companion, which serves as an interactive instruction sheet and a tool for data analysis.
The task is suitable for beyond-first-year university physics students with some prior
experience in uncertainty analysis, data representation, and data analysis. Based on
our observations the experiment is very engaging. Students have consistently reported
interest in the experiment and they have found it a good demonstration of the DFT
method.
{"title":"Exploring digital signal processing using an interactive Jupyter notebook and smartphone accelerometer data","authors":"Pekka Pirinen, Pascal Klein, Simon Zacharias Lahme, Antti Lehtinen, Lucija Rončević, Ana Susac","doi":"10.1088/1361-6404/ad0790","DOIUrl":"https://doi.org/10.1088/1361-6404/ad0790","url":null,"abstract":"Abstract Digital signal processing is a valuable practical skill for the contemporary
physicist, yet in physics curricula, its central concepts are often introduced either
in method courses in a highly abstract and mathematics-oriented manner or in lab
work with little explicit attention. In this paper, we present an experimental task in
which we focus on a practical implementation of the discrete Fourier transform (DFT)
in an everyday context of vibration analysis using data collected by a smartphone
accelerometer. Students are accompanied in the experiment by a Jupyter notebook
companion, which serves as an interactive instruction sheet and a tool for data analysis.
The task is suitable for beyond-first-year university physics students with some prior
experience in uncertainty analysis, data representation, and data analysis. Based on
our observations the experiment is very engaging. Students have consistently reported
interest in the experiment and they have found it a good demonstration of the DFT
method.","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-27DOI: 10.1088/1361-6404/ad0188
Hollis Williams
Abstract The relativistic correction to the precession of the perihelion of Mercury provided key evidence for the accuracy of general relativity as a theory of gravity. This example still has a large amount of potential to introduce students to the power of numerical simulations in theoretical physics, but existing approaches may be too detailed for many students and involve them beginning to learn a programming language at the same time. In this article, we take a simpler approach which uses as little coding as possible. The equation for the orbit of a planet is solved with and without relativistic corrections. It is shown that there is precession of the perihelion in the relativistic case, whereas in the Newtonian case, the orbit does not rotate about the origin. Quantitative information is extracted on the precession of the perihelion of Mercury and shown to match with observations.
{"title":"An elementary approach to simulating the perihelion of Mercury","authors":"Hollis Williams","doi":"10.1088/1361-6404/ad0188","DOIUrl":"https://doi.org/10.1088/1361-6404/ad0188","url":null,"abstract":"Abstract The relativistic correction to the precession of the perihelion of Mercury provided key evidence for the accuracy of general relativity as a theory of gravity. This example still has a large amount of potential to introduce students to the power of numerical simulations in theoretical physics, but existing approaches may be too detailed for many students and involve them beginning to learn a programming language at the same time. In this article, we take a simpler approach which uses as little coding as possible. The equation for the orbit of a planet is solved with and without relativistic corrections. It is shown that there is precession of the perihelion in the relativistic case, whereas in the Newtonian case, the orbit does not rotate about the origin. Quantitative information is extracted on the precession of the perihelion of Mercury and shown to match with observations.","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":"80 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136233036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.1088/1361-6404/ad06bf
Matteo Zaegel, Mathis Vehils-Vinals, Hugo Guastalla, Benjamin Benabou, Auguste Gires
Abstract We have all wondered once whether we should walk, run or sprint under the rain in order to stay as dry as possible. Previous publications already addressed this subject using simple models, as for the shape of the body and the description of the rain and wind. This paper presents a detailed approach which relies on a more realistic 'human body' shape and accounts of the variability in time of both the wind and the rain drop size and velocity distributions. It appears that in some seldom cases with tailwind and light rain, there is an optimum velocity, but in general it is better to run as fast as possible. While 'running' instead of 'walking' yields significant gain, the extra effort required to 'sprint' is not always worth it.
{"title":"Should you walk, run or sprint in the rain to get less wet?","authors":"Matteo Zaegel, Mathis Vehils-Vinals, Hugo Guastalla, Benjamin Benabou, Auguste Gires","doi":"10.1088/1361-6404/ad06bf","DOIUrl":"https://doi.org/10.1088/1361-6404/ad06bf","url":null,"abstract":"Abstract We have all wondered once whether we should walk, run or sprint under the rain in order to stay as dry as possible. Previous publications already addressed this subject using simple models, as for the shape of the body and the description of the rain and wind. This paper presents a detailed approach which relies on a more realistic 'human body' shape and accounts of the variability in time of both the wind and the rain drop size and velocity distributions. It appears that in some seldom cases with tailwind and light rain, there is an optimum velocity, but in general it is better to run as fast as possible. While 'running' instead of 'walking' yields significant gain, the extra effort required to 'sprint' is not always worth it.","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":"4 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134973479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.1088/1361-6404/ad06be
Francesco Bernardini, Abhijit Chakraborty, Carlos R Ordonez
Abstract This pedagogical article elucidates the fundamentals of trapped-ion quantum computing, which is one of the potential platforms for constructing a scalable quantum computer. The evaluation of a trapped-ion system's viability for quantum computing is conducted in accordance with DiVincenzo's criteria.
{"title":"Quantum computing with trapped ions: a beginner's guide","authors":"Francesco Bernardini, Abhijit Chakraborty, Carlos R Ordonez","doi":"10.1088/1361-6404/ad06be","DOIUrl":"https://doi.org/10.1088/1361-6404/ad06be","url":null,"abstract":"Abstract This pedagogical article elucidates the fundamentals of trapped-ion quantum computing, which is one of the potential platforms for constructing a scalable quantum computer. The evaluation of a trapped-ion system's viability for quantum computing is conducted in accordance with DiVincenzo's criteria.","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":"50 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134973345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-19DOI: 10.1088/1361-6404/acfd23
Yuchen Jiang, Jin Wang, Xiaojie Wang
Abstract We investigated the problem of ‘magnetic levitation’ originating from the 33rd International Young Physicists’ Tournament (IYPT). The problem was first investigated by a PRL paper (Baldwin et al. ) in 2018, which states that the flea of a magnetic stirrer spinning fast enough in a liquid with a high viscosity coefficient can jump from the bottom and levitate stably. The magnetic force and gravity balance periodically. This phenomenon includes several concepts: magnetic dipoles, rigid-body rotation, fluid mechanics and magnetic levitation. They are more or less unfamiliar to undergraduate students. However, the movement of the flea could be described with a concise forced vibration equation, which is familiar in textbooks. The phenomenon could be divided into two stages: synchronous movement and levitation state. The transition is the jumping of the flea. We demonstrated this process and presented several equations to build this physical model. The progression of the phenomenon is due to the increase in the drive magnet angular velocity called the drive velocity. We verified our theory by simulation and experiments. Several parameters are experimentally verified to influence the phenomenon. We also discussed the origin of dynamic stabilization, which would be slightly complicated but worthy for students. In short, we introduce an interesting problem originating from the PRL paper that can be easily achieved under laboratory conditions. We extend some content in a pedagogical way that would be helpful for students to understand the related physical concepts, such as the influence of the viscosity coefficient of the liquid on the flea’s motion, which is not discussed in the PRL paper.
摘要:我们研究了起源于第33届国际青年物理学家锦标赛(IYPT)的“磁悬浮”问题。2018年，PRL的一篇论文(Baldwin et al.)首先研究了这个问题，该论文指出，磁性搅拌器的跳蚤在高粘度系数的液体中旋转得足够快，可以从底部跳起并稳定悬浮。磁力和重力周期性地平衡。这种现象包括几个概念:磁偶极子、刚体旋转、流体力学和磁悬浮。对于本科生来说，它们或多或少有些陌生。然而，跳蚤的运动可以用一个简明的强迫振动方程来描述，这在教科书中是熟悉的。这种现象可分为两个阶段:同步运动和悬浮状态。过渡是跳蚤的跳跃。我们演示了这个过程，并提出了几个方程来建立这个物理模型。这种现象的进展是由于驱动磁体角速度的增加，称为驱动速度。我们通过仿真和实验验证了我们的理论。实验验证了几个参数对这一现象的影响。我们还讨论了动态稳定的起源，这可能有点复杂，但值得学生学习。简而言之，我们引入了一个有趣的问题，起源于PRL论文，可以很容易地在实验室条件下实现。我们以教学的方式扩展了一些有助于学生理解相关物理概念的内容，例如液体粘度系数对跳蚤运动的影响，这在PRL论文中没有讨论。
{"title":"Study on the Magnetic Levitation of a Magnetic Flea","authors":"Yuchen Jiang, Jin Wang, Xiaojie Wang","doi":"10.1088/1361-6404/acfd23","DOIUrl":"https://doi.org/10.1088/1361-6404/acfd23","url":null,"abstract":"Abstract We investigated the problem of ‘magnetic levitation’ originating from the <?CDATA $33mathrm{rd}$?> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <mml:mn>33</mml:mn> <mml:mi>rd</mml:mi> </mml:math> International Young Physicists’ Tournament (IYPT). The problem was first investigated by a PRL paper (Baldwin et al. ) in 2018, which states that the flea of a magnetic stirrer spinning fast enough in a liquid with a high viscosity coefficient can jump from the bottom and levitate stably. The magnetic force and gravity balance periodically. This phenomenon includes several concepts: magnetic dipoles, rigid-body rotation, fluid mechanics and magnetic levitation. They are more or less unfamiliar to undergraduate students. However, the movement of the flea could be described with a concise forced vibration equation, which is familiar in textbooks. The phenomenon could be divided into two stages: synchronous movement and levitation state. The transition is the jumping of the flea. We demonstrated this process and presented several equations to build this physical model. The progression of the phenomenon is due to the increase in the drive magnet angular velocity called the drive velocity. We verified our theory by simulation and experiments. Several parameters are experimentally verified to influence the phenomenon. We also discussed the origin of dynamic stabilization, which would be slightly complicated but worthy for students. In short, we introduce an interesting problem originating from the PRL paper that can be easily achieved under laboratory conditions. We extend some content in a pedagogical way that would be helpful for students to understand the related physical concepts, such as the influence of the viscosity coefficient of the liquid on the flea’s motion, which is not discussed in the PRL paper.","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135667264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-17DOI: 10.1088/1361-6404/acfdd9
Lachezar Slavchev Simeonov
Abstract When considering the motion of an incompressible fluid, it is common practice to take the curl on both sides of the Navier–Stokes (or Euler) equations and cancel the pressure force. The governing equations are sufficient to derive the velocity field of the fluid without any knowledge of the pressure. In fact, the pressure is only calculated after obtaining the velocity field. This raises a number of conceptual problems. For instance, why is the pressure unnecessary for obtaining the velocity field? Traditionally, forces have been considered as the ‘causes’ of motion, and the resulting acceleration as the ‘effect’. However, the acceleration (the effect) and the resulting velocity field can be obtained without any recourse to the pressure (the cause), seemingly violating the principle of ‘cause’ and ‘effect’. We address these questions by deriving the pressure force of an incompressible fluid, starting from d’Alembert’s principle of virtual work, as a ‘reaction force’ that maintains the incompressibility condition. Next, we show that taking the curl on both sides of the Navier–Stokes (or Euler) equations is equivalent to using d’Alembert’s principle of virtual work, which cancels out the virtual work of the pressure gradient. This shows that abstract procedures, such as taking the curl on both sides of an equation, can actually be tacit applications of rich physical principles, without one realizing it. This can be quite instructive in a classroom of undergraduate students.
{"title":"Pressure Gradient in an Incompressible Fluid as a Reaction Force and the Preservation of the Principle of `Cause and Effect`","authors":"Lachezar Slavchev Simeonov","doi":"10.1088/1361-6404/acfdd9","DOIUrl":"https://doi.org/10.1088/1361-6404/acfdd9","url":null,"abstract":"Abstract When considering the motion of an incompressible fluid, it is common practice to take the curl on both sides of the Navier–Stokes (or Euler) equations and cancel the pressure force. The governing equations are sufficient to derive the velocity field of the fluid without any knowledge of the pressure. In fact, the pressure is only calculated after obtaining the velocity field. This raises a number of conceptual problems. For instance, why is the pressure unnecessary for obtaining the velocity field? Traditionally, forces have been considered as the ‘causes’ of motion, and the resulting acceleration as the ‘effect’. However, the acceleration (the effect) and the resulting velocity field can be obtained without any recourse to the pressure (the cause), seemingly violating the principle of ‘cause’ and ‘effect’. We address these questions by deriving the pressure force of an incompressible fluid, starting from d’Alembert’s principle of virtual work, as a ‘reaction force’ that maintains the incompressibility condition. Next, we show that taking the curl on both sides of the Navier–Stokes (or Euler) equations is equivalent to using d’Alembert’s principle of virtual work, which cancels out the virtual work of the pressure gradient. This shows that abstract procedures, such as taking the curl on both sides of an equation, can actually be tacit applications of rich physical principles, without one realizing it. This can be quite instructive in a classroom of undergraduate students.","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135945122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-17DOI: 10.1088/1361-6404/acfbc5
Allan Lambit Alinea, Cedrix Jadrin
Abstract In this work, we consider the spread of a ‘civilisation’ in an idealised homogeneous isotropic universe where all the planets of interest are habitable. Following a framework that goes beyond the usual idea of percolation in common undergraduate computational physics textbooks, we investigate the behaviour of the number of colonised planets with time, and the total colonisation time for three types of universes. These include static, dark energy-dominated, and matter-dominated universes. For all these types of universes, we find a remarkable fit with the Logistic Growth Function for the number of colonised planets with time. This is in spite of the fact that for the matter- and dark-energy dominated universes, the space itself is expanding. For the total colonisation time, T , the case for a dark energy-dominated universe is marked with divergence beyond the linear regime characterised by small values of the Hubble parameter, H . Not all planets in a spherical section of this universe can be ‘colonised’ due to the presence of a shrinking Hubble sphere. In other words, the recession speeds of other planets go beyond the speed of light making them impossible to reach. On the other hand, for a matter-dominated universe, while there is an apparent horizon, the Hubble sphere is growing instead of shrinking. This leads to a finite total colonisation time that depends on the Hubble parameter characterising the Universe; in particular, we find T ∼ H for small H and T ∼ H 2 for large H .
{"title":"Percolation of 'Civilisation' in a Homogeneous Isotropic Universe","authors":"Allan Lambit Alinea, Cedrix Jadrin","doi":"10.1088/1361-6404/acfbc5","DOIUrl":"https://doi.org/10.1088/1361-6404/acfbc5","url":null,"abstract":"Abstract In this work, we consider the spread of a ‘civilisation’ in an idealised homogeneous isotropic universe where all the planets of interest are habitable. Following a framework that goes beyond the usual idea of percolation in common undergraduate computational physics textbooks, we investigate the behaviour of the number of colonised planets with time, and the total colonisation time for three types of universes. These include static, dark energy-dominated, and matter-dominated universes. For all these types of universes, we find a remarkable fit with the Logistic Growth Function for the number of colonised planets with time. This is in spite of the fact that for the matter- and dark-energy dominated universes, the space itself is expanding. For the total colonisation time, T , the case for a dark energy-dominated universe is marked with divergence beyond the linear regime characterised by small values of the Hubble parameter, H . Not all planets in a spherical section of this universe can be ‘colonised’ due to the presence of a shrinking Hubble sphere. In other words, the recession speeds of other planets go beyond the speed of light making them impossible to reach. On the other hand, for a matter-dominated universe, while there is an apparent horizon, the Hubble sphere is growing instead of shrinking. This leads to a finite total colonisation time that depends on the Hubble parameter characterising the Universe; in particular, we find T ∼ H for small H and T ∼ H 2 for large H .","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135944288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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