Pub Date : 2024-03-12DOI: 10.1088/1361-6404/ad2c2d
Rod Cross
Calculations are presented on the trajectory of a golf ball that rolls across the inclined surface of a golf green. The ball follows a curved path and comes to a stop at a point displaced at an angle to the initial launch direction. It is shown that the displaced angle is independent of the launch speed but depends on the launch angle and the ratio of the incline angle to the coefficient of rolling friction. The stopping distance is proportional to the launch speed squared. A simple experiment is described to check the calculations.
{"title":"Trajectory of a golf ball on a sloping green","authors":"Rod Cross","doi":"10.1088/1361-6404/ad2c2d","DOIUrl":"https://doi.org/10.1088/1361-6404/ad2c2d","url":null,"abstract":"Calculations are presented on the trajectory of a golf ball that rolls across the inclined surface of a golf green. The ball follows a curved path and comes to a stop at a point displaced at an angle to the initial launch direction. It is shown that the displaced angle is independent of the launch speed but depends on the launch angle and the ratio of the incline angle to the coefficient of rolling friction. The stopping distance is proportional to the launch speed squared. A simple experiment is described to check the calculations.","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140311721","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 : 2024-03-12DOI: 10.1088/1361-6404/ad2c2f
Spyros Efthimiades
The Schrödinger equation relates the emergent quantities of wavefunction and electric potential and is postulated as a principle of quantum physics or obtained heuristically. However, physical consistency requires that the Schrödinger equation is a low-energy dynamical condition we can derive from the foundations of quantum electrodynamics. Due to the small value of the electromagnetic coupling constant, we show that the electric potential accurately represents the contributions of intermediate low-energy photon exchanges. Then, from the total nonrelativistic energy relation, we see that the dominant term of the electron wavefunction is a superposition of plane waves that satisfies the Schrödinger equation. Our derivation shows that the Schrödinger equation is not an energy conservation relation because its middle term does not represent the electron kinetic energy as assumed. We analyze the physical content of the Schrödinger equation and verify our assessments by calculating and evaluating the physical quantities in the ground state of the hydrogen atom. Furthermore, we explain why nonrelativistic quantum dynamics differs from classical dynamics. Undergraduate students can follow the derivation because it involves fundamental physical concepts and mathematical expressions, and we explain every step.
{"title":"Derivation of the Schrödinger equation from QED","authors":"Spyros Efthimiades","doi":"10.1088/1361-6404/ad2c2f","DOIUrl":"https://doi.org/10.1088/1361-6404/ad2c2f","url":null,"abstract":"The Schrödinger equation relates the emergent quantities of wavefunction and electric potential and is postulated as a principle of quantum physics or obtained heuristically. However, physical consistency requires that the Schrödinger equation is a low-energy dynamical condition we can derive from the foundations of quantum electrodynamics. Due to the small value of the electromagnetic coupling constant, we show that the electric potential accurately represents the contributions of intermediate low-energy photon exchanges. Then, from the total nonrelativistic energy relation, we see that the dominant term of the electron wavefunction is a superposition of plane waves that satisfies the Schrödinger equation. Our derivation shows that the Schrödinger equation is not an energy conservation relation because its middle term does not represent the electron kinetic energy as assumed. We analyze the physical content of the Schrödinger equation and verify our assessments by calculating and evaluating the physical quantities in the ground state of the hydrogen atom. Furthermore, we explain why nonrelativistic quantum dynamics differs from classical dynamics. Undergraduate students can follow the derivation because it involves fundamental physical concepts and mathematical expressions, and we explain every step.","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140311718","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 : 2024-03-12DOI: 10.1088/1361-6404/ad2c2e
Milan Batista
This short paper presents a simple analytical model for the abrupt termination of circular motion, as discussed in the ‘The Most Mind-Blowing Aspect of Circular Motion’. The model confirms that when a string is released, a ball at the far end of the string continues to move in a near-circular motion for a short time.
{"title":"The initial trajectory of a ball released from uniform circular motion","authors":"Milan Batista","doi":"10.1088/1361-6404/ad2c2e","DOIUrl":"https://doi.org/10.1088/1361-6404/ad2c2e","url":null,"abstract":"This short paper presents a simple analytical model for the abrupt termination of circular motion, as discussed in the ‘<italic toggle=\"yes\">The Most Mind-Blowing Aspect of Circular Motion’</italic>. The model confirms that when a string is released, a ball at the far end of the string continues to move in a near-circular motion for a short time.","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140311841","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 : 2024-03-08DOI: 10.1088/1361-6404/ad2aa2
Hollis Williams
How far can a person sink downwards in quicksand? Experience would seem to suggest that there is low risk of submerging completely, but it is not easy to demonstrate this because of the complex rheology of granular suspensions. We study several mathematical models for the sinking of a vertical cylinder downwards into quicksand, finding that an approach with a buoyancy equation modified by drag force gives an unphysical answer. We instead argue that our proposed conclusion is supported by considering the dynamics of vibration-induced compactification of liquid-saturated granular suspensions. We compare quicksand with other non-Newtonian fluids, emphasising that in this case the same model does not apply and that the risk of drowning could be much more significant. We finish by suggesting some relevant experiments that can be performed in a classroom setting.
{"title":"Maximal submergence in dense granular suspensions","authors":"Hollis Williams","doi":"10.1088/1361-6404/ad2aa2","DOIUrl":"https://doi.org/10.1088/1361-6404/ad2aa2","url":null,"abstract":"How far can a person sink downwards in quicksand? Experience would seem to suggest that there is low risk of submerging completely, but it is not easy to demonstrate this because of the complex rheology of granular suspensions. We study several mathematical models for the sinking of a vertical cylinder downwards into quicksand, finding that an approach with a buoyancy equation modified by drag force gives an unphysical answer. We instead argue that our proposed conclusion is supported by considering the dynamics of vibration-induced compactification of liquid-saturated granular suspensions. We compare quicksand with other non-Newtonian fluids, emphasising that in this case the same model does not apply and that the risk of drowning could be much more significant. We finish by suggesting some relevant experiments that can be performed in a classroom setting.","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140316982","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 : 2024-03-07DOI: 10.1088/1361-6404/ad312c
N. A. Machado, Frederico Cruz, M. Hahn, Paulo Simeao Carvalho
Traditional problems in mechanics usually involve forces that regenerate motions described by analytical solutions, easily obtained from differential equations. However, when magnetic forces are involved, problems can be very complex to describe analytically. Students of introductory Physics courses can benefit from solving questions of unusual complexity, as this broadens their view in more general contexts in Physics. In this paper, we describe the problematic situation of magnets that attract a coin of a magnetizable alloy. The problem is solved analytically, assuming that the force exerted on the coin varies exponentially with the distance between them. The experiment was filmed with a camera at a high rate frame and the experimental values compared with those predicted by the theoretical model. The results confirm that the theoretical model is adequate to describe the kind of interaction involved, showing that these are very different from those usually found in problems with gravitational, electrical or elastic interactions. This work opens up future perspectives for investigations with attractive and repulsive forces between magnets.
{"title":"Exploring Magnetic Interactions between Magnets and a Magnetizable Alloy","authors":"N. A. Machado, Frederico Cruz, M. Hahn, Paulo Simeao Carvalho","doi":"10.1088/1361-6404/ad312c","DOIUrl":"https://doi.org/10.1088/1361-6404/ad312c","url":null,"abstract":"\u0000 Traditional problems in mechanics usually involve forces that regenerate motions described by analytical solutions, easily obtained from differential equations. However, when magnetic forces are involved, problems can be very complex to describe analytically. Students of introductory Physics courses can benefit from solving questions of unusual complexity, as this broadens their view in more general contexts in Physics. In this paper, we describe the problematic situation of magnets that attract a coin of a magnetizable alloy. The problem is solved analytically, assuming that the force exerted on the coin varies exponentially with the distance between them. The experiment was filmed with a camera at a high rate frame and the experimental values compared with those predicted by the theoretical model. The results confirm that the theoretical model is adequate to describe the kind of interaction involved, showing that these are very different from those usually found in problems with gravitational, electrical or elastic interactions. This work opens up future perspectives for investigations with attractive and repulsive forces between magnets.","PeriodicalId":50480,"journal":{"name":"European Journal of Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140077973","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 : 2024-03-05DOI: 10.1088/1361-6404/ad242a
Frédéric Perrier, Frédéric Girault
Resistor networks, used to model new types of natural or artificial matter, also provide generic examples for practising the methods of physics for obtaining estimates, revealing the main properties of a system and deriving exact expressions. Symmetric bracelet resistor networks are constructed by connecting n identical resistors in a circle, and then connecting two such circles by another set of n identical resistors. First, using van Steenwijk’s method, we establish that the equivalent resistance or two-point resistance (TPR) between any two nodes is derived when the layer-to-layer resistance R