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Trajectory of a golf ball on a sloping green 高尔夫球在倾斜果岭上的运动轨迹
IF 0.7 4区 教育学 Q3 Physics and Astronomy Pub Date : 2024-03-12 DOI: 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.
计算高尔夫球在高尔夫果岭倾斜表面上滚动的轨迹。高尔夫球沿着弯曲的轨迹移动,最后停在一个与初始发射方向成一定角度的位移点上。实验表明,位移角与发射速度无关,但取决于发射角和倾斜角与滚动摩擦系数之比。停止距离与发射速度的平方成正比。描述了一个简单的实验来检验计算结果。
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引用次数: 0
Derivation of the Schrödinger equation from QED 从 QED 推导出薛定谔方程
IF 0.7 4区 教育学 Q3 Physics and Astronomy Pub Date : 2024-03-12 DOI: 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.
薛定谔方程关系到波函数和电动势这两个突现量,它被假定为量子物理学的一个原理,或通过启发式方法获得。然而,物理一致性要求薛定谔方程是我们可以从量子电动力学基础中推导出的低能动力学条件。由于电磁耦合常数的值很小,我们证明电势准确地代表了中间低能光子交换的贡献。然后,从总的非相对论能量关系中,我们发现电子波函数的主导项是平面波的叠加,满足薛定谔方程。我们的推导表明,薛定谔方程并不是能量守恒关系,因为它的中间项并不像假设的那样代表电子动能。我们分析了薛定谔方程的物理内容,并通过计算和评估氢原子基态的物理量验证了我们的评估。此外,我们还解释了非相对论量子动力学不同于经典动力学的原因。由于推导过程涉及基本的物理概念和数学表达式,而且我们对每一步都进行了解释,因此本科生可以跟上推导过程。
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引用次数: 0
The initial trajectory of a ball released from uniform circular motion 从匀速圆周运动中释放出来的小球的初始轨迹
IF 0.7 4区 教育学 Q3 Physics and Astronomy Pub Date : 2024-03-12 DOI: 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.
这篇短文介绍了 "圆周运动最令人震惊的方面 "中讨论的圆周运动突然终止的简单分析模型。该模型证实,当松开绳子时,绳子远端的小球会在短时间内继续做近似圆周运动。
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引用次数: 0
Maximal submergence in dense granular suspensions 在致密颗粒悬浮液中的最大浸没度
IF 0.7 4区 教育学 Q3 Physics and Astronomy Pub Date : 2024-03-08 DOI: 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.
人在流沙中能向下沉入多深?经验似乎表明,完全沉入水中的风险很低,但由于颗粒悬浮物的流变复杂,要证明这一点并不容易。我们研究了几个垂直圆柱体向下沉入流沙的数学模型,发现用阻力修正浮力方程的方法给出了一个不符合实际的答案。相反,我们认为,考虑到液体饱和颗粒悬浮物的振动诱导压实动力学,可以支持我们提出的结论。我们将流沙与其他非牛顿流体进行了比较,强调在这种情况下,相同的模型并不适用,溺水的风险可能要大得多。最后,我们提出了一些可以在课堂上进行的相关实验。
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引用次数: 0
Exploring Magnetic Interactions between Magnets and a Magnetizable Alloy 探索磁铁与可磁化合金之间的磁性相互作用
IF 0.7 4区 教育学 Q3 Physics and Astronomy Pub Date : 2024-03-07 DOI: 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.
力学中的传统问题通常涉及由分析解描述的再生运动的力,这些分析解很容易从微分方程中获得。然而,当涉及磁力时,问题的分析描述可能会非常复杂。物理入门课程的学生可以从解决异常复杂的问题中获益,因为这可以拓宽他们在物理学更广泛背景下的视野。在本文中,我们描述了磁铁吸引可磁化合金硬币的问题情况。假设施加在硬币上的力随它们之间的距离呈指数变化,这个问题可以通过分析来解决。实验用摄像机以高速帧拍摄,并将实验值与理论模型预测值进行比较。结果证实,理论模型足以描述所涉及的相互作用类型,表明这些类型与通常在引力、电或弹性相互作用问题中发现的类型截然不同。这项工作为研究磁体之间的吸引力和排斥力开辟了未来的前景。
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引用次数: 0
Two-point resistances in symmetric bracelet resistor networks: accurate estimates and exact expressions 对称支架电阻器网络中的两点电阻:精确估算和确切表达式
IF 0.7 4区 教育学 Q3 Physics and Astronomy Pub Date : 2024-03-05 DOI: 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 R0n is known. We then determine R0n by an elementary recurrence relation which converges rapidly to its large n limit. Using this reference value of R0n, accurate estimates of other TPRs follow for all values of n, characterised by a leading 1/n variation. In addition, exact explicit expressions of the TPRs can be calculated for any value of n. These networks, prototypes of three-dimensional networks considered in research, can be used to illustrate the diversity of the physical approach, the power of elementary methods, and to learn to be comfortable with approximations. Easy to make and use for experimental tests, they can support hands-on activities and conceptual changes.
用于模拟新型自然或人造物质的电阻网络,也为实践物理学方法提供了通用范例,以获得估计值、揭示系统的主要特性并推导出精确的表达式。对称手镯电阻器网络是通过在一个圆中连接 n 个相同的电阻器,然后用另一组 n 个相同的电阻器连接两个这样的圆来构建的。首先,利用 van Steenwijk 方法,我们确定了当层间电阻 R0n 已知时,任意两个节点之间的等效电阻或两点电阻 (TPR) 即可求出。然后,我们通过一个基本的递推关系确定 R0n,该递推关系可快速收敛至大 n 极限。利用 R0n 的这一参考值,我们可以对所有 n 值的其他 TPR 进行精确估算,其特点是领先的 1/n 变化。此外,对于任何 n 值,都可以计算出 TPR 的精确明确表达式。这些网络是研究中考虑的三维网络的原型,可用于说明物理方法的多样性、基本方法的威力,以及学会适应近似值。这些网络易于制作和用于实验测试,可以支持实践活动和概念变革。
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引用次数: 0
Student understanding of the Bloch sphere 学生了解布洛赫球
IF 0.7 4区 教育学 Q3 Physics and Astronomy Pub Date : 2024-02-28 DOI: 10.1088/1361-6404/ad2393
Peter Hu, Yangqiuting Li, Roger S K Mong, Chandralekha Singh
Quantum information science is a rapidly growing interdisciplinary field that is attracting the attention of academics and industry experts alike. It requires talent from a wide variety of traditional fields, including physics, engineering, chemistry, and computer science, to name a few. To prepare students for such opportunities, it is important to give them a strong foundation in the basics of quantum information science, in which quantum computing plays a central role. In this study, we discuss the development, validation, and evaluation of a tutorial on the Bloch sphere, a useful visual tool for developing intuition about single quantum bits (qubits), which are the basic building block of any quantum computer. Students’ understanding was evaluated after they received traditional lecture-based instruction on the requisite topics, and again after engaging with the tutorial. We observe, analyze, and discuss their improvement in performance on concepts covered in the tutorial.
量子信息科学是一个快速发展的跨学科领域,吸引着学术界和业界专家的关注。它需要各种传统领域的人才,包括物理学、工程学、化学和计算机科学等等。为了让学生做好准备,迎接这样的机遇,必须为他们打下坚实的量子信息科学基础,而量子计算在其中发挥着核心作用。在本研究中,我们讨论了布洛赫球教程的开发、验证和评估。布洛赫球是一种有用的可视化工具,用于培养学生对单个量子比特(量子比特)的直觉,而单个量子比特是任何量子计算机的基本构件。学生们在接受了有关必要主题的传统讲授式教学后,又在学习了教程后,对他们的理解能力进行了评估。我们观察、分析并讨论了他们在教程所涉及的概念方面的成绩提高情况。
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引用次数: 0
Wilks’s theorem, global fits, and neutrino oscillations 威尔克斯定理、全局拟合和中微子振荡
IF 0.7 4区 教育学 Q3 Physics and Astronomy Pub Date : 2024-02-26 DOI: 10.1088/1361-6404/ad261d
J M Hardin
Tests of models for new physics appearing in neutrino experiments often involve global fits to a quantum mechanical effect called neutrino oscillations. This paper introduces students to methods commonly used in these global fits starting from an understanding of more conventional fitting methods using log-likelihood and χ2 minimization. Specifically, we discuss how the Δχ2, which compares the χ2 of the fit with the new physics to the χ2 of the Standard Model prediction, is often interpreted using Wilks’s theorem. This paper uses toy models to explore the properties of Δχ2 as a test statistic for oscillating functions. The statistics of such models are shown to deviate from Wilks’s theorem. Tests for new physics also often examine data subsets for ‘tension’ called the ‘parameter goodness of fit’. In this paper, we explain this approach and use toy models to examine the validity of the probabilities from this test also. Although we have chosen a specific scenario—neutrino oscillations—to illustrate important points, students should keep in mind that these points are widely applicable when fitting multiple data sets to complex functions.
对中微子实验中出现的新物理模型的检验,往往涉及对一种称为中微子振荡的量子力学效应的全局拟合。本文从了解使用对数似然法和 χ2 最小化的常规拟合方法入手,向学生介绍这些全局拟合中常用的方法。具体来说,我们讨论了如何使用威尔克斯定理来解释Δχ2,Δχ2是将新物理拟合的χ2与标准模型预测的χ2进行比较的结果。本文利用玩具模型来探索Δχ2 作为振荡函数检验统计量的特性。结果表明,这些模型的统计量偏离了 Wilks 定理。对新物理学的检验还经常检查数据子集的 "张力",称为 "参数拟合度"。在本文中,我们将解释这种方法,并使用玩具模型来检验这种检验的概率的有效性。虽然我们选择了中微子振荡这一特定场景来说明重要观点,但学生们应牢记,这些观点广泛适用于将多个数据集拟合为复杂函数。
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引用次数: 0
A research-informed graphical tool to visually approach Gauss’ and Stokes’ theorems in vector calculus 以研究为基础的图形工具,用于直观地处理向量微积分中的高斯定理和斯托克斯定理
IF 0.7 4区 教育学 Q3 Physics and Astronomy Pub Date : 2024-02-21 DOI: 10.1088/1361-6404/ad2390
L Hahn, S A Blaue, P Klein
Gauss’ and Stokes’ theorems are fundamental results in vector calculus and important tools in physics and engineering. When students are asked to describe the meaning of Gauss’ divergence theorem, they often use statements like this: ‘The sum of all sources of a vector field in a region gives the net flux out of the region’. In order to raise this description to a mathematically sound level of understanding, we present an educational approach based on the visual interpretation of the vector differential operators, i.e. divergence and curl. As a starting point, we use simple vector field diagrams for a qualitative approach to connect both sides of the integral theorems, and present an interactive graphical tool to support this connection. The tool allows to visualise two-dimensional vector fields, to specify vector decomposition, to evaluate divergence and curl point wise, and to draw rectangles to determine surface and line integrals. From a meta-perspective, we situate this educational approach into learning with (multiple) representations. Based on prior research, the graphical tool addresses various learning difficulties of vector fields that are connected to divergence and curl. The tool was incorporated into the weekly lecture-based recitations of Physics II (electromagnetism) in 2022 and 2023, and we assessed various educational outcome measures. The students overall reported the tool to be intuitive and user-friendly (level of agreement 76%, N=125), considered it helpful for understanding and recommended its use for introductory physics courses (level of agreement 65%, N=65).
高斯定理和斯托克斯定理是向量微积分的基本结果,也是物理学和工程学的重要工具。当要求学生描述高斯发散定理的含义时,他们经常使用这样的语句:一个区域内矢量场的所有来源之和给出了流出该区域的净通量"。为了将这一描述提升到数学理解的高度,我们提出了一种基于矢量微分算子(即发散和卷曲)视觉解释的教育方法。作为起点,我们使用简单的向量场图作为连接积分定理两边的定性方法,并提供了一个交互式图形工具来支持这种连接。该工具允许将二维向量场可视化,指定向量分解,以点为单位评估发散和卷曲,以及绘制矩形以确定面积分和线积分。从元视角来看,我们将这种教育方法定位为利用(多重)表征进行学习。基于先前的研究,图形工具解决了与发散和卷曲有关的向量场的各种学习困难。该工具被纳入了2022年和2023年物理II(电磁学)的每周讲授式背诵课程,我们对各种教育成果进行了评估。总体而言,学生们认为该工具直观易用(同意率 76%,N=125),认为它有助于理解,并推荐在物理入门课程中使用(同意率 65%,N=65)。
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引用次数: 0
How can the global temperature increase be explained to students? 如何向学生解释全球气温升高?
IF 0.7 4区 教育学 Q3 Physics and Astronomy Pub Date : 2024-02-16 DOI: 10.1088/1361-6404/ad2429
R De Luca, V Lamberti
The temperature increase of the system Earth-atmosphere can be described by means of a parallelism between the leaking bucket’s dynamics and the energy balance between the incoming radiant power from the Sun and the outcoming net heat flow rate to outer space. The analogy is interesting from a didactical point of view, since the stable fixed point of the temperature can be related to the height of the water inside the leaking bucket in the presence of a fixed fluid flow rate. Qualitative predictions of global temperature increase for increasing concentration of greenhouse gases in the atmosphere can be obtained by reducing the outcoming net heat flow rate, using buckets with smaller orifices.
地球-大气层系统的温度升高可以通过漏斗的动态与太阳辐射功率和流向外太空的净热流量之间的能量平衡之间的平行关系来描述。从教学的角度来看,这种类比非常有趣,因为在固定的流体流速下,温度的稳定固定点可以与漏斗内水的高度相关联。如果使用孔径较小的水桶,降低流出的净热流量,就可以对大气中温室气体浓度增加时全球气温升高的情况做出定性预测。
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引用次数: 0
期刊
European Journal of Physics
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