Issues

Editor's Note: When a ball is dropped on a horizontal surface with no initial spin, previous studies have found that its bouncing behavior can be simply described using a coefficient of restitution, which gives the ratio of the velocity after the bounce to before the bounce. The value of this coefficient is −1 for a perfectly elastic ball/surface, with a smaller magnitude for any real ball. In many sports, like golf, basketball, or bowling, balls are thrown at an angle and are often given some initial spin by the player. Depending on initial conditions, these balls can bounce, roll, or start by bouncing and then transition to rolling. Here, the transition from bouncing to rolling is shown to be described by using both a vertical and horizontal coefficient of restitution, with the horizontal velocity defined at the spinning edge of the ball rather than its center. Videos of undergraduate level experiments are included, with results used to validate the model.

Editor's Note: Have you heard about tractor magnets? Certain fixed arrangements of three cylindrical magnets on a tabletop surface result in a stable equilibrium for a fourth follower magnet. In this article you will find the theoretical basis for this curious stability, a thorough experimental analysis of the stability of different possible magnet arrangements, and supplementary videos and 3D printing plans so you and your students can create and study your own tractor magnets.

Editor's Note: Many accepted “truths” about field lines are not entirely correct. Magnetic field lines can start and stop, they do not ordinarily form closed loops, and there is generally no relation between their density and the strength of the magnetic field. These points are made in the more complicated electric current configurations that are presented here and they can easily be demonstrated in upper-level undergraduate electromagnetism courses. The linked Mathematica techniques will enable instructors to construct their own innovative models.

Editor's Note: Did you know that you could simulate a single-slit diffraction experiment using a lock-in amplifier? If not, you can read through this paper, which shows how using a rectangular time window when playing with a lock-in is the time-equivalent of the rectangular aperture in the optics experiment: The properties of Fourier transform guarantee that you will obtain a sinc function either in frequency for the lock-in experiment or in wave vector for the light diffraction experiment. Appropriate for undergraduate optics or instrumentation class.

Editor's Note: Rutherford scattering, named after the famous gold foil experiment designer, is the scattering of one charged particle by another charged particle of fixed position. The authors here present a new approximation for this quantum scattering process, discuss the shortcomings of the traditional Born approximation approach found in many quantum mechanics textbooks, and show how their new techniques may be applied to the problem of the scattering of classical waves by black holes.

Editor's note: Entanglement is often an elusive property for students. This paper, appropriate for advanced quantum mechanics class, shows on a practical example how to calculate entanglement. It considers four spins-1/2 that are evenly distributed on a ring and coupled to one another though competing nearest- and next-nearest-neighbor interactions. After determining the eigenstates and their energies, which is in itself a nice example of addition of more than two angular momenta, one can determine the entanglement of two subsystems. Spoiler: it depends on how you partition the system.

Editor's Note: Did you know that state-of-the-art experiments could measure timescales down to a few rontoseconds (such things do actually exist and correspond to 10  $− 27$ s)? In the spirit of Physics for Future Presidents, this paper uses the theme of fast timescales to build a semester-long syllabus to introduce undergraduate science majors to various physics concepts, ranging from condensed matter to particle physics, and to examples of commonly used technology (ink-jet printing, ultrasound imaging, high-speed cameras, etc.). Such a course could also be used for outreach, and it emphasizes the use of dimensional analysis and orders of magnitude.

Editor's note: The behavior of an electron confined to move in two dimensions in the presence of a perpendicular magnetic field underlies the quantum Hall effect. This is also an analytically solvable problem, so it is valuable to quantum mechanics pedagogy. This paper reviews the solution and then shows a neat trick for solving the problem when the electron's effective mass is anisotropic, a situation that is fairly common in solid-state systems.

Editor's Note: Readers who are familiar with the “barn door” paradox will enjoy this 2D version of it. Imagine a rod, thrown like a javelin, moving at a constant horizontal speed and also falling at a constant vertical speed. As it reaches ground level, it falls precisely through a javelin-length slit in the floor. However, in the reference frame of the javelin, the slit will be length-contracted and the javelin will have its (longer) proper length. So as the ground rises up to meet it, how does it pass through? The authors revisit this classic paradox and present new ideas on its resolution, including the concept of relativistic boost-induced rotations.

Editor's Note: This Comment expands on a recent article on the Lorentz transformation, clearly explaining the roles in the transformation of the light principle, the limiting speed c, and linearity.