Within a vehicle in free fall, such as a spacecraft orbiting the Earth, everything appears as if there was no force of gravity: objects float and remain stationary with respect to the walls, a situation commonly described as “weightlessness.” In relativity, such a vehicle is considered an inertial reference system. In fact, this is not exactly true because of the non-uniformity of the gravitational

Conventional calculations of the inertia tensor in undergraduate physics course are usually done for highly symmetrical bodies. Students might therefore get the impression that the moment of inertia about any axis through the center of mass is the same only for bodies with the highest degree of symmetry relative to this point, e.g., for spheres. A simple, seemingly counterintuitive example is presented

A well-known problem in classical mechanics that is often presented for pedagogical purposes involves a small mass that slides without friction under a gravitational force on the surface of a sphere. Commonly, students are asked to find the angular position where a mass with no azimuthal motion leaves the spherical surface, and this question is easily within the reach of most intermediate physics students

A novel graphical depiction of the relativistic barn and pole paradox is constructed in such a way that the space-time plot axes are dimensionless for both observers. Times of events over the entire range of pole-to-barn length ratio and relative speeds can be displayed in one plot for each observer. That for the barn frame clearly depicts the range of pole-to-barn length ratio over which the paradox

It is anticipated that future skies over urban areas will be busy with drones flying back and forth delivering packages. Taking New York City as an extreme example, it is estimated that by 2026, 2600 delivery drones could simultaneously populate the city's airspace. The drone–drone collision rate of “dumb” drones can be calculated by treating them as a gas of large, randomly moving, spherical molecules

Quantum algorithms offer efficient solutions to computational problems that are expensive to solve classically. Publicly available quantum computers, such as those provided by IBM, can now be used to run small quantum circuits that execute quantum algorithms. However, these quantum computers are highly prone to noise. Here, we introduce important concepts of quantum circuit noise and connectivity that

How many ways can we explore the Sun? We have images in many wavelengths and squiggly lines of many parameters that we can use to characterize the Sun. We know that while the Sun is blindingly bright to the naked eye, it also has regions that are dark in some wavelengths of light. All of those classifications are based on vision. Hearing is another sense that can be used to explore solar data. Some

Mitchell Feigenbaum discovered an intriguing property of viewing images through cylindrical mirrors or looking into water. Because the eye is a lens with an opening of about 5 mm, many different rays of reflected images reach the eye and need to be interpreted by the visual system. This has the surprising effect that what one perceives depends on the orientation of the head, whether it is tilted or

In many optics applications, it is important to use well-polarized light. However, there are situations in which randomly polarized light has distinct advantages. We demonstrate two approaches by which a polarized light beam can be totally depolarized, each using a simple setup and inexpensive components. The first method, designed for narrow spectrum light, works by combining the horizontal polarization

This work outlines a new instructional laboratory experiment focused on the photoelectric effect and the determination of Planck's constant. The described laboratory system employs contemporary experimental techniques, including real-time data acquisition based on the use of Arduino boards. The basis of this experiment is to measure the associated turn-on voltages of a small neon bulb as it is illuminated

Synchronization plays an important role in many physical processes. We discuss synchronization in a molecular dynamics simulation of a single particle moving through a viscous liquid while being driven across a washboard potential energy landscape. Our results show many dynamical patterns as the landscape and driving force are altered. For certain conditions, the particle's velocity and location are

We explore the forces and conservation laws that govern the motion of a hockey puck that slides without friction on a smooth, rotating, self-gravitating spheroid. The earth's oblate spheroidal shape (apart from small-scale surface features) is determined by balancing the gravitational forces that hold it together against the centrifugal forces that try to tear it apart. The earth achieves this shape

Magnus gliders are spinning toys displaying spectacular looped trajectories when launched at large velocity. These trajectories originate from the large amplitude of the Magnus force due to translational velocities of a few meters per second combined with a backspin of a few hundred radians per seconds. In this article, we analyse the trajectories of Magnus gliders built from paper cups, easily reproducible

The Newcomb–Benford law, also known as the first-digit law, gives the probability distribution associated with the first digit of a dataset so that, for example, the first significant digit has a probability of 30.1% of being 1 and 4.58% of being 9. This law can be extended to the second and next significant digits. This article presents an introduction to the discovery of the law and its derivation

We study dynamical Newton-ring like fringes created by interfering Fresnel reflections of an evaporating sessile liquid droplet, which acts as a miniature convex lens. We show that conventional thin-film interference theory cannot be applied to explain the physical phenomenon. Because of the large thickness and curvature of the liquid droplet, the geometrical light paths of the reflected beams become

Normal shocks are generally treated by defining a stationary adiabatic shock discontinuity with a supersonic upstream velocity. A fundamental feature of this approach is that the problem is steady, and the stagnation enthalpy is constant across the shock, greatly simplifying the analysis. However, shocks are generally not stationary, but rather are unsteady flow features often translating into a quiescent

Reflection of plane electromagnetic waves at a planar surface separating two media is typically discussed at length in undergraduate texts. However, a similar analysis for the case of evanescent electromagnetic waves, although illustrative and interesting, is rarely discussed. In this paper, we present such an analysis for the reflection of an evanescent wave at a plane dielectric surface and highlight

With the successes of the laser interferometer gravitational-wave observatory, we anticipate increased interest in working with squeezed states in the undergraduate and graduate quantum-mechanics classroom. Because squeezed-coherent states are minimum uncertainty states, their wavefunctions in position and momentum space must be Gaussians. But this result is rarely discussed in treatments of squeezed

We calculate the minimum distance at which one may approach a black hole in a free flyby. It corresponds to r = 4 m for the Schwarzschild black hole for a probe that was non-relativistic at infinity. The problem is formulated in a way that is useful for teaching introductory general relativity.

This Resource Letter introduces theory, frameworks, research, and historical information that connect the experiences of people from ethnic/racial groups that have been minoritized and marginalized in science with a focus on physics. This resource is intended for all physicists who teach, advise, work in research groups with, and interact with other people. We start this report with an introduction

The French press is a popular device for brewing coffee, comprising a cylindrical beaker—or “jug”—fitted with a lid and plunger with a fine wire mesh filter. The plunger is used to drive the solid coffee particles to the bottom of the jug, separating these grounds from hot liquid above. When using the French press in this way, a growing permeable pack of ground coffee is pushed through hot water by

Problems involving rolling without slipping or no sideways skidding, to name a few, introduce velocity-dependent constraints that can be efficiently treated by the method of Lagrange multipliers in the Lagrangian formulation of the classical equations of motion. In doing so, one finds, as a bonus, the constraint forces, which must be independent of the solution of the equations of motion and can only

The equilibrium and fluctuations of an ideal gas in a rigid container are studied by every student of statistical mechanics. Here, we examine the less well-known case when the box is floating freely; in particular, we determine the fluctuations of the box in velocity and position due to interactions with the gas it contains. This system is a toy model for the fluctuations in velocity and position of

Conditions under which a quantum particle can be described using classical quantities are studied. We investigate the wavefunction of a quantum particle submitted to a potential field for which all quantum effects vanish, even if Planck's constant is non-negligible. This problem is equivalent to the problem of the motion of a particle in a refringent medium. The indices of refraction of such media

We discuss three approaches that have been used in obtaining the Born approximation for Coulomb scattering: the standard approach, making use of a convergence factor (“screening”), Oppenheimer's approach using cylindrical (instead of spherical) coordinates, and finally the Landau and Lifshitz approach. We simplify and clarify Oppenheimer's calculation, which has been rarely used because of its complications

Digital computation is central to almost all scientific endeavors and has become integral to university physics education. Students collect experimental data using digital devices, process data using spreadsheets and graphical software, and develop scientific programming skills for modeling, simulation, and computational work. Issues associated with the floating-point representation of numbers are

The implicit midpoint method described by Chambliss and Franklin [Am. J. Phys. 88, 1075 (2020)] is known not to exactly conserve energy for a spatially varying magnetic field, nor does it conserve angular momentum in a constant magnetic field. It is, therefore, not competitive with the widely used Boris solver in plasma physics, which conserves both.

This paper theoretically studies a simple system of two identical linear springs connected symmetrically to a mass in a V-shaped configuration, with an additional adjustable external force applied to the mass. As this force is varied, under certain conditions the equilibrium position of the mass demonstrates strong dependence on the history of changes in the external force, exhibiting hysteresis. Mathematically

An ellipse on a non-slip inclined plane can rock, roll, or jump. Below a threshold energy, it rocks about its static equilibrium configuration. Above this energy, it rolls, and, if it rolls, it will eventually jump off the incline after some number of rolls. It is shown that jumping can only occur in certain configurations of the ellipse.

The static and dynamic properties of an asymmetric rigid body are studied using a Lagrangian approach. The body consists of a thin-walled cylinder balanced by a point mass attached to its inner surface. When resting on an inclined plane, stable and unstable equilibrium points are found, and oscillations about the stable point are studied for different angles of inclination and different values of the

Physics textbooks generally present the equation for universal gravitation as if it sprang fully formed from the head of Isaac Newton, which it most certainly did not. This analysis traces the development of Classical/Newtonian gravity as it evolved from Aristotle to Copernicus to Kepler, Horrocks, Roberval, Boulliau, Baliani, Borelli, Hooke, Halley, Wren, Newton, and Cavendish, and then finally on

In this paper, we use a straightforward numerical method to solve scattering models in one-dimensional lattices based on a tight-binding band structure. We do this by using the wave packet approach to scattering, which presents a more intuitive physical picture than the traditional plane wave approach. Moreover, a general matrix diagonalization method that is easily accessible to undergraduate students

In 2015, the first observation of gravitational waves marked a breakthrough in astrophysics and in technological research and development. The discovery of a gravitational-wave signal from the collision of two black holes, a billion light-years away, received considerable interest from the media and public. We describe the development of a purpose-built exhibit explaining this new area of research

Traditional introductory physics laboratories typically focus on guided experimentation while neglecting essential aspects of the scientific practice including computational reasoning. While there is a broad consensus among physics faculty on the need to introduce undergraduate students to computation, its integration in physics curricula is far from adequate. In this article, we document the process

We use coherence theory to explain why it is necessary to modify the conventional setup of a Michelson interferometer to obtain Haidinger rings with an extended source of white light. The modification consists of introducing a glass slide into one of the two arms of the interferometer. This insertion circumvents the drastic restriction imposed by the low temporal coherence of white light, which prevents

The data that students collect from the typical advanced undergraduate experiment on saturated absorption spectroscopy (SAS) of rubidium can be used to measure the isotope shift and thus lead to an estimate of the isotopic ground-state energy shift. This helps students refine their “picture” of the atomic ground state. We describe theoretically why SAS works well with free-running laser diodes, demonstrate

The naïve Bohr quantization condition mvr = n ℏ is applied to arbitrary spherically symmetric power-law potentials. The dependence of the energy eigenvalues on the principal quantum number n agrees with fully quantum mechanical results either exactly (for the hydrogen atom, the harmonic oscillator, and the infinite square well) or asymptotically (the linear potential). This naïve treatment can be used

The problem of rendezvous, the meeting of spacecraft in orbit, is an important aspect of mission planning. We imagine a situation where a chaser craft, initially traveling on the same circular orbit as its target and separated from it by a known distance, must select an initial thrust vector that will allow it to meet the target (interception) followed by a second thrust vector that will allow it to

It is a common experience that water shoots higher when we block a garden hose outlet by our thumb. But what causes this? How high does the water go? Does water from our neighbor's garden hose reach the same height? Is there an optimum outlet blockage that results in the greatest height that water can reach? Here, we show that a competition between viscous friction along the hose and the viscous dissipation

We consider sinusoidal undulations which appear on certain garden hoses under normal use. We propose a model, using linear elasticity, explaining this phenomenon, and make a connection with biological structures as well as self-buckling. We compare observations with model predictions and suggest potential applications in the area of shape-changing materials.

In its usual form, the loop-the-loop (LtL) problem involves a uniform solid sphere rolling from rest down a linear ramp that transitions into a circular loop. The task is to find the minimum height from which the ball must be released in order to roll completely around the loop without breaking contact. The answer, found using the conservation of mechanical energy and Newton's second law, is invariably

The pandemic-induced lock-down has resulted in better air quality and visibility, with reported sightings of Himalayan peaks from hundreds of kilometers away. During the 18th and 19th centuries, good visibility was commonplace and there exist many such accounts by reputable Orientalists such as Henry Colebrooke and Sir William Jones. These sightings invite two questions. Is a line of sight to the peak

Modeling is an essential skill for the career physical scientist or engineer. One approach that attempts to integrate this element of critical thinking into the laboratory environment is the modeling framework lab. In this style of lab exercise, students construct a model of both the measurement apparatus and the phenomenon under investigation. A key challenge to implementing this approach in introductory

The operational principle of twisted nematic displays involves the dielectric anisotropy of nematics. This crucial property was discovered about a hundred years ago by Jeżewski and Kast who used a so-called resonance method in which the frequency of an LC tank circuit was set by the capacitance of a capacitor filled with a nematic liquid crystal. Jeżewski and Kast observed that the resonance frequency

Of the many topics generally taught in undergraduate or graduate optics courses, the propagation of light through crystals with electrical anisotropy is one of the most difficult topics for students and for teachers who do not work in this field. In particular, the mathematics and equations are complex vectorial equations, and this complicates the visualization of the implications of the theory. Also

The development of programming languages for quantum computing has increased rapidly over the last few years, making it practical for a hands-on approach in teaching quantum computation. In this paper, I introduce the standard textbook example of searching for one item out of four using Grover's search algorithm and extend it by including a quantum database. In addition to explaining how to include

Time evolution of quantum systems is normally taught using the Schrödinger equation. Here, we use the Heisenberg approach to consider three problems that are exactly solvable in the Schrödinger picture when the initial wavepacket is Gaussian: the free particle, the simple harmonic oscillator and a free particle with a spin-orbit interaction. We show that complete solutions can be obtained for the free

We demonstrate that the de Broglie wave of a particle in a gravitational field turns toward the region of lower gravitational potential, causing the particle to fall. This turning is caused by clocks running slower in the smaller potential. We use the analogy of ocean waves that are slower in shallower water and turn toward beaches. This approach implies that the motion is along a geodesic and explains

The detection of gravitational waves is possible thanks to a multidisciplinary approach involving different disciplines such as astrophysics, physics, engineering, and quantum optics. Consequently, it is important today for teachers to introduce the basic features of gravitational waves science in the undergraduate curriculum. The usual approach to gravitational wave physics is based on the use of

We describe an undergraduate physics laboratory experiment that uses a 2-inch diameter NaI gamma detector to measure natural radiation in soils. Students first calibrate the detector for the source-detector geometry, energy-dependent detection efficiency, and sample self-absorption. Then, the activity of the 238U and 232Th decay series, as well as the activity of 40K in the soil, are measured. The

We develop a direct geometric method to determine the orbital parameters and mass of a planet, and we then apply the method to Neptune using high-precision data for the other planets in the solar system. The method is direct in the sense that it does not involve curve fitting. This paper, thereby, offers a new pedagogical approach to orbital mechanics that could be valuable in a physics classroom.

A ball placed inside a hollow cylinder and spun around an axis perpendicular to the cylinder wall will roll back and forth along the cylinder wall, as expected, but it will also oscillate along the axial direction of the cylinder. These axial oscillations occur in horizontal, tilted, and even vertical cylinders. Here, the equations of motion for this setup are derived and discussed, and experimental

In this work, we present in detail, in an accessible manner for undergraduate and graduate physics students, the model of spontaneous curvature, due to Helfrich, that quantitatively explains why the red blood cells in their natural state adopt a biconcave shape. The main hypothesis is that the equilibrium cell shape satisfies the principle of minimum free energy. Therefore, in the model, an expression

Science students must deal with the errors inherent to all physical measurements and be conscious of the need to express them as a best estimate and a range of uncertainty. Errors are routinely classified as statistical or systematic. Although statistical errors are usually dealt with in the first years of science studies, the typical approaches are based on manually performing repetitive observations

We present a computational method to directly calculate and visualize the directional components of the Coulomb, radiation, and total electromagnetic fields, as well as the scalar and vector potentials generated by moving point charges in arbitrary motion with varying speeds. Our method explicitly calculates the retarded time of the point charge along a discretized grid, which is then used to determine

In the quasistatic limit, a time-varying magnetic field inside a conductor is governed by the diffusion equation. Despite the occurrence of this scenario in many popular physics demonstrations, the concept of magnetic diffusion appears not to have garnered much attention itself as a subject for teaching. We employ the model of a thin conducting tube in a time-varying axial field to introduce magnetic

Schrödinger's equation can be considered as a diffusion equation with a diffusion coefficient β 2 = ℏ / 2 m. In this article, we explore the implications of this view. Rewriting the wave function in a polar form and transforming Schrödinger's equation into two real equations, we show that one of them reduces to the continuity equation, and the other, a nonlinear dynamical equation for the probability

We present and discuss numerical solutions to a two-body quantum bound state problem closely related to that of the hydrogen atom and the deuterium nucleus. The forces binding the particles of our system are Yukawa forces, which fall off with distance faster than the Coulomb force and arise for non-relativistic particles whose interactions are mediated by massive scalar or vector particles; the Coulomb

The identification of particle velocity with the velocity of a wave group was a crucial assumption that resolved important inconsistencies in the theoretical developments of Louis de Broglie and Erwin Schrödinger. Interestingly, this was one of the few common aspects of their work. In this paper, we present a reconstruction of how group velocity became essential for both de Broglie and Schrödinger

The finite rectangular well (FRW) has been a staple of quantum mechanics texts and classes for decades and is the subject of a rich literature. Despite being a problem about which there would apparently be not much more to be said, the FRW continues to serve as a system for introducing students to various analytic techniques and has numerous connections to current technology and research. This paper