A highly reflective sail provides a way to propel a spacecraft out of the solar system using solar radiation pressure. The closer the spacecraft is to the Sun when it starts its outward journey, the larger the radiation pressure and so the larger the final velocity. For a spacecraft starting on the Earth's orbit, closer proximity can be achieved via a retrograde impulse from a rocket engine. The sail

We discuss a simple experiment investigating the shrinkage of surface soap bubbles sitting on a thin solid plate with a circular orifice located under the apex of the bubble. We identify three different shrinking regimes, the occurrence of which depends on a combination of key parameters that include the ratio between initial bubble and orifice sizes and physicochemical properties of the fluid system

High-speed (2873 fps) visualization of soap film bursting dynamics has been performed by means of equipment available to a high school laboratory. This paper presents the geometry of the experiment and description of the analysis method of the video frames. We carried out the experiments on the bursting of vertical plane soap films using a circular frame with single central and non-central puncture

Many textbooks dealing with surface tension favor the thermodynamic approach (minimization of some thermodynamic potential such as free energy) over the mechanical approach (balance of forces) to describe capillary phenomena, stating that the latter is flawed and misleading. Yet, a mechanical approach is more intuitive for students than free energy minimization, and does not require any knowledge of

When a falling chain strikes a surface, it can accelerate downwards faster than free-fall. This counterintuitive effect occurs when a tension is created in the chain above where it strikes the surface. The size of this tension, and how it is produced, depend on the type of chain used. For a chain made of rods that are slightly tilted from horizontal, the impact-induced tension is readily observable

The distribution of string tension on the contact line between an ideal string and a massive pulley is a frequently discussed but incompletely posed problem that confronts students in introductory mechanics. We highlight ambiguities in the usual presentation of this problem by the massive Atwood's machine and discuss two compact resolutions that treat situations where the pulley or the string elastically

This article considers an example of eddy current behaviour that has not been described previously in textbooks and physics education journals. This phenomenon involves a sharp rise in the active resistance of a multilayer coil with increasing alternating current frequency. This effect is not explained by the classic skin effect and is not related to losses in a core. The results of measurements of

The question of how much surplus of electric charge (“surcharge”) fits on an object is generally very difficult to answer. Here, it is shown that it is easy to answer when the object is a failed white dwarf star (a brown dwarf in its ground state) made of protons and electrons: Given the number of protons, how many electrons can there be? Surprisingly, the answer (in the form: as few as A and as many

In this work, we present a master equation approach to simulating DC transport through single electron transistors and quantum dots suitable for an upper-division undergraduate computational physics project. After introducing the basic theory describing transport through quantum dots, we present a simulation of the simple case of a metallic dot including effects due to the finite temperature of the

Interactive simulations and visualizations augment the teaching and learning of quantum mechanics by making equations and concepts come to life. However, graphical visualizations are nearly always limited to a set of hard-coded functionalities. Text-based codes can offer a higher degree of flexibility, but only at the expense of steep learning curves or time investments. We introduce Quantum Composer

This article describes how the author successfully adapted techniques drawn from the literature on active learning for use in a graduate-level course on quantum field theory. Students completed readings and online questions ahead of each class and spent class time working through problems that required them to practice the decisions and skills typical of a theoretical physicist. The instructor monitored

Many different formalisms exist for computing the phase of a matter-wave interferometer. However, it can be challenging to develop physical intuition about what a particular interferometer is actually measuring or about whether a given classical measurement provides equivalent information. Here, we investigate the physical content of the interferometer phase through a series of thought experiments

Anyone who has visited a science museum has seen the demonstration where a beach (or ping-pong) ball is suspended in mid-air at a fixed position by constant blowing from below. After a while, the ball inevitably tumbles to the ground but can easily be rebalanced, by hand, again at the suspension point. Here, we ask a different more delicate question. Can we blow the ball from rest, starting at the

We analyze the motion of a point-mass projectile shot uphill above a slanted surface, which bounces from it without loss of energy. We show that consecutive bounces occur in equal time intervals and obtain an explicit formula for the number of uphill bounces before the motion reverses and the projectile heads downhill. Additionally, we show that the projectile rises to the same height above the surface

A simple experiment carried out with readily available equipment is used to investigate the interesting physics that occurs when a ball is bounced on a flat surface and left to continue bouncing until it stops. A solution well known from introductory physics courses is that if a ball loses a fraction of its energy on each bounce, then it will bounce infinitely many times in a finite time interval.

The wave equation of sound is developed starting from the equations of fluid mechanics for the velocity and pressure variation in air. The equations for sound are analogous to Maxwell's equations for linear polarized electromagnetic plane waves: the pressure variation and the velocity correspond to the electric and magnetic fields, respectively. The order of magnitude of sound properties is explained

An alternative way of visualizing electromagnetic waves in matter and of deriving the Finite Difference Time Domain (FDTD) method for simulating Maxwell's equations for one-dimensional systems is presented. The method uses d'Alembert's splitting of waves into forward and backward pulses of arbitrary shape and allows for grid spacing and material properties that vary with the position. Constant velocity

We present a framework for investigating effective dynamics of SU(3) color charge. Two- and three-body effective interaction terms inspired by the Heisenberg spin model are considered. In particular, a toy model for a three-source “baryon” is constructed and investigated analytically and numerically for various choices of interactions. vpython is used to visualize the nontrivial color charge dynamics

It is regrettable that the quantum length of an object is rarely if ever discussed, because it provides an ideal pedagogical paradigm for understanding how a physicist uses classical intuition to define quantum properties and how such quantum properties behave as one would expect in the classical limit. It also provides for a way to understand many-particle states, and leads to interesting quantum

Learning vector calculus techniques is one of the major hurdles faced by physics undergraduates. However, beginners report various difficulties dealing with the index notation due to its bulkiness. Meanwhile, there have been graphical notations for tensor algebra that are intuitive and effective in calculations and can serve as a quick mnemonic for algebraic identities. Although they have been introduced

In recent times, spatial light modulators have become a common tool in optics laboratories as well as industrial environments for shaping the spatial structure of a light beam. Although these devices are often easy to use, their high cost has limited their use in many undergraduate laboratories. However, in recent years the progress in developing more cost-effective projectors has led to affordable

Computerized tomography (CT) has been used for decades by medical professionals to detect and diagnose injuries and ailments. CT scanners are based on interesting physics, but due to their bulk, cost, and safety, hands on experience with a medical CT scanner are unrealistic for undergraduate students. Therefore, operationally similar, yet small, safe, and inexpensive CT scanners are desirable teaching

Raised menisci around small discs positioned to pull up a water-air interface provide a highly controllable experimental setup capable of reproducing much of the rich phenomenology of gravitational lensing (or microlensing) by n-body clusters. Results are shown for single, binary, and triple mass systems. The scheme represents a versatile testbench for the (astro)physics of general relativity's gravitational

A rather delightful optical phenomenon, related to sun glints observed by the seashore, is illustrated: tiny spots of lights can be seen to move randomly in front of an observer wearing sunglasses with polarized lenses. On the seabed beneath the shallow waters, luminous wriggles can also be detected with or without sunglasses. By means of Fresnel equations, a plausible explanation of these two optical

Rabi oscillations and Floquet states are likely the most familiar concepts associated with a periodically time-varying Hamiltonian. Here, we present an exactly solvable model of a two-level system coupled to both a continuum and a classical field that varies sinusoidally with time, which sheds light on the relationship between the two problems. For a field of the rotating-wave-approximation form, results

We calculate spin correlation functions using IBM quantum processors, accessed online. We demonstrate the rotational invariance of the singlet state, interesting properties of the triplet states, and surprising features of a state of three entangled qubits. This exercise is ideal for remote learning and generates data with real quantum mechanical systems that are impractical to investigate in the local

The phenomenon of molecular binding, where two molecules, referred to as a receptor and a ligand, bind together to form a ligand-receptor complex, is ubiquitous in biology and essential for the accurate functioning of all life-sustaining processes. The probability of a single receptor forming a complex with any one of L surrounding ligand molecules at thermal equilibrium can be derived from a partition

This article discusses several erroneous claims which appear in textbooks on numerical methods and computational physics. These are not typos or mistakes an individual author has made, but widespread misconceptions. In an attempt to stop these issues from further propagating, we discuss them here, along with some background comments. In each case, we also provide a correction, which is aimed at summarizing

College writing has conventionally been taught by faculty in the humanities, but many schools have begun to encourage writing instruction by faculty from a variety of disciplines. Our institution, Davidson College, has a college-wide writing requirement of all first-year students, and encourages a broad spectrum of faculty to contribute to the writing curriculum. We describe our recent experiences

Two identical objects are simultaneously projected vertically upward with the same initial speed in a uniform gravitational field and then return to their starting point. One object is subject to a resistive force proportional to the nth power of its speed where n ≥ 0, such as linear (Stokes) drag for n = 1 or quadratic (Newtonian) drag for n = 2. The other object moves through vacuum with no resistance

You have a rocket in a high circular orbit around a massive central body (a planet or the Sun) and wish to escape with the fastest possible speed at infinity for a given amount of fuel. In 1929, Hermann Oberth showed that firing two separate impulses (one retrograde and one prograde) can be more effective than a direct transfer that expends all the fuel at once. This is due to the Oberth effect, whereby

Experience teaches us that a large balloon takes longer to deflate than a small one of the same kind. But what is the quantitative relation between the deflation time τ and the radius R of a balloon? A simple analysis, depending only upon elementary physics, shows that τ ∼ R 7 / 2—a prediction that is surprisingly easy to illustrate with a party balloon, a tape measure, and a smart-phone app.

The theory and practical design of a monopole ion trapping system are presented, along with our experimental observations of nonlinear motion of a single charged polyethylene microsphere in the trap. For these observations, the trap was operated at a drive frequency of 70 Hz and easily produced voltages in the kV range. The resulting ion motion was captured using a high-speed camera acquiring images

We describe a simple method to experimentally determine the frequency dependencies of the per-unit-length resistance and conductance of transmission lines. The experiment is intended as a supplement to the classic measurement of the transient response of a transmission line to a voltage step or pulse. In the transient experiment, an ideal (lossless) model of the transmission line is used to determine

The basic physics of traveling waves and standing waves are fundamental topics in the education of science and engineering students. These concepts are found in many areas, including classical mechanics, quantum mechanics, and electromagnetism. Typically, laboratory demonstrations of traveling waves and standing waves are based on mechanical systems such as vibrating strings and sound waves. This paper

Now that fundamental quantum principles of indeterminacy and measurement have become the basis of new technologies that provide secrecy between two communicating parties, there is a need to provide teaching laboratories that illustrate how these technologies work. In this article, we describe a laboratory exercise in which students perform quantum key distribution with single photons, and see how the

Intuition suggests that twisting a cylinder will shorten it, but here it is shown that for a cylinder of an incompressible material, like rubber, twisting will always produce elongation.

We describe a simple classroom demonstration of a fluid-dynamic instability. The demonstration requires only a bucket of water, a piece of string, and some used tealeaves or coffee grounds. We argue that the mechanism for the instability, at least in its later stages, is two-dimensional barotropic (shear-flow) instability and we present evidence in support of this. We show results of an equivalent

The teaching of structural stiffness is one of the keystones of the undergraduate curriculum in mechanics and the strength of materials. Standard linear theory, going back to Hooke's law, has proven to be very successful in predicting the performance of elastic structures under load. Many courses in basic mechanics have a conventional laboratory component often involving a universal testing machine

The purpose of this work is to study the phenomenon of tidal locking in a pedagogical framework by analyzing the effective gravitational potential of a two-body system with two spinning objects. It is shown that the effective potential of such a system is an example of a fold catastrophe. In fact, the existence of a local minimum and saddle point, corresponding to tidally locked circular orbits, is

With developments in modern instrumentation such as microelectromechanical gyro/accelerometers, high speed video analysis, and precision shaft encoders, there is an increased interest in the study of the large angle oscillations of pendulums as an example of nonlinear dynamics. The solution to the equation of motion for the non-linear pendulum cannot be expressed in terms of elementary functions and

We discuss an extension of the velocity Verlet method that accurately approximates the kinetic-energy-conserving charged particle motion that comes from magnetic forcing. For a uniform magnetic field, the method is shown to conserve both particle kinetic energy and magnetic dipole moment better than midpoint Runge–Kutta. We then use the magnetic velocity Verlet method to generate trapped particle trajectories

The illuminating role of differential forms in electromagnetism (EM) is seldom discussed in the classroom. It is the aim of this article to bring forth some of the relevant insights that can be learnt from a differential forms approach to EM. The article is self-contained in that no previous knowledge of forms is needed to follow it. The effective polarization of the classical vacuum due to a uniform

We developed a modified version of a conventional (BB84) quantum key distribution protocol that can be understood and implemented by students at a pre-university level. We intentionally introduce a subtle but critical simplification to the original protocol, allowing the experiment to be assembled at the skill level appropriate for the students, at the cost of creating a security loophole. The security

Most introductory books on quantum mechanics discuss the particle-in-a-box problem through solutions of the Schrödinger equation, at least, in the one-dimensional case. When introducing the virial theorem, however, its discussion in the context of this simple model is not considered and students ponder the question of the validity of the virial theorem for a system with, apparently, no forces. In this

A prototypical model of a one-dimensional metallic monatomic solid containing noninteracting electrons is studied, where the argument of the cosine potential energy, periodic with the lattice, contains the first reciprocal lattice vector G 1 = 2 π / a, where a is the lattice constant. The time-independent Schrödinger equation can be written in reduced variables as a Mathieu equation for which numerically

Spatial filtering is a commonly deployed technique to improve the quality of laser beams by optically filtering the noise. In the “textbook” example, the noise is usually assumed to be high frequency and the laser beam, Gaussian. In this case, the filtering is achieved by a simple pinhole placed at the common focal plane of two lenses. Here, we explain how to generalize the concept of spatial filtering

Optical coherence tomography, or in short OCT, is a measurement technique established in the early 1990s for the non-invasive imaging of interfaces in the bulk of biological tissues or other samples. A full-field OCT setup is built from a microscope combined with a Michelson interferometer, where the mirror in one arm is replaced by the sample. Using white light, which is temporally partially coherent

Charles Édouard Guillaume (1861–1928) was a Swiss physicist who received the 1920 Nobel Prize in physics for his precision measurements and discovery of anomalies in nickel steel alloys. In this work, we present a complete and commented translation of his remarkable article of 1896 on the temperature of interstellar space. The importance of this work is that it is the oldest estimate known to us of

An elementary, analytically soluble example of a pair of isospectral potentials is discussed. This example was developed in response to an inquiry by an undergraduate student who was intrigued by the widely used (but not often discussed in undergraduate courses) Rydberg–Klein–Rees (RKR) procedure for inverting rovibrational data to determine potential functions of diatomic molecules, which is based

An approximate solution is presented for simple harmonic motion in the presence of damping by a force which is a general power-law function of the velocity. The approximation is shown to be quite robust, allowing for a simple way to investigate amplitude decay in the presence of a general type of weak, nonlinear damping.

The room temperature compatibility of the negatively charged nitrogen-vacancy (NV−) center in diamond makes it the ideal quantum system for a university teaching lab. Here, we describe a low-cost experimental setup for coherent control experiments on the electronic spin state of the NV− center. We implement spin-relaxation measurements, optically detected magnetic resonance, Rabi oscillations, and

The ability to control the frequency of an external-cavity diode laser (ECDL) is an essential component for undergraduate laboratories and atomic physics research. Typically, the housing for the ECDL's diffraction grating and piezoelectric transducer is either purchased commercially or machined from metal. Here, we present an alternative to these commonly used options that utilizes 3D printing, a tool

We describe the design, construction, and operation of a LEGOTM model of a dynamic force microscope, using magnetic forces as an analog of interatomic interactions. The macroscope provides key insights into the operating principles of frequency modulated non-contact atomic force microscopy—currently the scanning probe technique of choice for maximal (i.e., submolecular) spatial resolution—and is, therefore

We construct an electronic circuit for mimicking a single neuron's behavior in connection with the dynamics of the Hodgkin–Huxley mathematical model. Our results show that the electronic neuron, even though it contains binary-state circuitry components, displays a timing interplay between the ion channels, which is consistent with the corresponding timing encountered in the model equations. This is

We present results and lessons learned from a Learning Assistant (LA) program combined with flipped classroom delivery and team learning in our calculus-based, introductory physics (mechanics) courses at California State University, Sacramento. A primarily undergraduate institution, our university serves a largely underprivileged student population, with one half of students coming from low-income

Standard university or high-school physics teaching material on projectile motion is usually based on Newton's second law in vacuum, neglecting aerodynamics. We present a low-cost experiment for teaching projectile motion using the students' cell phones and sports equipment, which allows the students to test theory and numerical simulation against experimental data in the real world. For a shot put

We use videos taken with a mobile phone to study conservation of energy, conservation of momentum, and the work-energy theorem by analyzing the collision of a cue ball and the eight ball. A video of the full time sequence, starting from before the cue ball is struck until well after the collision, is recorded with a mobile phone. The video is imported into Origin (free to teachers and students taking

We give a detailed derivation of the equation describing the elastic properties of the helical springs in the case of their large deformations. It is shown that the Taylor expansion for the restoring force contains, in addition to a linear term, a term quadratic in the strain. We discuss the physical consequences of this term. A surprising example of this is that for large values of the initial pitch

A detailed analysis of pendular motion is presented. Inertial effects, self-oscillation, and memory, together with non-constant moment of inertia, hysteresis, and negative damping are shown to be required for the comprehensive description of the free pendulum oscillatory regime. The effects of very high initial amplitudes, friction in the roller bearing axle, drag, and pendulum geometry are also analyzed