Volume 79, Issue 6, June 2011
 PAPERS


Resource Letter PS2: Physics of Sports
View Description Hide DescriptionThis Resource Letter provides a guide to the literature on the physics of sports, updating Resource Letter PS1, published 25 years ago (Ref. 17). The intent is to suggest literature for anyone curious about the basic physics of particular sports, for physics teachers searching for sports examples to augment their teaching, and for physicists contemplating research on unsolved sportsrelated questions.

Corked bats, juiced balls, and humidors: The physics of cheating in baseball
View Description Hide DescriptionThree questions of relevance to Major League Baseball are investigated from a physics perspective. Can a baseball be hit farther with a corked bat? Is there evidence that the baseball is more lively today than in earlier years? Can storing baseballs in a temperature or humiditycontrolled environment significantly affect home run production? These questions are subjected to a physics analysis, including an experiment and an interpretation of the data. The answers to the three questions are no, no, and yes, respectively.

Demonstration and study of the dispersion of water waves with a computercontrolled ripple tank
View Description Hide DescriptionThe design of a ripple tank built in an undergraduate student project is described. Water waves are excited acoustically using computer programmable wave shapes. The projected wavepatterns are recorded with a video camera and analyzed quantitatively. From the propagation of wave packets in distilled water at three different depths, the phase and group velocities are measured in the frequency range from 2 to 50 Hz. Good agreement with theory is found. The propagation of wave trains of different shapes is recorded and explained on the basis of the stationary phase approximation. Various types of precursors are detected. For a depth slightly above the critical depth and thus nearly dispersionfree, the solitarylike propagation of a single pulse is observed. In shallow water, the compression of a chirped pulse is demonstrated. Circular waves produced by falling water drops are recorded and analyzed.

How Einstein confirmed
View Description Hide DescriptionThe equivalence of mass and restenergy is one of the great discoveries of all time. Despite the current wisdom, Einstein did not derive this relation from first principles. Having conceived the idea in the summer of 1905 he spent more than 40 years trying to prove it. We briefly examine all of Einstein’s conceptual demonstrations of , focusing on their limitations and his awareness of their shortcomings. Although he repeatedly confirmed the efficacy of , he never constructed a general proof. Leaving aside that it continues to be affirmed experimentally, a rigorous proof of the massenergy equivalence is probably beyond the purview of the special theory.

More about Bohm’s quantum
View Description Hide DescriptionI give a pedagogical introduction to Bohm’s interpretation of quantum theory, suitable for graduate students.

Decay of wave packet revivals in the asymmetric infinite square well
View Description Hide DescriptionIn the infinite square well, any wave function will return to its initial state at integer multiples of the revival time. Most quantum systems do not exhibit perfect revivals, but some exhibit partial revivals in which the wave function returns close to its initial state. Subsequent partial revivals usually deteriorate in quality. We discuss the reasons for the perfect revivals in the infinite square well and how a small change in the potential disrupts the revivals. As an example, we examine partial revivals of a Gaussian wave packet in an infinite square well with a step. Firstorder and secondorder perturbation theory show that the rate at which revivals decay depends on the location of the step.

Two and three slit electron interference and diffraction experiments
View Description Hide DescriptionCurrent nanotechnology techniques make possible the preparation of slits in the submicrometer range so that electron interference and diffraction experiments can be done even with a conventional electron microscope. If the instrument is also equipped with a field emission source, it is possible to follow almost in real time the transition from the image of the slits to their Fraunhofer pattern through the intermediate Fresnel diffraction images. We discuss our results for the twoslit experiment and illustrate them for the threeslit case.

Millimeterwave Bragg diffraction of microfabricated crystal structures
View Description Hide DescriptionA compact diffraction apparatus is developed with millimeterwave propagation between two parallel plates. Two types of microfabricated model crystals are individually mounted on a rotatable structure. In contrast to previous work, the experimental results agree well with Bragg’s predictions because multiple scattering is minimized in this configuration. Factors that affect the resolution and signal strength, such as the number of scatterers, cylinder radius, and the distance between the detector and the model crystal, are analyzed. The apparatus offers a visually accessible way to teach students about crystal structure as well as scattering and diffraction.

Duality of force laws and conformal transformations
View Description Hide DescriptionAs was first noted by Isaac Newton, the two most famous ellipses of classical mechanics, arising from the force laws and , can be mapped onto each other by changing the location of the center of force. Less well known is that this mapping can also be achieved by the complex transformation, . We derive this result and its generalization by writing the Gaussian curvature in its covariant form, and then changing the metric by a conformal transformation which mimics this mapping of the curves. We indicate how the conserved Laplace–Runge–Lenz vector for the force law transforms under this transformation, and compare it with the corresponding quantities for the linear force law. Our main aim is to present this duality by introducing concepts from differential geometry.

The apparent “superCarnot” efficiency of hurricanes: Nature’s steam engine versus the steam locomotive
View Description Hide DescriptionThe thermodynamics of the hurricane—Nature’s steam engine—presents surprising contrasts with that of the steam locomotive. The hurricane rejects not only its waste heat at the lowest available temperature (as all heat engines must do to maximize efficiency), but also its work (that is, the kinetic energy of its winds) via frictional dissipation at the highest available temperature. We show how the hurricane’s “superCarnot” efficiency is consistent with the laws of thermodynamics. We also show that even standard heat engines can achieve “superCarnot” efficiency, albeit via a different mechanism and to a far inferior degree than the hurricane.

Classical physics of thermal scalar radiation in two spacetime dimensions
View Description Hide DescriptionThermal scalar radiation in two spacetime dimensions is treated within relativistic classical physics. We first consider an inertial frame in which we give the analogues of Boltzmann’s derivation of the Stefan–Boltzmann law and Wien’s derivation of the displacement theorem using the scaling appropriate to relativistic radiationtheory. The spectrum of classical scalar zeropoint radiation in an inertial frame is derived both from scale invariance and from Lorentz invariance. We then consider the behavior of thermal radiation in a coordinate frame undergoing (relativistic) constant acceleration. The classical zeropoint radiation of inertial frames is transformed to the coordinates of an accelerating frame. Although the twofield correlation function for zeropoint radiation at different spatial points at a single time is the same for inertial and accelerating frames, the correlation function at two different times at a single spatial coordinate is different and, in an accelerating frame, has a natural extension to nonzero temperature. The thermal spectrum in the accelerating frame is then transferred back to an inertial frame, giving the familiar Planck result.

Exact nonHookean scaling of cylindrically bent elastic sheets and the largeamplitude pendulum
View Description Hide DescriptionA sheet of elastic foil rolled into a cylinder and deformed between two parallel plates acts as a nonHookean spring if deformed normally to the axis. For large deformations the elastic force shows an interesting inverse square dependence on the interplate distance. This phenomenon was used as the basis for an experimental problem at the 41st International Physics Olympiad. We show that the corresponding variational problem for the equilibrium energy of the deformed cylinder is equivalent to a minimum action description of a simple gravitational pendulum with an amplitude of 90°. We use this analogy to show that the powerlaw of the force is exact for distances less than a critical value. An analytical solution for the elastic force is found and confirmed by measurements over a range of deformations covering both linear and nonlinear behaviors.

Schwarzschild and de Sitter solutions from the argument by Lenz and Sommerfeld
View Description Hide DescriptionThe Lenz–Sommerfeld argument allows an ingenious and simple derivation of the Schwarzschild solution of Einstein equations of general relativity. We use the same reasoning to construct the de Sitter line element.

What is the boundary condition for the radial wave function of the Schrödinger equation?
View Description Hide DescriptionThere is much discussion in the mathematical physics literature as well as in quantum mechanicstextbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of the radial Schrödinger equation leads to the appearance of a delta function term when the Laplace operator is written in spherical coordinates. As a result, regardless of the behavior of the potential, an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.

Refraction in spacetime
View Description Hide DescriptionRefraction, interference, and diffraction are distinguishing features of wavelike phenomena. Although they are usually associated only with a purely spatial wavepropagation pattern, analogs to interference and diffraction involving the spatiotemporal dynamics of waves in one dimension have been discussed. We complete the triplet of analogies by discussing how spatiotemporal analogs to refraction are exhibited by a quantum particle in one dimension that is scattering off a step barrier. Similarly, birefringence in spacetime occurs for a spin1/2 particle in a magnetic field.

Simple derivations of the Hamilton–Jacobi equation and the eikonal equation without the use of canonical transformations
View Description Hide DescriptionThe Hamilton–Jacobi equation in classical mechanics and the related eikonal equation in geometrical optics are often described as the “point of closest approach” between classical and quantum mechanics. Most textbook treatments of Hamilton–Jacobi theory are aimed at graduate students and derive the equation only after a long introduction to canonical transformations. Most treatments of the eikonal equation only emphasize its use in geometrical optics. We show that both the Hamilton–Jacobi equation and the eikonal equation can be derived by a common procedure using only elementary aspects of the Lagrangian and Hamiltonian formalisms introduced in undergraduateclassical mechanics courses. Through this common approach, we hope to highlight to undergraduates the deep connections between classical mechanics, classical wave theory, and Schrödinger’s wave mechanics.
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 NOTES AND DISCUSSIONS


Comment on “A close examination of the motion of an adiabatic piston,” by Eric A. Gislason [Am. J. Phys.78 (10), 995–1001 (2010)]
View Description Hide DescriptionA recent paper by Gislason treats the adiabatic piston, a system of two ideal gases in a horizontal cylinder and separated by an insulating piston that moves without friction. The analysis in this paper is comprehensive and useful as a teaching tool, but is somewhat misleading if not understood in the appropriate context. The evolution to equilibrium involves two mechanisms, a faster one leading to the equalization of pressures, and a slower one bringing the system to identical temperatures. Gislason addressed only the first mechanism. We note that the eventual final state is described by thermodynamics. Therefore, a discussion of the adiabatic piston can be enriched to promote a proper and general view of thermodynamics.

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 BOOK REVIEWS


Pursuing Power and Light: Technology and Physics from James Watt to Albert Einstein
View Description Hide DescriptionThis article reviews Pursuing Power and Light: Technology and Physics from James Watt to Albert Einstein by Bruce J. Hunt
