Volume 82, Issue 4, April 2014

We report investigations of collisions between steel balls and freeended rods using an electromechanical apparatus combined with a Michelson interferometer. Our experimental methodology allows for the evaluation of several collisioninitiated parameters, such as collision duration, speed of the intermaterial wave pulse, change in rod length, and postcollision centerofmass velocity. The fractional loss of kinetic energy of the ballrod system is studied theoretically and experimentally as a function of the rod length. Results are compared to predictions obtained from the elasticbody collision theory. Several aspects of elastic waves are illustrated without recourse to complicated mathematics. The experiment provides significant insight into the physical behavior of colliding solids.
 PAPERS


Magnet traveling through a conducting pipe: A variation on the analytical approach
View Description Hide DescriptionWe present an analytical study of magnetic damping. In particular, we investigate the dynamics of a cylindrical neodymium magnet as it moves through a conducting tube. Owing to the very high degree of uniformity of the magnetization for neodymium magnets, we are able to provide completely analytical results for the electromotive force generated in the pipe and the consequent retarding force. Our analytical expressions are shown to have excellent agreement with experimental observations.

Compression waves and kinetic energy losses in collisions between balls and rods of different lengths
View Description Hide DescriptionWe report investigations of collisions between steel balls and freeended rods using an electromechanical apparatus combined with a Michelson interferometer. Our experimental methodology allows for the evaluation of several collisioninitiated parameters, such as collision duration, speed of the intermaterial wave pulse, change in rod length, and postcollision centerofmass velocity. The fractional loss of kinetic energy of the ballrod system is studied theoretically and experimentally as a function of the rod length. Results are compared to predictions obtained from the elasticbody collision theory. Several aspects of elastic waves are illustrated without recourse to complicated mathematics. The experiment provides significant insight into the physical behavior of colliding solids.

Circular orbits on a warped spandex fabric
View Description Hide DescriptionWe present a theoretical and experimental analysis of circularlike orbits made by a marble rolling on a warped spandex fabric. We show that the mass of the fabric interior to the orbital path influences the motion of the marble in a nontrivial way and can even dominate the orbital characteristics. We also compare a Keplerlike expression for such orbits to similar expressions for orbits about a spherically symmetric massive object in the presence of a constant vacuum energy, as described by general relativity.

The Schwarzschild metric: It's the coordinates, stupid!
View Description Hide DescriptionEvery general relativity textbook emphasizes that coordinates have no physical meaning. Nevertheless, a coordinate choice must be made in order to carry out real calculations, and that choice can make the difference between a calculation that is simple and one that is a mess. We give a concrete illustration of the maxim that “coordinates matter” using the exact Schwarzschild solution for a vacuum, static spherical spacetime. We review the standard textbook derivation, Schwarzschild's original 1916 derivation, and a derivation using the LandauLifshitz formulation of the Einstein field equations. The last derivation is much more complicated, has one aspect for which we have been unable to find a solution, and gives an explicit illustration of the fact that the Schwarzschild geometry can be described in infinitely many coordinate systems.

Beyond Clausius–Clapeyron: Determining the second derivative of a firstorder phase transition line
View Description Hide DescriptionWe obtain an expression for the second derivative of the line in a PT diagram denoting a firstorder phase transition for a pure hydrostatic system. Our result goes beyond the classical Clausius–Clapeyron equation, which provides only the first derivative of the pressure with respect to the temperature along the transition line. We present two pedagogical derivations suitable for an undergraduate thermodynamics class; the first one uses derivatives of the entropy while the second one uses derivatives of the enthalpy. The final expression for the second derivative involves only standard thermodynamic quantities such as the specific heats, the isothermal compressibilities, and the coefficients of thermal expansion of the two phases at the transition line. As an illustration, we compute the second derivatives of the freezing and vaporization lines of water at atmospheric pressure, and show that at this pressure the freezing line is concave down (negative second derivative) while the vaporization line is concave up (positive second derivative).

Advantages of using a logarithmic scale in pressurevolume diagrams for Carnot and other heat engine cycles
View Description Hide DescriptionWe demonstrate that plotting the PV diagram of an ideal gas Carnot cycle on a logarithmic scale results in a more intuitive approach for deriving the final form of the efficiency equation. The same approach also facilitates the derivation of the efficiency of other thermodynamic engines that employ adiabatic ideal gas processes, such as the Brayton cycle, the Otto cycle, and the Diesel engine. We finally demonstrate that logarithmic plots of isothermal and adiabatic processes help with visualization in approximating an arbitrary process in terms of an infinite number of Carnot cycles.

An experiment to measure the instantaneous distance to the Moon
View Description Hide DescriptionWe propose an experimental technique for determining the distance to the Moon. Our technique is based on measuring the change in angular size of the lunar disk due to the variation of the observerMoon distance, as caused by the rotation of the Earth over several hours. Using this method we obtained a value of 3.46 × 10^{5} km with a precision of 7%. Additionally, our technique allows for the determination of the Moon radius (1678 km ± 7%), and the instantaneous radial velocity with respect to the Earth (26.4 m/s ± 26%). A unique advantage of this method is that it is performed from a single location with a single observer, unlike the traditional parallaxbased measurements that require at least two observers with a large separation distance.

Measuring the Moon's orbit using a handheld camera
View Description Hide DescriptionThis paper describes a way to measure the Moon's distance and orbital eccentricity using a digital camera. The method consists of taking photographs of the Moon and measuring the size of the lunar disk in each picture. On a series of images taken on the same night, the effect of the Earth's size is evident and thus the distance to the Moon can be computed. A larger series of images, covering several weeks, demonstrates that the Moon's orbit is not perfectly circular.

Understanding current signals induced by drifting electrons
View Description Hide DescriptionConsider an electron drifting in a gas toward a collection electrode. A common misconception is that the electron produces a detectable signal only upon arrival at the electrode. In fact, the situation is quite the opposite. The electron induces a detectable current in the electrode as soon as it starts moving through the gas. This induced current vanishes when the electron arrives at the plate. To illustrate this phenomenon experimentally, we use a gasfilled parallelplate ionization chamber and a collimated ^{241}Am alpha source, which produces a track of a fixed number of ionization electrons at a constant distance from the collection electrode. We find that the detected signal from the ionization chamber grows with the electron drift distance, as predicted by the model of charge induction, and in conflict with the idea that electrons are detectable upon arrival at the collection plate.

Quick and easy escape from a metastable state
View Description Hide DescriptionAn elementary but complete calculation of metastable state decay from a piecewiseconstant onedimensional potential well is presented that does not invoke the concept of an attempt frequency or require a knowledge of the WentzelKramersBrillouin method. Connection is made with scattering resonances for advanced readers.

The pilotwave perspective on spin
View Description Hide DescriptionThe alternative pilotwave theory of quantum phenomena—associated especially with Louis de Broglie, David Bohm, and John Bell—reproduces the statistical predictions of ordinary quantum mechanics but without recourse to special ad hoc axioms pertaining to measurement. That (and how) it does so is relatively straightforward to understand in the case of position measurements and, more generally, measurements, whose outcome is ultimately registered by the position of a pointer. Despite a widespread belief to the contrary among physicists, the theory can also account successfully for phenomena involving spin. The main goal of this paper is to explain how the pilotwave theory's account of spin works. Along the way, we provide illuminating comparisons between the orthodox and pilotwave accounts of spin and address some puzzles about how the pilotwave theory relates to the important theorems of Kochen and Specker and Bell.

Accurate physical laws can permit new standard units: The two laws and the proportionality of weight to mass
View Description Hide DescriptionThree common approaches to are: (1) as an exactly true definition of force in terms of measured inertial mass m and measured acceleration ; (2) as an exactly true axiom relating measured values of and m; and (3) as an imperfect but accurately true physical law relating measured to measured , with m an experimentally determined, matterdependent constant, in the spirit of the resistance R in Ohm's law. In the third case, the natural units are those of and , where is normally specified using distance and time as standard units, and from a spring scale as a standard unit; thus mass units are derived from force, distance, and time units such as newtons, meters, and seconds. The present work develops the third approach when one includes a second physical law (again, imperfect but accurate)—that balancescale weight W is proportional to m—and the fact that balancescale measurements of relative weight are more accurate than those of absolute force. When distance and time also are more accurately measurable than absolute force, this second physical law permits a shift to standards of mass, distance, and time units, such as kilograms, meters, and seconds, with the unit of force—the newton—a derived unit. However, were force and distance more accurately measurable than time (e.g., time measured with an hourglass), this second physical law would permit a shift to standards of force, mass, and distance units such as newtons, kilograms, and meters, with the unit of time—the second—a derived unit. Therefore, the choice of the most accurate standard units depends both on what is most accurately measurable and on the accuracy of physical law.
