Volume 47, Issue 6, June 1979
Index of content:
 Papers


Physics of the tennis racket
View Description Hide DescriptionSeveral parameters concerning the performance of tennis rackets are examined both theoretically and experimentally. Information is obtained about the location of the center of percussion, the time a ball spends in contact with the strings, the period of oscillation of a tennis racket, and the coefficient of restitution of a tennis ball. From these data it may be possible to design a racket with improved playing characteristics.

Analogy between thermodynamics and mechanics
View Description Hide DescriptionWe note that equations of state—by which we mean identical relations among the thermodynamic variables characterizing a system—are actually first‐order partial differential equations for a function which defines the thermodynamics of the system. Like the Hamilton‐Jacobi equation, such equations can be solved along trajectories given by Hamilton’s equations, the trajectories being quasistatic processes which obey the given equation of state. This gives rise to the notion of thermodynamic functions as infinitesimal generators of quasistatic processes, with a natural Poisson bracket formulation. This formulation of thermodynamic transformations is invariant under canonical coordinate transformations, just as classical mechanics is, which is to say that thermodynamics and classical mechanics have the same formal structure, namely a symplectic structure.

Thermodynamic derivation of the equilibrium distribution functions of statistical mechanics
View Description Hide DescriptionA simplified derivation of the equilibrium distribution functions is presented. The derivation proceeds from the change in the Helmholtz free energy when a particle is added to a system of fixed temperature, volume, and chemical potential. Besides its simplicity, this form of the derivation offers a clear view of the relationship between statistical mechanics and macroscopic thermodynamics.

Spatial Bose‐Einstein condensation
View Description Hide DescriptionThree examples of s p a t i a l Bose‐Einstein condensations in which the particles macroscopically occupy the lowest localized state of an inhomogeneous external potential are analyzed. The three cases are (a) a small potential well in a large box which causes a spectrum with a gap, (b) a harmonic oscillator potential, and (c) randomly sized trapping potentials caused by ’’impurities.’’ All three cases are two dimensional so that no Bose condensation occurs without the inhomogeneous potentials. An attempt is made to keep the treatments as mathematically simple as possible. A review of much of the literature of spatialBose condensations is provided. The special problem of the form of the thermodynamic limit in an inhomogeneous potential is discussed for case (b). Numerical examples applying to monolayer^{4}He adsorbed on a surface are treated.

Diffraction of a laser beam
View Description Hide DescriptionThe effect of the nonuniform irradiance across a laser beam on diffraction of the beam is investigated. Calculations of Fraunhofer diffraction by a circular aperture show the deviations which occur from the usual Airy disk for several relative spot and aperture sizes.

Simple laser scattering experiment for biology‐oriented physics labs
View Description Hide DescriptionA low intensity laser light scattering laboratory exercise has been designed and introduced into the lower division physics course for biology students at the University of California, Santa Barbara. Students determine the diameter of micron‐sized latex spheres (simulated microbes) in water suspension. By using their own eyes as part of the measuring apparatus the students become personally involved in the experiment, which is relevant to their field of interest. Accuracies of 1%–2% are normal, and student response has been unusually positive.

A novel light trapping phenomenon in fluid media
View Description Hide DescriptionInjection of a thin layer of solution at the boundary of a moving solvent is utilized to create a thin fluid sheet having an index of refraction greater than that of the surrounding medium. Conditions are such that light trapping is easily observed in the thin fluid sheet. Diffusion effects give rise to a concentration profile which is quadratic at the boundary. As a consequence of this quadratic dependence, an interesting hopping and refocusing of a trapped beam is observed as it traverses the fluid sheet. The observed phenomenon is readily modeled by use of the Navier Stokes equation and the diffusionequation.

Oscillations of a particle attached to a heavy spring: An application of the Stieltjes integral
View Description Hide DescriptionThe normal‐mode motions of a massive particle and a spring of non‐negligible mass to whose end it is attached are shown to be described by ’’nonorthogonal’’ displacement functions. Two equivalent methods of establishing an orthogonality relation — a specially defined inner product and application of the Riemann‐Stieltjes integral — are introduced. The results lead to the complete solution for the general motion of the spring‐plus‐particle system. This solution is applied to provide rigorous justification of the intuitive method that leads to the familiar rule (’’... add one‐third the mass of the spring ...’’) for obtaining the period of a particle oscillating at the end of a uniform spring whose non‐negligible mass is small in comparison. It is shown that the presence/absence of gravity is irrelevant to the general conclusions reached.

The great beer bottle experiment
View Description Hide DescriptionWe describe a low budget experiment, using materials that are easily obtained, to measure the speed of sound with an error of less than 10%. The device consists of a Helmholtz resonator of cylindrical cross section (a standard 12‐oz beer bottle). The system is analogous to a mass vibrating on a spring. The resonant frequency is determined by the mass of the air plug located in and near the bottle neck and the spring constant associated with the elasticity of the air filling the remainder of the bottle. A simple procedure involving length measurements with a vernier caliper is outlined and a discussion of the approximations made in arriving at this procedure is presented.

Geometry of the Kepler initial value problem explored with two circles
View Description Hide DescriptionIt is shown, by a judicious exploitation of the Runge‐Lenz vector, that the Kepler initial value problem is endowed with a nice geometric structure which enables one to figure out, even mentally, after a single trivial arithmetic calculation, the orbit associated with any given set of initial data.

Use of ‖Ψ‖^{2} and flux to simplify analysis of transmission past rectangular barriers or wells
View Description Hide DescriptionThe transmission coefficient for a completely general one‐dimensional rectangular potential barrier or well, or series of steps, is analyzed simply by concentrating on ‖Ψ‖^{2} rather than Ψ. Graphs of ‖Ψ‖^{2} for all cases are given, whereas only two textbooks graph anything related to Ψ, and those plots of Re(Ψ) can be misinterpreted.

Nonrelativistic contribution to Mercury’s perihelion precession
View Description Hide DescriptionWe present here a calculation of the precession of the perihelion of Mercury due to the perturbations from the outer planets. The time‐average effect of each planet is calculated by replacing that planet with a ring of linear mass density equal to the mass of the planet divided by the circumference of its orbit. The calculation is easier than examples found in many undergraduate theoretical mechanics books and yields results which are in excellent agreement with more advanced treatments. The perihelion precession is seen to result from the fact that the outer planets slightly change the radial period of oscillation from the simple harmonic period usually calculated for small displacements from equilibrium. This new radial period therefore no longer matches the orbital period and the orbit consequently does not exactly retrace itself. The general question of whether a given perturbation will cause the perihelion to advance or regress is shown to have the following answer: if a perturbing force is central and repulsive and also becomes stronger as the distance from the force center increases, the perihelion will advance. If the central perturbing force is attractive and also becomes stronger as the distance from the force center increases, the perihelion will regress.

Development of canonical transformations from Hamilton’s principle
View Description Hide DescriptionIt is shown that the variations δq _{ j } and δQ _{ i } cannot both be simultaneously zero at the limits of integration in application of Hamilton’s principle to the development of the theory of canonical transformations. This means that δF _{1} does not vanish. However, the derivation requires only that the δq _{ j } be zero at the end points. This lack of symmetry between the q _{ j } and Q _{ i } in the formulaion is noted with respect to the Lagrangian and the fundamental commutator relations of quantum mechanics. Further, what is meant by an independent variation of p _{ j } and q _{ j } (or Q _{ i } and P _{ i }) in the formalism is discussed in the use of Hamilton’s principle in the derivation of Hamilton’s equations of motion.

Structure and vibrational behavior of interstitial atoms in metals: An air table demonstration
View Description Hide DescriptionMagnetic cylinders on an air table behave similarly to atoms in many face‐centered metals like aluminum or copper. They qualitatively exhibit the striking properties of a split‐interstitial point defect (dumbbell): the structure, the localized and resonance vibration modes of a symmetric dumbbell; its effect upon the shear modulus; and the structure of an unsymmetric (mixed) dumbbell, containing a foreign atom. All effects can be shown to a large audience by means of a transparent air table put on an overhead projector.

A rapid, convenient, and precise method for the absolute determination of the acceleration of gravity
View Description Hide DescriptionA compact and portable apparatus for the measurement of the value for the gravitational accelerationg is described and the results of an experiment is given. It consists of a falling mercury drop and an electronic timing circuit which conveniently and automatically generates a large number of data in a short period of time, yielding results with a high degree of precision. In an experiment on 47 consecutive trials, a value of 982.44 cm sec^{−2} was obtained with a standard error of the mean of ±0.02 cm sec^{−2}, and an accuracy of 0.22%.

Simplified derivation of the Fokker‐Planck equation
View Description Hide DescriptionAn alternative derivation of the Fokker‐Planck equation for the probability density of a random noise process is presented, starting from the Langevin equation. The derivation makes use of the first two derivatives of the Dirac delta function. The derivation of the Fokker‐Planck equation then becomes simpler and more transparent, at least for those willing to accept these singularity functions.

Group theoretical interpretation of von Neumann’s theorem on composite systems
View Description Hide DescriptionWe show that the physical meaning of a mathematical theorem on composite systems (described quantum mechanically) established by von Neumann becomes transparent with reference to a physical example, i.e., that of a composite system in a state of definite angular momentum. In this case the theorem is shown to reproduce the Clebsch‐Gordan expansion, a fact which has been universally ignored. The theorem has a more general validity, the only condition being the existence of a conservation law, or, in other terms, of an underlying group structure (at least for cases of physical interest). This circumstance, in the case of non‐Abelian groups, proves essential in the discussion of the Einstein, Podolsky, Rosen paradox and related problems.

 Notes and Discussions


Comment on ’’Simple model for the emission particles by black holes’’
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Reply to P. C. Peters ’’Comment on ’Simple model for the emission particles by black holes’’’
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