Volume 40, Issue 3, March 1972
 PAPERS


Classical Limit and the WKB Approximation
View Description Hide DescriptionWe prove that a wavepacket solution in the WKB approximation to the Schrödinger equation follows the classical trajectory, and we discuss the nature of this solution.

The Second Balance Equation Associated with the Telegrapher's Equations
View Description Hide DescriptionA possible physical interpretation is given for the second balance equation associated with the telegrapher's equations.

Computer Aided Physics Laboratory Instruction: Project caplin
View Description Hide DescriptionA project on computer assisted physics laboratory instruction (Projectcaplin) has led to the development of material used in a first semester introductory physics laboratory for nonscience majors. In the laboratory the student first obtains data from traditional laboratory experiments. The student then has the option to use parts (or all) of preprogrammed computermaterial designed to: (1) accept the experimental data and scan it for inconsistencies; (2) carry out an error analysis on the data; (3) perform calculations of physical parameters based on the experimental data; (4) yield tabular displays of calculated results; and (5) yield graphical displays of results. In some cases, computer programs allow simulation of experiments with the possibility of altering values of parameters or experimental details. It has been observed that the computer aided laboratories have considerably improved comprehension of data error analysis, of graphic representation of physical concepts, and have aided in carrying out calculational phases of the experiment left to the student. Furthermore, laboratory time saved through the use of the computer system has enabled expansion of the scope of individual laboratories.

On Lagrangians with Higher Order Derivatives
View Description Hide DescriptionClassical mechanics based on Lagrangians with higher order derivatives is investigated. It is found that this generalization does not lead to any new feature of the theory, apart from an increase of the dimensionality of phase space. Conserved quantities are also derived that reduce to the usual ones in the absence of higher derivatives.

Analog Computer Solution of the Electrodiffusion Equation for a Simple Membrane
View Description Hide DescriptionAn analog solution was obtained for the NernstPlanck and Poisson equations which describe the ion concentration across a simple membrane held at a potential difference V. These equations constitute a nonlinear system of differential equations. It was learned that even though electroneutrality was assumed as a boundary condition, it did not hold within the membrane itself. The electric field variation within the membrane was also determined.

An Electrostatic Oscillator Experiment
View Description Hide DescriptionAn experiment involving a conducting sphere that oscillates between two electrodes is described. Theory is presented that predicts a linear relationship between frequency and applied voltage for the system, regardless of electrode configuration. Data taken for several geometries is in good agreement with this theory. Some properties of the oscillator system, including its desirable characteristics as a high voltage voltmeter, are discussed.

The Gravitational Red Shift: A ThreeBody Effect
View Description Hide DescriptionThe nonlinearity of Einstein's gravitational equations produces threebody forces in the equations of motion. One of their consequences is that the motion of an electron around a nucleus (or a satellite around a planet) is slowed down by the existence of a distant massive body. This is the familiar gravitational red shift, which can thus be derived without invoking the equivalence principle.

Electron Optics: A Topic of a Computer Applications Course
View Description Hide DescriptionElectron optics is especially suited as a topic for independent study as part of a computer applications course. The principles are elementary, and the student gains greater understanding of the classical similarities between particles and waves. A method of solution is illustrated, and the results appear more reliable than those existing in prior literature.

Physics and Philosophy: A Problem for Education Today
View Description Hide DescriptionThe teaching of the process of science in physics courses is discussed with reference to recent intellectual trends. Arguments and examples indicate that “process courses” must necessarily involve philosophical questions and questions of social criticism currently raised. A description is given of an interdisciplinary upperdivision course for science and nonscience majors that incorporates some of the features advocated.

On the Ehrenfest Paradox
View Description Hide DescriptionCertain aspects of the socalled Ehrenfest paradox are discussed. It is pointed out that while the Ehrenfest description of uniform rotation is not in keeping with relativity, the “paradox” per se is independent of this fact, and rests solely upon an imprecise use of notation.

Complex Potentials in Classical Mechanics and Geometrical Optics
View Description Hide DescriptionIt has been shown that the method of complex potentials widely used in hydrodynamics and electrostatics can be used in twodimensional problems of particle mechanics and geometrical optics too.

Clarification on Two Important Questions in Rigid Body Mechanics
View Description Hide DescriptionIn most courses on rigid body mechanics, many of the important subtleties are only superficially stated or investigated. In this article, we give two such fundamental questions (with solutions) that most students ponder upon without apparent resolution.

On the Inversion of Noether's Theorem in Classical Dynamical Systems
View Description Hide DescriptionA simple and conceptually clear proof of the inverse Noether's theorem is presented, both for classical point mechanics and for classical field theory. We start from an analysis of the time dependence (of the four divergence, respectively), of an arbitrary dynamical quantity, to relate it to the variation of the action integral induced by a suitably defined infinitesimal transformation. When the dynamical quantity is chosen to be a constant of motion (a divergenceless quantity, respectively), this infinitesimal transformation is shown to act as an invariance transformation on the dynamical system.

A Demonstration of Momentum Conservation using Bow, Arrow, and Ballistic Pendulum
View Description Hide DescriptionThe measurement of the variation of the velocity of an arrow with the bending of the bow using either a ballistic pendulum or the freefall phenomenon provided in both cases coinciding linear dependences. These results create the possibility for the utilization of the bow, arrow, and ballistic pendulum for the demonstration and practical exercises of several important physical phenomena. In this paper greatest attention is paid to the demonstration of the momentum conservation principle.

Great Originals of Modern Physics
View Description Hide DescriptionEuropean travel can provide an intimate view of the implements and locales of great discoveries in physics for the knowledgeable traveler. The four museums at Cambridge, London, RemscheidLennep, and Munich display a full range of discovery apparatus in modern physics as outlined here.

A ComputerAssisted Course In Intermediate Mechanics
View Description Hide DescriptionThis article describes an intermediate level mechanics course based on the solution by computer of seven problems. The nature of these problems and their solution by computer is discussed, and the sources of the programs indicated.

Geometrical Approach to Torque Free Motion of a Rigid Body Having Internal Energy Dissipation
View Description Hide DescriptionA geometrical approach, which differs from the classical Poinsot solution, is developed for the moment free motion of a rigid body in order to discuss the case when energy is dissipated internally. Conservation of angular momentum and conservation of energy are used, respectively, to construct two surfaces, a sphere S and an ellipsoid E, which always intersect along a closed curve C; some typical intersections are shown. The motion of the body is such that the lip of the angular momentum vector remains on a curve C. Although less complete than Poinsot solution, this geometrical description provides a check on some classical results and is particularly useful in characterizing the motion when the angular momentum is kept constant while the energy decreases; then the sphere S remains the same while the ellipsoid E shrinks. The final state, corresponding to the minimum possible energy configuration, is a pure rotation about the principal axis of maximum inertia. Many interesting problems (e.g., the rotational stability of artificial satellites, the axial alignment of asteroids, and the damping of the Chandler wobble for the Earth and pulsars) are discussed in the context of this solution.

Mass Distribution and Frequencies of a Vertical Spring
View Description Hide DescriptionA vertical spring in a gravitational field has a density increase from top to bottom. The density variation, which is often not noticeable, depends on the ratio of the spring weight to the quantity , the spring constant times the unstretched length . Two springs prepared by prestretching 78 and 23 turn portions of a “Slinky” showed density increases by factors of about 10 and 4, respectively. The detailed density distribution of the smaller spring agrees very well with the results of an analysis based on Hooke's law behavior. The heavier spring shows nonHookean effects. The lowest few frequencies can be measured by driving the spring in resonance by hand, and for the lighter spring these also agree well with calculated values. The predicted nonuniform distribution of nodes is observable. This simple apparatus may provide stimulating experiments for intermediate laboratories.

The Context of Inquiry in Physics
View Description Hide DescriptionCollege physicstextbooks emphasize the conclusions of research at the expense of the context of inquiry in physics. In order to overcome this deficiency, at least one of the physics courses within a student's program should include specific consideration of the context of inquiry in physics. An excellent focus for this consideration would be the process of theory evaluation in physics whose character is indicated in this paper.

Solution of a OneDimensional ThreeBody Problem in Classical Mechanics
View Description Hide DescriptionA onedimensional problem of three equal particles interacting pairwise via a potential that is the sum of a quadratic and an inversely quadratic term is discussed from the point of view of classical mechanics. It is shown that the quantum energy spectrum can be obtained from the classical theory of actionangle variables and BohrSommerfeld rules.
