Volume 46, Issue 3, March 1978
 Papers


Galileo and the Tower of Pisa experiment
View Description Hide DescriptionA study of Galileo’s book O n M o t i o n shows that many modern textbooks misrepresent what he would have been trying to prove in the legendary Tower of Pisa experiment. Recently performed experiments and calculations based on the current theory of falling bodies also show that it is doubtful if Galileo could have obtained the results claimed in the legend.

Circumventing Newton: A study in scientific creativity
View Description Hide DescriptionIs there a parallel between the creative work of Newton and of Beethoven? This question provides the background for a specific case study in a natural sciences course for nonscience students. The case study shows how Kepler’s three laws can be derived either from the standard Newtonian concepts, or alternatively from conservation laws. The latter approach does not use concepts of force or inertial straight‐line motion. By demonstrating an alternative physical framework to explain certain basic celestial phenomena, Newton’s accomplishment becomes more personally his own. There is no necessity in nature that the Newtonian world system had to be found. Newton’s P r i n c i p i a is a personal achievement that places him in the same creative class as Beethoven or Shakespeare.

Personalized introductory courses: A longitudinal study
View Description Hide DescriptionGroups of 139 and 137 undergraduate students took three physics courses taught by either the personalized system of instruction or the lecture‐discussion method. Students from these groups were compared in two subsequent chemistry courses and a subsequent biology course. The PSI group achieved significantly higher grades in each case. The junior year major area grades of students who majored in applied mechanics and engineering sciences, biology, chemistry, and physics were also compared. In each case, the PSI group achieved significantly higher grades. For each comparison, the mean high school grade point average, SAT scores, and freshman university grade point average were used to verify that no selective attrition had occurred.

Measurement of acoustical second‐order Doppler effect as an introductory experiment to special relativity
View Description Hide DescriptionA different approach is suggested for introducing special relativity. This approach exploits the distinction between the Doppler effect for sound and for light. With the Doppler effect for sound the difference in the frequency when the observer is in motion is not the same as when the emitter is in motion. This is exhibited already in the second order for sound, but for light there is no such shift in any order. An experiment suitable for an advanced undergraduate laboratory has been designed which can detect this second‐order difference in the acoustical shift. The system oberver (microphone) or emitter (loudspeaker) is mounted on an air track and measurement is made with a specially built frequency meter. The theoretical expression for the shift was confirmed in our laboratory for speeds from 3 to 7 m/s at an emitted frequency of 40 kHz.

The Coulomb problem and the Maxwell fish‐eye problem
View Description Hide DescriptionThe potential related to the Maxwell fish‐eye problem by optico‐mechanical analogy is derived from mechanical considerations. The dynamics of this potential is shown to be related to the dynamics of the Coulomb potential by an exchange transformation. The effective potential obtained from the transformation for the Coulomb problem in momentum space is used to discuss the anomalous behavior of the Coulomb trajectories in momentum space. Derivations of the index of refraction in the original Maxwell problem as well as its generalization are given.

Heat engines and the performance of external work
View Description Hide DescriptionA simple model is proposed whereby the work per cycle done on an external agent by a reversible heat engine can be analyzed explicitly. The external agent is a sand reservoir with a specified initial vertical weight distribution. During each successive cycle of the engine, this distribution becomes more nonuniform, with sand being shifted from lower to higher levels. The increase in the sand’s gravitational potential energy is equal to the corresponding work done by the engine. Several different cycles are discussed and the change per cycle in the sand’s weight distribution is calculated and graphed for each. The model and its major results are suitable for presentation to a wide audience of physics students.

Hyperbolic motion and radiation
View Description Hide DescriptionThe problem of the radiation emitted by a charged particle undergoing (eternal) hyperbolic motion is considered. The early approach of Pauli whereby such a system does not radiate and the formulations of more recent authors under which the system does radiate are presented within a unified framework. From this perspective it becomes clear why Pauli’s approach is not valid for this unusual problem. The discussion hinges upon the demonstration of the Lorentz invariant result that, for eternal hyperbolic motion, the radiant energy flux is not given by the flux of the total Poynting vector, before the charge reaches the turning point.

The Gibbs paradox and quantum gases
View Description Hide DescriptionThe sudden drop in the isothermal entropy of mixing when two ideal gases are made identical is analyzed for ideal quantum gases. For fermions of large reduced chemical potentials μ/k T this drop can be made arbitrarily small and this leads to a new resolution of the paradox.

Transformation properties of electric and magnetic fields
View Description Hide DescriptionThe transformation properties of electric and magnetic fields as a consequence of the Lorentz or conformal covariance of Maxwell’sequations is considered. The customary transformation rules for E and B are shown to actually not be necessary consequences of, but only sufficient conditions for the above covariance. Conditions are then derived under which the customary transformation rules for E and B do follow from the covariance of Maxwell’sequations. The conditions needed for the Lorentz and conformal transformations are different, but apparently are compatible.

Mutual inertia
View Description Hide DescriptionThe mutual inertia concept is introduced through the study of the small oscillations of a periodic linear chain of particles gliding on an air‐track and joined by loaded strings. The term ’’mutual inertia’’ is clarified by considering the electrical analog of the chain. In this L C ladder network the neighboring inductors are coupled through mutual inductance, which is the electrical counterpart of ’’mutual inertia.’’ A pure asymmetrical mutual inertia coupling is then dealt with in order to provide a more encompassing description of the concept. Finally, the limit case of a perfect transformer is considered.

A mechanical model for teaching quantum mechanics
View Description Hide DescriptionThe wave function ψ in quantum mechanics is a rather abstract quantity with which the physics student often has trouble. It is instructionally advantageous therefore, to have a mechanical model which can be described both by the quantum‐mechanical formalism and also by derivations from ’’first principles.’’ It is shown that a particle traveling in a helical path gives rise to such a model.

Dumbbell model for the classical radiation reaction
View Description Hide DescriptionIn his treatise on electron theory, Lorentz showed that the classical radiation reaction is attributable to the breakdown of Newton’s Third Law within the structure of an accelerating charge. But Lorentz’s calculation, based on a spherical model, is so difficult as to obscure the underlying physical principle, and requires that the charge be instantaneously at rest. Noting that the answer must be structure independent, in the point‐charge limit, we develop Lorentz’s theory in the context of a more tractable model: the dumbbell. We also compute the self‐torque on a rotating dumbell.

Relativistics statics and F. W. Sears
View Description Hide DescriptionIn a preliminary manuscript, written May 1975, F. W. Sears has argued that the paradox of the right angle lever arises from the use of the noncovariant definition of length, with which the Lorentz contraction is connected. He suggested that the problems connected with relativistic formulations of statics would be removed by a manifestly covariant description, using tensor quantities only. In this article a covariant formulation of the torque/angular mementum relationship is discussed, and it is shown that this covariant formulation requires an asynchronous description in an arbitrary inertial frame.

Double‐group theory on the half‐shell and the two‐level system. I. Rotation and half‐integral spin states
View Description Hide DescriptionA geometrical construction by Hamilton is used to simplify the quantum mechanics of half‐integral spin. A slide rule is described which can be used to (a) compute products of half‐integral or integral spin rotation operators, (b) convert between the Euler‐angle and ’’axis‐angle’’ rotation operator parameters, and (c) calculate the time evolution of a spin‐1/2 state for a constant Hamiltonian operator. A type of nomogram is developed which suggests ways to simplify the ’’double‐group’’ theory of half‐integral spin in molecular point symmetry, as well as the ’’ordinary’’ group theory for integral spin systems. Cubic and icosahedral symmetry group characters are derived for half‐integral spin operators.

Double‐group theory on the half‐shell and the two‐level system. II. Optical polarization
View Description Hide DescriptionRelations are derived between several different descriptions of optical polarization by analogy to the theory of spin 1/2. The rotational slide rule developed in the preceding article (I) is used to (a) compute the final polarization state and phase of an optical beam given the optical wave matrix, initial state, and phase, (b) make conversions between various types of polarization parameters, and (c) find the output intensity for perfect polarizers. Other polarization problems and methods are discussed briefly.

Gauge invariance rediscovered
View Description Hide DescriptionA pedagogical approach to gauge invariance is presented which is based on the analogy between gauge transformations and relativity. By using the concept of an internal space, purely geometrical arguments are used to teach the physical ideas behind gauge invariance. Many of the results are applicable to general gauge theories.

Laboratory study of the radioactivity from fission products in microscopic fallout particles
View Description Hide DescriptionAn experiment is described wherein simple γ‐ray detection and analysis techniques were utilized to study the radioactivity in local precipitation. Gamma spectra from filtered particulate, evaporation residue, and water from a sample of freshly fallen snow were counted, and the relative amount of several radionuclides present were identified. These data show that several isotopes produced in the recent Chinese nuclear detonation (Lop Nor testing ground, Sinkiang Province, 17 November 1976) are observable. Furthermore the data are sufficient to estimate the date of the detonation and to determine some of the characteristics of the explosion.

Square‐wave model for a pendulum with oscillating suspension
View Description Hide DescriptionIf a sinusoidal oscillation of the point of support of a pendulum is approximated by a square wave, a matrix method may be used to discuss parametric resonance and the stability of the inverted pendulum.

Thermodynamic work on a harmonic oscillator
View Description Hide DescriptionThe harmonic oscillator is examined as an example of a thermodynamic system with properties analogous to the ideal gas at high temperatures, but better behaved as T→0° K. The concept of P d V work is examined in the generalized context of the oscillator ’’pressure.’’
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 Notes and Discussions


Virial theorem and H^{+} _{2} bonding
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