Volume 47, Issue 2, February 1979
 Papers


Doubling your sunsets or how anyone can measure the earth’s size with wristwatch and meterstick
View Description Hide DescriptionA method is described whereby, using primitive equipment, anyone can measure the size of the Earth, to an accuracy of order of magnitude 10% — by observing two sunsets in the space of a few seconds. The calculated result’s closeness to the truth is comparable to the best extant ancient estimates.

Electrostatic potential energy leading to a gravitational mass change for a system of two point charges
View Description Hide DescriptionA system consisting of two point charges has an energy contribution from the system electrostatic potential energy and accordingly a contribution to the system mass given by this energy divided by c ^{2}. Here we investigate the change in weight associated with the electrostatic potential energy for a system of two point charges supported side by side against a weak gravitational field. The gravitational distortion of the Coulomb field of a point charge is calculated using the equivalence principle, and it is then shown that the change in the two‐particle system weight can be understood precisely as the change in supporting force necessary to balance the electric force of each charge upon the other. The example provides a clear illustration and detailed mechanism for understanding the mass‐energy connection for weak gravitational fields.

Kinetic theory simulator for laboratory use
View Description Hide DescriptionAn apparatus is described which allows one to study the speed distribution, the gravitational distribution, and the mean free path of steel balls agitated into two‐dimensional motion through collisions with the moving walls of their enclosure.

A physics teacher looks at the new (1977) Medical College Admission Test
View Description Hide DescriptionIt appears, from the recently published ’’illustrative test’’ and the sample questions which were published previously and from the list of topics, that the new Medical College Admission Test contains a large number of physics related questions in the chemistry section in addition to those in the physics section. Also, many of the questions in the reading and quantitative analysis skill section are of concern to physics teachers who hope their students learn to read carefully and to analyze data. Although most questions are straightforward at a reasonable first year level, there are a disturbing number of questions that seem ambiguous. Physics teachers should be concerned about such shortcomings of a test that plays a major role in the career of many of their students.

Projectile motion revisited
View Description Hide DescriptionThe power of symmetry and simple geometrical arguments is demonstrated in this somewhat unusual treatment of projectile motion. Keeping calculus and trigonometry manipulations to a bare minimum, the motion is analyzed in its greatest generality. Direction of velocity at any point, range, time of flight, maximum height, safety parabola, and maximum range are deduced for a projectile launched upon a plane inclined at any angle with respect to the horizontal.

Dilemma of the primary colors
View Description Hide DescriptionIt is an enigma for students why artists use red, yellow, and blue as primary colors, whereas physicists use red, green, and blue. To answer this problem, the spectra of a series of mixtures of red, green, blue, and yellow tempera (poster paints) were obtained using a Carey spectrophotometer with an integrating sphere. Points on the chromaticity diagram were obtained from the spectra, showing that whereas the mixture of yellow and blue yielded a fairly saturated green, the path of mixtures of green and red traveled on the blue side of the white point. Hence yellow is preferable as a primary color than green for tempera.

Fraunhofer diffraction patterns from apertures illuminated with nonparallel light
View Description Hide DescriptionThe size of the Fraunhofer diffraction pattern produced when a circular aperture is illuminated with a diverging beam of light, in contrast with a parallel beam, is derived on the basis of simple lens theory by introducing the concept of a virtual diffraction pattern. The results of the simple analysis agree with the diffraction pattern size determined by a formal analysis of the light wave amplitude. The formal analysis reveals a general scale factor relating the Fraunhofer diffraction pattern size for an aperture illuminated with parallel light to the size for nonparallel illumination. The diffraction pattern intensity is independent of the position of the aperture in its plane, based on a property of the Fourier transform. The invariance in intensity for nonparallel illumination of the aperture is also shown by a simple argument based on the Abbe theory of image formation by diffraction. Finally, some experimental results are given.

Quantization of inequivalent classical Hamiltonians
View Description Hide DescriptionIt is well known that to quantize any dynamical system it is n e c e s s a r y for a generator of the classical motion to exist in the form of a Hamiltonian function. It is shown, by using the example of a damped harmonic oscillator, that a particular class of inequivalent classical Hamiltonians exist which make quantization of the system ambiguous. Hence it is conclued that it is n o t s u f f i c i e n t for the Hamiltonian to merely generate the motion, but it must also be necessarily related via a canonical transformation to the total energy of the system.

Catastrophe of the molecular field
View Description Hide DescriptionThe molecular field theory of ferromagnetism is used as an example to illustrate the use of elementary catastrophe theory. It is shown that the molecular‐field approach gives rise to a cusp catastrophe set. A brief examination is made of the reasons why the elementary catastrophe theory does not produce critical exponents which are in agreement with experimental values. It is suggested that the assumption of an equal areas rule denies the transversality upon which the generality of catastrophe theory is based. It is proposed that an approach to the problem of critical exponents through the much more sophisticated generalized catastrophes might prove fruitful.

Linear collisions with harmonic oscillator forces: The inverse scattering problem
View Description Hide DescriptionA particular kind of elasticscattering is studied between three bodies of equal mass. Two of the bodies are at rest and in contact with each other. The third body impinges on these two. The motion of the bodies is confined to a straight line as in a linear air track. The scattering is mediated by nearest‐neighbor Hooke’s law forces having force constants k _{1} and k _{2}. The final velocities of all three bodies are studied as a function of β (=k _{2}/k _{1}). It is shown that the possible final scattering states may be divided into two classes. In one class it is possible to determine β from a knowledge of the final velocities of the three bodies. In the other class no such determination is possible. A formula is derived that yields an infinite set of values of β, all of which produce the same velocities in the final state. Other measurements are proposed which allow a determination of β and the individual spring constants k _{1} and k _{2} as well as the size of the springs. These further measurements involve time correlations between incident and scattered particles. No measurements are permitted during the time of interaction. The correspondence principle indicates that the same ambiguity will be present in the analogous one‐dimensional quantum‐ mechanical problem necessitating time correlation measurements to solve the inverse scattering problem.

Permutation operator: An explicit representation
View Description Hide DescriptionAn explicit representation of the permutation operator in terms of coordinate and momentum operators is derived.

Exact solution of a Faraday’s law problem that includes a nonlinear term and its implication for perturbation theory
View Description Hide DescriptionAn exact solution of a Faraday’s law problem with a nonlinear term is presented. The results are compared with those obtained by the perturbation methods of Poisson and Poincaré and of Kryloff and Bogoliuboff. The reasons for the superiority of the latter method are enumerated.

Probabilistic description of radioactivity based on the good‐as‐new postulate
View Description Hide DescriptionThe good‐as‐new postulate applied to radioactivity says: The conditional probability that a radioactive atom will ’’live’’ past time s+t, given that it has lived past time s, is equal to the unconditional probability that it would have lived past time t, starting from time zero. This postulate leads directly to the conclusion that the decay time of a radioactive atom is an exponentially distributed random variable. Add the postulate that the decay times of individual atoms in an aggregrate of identical radioactive atoms are independent, and a complete description of the random behavior of an aggregrate can be constructed. This approach stresses from the outset the randomness inherent in the radioactive decay process and is offered as a complement to the deterministic approach that treats fluctuations about mean values. All results are exact within the context of the model and are applicable to aggregrates of any size—no assumptions about large numbers are required.

Electronic device of didactic and electrometric interest for the study of R L C circuits
View Description Hide DescriptionThis paper presents a method of studying R L C circuits with the help of the oscilloscope in the X Y Z mode, complemented by an electronic device which generates a marker‐trace on the screen and which is used to measure frequencies without needing any reference point on the screen. Although the electronic device can be used in a fairly quantitative manner, the authors have proved its special utility in the teaching of electric resonance, since it shows the phenomena associated with electric resonance in a very intuitive way.

Wigner distribution for angle coordinates in quantum mechanics
View Description Hide DescriptionThe method of Wigner distribution functions, and the Weyl correspondence between quantum and classical variables, are extended from the usual kind of canonically conjugate position and momentum operators to the case of an angle and angular momentum operator pair. The sense in which one has a description of quantum mechanics using classical phase‐space language is much clarified by this extension.

Quantum state determination: Quorum for a particle in one dimension
View Description Hide DescriptionIn this paper we wish to illuminate the axiom that the quantal state describes a statistical ensemble of similar systems identically prepared, and is not to be identified with any single system. The mathematical representative of the general quantum state is the density matrix or statistical operator in Hilbert space. We demonstrate how this operator may be determined empirically by calculations involving only the measured mean values of a set of observables we call a ’’quorum.’’ As an example of this approach a state determination procedure is described for a spinless particle moving in one dimension; the corresponding quorum turns out to involve only position data associated with various instants subsequent to the act of preparation.
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 Notes and Discussion



Note on the Harris cranking model calculation of the moment of inertia using the method of Dalgarno and Lewis
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Comment on: Hyperbolic mirrors, holography, and Fermat’s principle
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Recursive evaluation of Fourier series
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