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Gravitational waves from orbiting binaries without general relativity
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 86 (No. 3) |
| Halaman | : | 186-197 |
| Abstrak | : | Using analogies with electromagnetic radiation, we present a calculation of the properties of gravitational radiation emitted by orbiting binary objects. The calculation produces results that have the same dependence on the masses of the orbiting objects, the orbital frequency, and the mass separation as do the results from the linear version of general relativity (GR). However, the calculation yields polarization, angular distributions, and overall power results that differ from those of GR. Nevertheless, the calculation produces waveforms that are very similar to those observed by the Laser Interferometer Gravitational-Wave Observatory (LIGO-VIRGO) gravitational wave collaboration in 2015 up to the point at which the binary merger occurs. The details of the calculation should be understandable by upper-level physics students and physicists who are not experts in GR. |
Double well potentials with a quantum moat barrier or a quantum wall barrier give rise to similar entangled wave functions
| Pengarang | : | F. Marsiglio |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 86 (No. 3) |
| Halaman | : | 180-185 |
| Abstrak | : | The solution to a problem in quantum mechanics is generally a linear superposition of states. The solutions for double well potentials epitomize this property, and go even further than this: they can often be described by an effective model whose low energy features can be described by two states—one in which the particle is on one side of the barrier, and a second where the particle is on the other side. Then the ground state remains a linear superposition of these two macroscopic-like states. In this paper, we illustrate that this property is achieved similarly with an attractive potential that separates two regions of space, as opposed to the traditionally repulsive one. In explaining how this comes about we revisit the concept of “orthogonalized plane waves,” first discussed in 1940 to understand electronic band structure in solids, along with the accompanying concept of a pseudopotential. We show how these ideas manifest themselves in a simple double well potential, whose “barrier” consists of a moat instead of the conventional wall. |
A study of kinetic friction: The Timoshenko oscillator
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 86 (No. 3) |
| Halaman | : | 174-179 |
| Abstrak | : | Friction is a complex phenomenon that is of paramount importance in everyday life. We present an easy-to-build and inexpensive experiment illustrating Coulomb's law of kinetic friction. The so-called friction, or Timoshenko, oscillator consists of a plate set into periodic motion through the competition between gravity and friction on its rotating supports. The period of such an oscillator gives a measurement of the coefficient of kinetic friction μk between the plate and the supports. Our prototype is mainly composed of a motor, LEGO blocks, and a low-cost microcontroller, but despite its simplicity, the results obtained are in good agreement with values of μk found in the literature (enhanced online). |
A compact disc under skimming light rays
| Pengarang | : | A. Naddeo |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 86 (No. 3) |
| Halaman | : | 169-173 |
| Abstrak | : | The optical properties of a compact disc (CD) under “skimming” light rays have been analyzed. We have noticed that a clear green line can be detected when the disc is irradiated with light rays coming from a lamp in such a way that only those skimming the CD, held horizontally, are selected. We provide a physical interpretation of this phenomenon on the basis of elementary optics concepts. Extension of these concepts to digital versatile discs (DVDs) is given. |
Analytical Problem Solving Based on Causal, Correlational and Deductive Models
| Pengarang | : | Dacheng Tao,Hussain, Amir,Jonathan Wu,Dongbin Zhao |
| Nama Majalah/Jurnal | : | The American Statistician |
| Volume / Edisi | : | 77 (No. 1) |
| Halaman | : | 51-61 |
| Abstrak | : | Many approaches for solving problems in business and industry are based on analytics and statistical modeling. Analytical problem solving is driven by the modeling of relationships between dependent (Y) and independent (X) variables, and we discuss three frameworks for modeling such relationships: cause-and-effect modeling, popular in applied statistics and beyond, correlational predictive modeling, popular in machine learning, and deductive (first-principles) modeling, popular in business analytics and operations research. We aim to explain the differences between these types of models, and flesh out the implications of these differences for study design, for discovering potential X/Y relationships, and for the types of solution patterns that each type of modeling could support. We use our account to clarify the popular descriptive-diagnostic-predictive-prescriptive analytics framework, but extend it to offer a more complete model of the process of analytical problem solving, reflecting the essential differences between causal, correlational, and deductive models. |
Deriving mass-energy equivalence and mass-velocity relation without light
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 86 (No. 4) |
| Halaman | : | 284-289 |
| Abstrak | : | Relativity requires that a particle's momentum and energy are the same functions of the particle's velocity in all inertial frames. Using the fact that momentum and energy must transform linearly between reference frames, we present a novel derivation of the mass-energy equivalence, namely, the relation that the energy is proportional to the moving mass, with no postulate about the existence of light or its properties. We further prove the mass-velocity relation without relying on momentum and energy conservation or on the Lorentz transformation. It is demonstrated that neither conservation laws nor the Lorentz transformation are necessary to establish those relations, and that those relations have a wider scope of validity than that of the conservation laws and the invariance of the speed of light. |
The contrasting roles of Planck's constant in classical and quantum theories
| Pengarang | : | Timothy H. Boyer |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 86 (No. 4) |
| Halaman | : | 280-283 |
| Abstrak | : | We trace the historical appearance of Planck's constant in physics, and we note that initially the constant did not appear in connection with quanta. Furthermore, we emphasize that Planck's constant can appear in both classical and quantum theories. In both theories, Planck's constant sets the scale of atomic phenomena. However, the roles played in the foundations of the theories are sharply different. In quantum theory, Planck's constant is crucial to the structure of the theory. On the other hand, in classical electrodynamics, Planck's constant is optional, since it appears only as the scale factor for the (homogeneous) source-free contribution to the general solution of Maxwell's equations. Since classical electrodynamics can be solved while taking the homogenous source-free contribution in the solution as zero or non-zero, there are naturally two different theories of classical electrodynamics, one in which Planck's constant is taken as zero and one where it is taken as non-zero. The textbooks of classical electromagnetism present only the version in which Planck's constant is taken to vanish. |
Force and torque of a string on a pulley
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 86 (No. 4) |
| Halaman | : | 275-279 |
| Abstrak | : | Every university introductory physics course considers the problem of Atwood's machine taking into account the mass of the pulley. In the usual treatment, the tensions at the two ends of the string are offhandedly taken to act on the pulley and be responsible for its rotation. However, such a free-body diagram of the forces on the pulley is not a priori justified, inducing students to construct wrong hypotheses such as that the string transfers its tension to the pulley or that some symmetry is in operation. We reexamine this problem by integrating the contact forces between each element of the string and the pulley and show that although the pulley does behave as if the tensions were acting on its ends, this comes only as the final result of a detailed analysis. We also address the question of how much friction is needed to prevent the string from slipping over the pulley. Finally, we deal with the case in which the string is on the verge of sliding and show that this cannot happen unless certain conditions are met by the coefficient of static friction and the masses involved. |
Using the Lambert Function to Estimate Shared Frailty Models with a Normally Distributed Random Intercept
| Pengarang | : | Hadrien Charvat |
| Nama Majalah/Jurnal | : | The American Statistician |
| Volume / Edisi | : | 77 (No. 1) |
| Halaman | : | 41-50 |
| Abstrak | : | Shared frailty models, that is, hazard regression models for censored data including random effects acting multiplicatively on the hazard, are commonly used to analyze time-to-event data possessing a hierarchical structure. When the random effects are assumed to be normally distributed, the cluster-specific marginal likelihood has no closed-form expression. A powerful method for approximating such integrals is the adaptive Gauss-Hermite quadrature (AGHQ). However, this method requires the estimation of the mode of the integrand in the expression defining the cluster-specific marginal likelihood: it is generally obtained through a nested optimization at the cluster level for each evaluation of the likelihood function. In this work, we show that in the case of a parametric shared frailty model including a normal random intercept, the cluster-specific modes can be determined analytically by using the principal branch of the Lambert function, ????0. Besides removing the need for the nested optimization procedure, it provides closed-form formulas for the gradient and Hessian of the approximated likelihood making its maximization by Newton-type algorithms convenient and efficient. The Lambert-based AGHQ (LAGHQ) might be applied to other problems involving similar integrals, such as the normally distributed random intercept Poisson model and the computation of probabilities from a Poisson lognormal distribution. |
On the optical path length in refracting media
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 86 (No. 4) |
| Halaman | : | 268-274 |
| Abstrak | : | The path light follows as it travels through a substance depends on the substance's index of refraction. This path is commonly known as the optical path length (OPL). In geometrical optics, the laws of reflection and refraction are simple examples for understanding the path of light travel from source to detector for constant values of the traveled substances' refraction indices. In more complicated situations, the Euler equation can be quite useful and quite important in optics courses. Here, the well-known Euler differential equation (EDE) is used to obtain the OPL for several index of refraction models. For pedagogical completeness, the OPL is also obtained through a modified Monte Carlo (MC) method, versus which the various results obtained through the EDE are compared. The examples developed should be important in projects involving undergraduate as well as graduate students in an introductory optics course. A simple matlab script (program) is included that can be modified by students who wish to pursue the subject further. |