scholarly journals On the pendulum

1833 ◽  
Vol 2 ◽  
pp. 401-401
Keyword(s):  
Knife Edge ◽  
Philosophical Magazine ◽  
Simple Pendulum ◽  
First Approximation

The ingenious and beautiful application, made by Capt. Kater, of Huygens’s theorem respecting the convertibility of the centres of suspension and oscillation, to the determination of the length of the simple pendulum, is to be considered as a first approximation to the solution of this problem. The accuracy of this determination, however, may be affected by many circumstances which the theory does not take into account; and the object of the author in this paper is to investigate the limits of the errors that may arise from neglecting them. Laplace and Whewell have shown that when the knife-edges are considered as cylinders of small but of equal radii of curvature, their distance is still equal to the length of the simple pendulum. The author treats the question with the utmost generality, and discusses all the circumstances which may affect the accuracy of Capt. Kater’s method, including all possible deviations and positions of the axes. He takes, as an example, the pendulum used by Mr. Baily, and described by him in the Philosophical Magazine of last February; and investigates the errors which would arise in the length of the simple pendulum corresponding to given deviations of the knife-edges. He also considers the case in which the agate planes are fixed on the pendulum, and vibrate on a fixed knife-edge; and finds that the length of the simple pendulum is here also equal to the distance between the planes.

scholarly journals XIV . On the pendulum

1830 ◽  
Vol 120 ◽  
pp. 201-208
Keyword(s):  
Small Deviation ◽  
Moment Of Inertia ◽  
Knife Edge ◽  
Philosophical Magazine ◽  
Small Curvature ◽  
Simple Pendulum ◽  
Numerical Example ◽  
First Approximation ◽  
The Moment ◽  
Ingenious Method

Captain Kater was the first who made use of Huygens’s theorem with respect to the convertibility of the centres of suspension and oscillation to eliminate the moment of inertia, and to obtain the length of the simple pendulum by measuring the distance between the knife edges or axes of suspension. But this very ingenious method of determining the length of the simple pendulum must be considered as a first approximation, which is true only when many circumstances which might affect the truth of the result are not taken into account, but of which the following investigation shows that when the experiments are conducted with care, the effect is insensible. It is, however, desirable to ascertain carefully the limits of the errors which may rise from the circumstances to which I have alluded, and to render the theory of Captain Kater’s pendulum as perfect as the method of observation. Laplace has given a complete theory of the apparatus used by Borda in the Connaissance des Temps; and he has shown that in the apparatus of Captain Kater, the distance between the knife edges is equal to the length of the simple pendulum , when they are considered as cylinders of small curvature, provided their radii of curvature are equal; which theorem is also proved in Professor Whewell’s Dynamics. But no one I believe has yet discussed all the circumstances which affect the accuracy of Captain Kater’s method; and I have therefore attempted to do this in the following paper, in which I have treated the question with the utmost generality, taking the case of all possible deviations and of axes any how placed, provided only that they are synchronous. I have taken the pendulum used by Mr. Baily, and described by him in the Philosophical Magazine of last February, to afford a numerical example, and I have given the errors which would arise in the length of the simple pendulum corresponding to given deviations of the knife edges: it is difficult to make the results intelligible without the use of symbols; but I may add, that the effect of a small deviation of one of the knife edges in azimuth is quite insensible: this is not the case with a deviation in altitude: a deviation of a degree in altitude increases by 3 the vibrations in twenty-four hours: a deviation from horizontally in the agate planes has a more sensible influence than either of the former deviations : a deviation in horizontally in the agate planes of 10' increases by about 6 the vibrations in twenty-four hours: both these deviations have the effect of rendering the distance between the knife edges greater than the true length of the simple pendulum . I have also considered the case in which the agate planes are fixed on the pendulum and vibrate on a fixed knife edge; and I find, as might be expected, that the length of the simple pendulum is equal to the distance between the planes.

Simultaneous Determination of Several Elements by Spectrochemical Addition Method … (II) (Tabulation of Analytical Value)

1968 ◽  
Vol 22 (6) ◽  
pp. 749-752 ◽  
Author(s):  
Isoo Masuda ◽  
Tamon Inouye
Keyword(s):  
Simultaneous Determination ◽  
Order Approximation ◽  
Analytical Data ◽  
Zero Order ◽  
Dilution Factor ◽  
Improved Method ◽  
Addition Method ◽  
Zero Order Approximation ◽  
First Approximation

An improved method for the tabulation of analytical data, obtained by addition and successive dilution procedures for spectrochemical analysis, is presented. The author's previous work shows that the solution of the first approximation diverges at some dilution factor smaller than unity when the slope of the working curve of added series is greater than that of unadded series. By obtaining the distance between this position and the origin, and taking it as a correction factor for zero-order approximation, tabulation of the analytical value, in the case of β>α, is carried out. One parameter of the calculation is deleted by normalizing the spectral intensity; therefore, the tabulation can be simplified.

Graphical-Numerical Solution of Problems of Saint-Venant Torsion and Bending

1953 ◽  
Vol 20 (3) ◽  
pp. 321-326
Author(s):  
B. A. Boley
Keyword(s):  
Cross Section ◽  
Stress Function ◽  
Arbitrary Cross Section ◽  
Successive Approximations ◽  
Shear Stresses ◽  
Shearing Stress ◽  
First Approximation ◽  
Difference Form ◽  
Bending Of Beams

Abstract A simple successive-approximations procedure for the solution of the problems of Saint-Venant torsion and bending of beams of arbitrary cross section is presented. The shear stresses in a cross section of the beam are first calculated from the formulas valid for thin-walled sections, on the basis of an assumed set of lines of shearing stress. From these a first approximation to the stress function of either the torsion or the bending problem is found. The second approximation to the stress function is then obtained from the governing equation of the problem, expressed in finite-difference form; this in turn allows the determination of an improved set of lines of shearing stress, and hence of the shearing stress itself. The procedure can be repeated until the results of two successive steps are sufficiently close. Applications are presented for a beam cross section for which the exact solutions are known, and it is shown that no further difficulties arise in applications to more complicated shapes.

scholarly journals Water Price: Environment Sustainability and Resource Cost

Water ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 3176
Author(s):  
Sonia Sanabria ◽  
Joaquín Torres
Keyword(s):  
Environmental Sustainability ◽  
The European Union ◽  
Total Water ◽  
First Approximation ◽  
Pricing System ◽  
Current Water ◽  
Resource Cost ◽  
Open Discussion ◽  
Environment Sustainability

The determination of a price for water is an open discussion among related players, directly or indirectly, in water management. In the context of the recovery of water service costs, as referred to in Article 9 of the Water Framework Directive 2000/60/EC (WFD), legislation applicable in all member countries of the European Union, the total water cost is broken down into three blocks; financial, environmental, and resource. It is the last component that generates the most uncertainty both in its conceptualization and in its valuation. The need to establish a pricing system for water (water tariff) implies that the different concepts that make it up are correctly delimited. The main goal of this paper is to propose a first approximation to a new theoretical framework to establish a relationship between environmental sustainability and the valuation of the resource cost—given that current water consumption can provoke future water availability difficulties, making it a scarce commodity that resource cost must be correctly delimited. Taking into account the prospective nature of environmental sustainability, the measure of its value should be based on the use of stochastic models that reflect the associated uncertainty.

Investigation of Urtite as an Analog of Waste Slag from a Plasma Shaff Furnace

1995 ◽  
Vol 412 ◽  
Author(s):  
S. A. Dmitriev ◽  
S. V. Stefanovsky
Keyword(s):  
Radioactive Waste ◽  
Electron Probe ◽  
Nepheline Syenite ◽  
Shaft Furnace ◽  
Alkali Feldspar ◽  
Chemical Compositions ◽  
Natural Analog ◽  
First Approximation ◽  
Geochemical Investigation

abstractMineralogical-geochemical investigation of a sample of nepheline syenite (urtite) as a natural analog of final radioactive waste form has been performed. The specimen of urtite consists of nepheline, alkali feldspar, pyroxene, sphene, apatite and minor magnetite and amphibole. As a first approximation, urtite simulates the mineral composition of waste slag produced in a plasma shaft furnace at SIA “Radon”. Determination of chemical compositions of the minerals by electron-probe microanalysis has shown that the main phases that hosted radionuclides and their geochemical analogs are as follows: nepheline (Rb and probably Cs), feldspar (Ba), sphene (Zr, Nb, REE, and actinides) and apatite (Sr, REE, and actinides).

scholarly journals XL.—On the Mechanical Action of Heat

1853 ◽  
Vol 20 (4) ◽  
pp. 565-589 ◽  
Author(s):  
William John Macquorn Rankine
Keyword(s):  
Thermal Effects ◽  
Large Scale ◽  
Mechanical Action ◽  
Mechanical Theory ◽  
Philosophical Magazine ◽  
William Thomson ◽  
Series Of Experiments ◽  
Elastic Fluids ◽  
Intention To Continue

Section VI.—A Review of the Fundamental Principles of the Mechanical Theory of Heat; with Remarks on the Thermic Phenomena of Currents of Elastic Fluids, as illustrating those Principles.(Article 46.) I have been induced to write this Section, in continuation of a paper on the Mechanical Action of Heat, by the publication (in the Philosophical Magazine for December 1852, Supplementary Number) of a series of experiments by Mr Joule and Professor William Thomson, on the Thermal Effects experienced by Air in rushing through small Apertures. Although those authors express an intention to continue the experiments on a large scale, so as to obtain more precise results; yet the results already obtained are sufficient to constitute the first step towards the experimental determination of that most important function in the theory of the mechanical action of heat, which has received the name of Carnot's Function.

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