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Dimensional Analysis and the Theory of Natural Units
LIBRARY TECHNICAL REPORT SECTION SCHOOL NAVAL POSTGRADUATE MONTEREY, CALIFORNIA 93940 NPS-57Gn71101A NAVAL POSTGRADUATE SCHOOL Monterey, California DIMENSIONAL ANALYSIS AND THE THEORY OF NATURAL UNITS "by T. H. Gawain, D.Sc. October 1971 lllp FEDDOCS public This document has been approved for D 208.14/2:NPS-57GN71101A unlimited release and sale; i^ distribution is NAVAL POSTGRADUATE SCHOOL Monterey, California Rear Admiral A. S. Goodfellow, Jr., USN M. U. Clauser Superintendent Provost ABSTRACT: This monograph has been prepared as a text on dimensional analysis for students of Aeronautics at this School. It develops the subject from a viewpoint which is inadequately treated in most standard texts hut which the author's experience has shown to be valuable to students and professionals alike. The analysis treats two types of consistent units, namely, fixed units and natural units. Fixed units include those encountered in the various familiar English and metric systems. Natural units are not fixed in magnitude once and for all but depend on certain physical reference parameters which change with the problem under consideration. Detailed rules are given for the orderly choice of such dimensional reference parameters and for their use in various applications. It is shown that when transformed into natural units, all physical quantities are reduced to dimensionless form. The dimension- less parameters of the well known Pi Theorem are shown to be in this category. An important corollary is proved, namely that any valid physical equation remains valid if all dimensional quantities in the equation be replaced by their dimensionless counterparts in any consistent system of natural units. -
Chapter 5 Dimensional Analysis and Similarity
Chapter 5 Dimensional Analysis and Similarity Motivation. In this chapter we discuss the planning, presentation, and interpretation of experimental data. We shall try to convince you that such data are best presented in dimensionless form. Experiments which might result in tables of output, or even mul- tiple volumes of tables, might be reduced to a single set of curves—or even a single curve—when suitably nondimensionalized. The technique for doing this is dimensional analysis. Chapter 3 presented gross control-volume balances of mass, momentum, and en- ergy which led to estimates of global parameters: mass flow, force, torque, total heat transfer. Chapter 4 presented infinitesimal balances which led to the basic partial dif- ferential equations of fluid flow and some particular solutions. These two chapters cov- ered analytical techniques, which are limited to fairly simple geometries and well- defined boundary conditions. Probably one-third of fluid-flow problems can be attacked in this analytical or theoretical manner. The other two-thirds of all fluid problems are too complex, both geometrically and physically, to be solved analytically. They must be tested by experiment. Their behav- ior is reported as experimental data. Such data are much more useful if they are ex- pressed in compact, economic form. Graphs are especially useful, since tabulated data cannot be absorbed, nor can the trends and rates of change be observed, by most en- gineering eyes. These are the motivations for dimensional analysis. The technique is traditional in fluid mechanics and is useful in all engineering and physical sciences, with notable uses also seen in the biological and social sciences. -
Package 'Constants'
Package ‘constants’ February 25, 2021 Type Package Title Reference on Constants, Units and Uncertainty Version 1.0.1 Description CODATA internationally recommended values of the fundamental physical constants, provided as symbols for direct use within the R language. Optionally, the values with uncertainties and/or units are also provided if the 'errors', 'units' and/or 'quantities' packages are installed. The Committee on Data for Science and Technology (CODATA) is an interdisciplinary committee of the International Council for Science which periodically provides the internationally accepted set of values of the fundamental physical constants. This package contains the ``2018 CODATA'' version, published on May 2019: Eite Tiesinga, Peter J. Mohr, David B. Newell, and Barry N. Taylor (2020) <https://physics.nist.gov/cuu/Constants/>. License MIT + file LICENSE Encoding UTF-8 LazyData true URL https://github.com/r-quantities/constants BugReports https://github.com/r-quantities/constants/issues Depends R (>= 3.5.0) Suggests errors (>= 0.3.6), units, quantities, testthat ByteCompile yes RoxygenNote 7.1.1 NeedsCompilation no Author Iñaki Ucar [aut, cph, cre] (<https://orcid.org/0000-0001-6403-5550>) Maintainer Iñaki Ucar <[email protected]> Repository CRAN Date/Publication 2021-02-25 13:20:05 UTC 1 2 codata R topics documented: constants-package . .2 codata . .2 lookup . .3 syms.............................................4 Index 6 constants-package constants: Reference on Constants, Units and Uncertainty Description This package provides the 2018 version of the CODATA internationally recommended values of the fundamental physical constants for their use within the R language. Author(s) Iñaki Ucar References Eite Tiesinga, Peter J. Mohr, David B. -
Three-Dimensional Analysis of the Hot-Spot Fuel Temperature in Pebble Bed and Prismatic Modular Reactors
Transactions of the Korean Nuclear Society Spring Meeting Chuncheon, Korea, May 25-26 2006 Three-Dimensional Analysis of the Hot-Spot Fuel Temperature in Pebble Bed and Prismatic Modular Reactors W. K. In,a S. W. Lee,a H. S. Lim,a W. J. Lee a a Korea Atomic Energy Research Institute, P. O. Box 105, Yuseong, Daejeon, Korea, 305-600, [email protected] 1. Introduction thousands of fuel particles. The pebble diameter is 60mm High temperature gas-cooled reactors(HTGR) have with a 5mm unfueled layer. Unstaggered and 2-D been reviewed as potential sources for future energy needs, staggered arrays of 3x3 pebbles as shown in Fig. 1 are particularly for a hydrogen production. Among the simulated to predict the temperature distribution in a hot- HTGRs, the pebble bed reactor(PBR) and a prismatic spot pebble core. Total number of nodes is 1,220,000 and modular reactor(PMR) are considered as the nuclear heat 1,060,000 for the unstaggered and staggered models, source in Korea’s nuclear hydrogen development and respectively. demonstration project. PBR uses coated fuel particles The GT-MHR(PMR) reactor core is loaded with an embedded in spherical graphite fuel pebbles. The fuel annular stack of hexahedral prismatic fuel assemblies, pebbles flow down through the core during an operation. which form 102 columns consisting of 10 fuel blocks PMR uses graphite fuel blocks which contain cylindrical stacked axially in each column. Each fuel block is a fuel compacts consisting of the fuel particles. The fuel triangular array of a fuel compact channel, a coolant blocks also contain coolant passages and locations for channel and a channel for a control rod. -
Improving the Accuracy of the Numerical Values of the Estimates Some Fundamental Physical Constants
Improving the accuracy of the numerical values of the estimates some fundamental physical constants. Valery Timkov, Serg Timkov, Vladimir Zhukov, Konstantin Afanasiev To cite this version: Valery Timkov, Serg Timkov, Vladimir Zhukov, Konstantin Afanasiev. Improving the accuracy of the numerical values of the estimates some fundamental physical constants.. Digital Technologies, Odessa National Academy of Telecommunications, 2019, 25, pp.23 - 39. hal-02117148 HAL Id: hal-02117148 https://hal.archives-ouvertes.fr/hal-02117148 Submitted on 2 May 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Improving the accuracy of the numerical values of the estimates some fundamental physical constants. Valery F. Timkov1*, Serg V. Timkov2, Vladimir A. Zhukov2, Konstantin E. Afanasiev2 1Institute of Telecommunications and Global Geoinformation Space of the National Academy of Sciences of Ukraine, Senior Researcher, Ukraine. 2Research and Production Enterprise «TZHK», Researcher, Ukraine. *Email: [email protected] The list of designations in the text: l -
Certain Investigations Regarding Variable Physical Constants
IJRRAS 6 (1) ● January 2011 www.arpapress.com/Volumes/Vol6Issue1/IJRRAS_6_1_03.pdf CERTAIN INVESTIGATIONS REGARDING VARIABLE PHYSICAL CONSTANTS 1 2 3 Amritbir Singh , R. K. Mishra & Sukhjit Singh 1 Department of Mathematics, BBSB Engineering College, Fatehgarh Sahib, Punjab, India 2 & 3 Department of Mathematics, SLIET Deemed University, Longowal, Sangrur, Punjab, India. ABSTRACT In this paper, we have investigated details regarding variable physical & cosmological constants mainly G, c, h, α, NA, e & Λ. It is easy to see that the constancy of G & c is consistent, provided that all actual variations from experiment to experiment, or method to method, are due to some truncation or inherent error. It has also been noticed that physical constants may fluctuate, within limits, around average values which themselves remain fairly constant. Several attempts have been made to look for changes in Plank‟s constant by studying the light from quasars and stars assumed to be very distant on the basis of the red shift in their spectra. Detail Comparison of concerned research done has been done in the present paper. Keywords: Fundamental physical constants, Cosmological constant. A.M.S. Subject Classification Number: 83F05 1. INTRODUCTION We all are very much familiar with the importance of the 'physical constants‟, now in these days they are being used in all the scientific calculations. As the name implies, the so-called physical constants are supposed to be changeless. It is assumed that these constants are reflecting an underlying constancy of nature. Recent observations suggest that these fundamental Physical Constants of nature may actually be varying. There is a debate among cosmologists & physicists, whether the observations are correct but, if confirmed, many of our ideas may have to be rethought. -
Units of Measurement and Dimensional Analysis
Measurements and Dimensional Analysis POGIL ACTIVITY.2 Name ________________________________________ POGIL ACTIVITY 2 Units of Measurement and Dimensional Analysis A. Units of Measurement- The SI System and Metric System here are myriad units for measurement. For example, length is reported in miles T or kilometers; mass is measured in pounds or kilograms and volume can be given in gallons or liters. To avoid confusion, scientists have adopted an international system of units commonly known as the SI System. Standard units are called base units. Table A1. SI System (Systéme Internationale d’Unités) Measurement Base Unit Symbol mass gram g length meter m volume liter L temperature Kelvin K time second s energy joule j pressure atmosphere atm 7 Measurements and Dimensional Analysis POGIL. ACTIVITY.2 Name ________________________________________ The metric system combines the powers of ten and the base units from the SI System. Powers of ten are used to derive larger and smaller units, multiples of the base unit. Multiples of the base units are defined by a prefix. When metric units are attached to a number, the letter symbol is used to abbreviate the prefix and the unit. For example, 2.2 kilograms would be reported as 2.2 kg. Plural units, i.e., (kgs) are incorrect. Table A2. Common Metric Units Power Decimal Prefix Name of metric unit (and symbol) Of Ten equivalent (symbol) length volume mass 103 1000 kilo (k) kilometer (km) B kilogram (kg) Base 100 1 meter (m) Liter (L) gram (g) Unit 10-1 0.1 deci (d) A deciliter (dL) D 10-2 0.01 centi (c) centimeter (cm) C E 10-3 0.001 milli (m) millimeter (mm) milliliter (mL) milligram (mg) 10-6 0.000 001 micro () micrometer (m) microliter (L) microgram (g) Critical Thinking Questions CTQ 1 Consult Table A2. -
Variable Planck's Constant
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 29 January 2021 doi:10.20944/preprints202101.0612.v1 Variable Planck’s Constant: Treated As A Dynamical Field And Path Integral Rand Dannenberg Ventura College, Physics and Astronomy Department, Ventura CA [email protected] Abstract. The constant ħ is elevated to a dynamical field, coupling to other fields, and itself, through the Lagrangian density derivative terms. The spatial and temporal dependence of ħ falls directly out of the field equations themselves. Three solutions are found: a free field with a tadpole term; a standing-wave non-propagating mode; a non-oscillating non-propagating mode. The first two could be quantized. The third corresponds to a zero-momentum classical field that naturally decays spatially to a constant with no ad-hoc terms added to the Lagrangian. An attempt is made to calibrate the constants in the third solution based on experimental data. The three fields are referred to as actons. It is tentatively concluded that the acton origin coincides with a massive body, or point of infinite density, though is not mass dependent. An expression for the positional dependence of Planck’s constant is derived from a field theory in this work that matches in functional form that of one derived from considerations of Local Position Invariance violation in GR in another paper by this author. Astrophysical and Cosmological interpretations are provided. A derivation is shown for how the integrand in the path integral exponent becomes Lc/ħ(r), where Lc is the classical action. The path that makes stationary the integral in the exponent is termed the “dominant” path, and deviates from the classical path systematically due to the position dependence of ħ. -
Testable Criterion Establishes the Three Inherent Universal Constants
Applied Physics Research; Vol. 8, No. 6; 2016 ISSN 1916-9639 E-ISSN 1916-9647 Published by Canadian Center of Science and Education Testable Criterion Establishes the Three Inherent Universal Constants Gordon R. Kepner1 1 Membrane Studies Project, USA Correspondence: Gordon R. Kepner, Membrane Studies Project, USA. E-mail: [email protected] Received: September 11, 2016 Accepted: November 18, 2016 Online Published: November 24, 2016 doi:10.5539/apr.v8n6p38 URL: http://dx.doi.org/10.5539/apr.v8n6p38 Abstract Since the late 19th century, differing views have appeared on the nature of fundamental constants, how to identify them, and their total number. This concept has been broadly and variously interpreted by physicists ever since with no clear consensus. There is no analysis in the literature, based on specific criteria, for establishing what is or is not an inherent universal constant. The primary focus here is on constants describing a relation between at least two of the base quantities mass, length and time. This paper presents a testable criterion to identify these unique constants of our local universe. The value of a constant meeting these criteria can only be obtained from experimental measurement. Such constants set the scales of these base quantities that describe, in combination, the observable phenomena of our universe. The analysis predicts three such constants, uses the criterion to identify them, while ruling out Planck’s constant and introducing a new physical constant. Keywords: fundamental constant, inherent universal constant, new inherent constant, Planck’s constant 1. Introduction The analysis presented here is conceptually and mathematically simple, while going directly to the foundations of physics. -
Planck Dimensional Analysis of Big G
Planck Dimensional Analysis of Big G Espen Gaarder Haug⇤ Norwegian University of Life Sciences July 3, 2016 Abstract This is a short note to show how big G can be derived from dimensional analysis by assuming that the Planck length [1] is much more fundamental than Newton’s gravitational constant. Key words: Big G, Newton, Planck units, Dimensional analysis, Fundamental constants, Atomism. 1 Dimensional Analysis Haug [2, 3, 4] has suggested that Newton’s gravitational constant (Big G) [5] can be written as l2c3 G = p ¯h Writing the gravitational constant in this way helps us to simplify and quantify a long series of equations in gravitational theory without changing the value of G, see [3]. This also enables us simplify the Planck units. We can find this G by solving for the Planck mass or the Planck length with respect to G, as has already been done by Haug. We will claim that G is not anything physical or tangible and that the Planck length may be much more fundamental than the Newton’s gravitational constant. If we assume the Planck length is a more fundamental constant than G,thenwecanalsofindG through “traditional” dimensional analysis. Here we will assume that the speed of light c, the Planck length lp, and the reduced Planck constanth ¯ are the three fundamental constants. The dimensions of G and the three fundamental constants are L3 [G]= MT2 L2 [¯h]=M T L [c]= T [lp]=L Based on this, we have ↵ β γ G = lp c ¯h L3 L β L2 γ = L↵ M (1) MT2 T T ✓ ◆ ✓ ◆ Based on this, we obtain the following three equations Lenght : 3 = ↵ + β +2γ (2) Mass : 1=γ (3) − Time : 2= β γ (4) − − − ⇤e-mail [email protected]. -
Department of Physics Notes on Dimensional Analysis and Units
1 CSUC Department of Physics 301A Mechanics: Notes on Dimensional Analysis and Units I. INTRODUCTION Physics has a particular interest in those attributes of nature which allow comparison processes. Such qualities include e.g. length, time, mass, force, energy...., etc. By \comparison process" we mean that although we recognize that, say, \length" is an abstract quality, nonetheless, any two \lengths" may successfully be compared in a generally agreed upon manner yielding a real number which we sometimes, by custom, call their \ratio". It's crucial to realize that this outcome it isn't really a mathematical ratio since the participants aren't numbers at all, but rather completely abstract entities . viz. \Physical Extent in Space". Thus we undertake to represent physical qualities (i.e. certain aspects) of our system with real numbers. In general, we recognize that only `lengths' can be compared to `other lengths' etc - i.e. that the world divides into exclusive classes (e.g. `all lengths') of entities which can be compared to each other. We say that all the entities in any one class are `mutually comparable' with each other. These physical numbers will be our basic tools for expressing the relationships we observe in nature. As we proceed in our understanding of the physical world we will come to understand that any (and ultimately every !) properly expressed relationship (e.g. our theories) has several universal aspects of structure which we will study using what we may call \dimensional analysis". These structural aspects are of enormous importance and will be, as you progress, among the very first things you examine in any physical problem you encounter. -
5.3.43The First Relativistic Models of the Universe
AN INTRODUCTION TO GALAXIES AND COSMOLOGY 5.3.43The first relativistic models of the Universe According to general relativity, the distribution of energy and momentum determines the geometric properties of space–time, and, in particular, its curvature. The precise nature of this relationship is specified by a set of equations called the field equations of general relativity. In this book you are not expected to solve or even manipulate these equations, but you do need to know something about them, particularly how they led to the introduction of a quantity known as the cosmological constant. For this purpose, Einstein’s field equations are discussed in Box 5.1. BOX 5.13EINSTEIN’S FIELD EQUATIONS OF GENERAL RELATIVITY When spelled out in detail the Einstein field equations planetary motion in the Solar System and to predict the are vast and complicated, but in the compact and non-Newtonian deflection of light by the Sun. powerful notation used by general relativists they can Einstein published his first paper on relativistic be written with deceptive simplicity. Using this modern cosmology in the following year, 1917. In that paper he notation, the field equations that Einstein introduced in tried to use general relativity to describe the space–time 1916 can be written as geometry of the whole Universe, not just the Solar −8πG System. While working towards that paper he realized G = T (5.7) c 4 that one of the assumptions he had made in his 1916 paper was unnecessary and inappropriate in the broader Different authors may use different conventions for context of cosmology.