DOCSLIB.ORG

Regions of Central Configurations in a Symmetric 4+1 Body Problem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Regions of Central Configurations in a Symmetric 4+1 Body Problem CAPITAL UNIVERSITY OF SCIENCE AND TECHNOLOGY, ISLAMABAD Regions of Central Configurations in a Symmetric 4+1 Body Problem by Irtiza Ul Hassan A thesis submitted in partial fulfillment for the degree of Master of Philosophy in the Faculty of Computing Department of Mathematics 2020 i Copyright c 2020 by Irtiza Ul Hassan All rights reserved. No part of this thesis may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, by any information storage and retrieval system without the prior written permission of the author. ii To my parent, teachers, wife, friends and daughter Hoorain Fatima. iii CERTIFICATE OF APPROVAL Regions of Central Configurations in a Symmetric 4+1 Body Problem by Irtiza Ul Hassan MMT173025 THESIS EXAMINING COMMITTEE S. No. Examiner Name Organization (a) External Examiner Dr. Ibrar Hussain NUST, Islamabad (b) Internal Examiner Dr. Muhammad Afzal CUST, Islamabad (c) Supervisor Dr. Abdul Rehman Kashif CUST, Islamabad Dr. Abdul Rehman Kashif Thesis Supervisor December, 2020 Dr. Muhammad Sagheer Dr. Muhammad Abdul Qadir Head Dean Dept. of Mathematics Faculty of Computing December, 2020 December, 2020 iv Author's Declaration I, Irtiza Ul Hassan hereby state that my M.Phil thesis titled \Regions of Central Configurations in a Symmetric 4+1 Body Problem" is my own work and has not been submitted previously by me for taking any degree from Capital University of Science and Technology, Islamabad or anywhere else in the country/abroad. At any time if my statement is found to be incorrect even after my graduation, the University has the right to withdraw my M.Phil Degree. Irtiza Ul Hassan Registration No: MMT173025 v Plagiarism Undertaking I solemnly declare that research work presented in this thesis titled \Regions of Central Configurations in a Symmetric 4+1 Body Problem" is solely my research work with no significant contribution from any other person. Small contribution/help wherever taken has been duly acknowledged and that complete thesis has been written by me. I understand the zero tolerance policy of the HEC and Capital University of Science and Technology towards plagiarism. Therefore, I as an author of the above titled thesis declare that no portion of my thesis has been plagiarized and any material used as reference is properly referred/cited. I undertake that if I am found guilty of any formal plagiarism in the above titled thesis even after award of M.Phil Degree, the University reserves the right to withdraw/revoke my M.Phil degree and that HEC and the University have the right to publish my name on the HEC/University website on which names of students are placed who submitted plagiarized work. Irtiza Ul Hassan Registration No: MMT173025 vi Acknowledgements All praise to Almighty Allah, the most Benevolent and Merciful, the creator of Universe and man, who gave me the vision and courage to accomplish this work successfully. A research project at any level is very difficult to be accomplished alone. The contribution of many people have made it possible for me to complete this work. I would like to extend my appreciation especially to the following. First and foremost I extend my sincerest gratitude to my thesis supervisor Dr. Abdul Rehman Kashif who has supported me throughout my thesis work in every possible way. He presented the things in simplest way. His efforts and encouragement is really appreciable. He gave me self-belief, and confidence and provided me his support and guidance all way along. Under the supervision of him I never felt anything difficult in my thesis. Dr. Muhammad Sagheer, Dr. Shafqat Hussain, Dr. Dur-e-Shehwar, Dr. Samina Rashid, Dr. Rashid Ali and Dr. Muhammad Afzal provided me the friendly and comfortable environment to work in, without his guidance and support I would never be able to put this topic together. Last but not least, I would like to pay high regards to my parents and all family members for their sincere encouragement and inspiration throughout my research work and lifting me uphill this phase of life. I owe everything to them. Besides this, I would also like to thank all of my friends who supported me in writing, and incented me to strive towards my goal. Irtiza Ul Hassan vii Abstract In this review thesis, we model five body problem where four of the masses are placed at the vertices of an isosceles trapezoid and fifth mass is placed at the center of the system. We obtained the expression for m0 and M as a function of α and β which are distance parameters. The regions of the central configurations where positive masses can be selected are derived analytically as well as numerically. Also it is shown that no central configurations are possible in the complement of these regions. Contents Author's Declaration iv Plagiarism Undertakingv Acknowledgements vi Abstract vii List of Figuresx Abbreviations xi Physical Constants xii 1 Introduction1 1.1 Central Configuration (CC)......................2 1.2 Thesis Contribution...........................3 1.3 Dissertation Outlines..........................4 2 Preliminaries5 2.1 Basics Definitions............................5 2.1.1 Motion [25]...........................5 2.1.2 Mechanics [25]..........................5 2.1.3 Scalar [25]............................6 2.1.4 Vector [25]............................6 2.1.5 Field [25]............................6 2.1.6 Scalar Field [25].........................6 2.1.7 Vector Field [25]........................6 2.1.8 Conservative Vector Field [25].................7 2.1.9 Uniform Force Field [25]....................7 2.1.10 Central Force [25]........................7 2.1.11 Degree of Freedom [25].....................8 2.1.12 Center of Mass [25].......................8 2.1.13 Center of Gravity [25].....................8 2.1.14 Torque [25]...........................9 viii ix 2.1.15 Momentum [25].........................9 2.1.16 Point-like Particle [25]..................... 10 2.1.17 Angular Momentum [25].................... 10 2.1.18 Lorentz Transformation [25].................. 10 2.1.19 Lagrange Points [25]...................... 11 2.1.20 Equilibrium Solution [25].................... 11 2.1.21 Holonomic and Non Holonomic Constraints [25]....... 12 2.1.22 Galilean Transformation [25].................. 12 2.1.23 Celestial Mechanics [25].................... 13 2.1.24 Kepler's Laws of Planetary Motion [25]............ 14 2.1.25 Newton's Laws of Motion [25]................. 15 2.1.26 Newton's Universal Law of Gravitation [25]......... 15 2.2 Two Body Problem [26]........................ 16 2.3 The Solution to the Two-Body Problem [26]............. 16 2.3.1 Radial and Transverse Components of Velocity and Acceleration [26]...................... 19 2.4 N-Body Problem............................ 23 2.4.1 The Equations of Motion in the NBP............. 23 .............................. 23 3 Central Configuration Regions in Varying Central Mass With Four Equal Masses 25 3.1 Equation of Motion........................... 25 3.2 Varying Central Mass with Four Equal Masses............ 33 3.2.1 Positivity analysis of M and m0 ................ 35 4 Central Configuration Regions in Central Mass and Two Pairs of Same Masses 42 4.1 Special Case (α = β).......................... 56 5 Conclusion 59 Bibliography 60 List of Figures 2.1 Center of mass of two body system................... 16 2.2 Center of mass of two body system................... 21 3.1 Configurations of four and five body trapezoidal.......... 26 3.2 CC regions in trapezoidal 5BP varying central mass with four same masses................................... 38 3.3 M > 0 (Shaded Region)........................ 40 3.4 m0 > 0 (Shaded Region)........................ 40 3.5 M > 0 and m0 > 0 (Shaded Region)................. 41 4.1 (e∗ − f ∗)(α,β,C) > 0(colored)...................... 45 4.2 N(α,β,C) > 0(colored)......................... 47 4.3 D(α,β,C)> 0(colored).......................... 49 4.4 CC region in trapezoid 4 + 1 body problem.............. 50 4.5 M; m0 > 0 and C = 0.1......................... 51 4.6 M; m0 > 0 and C = 0.2......................... 51 4.7 M; m0 > 0 and C = 0.3......................... 52 4.8 M; m0 > 0 and C = 0.4......................... 52 4.9 M; m0 > 0 and C = 0.5......................... 53 4.10 M; m0 > 0 and C = 0.6......................... 53 4.11 M; m0 > 0 and C = 0.7......................... 54 4.12 M; m0 > 0 and C = 0.8......................... 54 4.13 M; m0 > 0 and C = 0.9......................... 55 4.14 M; m0 > 0 and C = 1.......................... 55 4.15 CC region in 5BP when α = β, m0 > 0............... 57 4.16 CC region in 5BP when α = β, M > 0............... 57 4.17 CC region in 5BP when α = β, m0 > 0 and M > 0......... 58 x Abbreviations 2BP Two-Body Ptoblem 3BP Three-Body Problem 4BP Four-Body Problem 5BP Five-Body Problem CC Central Configuration D Denominator Ms Mass of the Sun N Numerator NBP N-Body Problem SI System International xi Physical Constants Symbol Name Unit G Universal gravitational constant m3kg−1s−2 F Gravitational force Newton r Distance Meter P Linear momentum kgms−1 L Angular momentum kgm2s−1 mi Point masses kg µ Ratio of the masses R Real number 3 Such that 8 For all 2 Belongs to xii Chapter 1 Introduction In classical mechanics, the 2-body problem (2BP) is to predict the motion of two massive objects which are abstractly viewed as point particles. The 2BP is most common in the case of a gravity that occurs in astronomy to determine orbits of objects such as satellites, planets and stars. Newton solved 2BP by his fun- damental law of gravity. Newtonian mechanics is a mathematical model whose purpose is to account for the motions of the various objects in the universe. The basic concept of this model were first enunciated by Sir Isaac Newton in a work entitled \Philosophiae Naturalis Principia Mathematica"[1]. This work, which was published in 1687.
Load more

Recommended publications
  • Leibniz' Dynamical Metaphysicsand the Origins
    LEIBNIZ’ DYNAMICAL METAPHYSICSAND THE ORIGINS OF THE VIS VIVA CONTROVERSY* by George Gale Jr. * I am grateful to Marjorie Grene, Neal Gilbert, Ronald Arbini and Rom Harr6 for their comments on earlier drafts of this essay. Systematics Vol. 11 No. 3 Although recent work has begun to clarify the later history and development of the vis viva controversy, the origins of the conflict arc still obscure.1 This is understandable, since it was Leibniz who fired the initial barrage of the battle; and, as always, it is extremely difficult to disentangle the various elements of the great physicist-philosopher’s thought. His conception includes facets of physics, mathematics and, of course, metaphysics. However, one feature is essential to any possible understanding of the genesis of the debate over vis viva: we must note, as did Leibniz, that the vis viva notion was to emerge as the first element of a new science, which differed significantly from that of Descartes and Newton, and which Leibniz called the science of dynamics. In what follows, I attempt to clarify the various strands which were woven into the vis viva concept, noting especially the conceptual framework which emerged and later evolved into the science of dynamics as we know it today. I shall argue, in general, that Leibniz’ science of dynamics developed as a consistent physical interpretation of certain of his metaphysical and mathematical beliefs. Thus the following analysis indicates that at least once in the history of science, an important development in physical conceptualization was intimately dependent upon developments in metaphysical conceptualization. 1.
    [Show full text]
  • THE EARTH's GRAVITY OUTLINE the Earth's Gravitational Field
    GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY OUTLINE The Earth's gravitational field 2 Newton's law of gravitation: Fgrav = GMm=r ; Gravitational field = gravitational acceleration g; gravitational potential, equipotential surfaces. g for a non–rotating spherically symmetric Earth; Effects of rotation and ellipticity – variation with latitude, the reference ellipsoid and International Gravity Formula; Effects of elevation and topography, intervening rock, density inhomogeneities, tides. The geoid: equipotential mean–sea–level surface on which g = IGF value. Gravity surveys Measurement: gravity units, gravimeters, survey procedures; the geoid; satellite altimetry. Gravity corrections – latitude, elevation, Bouguer, terrain, drift; Interpretation of gravity anomalies: regional–residual separation; regional variations and deep (crust, mantle) structure; local variations and shallow density anomalies; Examples of Bouguer gravity anomalies. Isostasy Mechanism: level of compensation; Pratt and Airy models; mountain roots; Isostasy and free–air gravity, examples of isostatic balance and isostatic anomalies. Background reading: Fowler §5.1–5.6; Lowrie §2.2–2.6; Kearey & Vine §2.11. GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY FIELD Newton's law of gravitation is: ¯ GMm F = r2 11 2 2 1 3 2 where the Gravitational Constant G = 6:673 10− Nm kg− (kg− m s− ). ¢ The field strength of the Earth's gravitational field is defined as the gravitational force acting on unit mass. From Newton's third¯ law of mechanics, F = ma, it follows that gravitational force per unit mass = gravitational acceleration g. g is approximately 9:8m/s2 at the surface of the Earth. A related concept is gravitational potential: the gravitational potential V at a point P is the work done against gravity in ¯ P bringing unit mass from infinity to P.
    [Show full text]
  • A Conjecture of Thermo-Gravitation Based on Geometry, Classical
    A conjecture of thermo-gravitation based on geometry, classical physics and classical thermodynamics The authors: Weicong Xu a, b, Li Zhao a, b, * a Key Laboratory of Efficient Utilization of Low and Medium Grade Energy, Ministry of Education of China, Tianjin 300350, China b School of mechanical engineering, Tianjin University, Tianjin 300350, China * Corresponding author. Tel: +86-022-27890051; Fax: +86-022-27404188; E-mail: [email protected] Abstract One of the goals that physicists have been pursuing is to get the same explanation from different angles for the same phenomenon, so as to realize the unity of basic physical laws. Geometry, classical mechanics and classical thermodynamics are three relatively old disciplines. Their research methods and perspectives for the same phenomenon are quite different. However, there must be some undetermined connections and symmetries among them. In previous studies, there is a lack of horizontal analogical research on the basic theories of different disciplines, but revealing the deep connections between them will help to deepen the understanding of the existing system and promote the common development of multiple disciplines. Using the method of analogy analysis, five basic axioms of geometry, four laws of classical mechanics and four laws of thermodynamics are compared and analyzed. The similarity and relevance of basic laws between different disciplines is proposed. Then, by comparing the axiom of circle in geometry and Newton’s law of universal gravitation, the conjecture of the law of thermo-gravitation is put forward. Keywords thermo-gravitation, analogy method, thermodynamics, geometry 1. Introduction With the development of science, the theories and methods of describing the same macro system are becoming more and more abundant.
    [Show full text]
  • Universal Gravitational Constant EX-9908 Page 1 of 13
    Universal Gravitational Constant EX-9908 Page 1 of 13 Universal Gravitational Constant EQUIPMENT 1 Gravitational Torsion Balance AP-8215 1 X-Y Adjustable Diode Laser OS-8526A 1 45 cm Steel Rod ME-8736 1 Large Table Clamp ME-9472 1 Meter Stick SE-7333 INTRODUCTION The Gravitational Torsion Balance reprises one of the great experiments in the history of physics—the measurement of the gravitational constant, as performed by Henry Cavendish in 1798. The Gravitational Torsion Balance consists of two 38.3 gram masses suspended from a highly sensitive torsion ribbon and two1.5 kilogram masses that can be positioned as required. The Gravitational Torsion Balance is oriented so the force of gravity between the small balls and the earth is negated (the pendulum is nearly perfectly aligned vertically and horizontally). The large masses are brought near the smaller masses, and the gravitational force between the large and small masses is measured by observing the twist of the torsion ribbon. An optical lever, produced by a laser light source and a mirror affixed to the torsion pendulum, is used to accurately measure the small twist of the ribbon. THEORY The gravitational attraction of all objects toward the Earth is obvious. The gravitational attraction of every object to every other object, however, is anything but obvious. Despite the lack of direct evidence for any such attraction between everyday objects, Isaac Newton was able to deduce his law of universal gravitation. Newton’s law of universal gravitation: m m F G 1 2 r 2 where m1 and m2 are the masses of the objects, r is the distance between them, and G = 6.67 x 10-11 Nm2/kg2 However, in Newton's time, every measurable example of this gravitational force included the Earth as one of the masses.
    [Show full text]
  • RELATIVISTIC GRAVITY and the ORIGIN of INERTIA and INERTIAL MASS K Tsarouchas
    RELATIVISTIC GRAVITY AND THE ORIGIN OF INERTIA AND INERTIAL MASS K Tsarouchas To cite this version: K Tsarouchas. RELATIVISTIC GRAVITY AND THE ORIGIN OF INERTIA AND INERTIAL MASS. 2021. hal-01474982v5 HAL Id: hal-01474982 https://hal.archives-ouvertes.fr/hal-01474982v5 Preprint submitted on 3 Feb 2021 (v5), last revised 11 Jul 2021 (v6) 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. Distributed under a Creative Commons Attribution| 4.0 International License Relativistic Gravity and the Origin of Inertia and Inertial Mass K. I. Tsarouchas School of Mechanical Engineering National Technical University of Athens, Greece E-mail-1: [email protected] - E-mail-2: [email protected] Abstract If equilibrium is to be a frame-independent condition, it is necessary the gravitational force to have precisely the same transformation law as that of the Lorentz-force. Therefore, gravity should be described by a gravitomagnetic theory with equations which have the same mathematical form as those of the electromagnetic theory, with the gravitational mass as a Lorentz invariant. Using this gravitomagnetic theory, in order to ensure the relativity of all kinds of translatory motion, we accept the principle of covariance and the equivalence principle and thus we prove that, 1.
    [Show full text]
  • PPN Formalism
    PPN formalism Hajime SOTANI 01/07/2009 21/06/2013 (minor changes) University of T¨ubingen PPN formalism Hajime Sotani Introduction Up to now, there exists no experiment purporting inconsistency of Einstein's theory. General relativity is definitely a beautiful theory of gravitation. However, we may have alternative approaches to explain all gravitational phenomena. We have also faced on some fundamental unknowns in the Universe, such as dark energy and dark matter, which might be solved by new theory of gravitation. The candidates as an alternative gravitational theory should satisfy at least three criteria for viability; (1) self-consistency, (2) completeness, and (3) agreement with past experiments. University of T¨ubingen 1 PPN formalism Hajime Sotani Metric Theory In only one significant way do metric theories of gravity differ from each other: ! their laws for the generation of the metric. - In GR, the metric is generated directly by the stress-energy of matter and of nongravitational fields. - In Dicke-Brans-Jordan theory, matter and nongravitational fields generate a scalar field '; then ' acts together with the matter and other fields to generate the metric, while \long-range field” ' CANNOT act back directly on matter. (1) Despite the possible existence of long-range gravitational fields in addition to the metric in various metric theories of gravity, the postulates of those theories demand that matter and non-gravitational fields be completely oblivious to them. (2) The only gravitational field that enters the equations of motion is the metric. Thus the metric and equations of motion for matter become the primary entities for calculating observable effects.
    [Show full text]
  • The Confrontation Between General Relativity and Experiment
    The Confrontation between General Relativity and Experiment Clifford M. Will Department of Physics University of Florida Gainesville FL 32611, U.S.A. email: [email protected]fl.edu http://www.phys.ufl.edu/~cmw/ Abstract The status of experimental tests of general relativity and of theoretical frameworks for analyzing them are reviewed and updated. Einstein’s equivalence principle (EEP) is well supported by experiments such as the E¨otv¨os experiment, tests of local Lorentz invariance and clock experiments. Ongoing tests of EEP and of the inverse square law are searching for new interactions arising from unification or quantum gravity. Tests of general relativity at the post-Newtonian level have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, the Nordtvedt effect in lunar motion, and frame-dragging. Gravitational wave damping has been detected in an amount that agrees with general relativity to better than half a percent using the Hulse–Taylor binary pulsar, and a growing family of other binary pulsar systems is yielding new tests, especially of strong-field effects. Current and future tests of relativity will center on strong gravity and gravitational waves. arXiv:1403.7377v1 [gr-qc] 28 Mar 2014 1 Contents 1 Introduction 3 2 Tests of the Foundations of Gravitation Theory 6 2.1 The Einstein equivalence principle . .. 6 2.1.1 Tests of the weak equivalence principle . .. 7 2.1.2 Tests of local Lorentz invariance . .. 9 2.1.3 Tests of local position invariance . 12 2.2 TheoreticalframeworksforanalyzingEEP. ....... 16 2.2.1 Schiff’sconjecture ................................ 16 2.2.2 The THǫµ formalism .............................
    [Show full text]
  • Newton As Philosopher
    This page intentionally left blank NEWTON AS PHILOSOPHER Newton’s philosophical views are unique and uniquely difficult to categorize. In the course of a long career from the early 1670s until his death in 1727, he articulated profound responses to Cartesian natural philosophy and to the prevailing mechanical philosophy of his day. Newton as Philosopher presents Newton as an original and sophisti- cated contributor to natural philosophy, one who engaged with the principal ideas of his most important predecessor, René Descartes, and of his most influential critic, G. W. Leibniz. Unlike Descartes and Leibniz, Newton was systematic and philosophical without presenting a philosophical system, but, over the course of his life, he developed a novel picture of nature, our place within it, and its relation to the creator. This rich treatment of his philosophical ideas, the first in English for thirty years, will be of wide interest to historians of philosophy, science, and ideas. ANDREW JANIAK is Assistant Professor in the Department of Philosophy, Duke University. He is editor of Newton: Philosophical Writings (2004). NEWTON AS PHILOSOPHER ANDREW JANIAK Duke University CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521862868 © Andrew Janiak 2008 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.
    [Show full text]
  • Section 4.9: Projectile Motion. This Is the Reason Calculus Was Invented! You’Ve Heard the Story About the Apple Falling on Isaac’S Head
    Section 4.9: Projectile motion. This is the reason Calculus was invented! You've heard the story about the apple falling on Isaac's head. The apple is a projectile (an object subjected only to external forces, like gravity). The starting point is Newton's Second Law: F = ma. F is the set of forces acting on an object (right now, it's only gravity), m is the mass of the object, and a is the acceleration of the object. That's right, forces cause objects to move. Without forces, there is no motion. 1. Standard Notation. The position of an object is denoted s(t). The velocity of an object is denoted v(t), which is equivalent to s0(t). The acceleration of an object is denoted a(t). It is the rate of change of velocity, so a(t) = v0(t) = s00(t) 2. The force of gravity The force of gravity is the mass m times the gravitational constant g, which equals 9:812 m=s2 or 32:12 ft=s2. We put a minus sign in front to signify that gravity pulls objects down, in the negative direction. This means that F = ma −mg = ma −g = a Ah ha, objects falling under the influence of gravity are subjected to a constant force, −g. We will usually use g = 9:8 (or sometimes g = 10 to make the number simple) in MKS units, or g = 32 in imperial units, because we are more concerned with the process rather than huge numbers of significant digits. 3. Projectile Motion problems The goal of these problems is to find the expression for the position s(t) of an object.
    [Show full text]
  • Variation in Value of Universal Gravitational Constant - ‘G’ And
    Variation in Value of Universal Gravitational Constant - ‘G’ And Rise of Current Universe Rohit Kumar AT + PO – Matihani, District – Begusarai, Bihar -851129 [email protected] Abstract: In 1686 Newton gave his famous gravitational equation describing movement of planets around star. He left from the conventional notion of equilibrium of mass in which all the particles revolved around their center of mass. This led to creation of 4 fundamental forces of nature. Value of ‘G’ thus created in due process is in the foundation of every astrophysical analysis whether it be Newtonian or Einstein’s gravitational field. Here, it is shown that the value of ‘G’ varies with difference position, velocity and mass. Calculation is made for difference in value of ‘G’ for each planet with variation in distance ie; apogee and perigee. Change in monthly and daily variation of “G” is also shown, along with effect of height and depth from mean sea level. Force of gravity is equivalent to Centripetal force from center of mass of planet-star system. Calculation for value of G is made for an atom which is found to be 1029, thus cancelling the concept of the separation of forces of nature on basis of size. Such concept will be very useful for space exploration activity. KEY WORDS: Centripetal force, Gravitational constant -G, Gravitational Force, Hydrostatic equilibrium, Linear acceleration, Mass, Orbital and escape velocity, Time. 1. Introduction: Inverse square is at the center of gravitational and electromagnetic theory. Squaring the distance between the center of particles are considered to compare the movement w.r.t.
    [Show full text]
  • Universal Gravitation 6.3
    Name: Date: Universal Gravitation 6.3 The law of universal gravitation allows you to calculate the gravitational force between two objects from their masses and the distance between them. The law includes a value called the gravitational constant, or “G.” This value is the same everywhere in the universe. Calculating the force between small objects like grapefruits or huge objects like planets, moons, and stars is possible using this law. What is the law of universal gravitation? The force between two masses m1 and m2 that are separated by a distance r is given by: So, when the masses m1 and m2 are given in kilograms and the distance r is given in meters, the force has the unit of newtons. Remember that the distance r corresponds to the distance between the center of gravity of the two objects. For example, the gravitational force between two spheres that are touching each other, each with a radius of 0.3 meter and a mass of 1,000 kilograms, is given by: 11– 2 2 1,000 kg× 1,000 kg F = 6.67× 10 N-m ⁄ kg --------------------------------------------------= 0.000185 N ()0.3 m+ 0.3 m 2 Note: A small car has a mass of approximately 1,000 kilograms. Try to visualize this much mass compressed into a sphere with a diameter of 0.3 meters (30 centimeters). If two such spheres were touching one another, the gravitational force between them would be only 0.000185 newtons. On Earth, this corresponds to the weight of a mass equal to only 18.9 milligrams.
    [Show full text]
  • Newton's Gravitational Constant Big G Is Not a Constant
    Journal of Modern Physics, 2016, 7, 510-522 Published Online March 2016 in SciRes. http://www.scirp.org/journal/jmp http://dx.doi.org/10.4236/jmp.2016.76053 Newton’s Gravitational Constant Big G Is Not a Constant Ilija Barukčić Horandstrasse, Jever, Germany Received 7 February 2016; accepted 25 March 2016; published 28 March 2016 Copyright © 2016 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract Newton’s gravitational constant Big OG, appearing in both Newton’s law of gravity and Einstein’s general relativity, is essential for our today’s understanding of gravity. Thus far, despite of the very long history of measurements of Newton’s gravitational constant Big OG, a definite value is still not in sight. Surprisingly, the results of different experiments have varied by much more than would be expected due to systematic or random errors. Why do measurements of Newton’s gravi- tational constant Big OG vary so much? The purpose of this publication is to provide a logically and mathematically self-consistent theoretical proof that Newton’s gravitational constant Big OG is not a constant. Keywords Quantum Theory, Relativity Theory, Unified Field Theory, Causality 1. Introduction Today, Isaac Newton (1642-1727) is best known for his important contribution to the formulation of the theory of gravity. Newton’s Principia from a historical point of view was published in a philosophically tumultuous and rich time. As is well known, Philosophers at Newton’s time were “studying nature” often by studying texts (i.e.
    [Show full text]