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Andre Koch Torres Assis - - - - - . -- - --- Archimedes, the Center of Gravity, and the First of Mechanics: Zndedition The Law of the Lever Archimedes, the Center of Gravity, and the First Law of Mechanics 2nd edition The Law of the Lever Andre K.T. Assis Apeiron Montreal Published by C. Roy Keys Inc. 4405, rue St-Dominique Montreal, Quebec H2W 2B2 Canada http://redshift.vif.com © Andre K.T. Assis 2010 First Published 2010 Library and Archives Canada Cataloguing in Publication Assis, André Koch Torres, 1962- Archimedes, the center of gravity, and the first law of mechanics / Andre K.T. Assis. – 2nd ed. Includes bibliographical references. ISBN 978-0-9864926-4-8 1. Center of mass--Textbooks. 2. Center of mass--Experiments. 3. Mechanics--Textbooks. 4. Mechanics--Experiments. I. Title. QA839.A87 2008 531'.14 C2007-907265-8 Front cover: Engraving from Mechanics Magazine published in London in 1824. Back cover: Photos of a few of the experiments described in this book. A hori- zontal pasteboard triangle supported at the barycenter by a vertical stick. A rec- tangle and a plumb line suspended by a needle. An equilibrist upside down supported at the head, with modeling clay on his hands. A lever in equilibrium with different weights on each arm. To all those who, down through the centuries, have worked to preserve, translate, interpret, and disseminate the works of Archimedes. 3 4 Contents Preface to the Second Edition 9 Acknowledgments 11 IIntroduction 13 1TheLifeofArchimedes 17 2TheWorksofArquimedes 27 2.1 Extant Works . 27 2.2 Fragmentary Works . 34 2.3 The Method .............................. 37 II The Center of Gravity 41 3Geometry 43 3.1 Finding the Centers of Circles, Rectangles and Parallelograms . 43 3.2 The Triangle Centers . 44 4ExperimentsandDefinitionoftheCenterofGravity 49 4.1 Definitions............................... 49 4.2 Support for the Experiments . 51 4.3 First Experimental Procedure to Find the CG ........... 53 4.3.1 Provisional Definition CG1 ................. 54 4.3.2 Provisional Definition CG2 ................. 58 4.3.3 Provisional Definition CG3 ................. 58 4.4 Experiments with Concave Bodies or Pierced Bodies . .. 60 4.4.1 Provisional Definition CG4 ................. 62 4.4.2 Provisional Definition CG5 ................. 65 4.5 Experiments with Three-Dimensional Bodies . .66 4.6 Plumb Line, Vertical and Horizontal . 67 4.7 Second Experimental Procedure to Find the CG ......... 71 4.7.1 Practical Definition CG6 .................. 76 5 4.8 Third Experimental Procedure to Find the CG .......... 77 4.8.1 Practical Definition CG7 .................. 78 4.9 Conditions of Equilibrium for Supported Bodies . ... 79 4.9.1 Definitions of Stable, Unstable and Neutral Equilibrium . 83 4.10 Stability of a Body . 83 4.11 Conditions of Equilibrium for Suspended Bodies . ... 88 4.11.1 Stable and Neutral Equilibrium . 90 4.12 Cases in which the CG Coincides with the PS .......... 91 4.12.1 Definitive Definition CG8 .................. 94 4.13 Cases in which the CG does Not Change its Height by Rotating the Body, although the CG is Above the PS ........... 96 4.14 Summary . 100 5ExploringthePropertiesoftheCenterofGravity 103 5.1 Fun Activities with the Equilibrist . 103 5.2 Equilibrium Toys . 111 5.3 Equilibrium Games in the Pub . 115 5.4 Equilibrium of the Human Body . 117 5.5 The Extra-Terrestrial, ET ......................121 6HistoricalAspectsoftheCenterofGravity 123 6.1 Comments of Archimedes, Heron, Pappus, Eutocius and Simpli- ciusabouttheCenterofGravity . 123 6.2 Theoretical Values of Center of Gravity Obtained by Archimedes 132 6.2.1 One-dimensional Figures . 132 6.2.2 Two-dimensional Figures . 132 6.2.3 Three-dimensional Figures . 134 III Balances, Levers, and the Oldest Law of Mechanics137 7BalancesandtheMeasurementofWeight 141 7.1 Building a Balance . 141 7.2 Measurement of Weight . 148 7.2.1 Definitions of Equal Weights, of Heavier, and of Lighter.148 7.2.2 Definition of a Multiple Weight . 151 7.2.3 The Weight does Not Depend upon the Height of the Body153 7.3 Improving Balance Sensitivity . 154 7.4 Some Special Situations . 162 7.4.1 Condition of Equilibrium of a Suspended Body . 162 7.4.2 Balances with the Center of Gravity Above the Fulcrum . 165 7.4.3 Other Types of Balance . 166 7.5 Using Weight as a Standard of Force . 166 6 8TheLawoftheLever 169 8.1 Building and Calibrating Levers . 169 8.2 Experiments with Levers and the Oldest Law of Mechanics . ..170 8.2.1 First Part of the Law of the Lever . 175 8.2.2 Experimental Mistakes which Prevent the Verification of theLawoftheLever..................... 176 8.2.3 Second Part of the Law of the Lever . 178 8.3 Types of levers . 182 9MathematicalDefinitionofCenterofGravity 185 9.1 Algebraic Expression of the CG in Cartesian Coordinates . 185 9.2 Mathematical Definition CG9 ....................188 9.3 Theorems to Simplify the Calculation of the CG .........189 10 Explanations of and Deductions from the Law of the Lever 191 10.1 Law of the Lever as an Experimental Result . 191 10.2 Deriving the Law of the Lever from the Torque Concept . ..193 10.3 Law of the Lever Derived from the Experimental Result that a Weight 2P Acting at a Distance d from the Fulcrum is Equivalent to a Weight P Acting at a Distance d x,TogetherwithAnother Weight P Acting at a Distance d + x−fromtheFulcrum . 196 10.4 Law of the Lever as Derived by Duhem Utilizing a Modification of Work Attributed to Euclid . 199 10.5 Proof of the Law of the Lever by an Experimental Procedure Suggested by a Work Attributed to Euclid . 202 10.6 Theoretical Proof of the Law of the Lever Attributed to Euclid . 207 10.7 Archimedes’s Proof of the Law of the Lever and Calculation of theCenterofGravityofaTriangle . 209 10.7.1 Archimedes’s Proof of the Law of the Lever . 209 10.7.2 Archimedes’s Calculation of the CG of a Triangle . 215 Bibliography 219 7 8 Preface to the Second Edition This second edition is an improved and updated version of the book published in 2008, in English and in Portuguese.1 The Figures for the present edition were prepared by Daniel Robson Pinto. The present version has a better division of Chapters, Sections and Subsections. It has also an increased number of references. Misprints havebeencorrected. Some portions of the book have been clarified and better explained. 1[Ass08a] and [Ass08b]. 9 10 Acknowledgments The motivation to write this book arose from courses we gave tohighschoolsci- ence teachers over the past few years. The exchange of ideas with these teachers and with our collaborators at the University were very rich and stimulating. The inspiration for the majority of the experiments on equilibrium and the center of gravity (CG)ofbodiescamefromtheexcellentworksofNorberto Ferreira and Alberto Gaspar.2 We also thank many friends for suggestions, references and ideas: Nor- berto Ferreira, Alberto Gaspar, Rui Vieira, Emerson Santos,DicesarLassFer- nandez, Silvio Seno Chibeni, César José Calderon Filho, Pedro Leopoldo e Silva Lopes, Fábio Miguel de Matos Ravanelli, Juliano Camillo, Lucas An- gioni, Hugo Bonette de Carvalho, Ceno P. Magnaghi, Caio Ferrari de Oliveira, J. Len Berggren, Henry Mendell, Steve Hutcheon and GuilhermeSilvaMel. We thank as well our students at the Institute of Physics with whom we dis- cussed these ideas. My daughter and Eduardo Meirelles helpedwiththeFig- ures of the first version in English.3 The Figures for the present version of this book were prepared by Daniel Robson Pinto, through a fellowship awarded by SAE/UNICAMP, which we thank for this support. Daniel helped also to obtain old figures and references. Special thanks are due to the Institute of Physics, to the Institute of Mathe- matics, to the Grupo Gestor de Projetos Educacionais (GGPE) and to FAEPEX of the University of Campinas—UNICAMP, in Brazil, which provided the nec- essary conditions for the preparation of this book. Roy Keys, the Editor of Apeiron, has been a supporter for many years. He made an excellent editorial work for this book. Andre Koch Torres Assis Institute of Physics University of Campinas—UNICAMP 13083-859 Campinas - SP, Brazil Homepage: www.ifi.unicamp.br/˜assis E-mail: [email protected] 2[Fer], [Fer06] and [Gas03]. 3[Ass08a]. 11 12 Part I Introduction 13 One of the goals of this work is to present the basic phenomena of me- chanics through simple experiments performed with inexpensive materials. We present the fundamental experiments on falling bodies, equilibrium and oscilla- tions around equilibrium positions. We also show how the theoretical concepts are formed and modified during this process, just as occurred in the formulation of the basic laws of mechanics. We show how more complex phenomena can be explained and clarified by means of elementary experiments. Playful and curious experiments are also presented. They stimulate creativity, critical thinking and a sense of humour in science. They also relate everyday phenomena to the fundamental laws of physics. The emphasis is placed on experimental activities. After theexperimentswe formulate the definitions, concepts, postulates, principles, and laws describing the phenomena. The materials utilized are very simple, easily found at home or in stores, all of them very inexpensive. Even so, we can carry out very precise experiments and construct sensitive scientific equipment. The reader need not depend on any school or research laboratory, as he can build his own equipment and perform all the measurements. If the experiments presented here are performed in the classroom, each stu- dent should ideally perform all the tasks, even when working in a group. Each one should build his own equipment (support, plumb line, lever, etc.), cut out his geometric figures and then take all this personal materialhome.Thisproce- dure is richer in lessons than simple demonstrations of the effects by a teacher. It is essential that all students put their hands to the plough.

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