Archimedes, the Center of Gravity, and the First of Mechanics

225 Pages·2012·2.9 MB·English

Andre Koch Torres Assis
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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.
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Contents
Preface to the Second Edition 9
Acknowledgments 11
I Introduction 13
1 The Life of Archimedes 17
2 The Works of Arquimedes 27
2.1 Extant Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Fragmentary Works . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
II The Center of Gravity 41
3 Geometry 43
3.1 Finding the Centers of Circles, Rectangles and Parallelograms . . 43
3.2 The Triangle Centers . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Experiments and Definition of the Center of Gravity 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 ProvisionalDefinition CG1 . . . . . . . . . . . . . . . . . 54
4.3.2 ProvisionalDefinition CG2 . . . . . . . . . . . . . . . . . 58
4.3.3 ProvisionalDefinition CG3 . . . . . . . . . . . . . . . . . 58
4.4 Experiments with Concave Bodies or Pierced Bodies . . . . . . . 60
4.4.1 ProvisionalDefinition CG4 . . . . . . . . . . . . . . . . . 62
4.4.2 ProvisionalDefinition 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
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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
5 Exploring the Properties of the Center of Gravity 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
6 Historical Aspects of the Center of Gravity 123
6.1 Comments of Archimedes, Heron, Pappus, Eutocius and Simpli-
cius about the Center of Gravity . . . . . . . . . . . . . . . . . . 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
7 Balances and the Measurement of Weight 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
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8 The Law of the Lever 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
the Law of the Lever . . . . . . . . . . . . . . . . . . . . . 176
8.2.3 Second Part of the Law of the Lever . . . . . . . . . . . . 178
8.3 Types of levers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9 Mathematical Definition of Center of Gravity 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
Weight2P ActingataDistancedfromtheFulcrumisEquivalent
toaWeightP ActingataDistanced x,TogetherwithAnother
−
Weight P Acting at a Distance d+x from the Fulcrum . . . . . 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
the Center of Gravity of a Triangle . . . . . . . . . . . . . . . . . 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
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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.
ThepresentversionhasabetterdivisionofChapters,SectionsandSubsections.
It has also an increased number of references. Misprints have been corrected.
Some portions of the book have been clarified and better explained.
1[Ass08a]and[Ass08b].
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