Archimedes Cover Layout1.pub

200 Pages·2008·1.91 MB·English

Archimedes, the Center of Gravity, and the First Law of Me-

Andre Koch Torres Assis
chanics deals with the most fundamental aspects of physics.
A
The book describes the main events in the life of Archimedes r
c
and the content of his works. It goes on to discuss a large num- h
i
ber of experiments relating to the equilibrium of suspended m
bodies under the influence of e
d
Earth’s gravitational force. All ex- e
periments are clearly described s
,

and performed with simple, inex- t
h
pensive materials. These experi- e

ments lead to a clear conceptual C
e
definition of the center of gravity
n
of material bodies and illustrate t
e
practical procedures for locating it r

precisely. The conditions of sta- o
f
ble, neutral, and unstable equilibrium are analyzed. G
Many equilibrium toys and games are described and r
a
explained. Historical aspects of the concept are pre- v
i
sented, together with the theoretical values of center of t
y
gravity obtained by Archimedes. The book also explains how to build and calibrate ,

a
precise balances and levers. Several experiments are performed leading to a mathe-
n
matical definition of the center of gravity and the first law of mechanics, also called d

the law of the lever. Consequences of this law and different explanations of it are t
h
described at the end of the book, together with an exhaustive analysis of the works e

of Euclid and Archimedes. F
i
r
s
t

L
a
About the Author w
Andre Koch Torres Assis was born in Brazil (1962) and educated at the
o
State University of Campinas – UNICAMP, BS (1983), PhD (1987). He f

spent the academic year of 1988 in England with a post-doctoral position M
at the Culham Laboratory (United Kingdom Atomic Energy Authority). He e
c
spent one year in 1991-92 as a Visiting Scholar at the Center for h
Electromagnetics Research of Northeastern University (Boston, USA). a
n
From August 2001 to November 2002 he worked at the Institute for the
i
History of Natural Sciences, Hamburg University (Hamburg, Germany) c
s
with a research fellowship awarded by the Alexander von Humboldt

Foundation of Germany. He is the author of Weber’s Electrodynamics
(1994), Relational Mechanics (1999),
Inductance and Force Calculations in A
Electrical Circuits (with M. A. Bueno, s
s
2001), and The Electric Force of a i
s
Current (with J. A. ISBN 978-0-9732911-6-2 Archimedes, the Center
Hernandes, 2007) . He has
been professor of physics at ,!7IA9H3-cjbbgc!
UNICAMP since 1989,
working on the foundations A of Gravity, and the First
p
of electromagnetism, e
gravitation, and cosmology. ir
on Law of Mechanics
Archimedes, the Center of
Gravity, and the First Law
of Mechanics

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 2008
First Published 2008

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.

Includes bibliographical references.
ISBN 978-0-9732911-6-2

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: Oil painting of Archimedes by the Italian artist Domenico Fetti
(1589-1624), painted about 1620.

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.
Table of Contents
Acknowledgments..............................................................................................i
Chapter 1
The Life of Archimedes....................................................................................1
Chapter 2
The Works of Archimedes 11
2.1 Extant works.........................................................................................12
2.2 The Method...........................................................................................20
Chapter 3
The Center of Gravity....................................................................................25
3.1 Geometry..............................................................................................25
3.1.1 Materials........................................................................................25
3.1.2 Finding the centers of circles, rectangles
and parallelograms........................................................................25
3.1.3 The triangle centers.......................................................................26
3.2 Experiments on and definition of the center of
gravity..................................................................................................30
3.2.1 Definitions.....................................................................................31
3.2.2 Support for the experiments..........................................................32
3.2.3 First experimental procedure to find the
CG.................................................................................................34
3.2.4 Experiments with concave bodies or
pierced bodies................................................................................41
3.2.5 Experiments with three-dimensional
bodies............................................................................................45
3.2.6 Plumb line, vertical and horizontal................................................47
3.2.7 Second experimental procedure to find the
CG.................................................................................................50
3.2.8 Third experimental procedure to find the
CG.................................................................................................56
3.2.9 Conditions of equilibrium for supported
bodies............................................................................................58
3.2.10 Definitions of stable, unstable and neutral
equilibrium....................................................................................62
3.2.11 Conditions of equilibrium for suspended
bodies............................................................................................66
3.2.12 Definitive definition of the center of
gravity...........................................................................................69
3.2.13 Summary.......................................................................................74
3.3 Fun activities with the equilibrist.........................................................75
3.4 Equilibrium toys...................................................................................84
3.4.1 Equilibrium games in the pub.......................................................87
3.4.2 Equilibrium of the human body.....................................................89
3.4.3 The ET...........................................................................................94
Chapter 4
Historical Aspects of the Center of Gravity..................................................97
Chapter 5
Theoretical Values of Center of Gravity Obtained by Archimedes..........107
5.1 One-dimensional figures....................................................................107
5.2 Two-dimensional figures....................................................................107
5.3 Three-dimensional figures..................................................................109
Chapter 6
Balances and the Measurement of Weight..................................................111
6.1 Building a balance..............................................................................111
6.2 Measurement of weight......................................................................119
6.3 Improving balance sensitivity.............................................................123
6.4 Condition of equilibrium of a suspended body...................................132
6.5 Balances with the center of gravity above the
fulcrum...............................................................................................135
6.6 Other types of balance........................................................................136
6.7 Using weight as a standard of force....................................................136
Chapter 7
Levers and the First Law of Mechanics......................................................139
7.1 Building and calibrating levers...........................................................139
7.2 Experiments with levers and the first law of
mechanics...........................................................................................141
7.2.1 First part of the law of the lever..................................................144
7.2.2 Second part of the law of the lever..............................................148
7.3 Types of levers...................................................................................151
Chapter 8
Mathematical Definition of Center of Gravity...........................................153
Chapter 9
Explanations of and Deductions from the Law of the Lever.....................159
9.1 Law of the lever as an experimental result.........................................159
9.2 Deriving the law of the lever from the torque
concept...............................................................................................161
9.3 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, together
with another weight P acting at a distance
d + x from the fulcrum........................................................................164
9.4 Law of the lever as derived by Duhem
utilizing a modification of work attributed to
Euclid.................................................................................................168
9.5 Proof of the law of the lever by an
experimental procedure suggested by a work
attributed to Euclid.............................................................................170
9.6 Theoretical proof of the law of the lever
attributed to Euclid.............................................................................176
9.7 Archimedes’s proof of the law of the lever and
calculation of the center of gravity of a
triangle...............................................................................................177
9.7.1 Law of the lever...........................................................................177
9.7.2 CG of a triangle...........................................................................182
Bibliography..................................................................................................187
Introduction
One of the goals of this book is to present the basic phenomena of
mechanics through simple experiments performed with inexpen-
sive materials. We present the fundamental experiments on falling
bodies, equilibrium and oscillations around equilibrium positions.
We also show how the theoretical concepts are formed and modi-
fied 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 cu-
rious 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 the
experiments we formulate the definitions, concepts, postulates,
principles, and laws describing the phenomena. The materials util-
ized are very simple, easily found at home or in stores, all of them
very inexpensive. Even so, we can carry out very precise experi-
ments 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 class-
room, each student should ideally perform all the tasks, even
when working in a group. Each one should build his own equip-
ment (support, plumb line, lever, etc.), cut out his geometric fig-
ures and then take all this personal material home. This procedure
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.
The book is also rich in historical information, which gives
the context in which some laws were discovered, and also differ-
ent approaches taken in discovering them. We are careful about in
formulating concepts and physical principles. It will be seen, for
example, how difficult is to find the correct words to precisely de-
fine the center of gravity so that this concept can encompass a
whole series of experiments. We distinguish clearly between defi-
nitions, postulates, experimental results, and physical laws. We
also distinguish explanations from descriptions of phenomena.
These aspects illustrate the sociological and human aspects of the
formulation of physical laws.
This book is written for students and teachers of science,
physics, and mathematics. It can be utilized at High Schools or at
Universities, depending on the level at which each aspect is ana-
lyzed and explored. It has enough experimental and theoretical
material to be employed in all levels of teaching. Each teacher
should adapt the contents presented here to his own school envi-
ronment. It can also be utilized in courses on the history and phi-
losophy of science.
The best way to grasp the contents of the book is to perform
the majority of the experiments described here in parallel with the
reading. There are many philosophical, theoretical, and mathe-
matical approaches relating to physical science. But physics is es-
sentially an experimental science. It is the combination of all these
aspects that make it so fascinating. For this reason we strongly
recommend that the experiments presented in the book be re-
peated and improved. We hope that the reader will have the same
pleasure in performing these experiments as we had in developing
them.
When necessary we employ the sign ≡ as a symbol of defini-
tion. We utilize the SI international system of units.
To all those who, down through the centuries, have
worked to preserve, translate, interpret, and dis-
seminate the works of Archimedes.
Acknowledgments
The motivation to write this book arose from courses we gave to high
school science 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) of bodies came from the excellent works
of Norberto Ferreira and Alberto Gaspar.1
We also thank for suggestions, references and ideas: Norberto
Ferreira, Alberto Gaspar, Rui Vieira, Emerson, Dicesar Lass Fernandez,
Silvio Seno Chibeni, César José Calderon Filho, Pedro Leopoldo e Silva
Lopes, Fábio Miguel de Matos Ravanelli, Juliano Camillo, Lucas An-
gioni, J. Len Berggren, Henry Mendell, and Roy Keys, as well as our
students at the Institute of Physics with whom we discussed these ideas.
Steve Hutcheon read the whole manuscript and suggested many im-
provements. My daughter and Eduardo Meirelles helped with the fig-
ures.
Special thanks are due to the Institute of Physics, to the Institute of
Mathematics, to the Grupo Gestor de Projetos Educacionais (GGPE)
and to FAEPEX of the State University of Campinas, in Brazil, which
provided the necessary conditions for the preparation of this essay.
Andre K.T. Assis
Institute of Physics, State University of Campinas
13083-970 Campinas-SP, Brazil
E-mail: [email protected]
homepage: http://www.ifi.unicamp.br/~assis/

1 N . F e r r e i r a , M e c â n i c a , P r o j e t o R I P E – R ede de Instrumentação para o Ensino (São Paulo: In-
stituto de Física, USP.); N. Ferreira, Equilíbrio (São Paulo: Instituto de Física, USP, 2007),
http://www.ludoteca.if.usp.br/; A. Gaspar, Experiências de Ciências para o Ensino Fundamental
(São Paulo: Ática, 2003).
Archimedes, the Center of Gravity, and the First Law of Mechanics i
Andre K.T. Assis (Montreal: Apeiron 2008)