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


(paperback, 187 pages; ISBN 978-0-9732911-6-2)

Andre Koch Torres Assis

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From the Foreword

About the Author

Table of Contents  


Introduction


One of the goals of this book is to present the basic phenomena of mechanics through simple experiments performed with cheap materials. We present the fundamental experiments about falling bodies, equilibrium and oscillations around the equilibrium positions. Moreover, it is shown how the theoretical concepts are formed and modified during this process, the same having occurred with the formulation of the basic laws of mechanics. It is shown how more complex phenomena can be explained and clarified based on the elementary experiments. Playful and curious experiments are also presented. They stimulate the creativity, the critical thinking and the sense of humor in science. They also relate everyday phenomena with 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 utilized are very simple, easily found at home or in commerce, all of them very cheap. Despite this fact, we realize very precise experiments and construct sensitive scientific equipment. In this way the reader will not depend on any school or research laboratory, as he will build his own equipment and will perform all the measurements. In order to reach this goal, several different assemblies or materials are presented for each set up. If the experiments presented here are performed in the classroom, the ideal is that each student should perform all the tasks, even when they work in groups. Each one should build his own equipment (support, plumb line, lever, etc.), cut out his geometric figures and then take home all this personal material. This procedure is richer than the simple demonstration of the effects by the teacher. It is essential that all students put their hands to the plough. The book is also rich in historical information which gives the con-text of the appearance of some laws and also the different approaches related to them. We are careful about the formation of the concepts and physical principles. It will be seen, for example, how difficult is to find the correct words to define precisely the center of gravity so that this concept can encompass a whole set of experiments. We distinguish clearly what are the definitions, the postulates, the experimental results and the physical laws. We also distinguish the explanations from the descriptions of phenomena. These aspects illustrate the sociological and human aspects embodied in 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 in which each aspect is analyzed 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 environment. It can also be utilized in courses of history of science and philosophy. 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 mathematical aspects related to the physical science. But physics is essentially an experimental science. It is the combination of all these aspects that make it so fascinating. For this reason we recommend strongly that the experiments presented in the book should be repeated and improved. We hope that the reader will have the same pleasure in performing these experiments as we had in implementing them. When necessary we employ the sign as a symbol of definition. We utilize the international SI system of units.

 About the Author

About the Author Andre Koch Torres Assis was born in Brazil (1962) and educated at the State University of Campinas – UNICAMP, BS (1983), PhD (1987). He spent the academic year of 1988 in England with a post-doctoral position at the Culham Laboratory (United Kingdom Atomic Energy Authority). He spent one year in 1991-92 as a Visiting Scholar at the Center for Electromagnetics Research of Northeastern University (Boston, USA). From August 2001 to November 2002 he worked at the Institute for the History of Natural Sciences, Hamburg University (Hamburg, Germany) 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 Electrical Circuits (with M. A. Bueno, 2001), and The Electric Force of a Current (with J. A. Hernandes, 2007) . He has been professor of physics at UNICAMP since 1989, working on the foundations of electromagnetism, gravitation, and cosmology..

 

 

Contents

Acknowledgments

Chapter 1

The Life of Archimedes

Chapter 2

The Works of Archimedes

2.1 Extant works

2.2 The Method

Chapter 3

The Center of Gravity

3.1 Geometry

3.1.1 Materials

3.1.2 Finding the centers of circles, rectangles and parallelograms 

3.1.3 The triangle centers 

3.2 Experiments on and definition of the center of gravity 

3.2.1 Definitions

3.2.2 Support for the experiments 

3.2.3 First experimental procedure to find the CG

3.2.4 Experiments with concave bodies or pierced bodies

3.2.5 Experiments with three-dimensional bodies 

3.2.6 Plumb line, vertical and horizontal

3.2.7 Second experimental procedure to find the CG

3.2.8 Third experimental procedure to find the CG

3.2.9 Conditions of equilibrium for supported bodies

3.2.10 Definitions of stable, unstable and neutral equilibrium 

3.2.11 Conditions of equilibrium for suspended bodies 

3.2.12 Definitive definition of the center of gravity 

3.2.13 Summary 

3.3 Fun activities with the equilibrist 

3.4 Equilibrium toys

3.4.1 Equilibrium games in the pub 

3.4.2 Equilibrium of the human body

3.4.3 The ET

Chapter 4

Historical Aspects of the Center of Gravity

Chapter 5

Theoretical Values of Center of Gravity Obtained by Archimedes

5.1 One-dimensional figures

5.2 Two-dimensional figures

5.3 Three-dimensional figures

Chapter 6

Balances and the Measurement of Weight

6.1 Building a balance

6.2 Measurement of weight 

6.3 Improving balance sensitivity

6.4 Condition of equilibrium of a suspended body

6.5 Balances with the center of gravity above the fulcrum

6.6 Other types of balance

6.7 Using weight as a standard of force

Chapter 7

Levers and the First Law of Mechanics 

7.1 Building and calibrating levers

7.2 Experiments with levers and the first law of mechanics

7.2.1 First part of the law of the lever 

7.2.2 Second part of the law of the lever 

7.3 Types of levers 

Chapter 8

Mathematical Definition of Center of Gravity 

Chapter 9

Explanations of and Deductions from the Law of the Lever

9.1 Law of the lever as an experimental result 

9.2 Deriving the law of the lever from the torque concept 

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

9.4 Law of the lever as derived by Duhem utilizing a modification of work attributed to Euclid 

9.5 Proof of the law of the lever by an experimental procedure suggested by a work attributed to Euclid 

9.6 Theoretical proof of the law of the lever attributed to Euclid 

9.7 Archimedes’s proof of the law of the lever and calculation of the center of gravity of a triangle 

9.7.1 Law of the lever

9.7.2 CG of a triangle 

Bibliography