Classical Mechanics: a Critical Introduction Michael Cohen, Professor Emeritus Department of Physics and Astronomy University of Pennsylvania Philadelphia, PA 19104-6396 Copyright 2011, 2012 with Solutions Manual by Larry Gladney, Ph.D. Edmund J. and Louise W. Kahn Professor for Faculty Excellence Department of Physics and Astronomy University of Pennsylvania "Why, a four-year-old child could understand this... Run out and find me a four-year-old child." - GROUCHO i REVISED PREFACE (Jan. 2013) Anyone who has taught the \standard" Introductory Mechanics course more than a few times has most likely formed some fairly definite ideas re- garding how the basic concepts should be presented, and will have identified (rightly or wrongly) the most common sources of difficulty for the student. An increasing number of people who think seriously about physics peda- gogy have questioned the effectiveness of the traditional classroom with the Professor lecturing and the students listening (perhaps). I take no position regarding this question, but assume that a book can still have educational value. The first draft of this book was composed many years ago and was intended to serve either as a stand-alone text or as a supplementary \tutor" for the student. My motivation was the belief that most courses hurry through the basic concepts too quickly, and that a more leisurely discussion would be helpful to many students. I let the project lapse when I found that publishers appeared to be interested mainly in massive textbooks covering all of first-year physics. Now that it is possible to make this material available on the Internet to students at the University of Pennsylvania and elsewhere, I have revived and reworked the project and hope the resulting document may be useful to some readers. I owe special thanks to Professor Larry Gladney, who has translated the text from its antiquated format into modern digital form and is also preparing a manual of solutions to the end-of-chapter problems. Professor Gladney is the author of many of these problems. The manual will be on the Internet, but the serious student should construct his/her own solutions before reading Professor Gladney's discussion. Conversations with my colleague David Balamuth have been helpful, but I cannot find anyone except myself to blame for errors or defects. An enlightening discussion with Professor Paul Soven disabused me of the misconception that Newton's First Law is just a special case of the Second Law. The Creative Commons copyright permits anyone to download and re- produce all or part of this text, with clear acknowledgment of the source. Neither the text, nor any part of it, may be sold. If you distribute all or part of this text together with additional material from other sources, please identify the sources of all materials. Corrections, comments, criticisms, ad- ditional problems will be most welcome. Thanks. Michael Cohen, Dept. of Physics and Astronomy, Univ. of Pa., Phila, PA 19104-6396 email: [email protected] ii 0.1. INTRODUCTION 0.1 Introduction Classical mechanics deals with the question of how an object moves when it is subjected to various forces, and also with the question of what forces act on an object which is not moving. The word \classical" indicates that we are not discussing phenomena on the atomic scale and we are not discussing situations in which an object moves with a velocity which is an appreciable fraction of the velocity of light. The description of atomic phenomena requires quantum mechanics, and the description of phenomena at very high velocities requires Einstein's Theory of Relativity. Both quantum mechanics and relativity were invented in the twentieth century; the laws of classical mechanics were stated by Sir Isaac Newton in 1687[New02]. The laws of classical mechanics enable us to calculate the trajectories of baseballs and bullets, space vehicles (during the time when the rocket engines are burning, and subsequently), and planets as they move around the sun. Using these laws we can predict the position-versus-time relation for a cylinder rolling down an inclined plane or for an oscillating pendulum, and can calculate the tension in the wire when a picture is hanging on a wall. The practical importance of the subject hardly requires demonstration in a world which contains automobiles, buildings, airplanes, bridges, and ballistic missiles. Even for the person who does not have any professional reason to be interested in any of these mundane things, there is a compelling intellectual reason to study classical mechanics: this is the example par excellence of a theory which explains an incredible multitude of phenomena on the basis of a minimal number of simple principles. Anyone who seriously studies mechanics, even at an elementary level, will find the experience a true intellectual adventure and will acquire a permanent respect for the subtleties involved in applying \simple" concepts to the analysis of \simple" systems. I wish to distinguish very clearly between \subtlety" and \trickery". There is no trickery in this subject. The subtlety consists in the necessity of using concepts and terminology quite precisely. Vagueness in one's think- ing and slight conceptual imprecisions which would be acceptable in every- day discourse will lead almost invariably to incorrect solutions in mechanics problems. In most introductory physics courses approximately one semester (usu- ally a bit less than one semester) is devoted to mechanics. The instructor and students usually labor under the pressure of being required to \cover" a iii 0.1. INTRODUCTION certain amount of material. It is difficult, or even impossible, to \cover" the standard topics in mechanics in one semester without passing too hastily over a number of fundamental concepts which form the basis for everything which follows. Perhaps the most common area of confusion has to do with the listing of the forces which act on a given object. Most people require a considerable amount of practice before they can make a correct list. One must learn to distinguish between the forces acting on a thing and the forces which it exerts on other things, and one must learn the difference between real forces (pushes and pulls caused by the action of one material object on another) and demons like \centrifugal force" (the tendency of an object moving in a circle to slip outwards) which must be expunged from the list of forces. An impatient reader may be annoyed by amount of space devoted to discussion of \obvious" concepts such as \force", \tension", and \friction". The reader (unlike the student who is trapped in a boring lecture) is, of course, free to turn to the next page. I believe, however, that life is long enough to permit careful consideration of fundamental concepts and that time thus spent is not wasted. With a few additions (some discussion of waves for example) this book can serve as a self-contained text, but I imagine that most readers would use it as a supplementary text or study guide in a course which uses another textbook. It can also serve as a text for an online course. Each chapter includes a number of Examples, which are problems re- lating to the material in the chapter, together with solutions and relevant discussion. None of these Examples is a \trick" problem, but some contain features which will challenge at least some of the readers. I strongly recom- mend that the reader write out her/his own solution to the Example before reading the solution in the text. Some introductory Mechanics courses are advertised as not requiring any knowledge of calculus, but calculus usually sneaks in even if anonymously (e.g. in the derivation of the acceleration of a particle moving in a circle or in the definition of work and the derivation of the relation between work and kinetic energy). Since Mechanics provides good illustrations of the physical meaning of the \derivative" and the \integral", we introduce and explain these mathe- matical notions in the appropriate context. At no extra charge the reader who is not familiar with vector notation and vector algebra will find a dis- cussion of those topics in Appendix A. iv Contents 0.1 Introduction . iii 1 KINEMATICS: THE MATHEMATICAL DESCRIPTION OF MOTION 1 1.1 Motion in One Dimension . .1 1.2 Acceleration . .6 1.3 Motion With Constant Acceleration . .7 1.4 Motion in Two and Three Dimensions . 10 1.4.1 Circular Motion: Geometrical Method . 12 1.4.2 Circular Motion: Analytic Method . 14 1.5 Motion Of A Freely Falling Body . 15 1.6 Kinematics Problems . 20 1.6.1 One-Dimensional Motion . 20 1.6.2 Two and Three Dimensional Motion . 21 2 NEWTON'S FIRST AND THIRD LAWS: STATICS OF PARTICLES 25 2.1 Newton's First Law; Forces . 25 2.2 Inertial Frames . 28 2.3 Quantitative Definition of Force; Statics of Particles . 30 2.4 Examples of Static Equilibrium of Particles . 32 2.5 Newton's Third Law . 38 2.6 Ropes and Strings; the Meaning of \Tension" . 43 2.7 Friction . 52 2.8 Kinetic Friction . 63 2.9 Newton's First Law of Motion Problems . 67 3 NEWTON'S SECOND LAW; DYNAMICS OF PARTICLES 69 3.1 Dynamics Of Particles . 69 3.2 Motion of Planets and Satellites; Newton's Law of Gravitation 94 v CONTENTS CONTENTS 3.3 Newton's 2nd Law of Motion Problems . 101 4 CONSERVATION AND NON-CONSERVATION OF MO- MENTUM 105 4.1 PRINCIPLE OF CONSERVATION OF MOMENTUM . 105 4.2 Center of Mass . 111 4.3 Time-Averaged Force . 115 4.4 Momentum Problems . 122 5 WORK AND ENERGY 125 5.1 Definition of Work . 125 5.2 The Work-Energy Theorem .