Vectorial Mechanics

VECTOEIAL MECHANICS PREFACE This book has been designed as an introductory textbook in mechanics for students of engineering and of physics. It is hoped, moreover, that it will serve as a book of reference to those who, not content with merely passing the couree, wish to gain a fundamental understanding of a fundamental science. Also those students have been kept in mind who, Treed from the necessity of making grades, wish to review mechanics thoughtfully and thor oughly, beyond the point of having a few pat rules for solving special types of problems. The entire subject has been developed from three general principles, and no pains have been spared to show how they support the superstructure. I have aimed at an exposition as simple and direct as is consistent with a respectable standard of rigor. More over every part of the theory is fully illustrated by examples and accompanied by a large and varied collection of problems. Each chapter is followed by a concise summary of the principal results this should enable the student to see the woods in spite of the trees. The subject matter has been chosen with a view to its applica tions, especially in engineering. A narrow utilitarianism, how ever, has been avoided for as John Dewey has said, It does not pay to tether ones thought to the post of usefulness with too short a rope. Finally, while fitting the beams and columns into the structure of mechanics, my first concern has been their rigidity and strength but I have not been totally unmindful of the architecture. The order of the book is Statics, Kinematics, Kinetics. This is roughly the historical order of development. Possibly the individual learns mechanics in the same way the race hasacquired it. At any rate this order, though not the best on purely logical grounds, leads the student by easy stages into the more difficult parts of the subject. Statics is founded upon four basic principles, kinetics upon three. Chapter XIV closes by showing that the principles of statics are contained in the principles of kinetics Force and Acceleration, Vector Addition of Forces, Action and Reaction. iii IV PREFACE There are some departures from tradition in the material in cluded in the text, The method of index stresses J is developed in Chapter IV. Flexible cables are treated from a uniform point of view in Chapter VII. Chapter X in plane kinematics is fairly complete and could form the substance of a course on the kin ematics of machinery. Attention is called to the simple proof of the Theorem of Coriolis, a proof so worded that it also applies to the most general case. In Chapter XII a thorough treatment of free, damped, and forced vibrations is given without presupposing a knowledge of differential equations. The importance of these topics in engineering led to their inclusion. Here also Newtons induction of the Law of Universal Gravitation finds a place, to gether with a very brief deduction of Keplers Laws, As this deduction is perhaps the greatest single achievement in classical mechanics, it is hoped that its inclusion will not be taken amiss. In Chapter XIV, on rigid dynamics, the essential facts on the balancing of both revolving and reciprocating masses are simply obtained. A brief discussion of the kinematics of a rigid body is then followed by a treatment of gyroscopic motion, leading to the result of greatest technical importance. Since some of the greatest minds ofall time have contributed to the development of mechanics, it is hoped that this book shad ows forth a little of the beauty and profound imagination in their work. In the graceful words of Professor F. G. Donnan 11 The power of rigorous deductive logic in the hands of a mathematician of insight and imagination has always been one of the greatest aids in mans effort to understand that mysterious universe in which he lives...