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Vectorial mechanics - Brand
VECTORIAL MECHANICS
BY LOUIS BRAND
Professor of Mathematics University of Cincinnati
NEW YORK, JOHN WILEY & SONS, INC., 1930
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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 course," 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 thoroughly, 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. Moreover 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 applications, especially in engineering. A narrow utilitarianism, however, has been avoided; for as John Dewey has said, "It does not pay to tether one's 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 has acquired 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.
There are some departures from tradition in the material included in the text, The method of “index stresses” 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 kinematics 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 Newton's induction of the Law of Universal Gravitation finds a place, together with a very brief deduction of Kepler's 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 of all time have contributed to the development of mechanics, it is hoped that this book shadows forth a little of the beauty and profound imagination in their work. In the graceful words of Professor F. G. Donnan:
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 man's effort to understand that mysterious universe in which he lives. Without the presence of this power, the experimental discoverer might wander in the fields and pick the wild flowers of knowledge, but there would be no beautiful garden of understanding wherein the mind of man can find a serene delight.
The book contains more than can be given in a single course; but it is easy to arrange a course of any length from its subject matter. The introductory chapter in vector algebra may be studied as the course progresses; the summary of this chapter shows how little is actually needed for a working knowledge a knowledge which will prove of service in other parts of physics.
Although considerations of space have necessitated a brief treatment (or even the omission) of some important topics in mechanics, the author hopes that this book will also prove of service to the student in the years following his formal education.
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. Moreover 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 applications, especially in engineering. A narrow utilitarianism, however, has been avoided; for as John Dewey has said, "It does not pay to tether one's 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 has acquired 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.
There are some departures from tradition in the material included in the text, The method of “index stresses” 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 kinematics 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 Newton's induction of the Law of Universal Gravitation finds a place, together with a very brief deduction of Kepler's 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 of all time have contributed to the development of mechanics, it is hoped that this book shadows forth a little of the beauty and profound imagination in their work. In the graceful words of Professor F. G. Donnan:
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 man's effort to understand that mysterious universe in which he lives. Without the presence of this power, the experimental discoverer might wander in the fields and pick the wild flowers of knowledge, but there would be no beautiful garden of understanding wherein the mind of man can find a serene delight.
The book contains more than can be given in a single course; but it is easy to arrange a course of any length from its subject matter. The introductory chapter in vector algebra may be studied as the course progresses; the summary of this chapter shows how little is actually needed for a working knowledge a knowledge which will prove of service in other parts of physics.
Although considerations of space have necessitated a brief treatment (or even the omission) of some important topics in mechanics, the author hopes that this book will also prove of service to the student in the years following his formal education.
INTRODUCTION
Mechanics is the science which deals with the motion of bodies, including the special circumstances in which they remain at rest. The motion of a body is always referred to a frame of reference fixed in another body. Thus we would refer the motion of the connecting rod in a locomotive to the frame of the locomotive, the motion of a bullet to the earth, the motion of the earth to the sun, and the motion of the sun to a hypothetical rigid body supposed to be absolutely at rest. As nobody has ever been found in a state of "absolute rest," this term indeed being probably without any physical meaning, it is seen that mechanics is really concerned with the relative motions of bodies.
The motion of a body, consisting of a certain portion of matter, is described in terms of space and time, and regarded as being the result of certain forces acting on the body. Matter, space, time and force are the primitive and essentially undefinable concepts upon which the science of mechanics is based. Our experience and intuition form the source of our knowledge as to the nature of these concepts, and definitions and explanations only serve to give our ideas about them a greater precision.
Mechanics may be divided into two great branches: Kinematics and Dynamics.
That branch of mechanics which deals with the motion of bodies without reference to the forces acting on them is called Kinematics. Thus kinematics deals only with space and time and the concepts, such as velocity and acceleration, which are derived from them. Kinematics is geometry with the added element of time, and may therefore be described as the geometry of motion.
In Dynamics, however, the motion of bodies is considered in relation to the forces acting on them. Dynamics in turn is subdivided into Statics and Kinetics. Statics deals with the relation between the forces when the bodies considered remain at rest (relative to the frame of reference). If the bodies are moving or are set in motion by the forces acting on them the problem falls in the province of Kinetics.
Statics, although but a special case of kinetics, will be considered first in this book, as it forms, perhaps, the simplest part of mechanics, and leads the student by easy stages into the subject. Then kinematics, and finally kinetics, will be dealt with.
Both in kinematics and dynamics we have constantly to consider quantities that vary in time and space. Thus rates of change are of fundamental importance in mechanics, and require for their computation that branch of mathematics which is specially devoted to the study of continuous change the Calculus.
Finally the quantities that occur in mechanics are of two kinds, known as scalars and vectors. Scalar quantities are measured by the ordinary positive and negative numbers of arithmetic and algebra. Vector quantities, however, involve the idea of direction as well as magnitude; and in order to treat them simply and directly we shall begin with a chapter in the elements of vector algebra.
As very full cross references are given in this book, an article as well as a page number is given at the top of each page. Equations are referred to by article and number; thus ( 17, 2) means equation (2) in 17. Figures are given the number of the article in which they first appear, followed by a letter in case there is more than one in the article in question; Fig. 176, for example, is the second figure in 17. In the text all letters in heavy type denote vector quantities. In a figure, a heavy letter denotes the vector adjacent; a light letter near a vector denotes its length or magnitude.
After reading an article and carefully studying the solved examples, the student should next attack the problems as a test of his mastery of the subject-matter. Nearly all the answers of the problems are given in a list at the end of the book; this should be consulted after the solution is complete.
The following directions should be observed in solving problems.
1. Read the problem through carefully. If no figure is given, draw one, paying strict attention to the wording of the problem statement and lettering the drawing accordingly. (Draw, for example, the figure for Problem 2, 53.) If the lettering is not given, supply letters so that (a) the figure may be briefly referred to, (6) all unknown quantities have designations. It is frequently advisable to denote also the known quantities with letters and to solve the problem in general terms.
2. Plan a method of solution. If there are alternative methods, try to choose the shortest and most direct. Remember, however, that solving a problem in two ways affords a valuable check for correctness.
3. Draw free-body diagrams for all problems in dynamics. All forces acting on the body should be indicated by arrows pointing in the right directions. Do not forget the reactions of other bodies on the body in question. Frictional forces oppose impending slippage. In this connection the words smooth and rough mean respectively that friction is neglected or taken into account.
4. State briefly the principles of mechanics involved in the solution. The examples solved in the text show how this may be simply done. If several equations are involved it is well to number them for convenient reference.
5. If the problem consists of several parts, label each part of the solution.
6. Make a practice of solving problems in general terms. All known and unknown quantities are then designated by letters and the solution is effected algebraically. The solution should then be tested by the check of dimensions. Besides affording this valuable check, this method shows the structure of the final results and facilitates the numerical calculations effected by substituting numerical values in the formulas of the general solution. The arithmetic is thus postponed to the end of the problem, cancellations are readily made, and the final calculations carried out by slide-rule or four-place logarithms. The student may thus concentrate on mechanical principles in solving the problem, unhampered by the details of arithmetic. Although the student will not believe this at the outset, a general problem is often more readily solved than a specific one. To quote a saying of a great American scientist, J. Willard Gibbs The whole is simpler than its parts.
7. Carry out the solution neatly and systematically to the end. Careless or badly arranged work frequently leads to serious errors and is difficult to retrace in checking.
The motion of a body, consisting of a certain portion of matter, is described in terms of space and time, and regarded as being the result of certain forces acting on the body. Matter, space, time and force are the primitive and essentially undefinable concepts upon which the science of mechanics is based. Our experience and intuition form the source of our knowledge as to the nature of these concepts, and definitions and explanations only serve to give our ideas about them a greater precision.
Mechanics may be divided into two great branches: Kinematics and Dynamics.
That branch of mechanics which deals with the motion of bodies without reference to the forces acting on them is called Kinematics. Thus kinematics deals only with space and time and the concepts, such as velocity and acceleration, which are derived from them. Kinematics is geometry with the added element of time, and may therefore be described as the geometry of motion.
In Dynamics, however, the motion of bodies is considered in relation to the forces acting on them. Dynamics in turn is subdivided into Statics and Kinetics. Statics deals with the relation between the forces when the bodies considered remain at rest (relative to the frame of reference). If the bodies are moving or are set in motion by the forces acting on them the problem falls in the province of Kinetics.
Statics, although but a special case of kinetics, will be considered first in this book, as it forms, perhaps, the simplest part of mechanics, and leads the student by easy stages into the subject. Then kinematics, and finally kinetics, will be dealt with.
Both in kinematics and dynamics we have constantly to consider quantities that vary in time and space. Thus rates of change are of fundamental importance in mechanics, and require for their computation that branch of mathematics which is specially devoted to the study of continuous change the Calculus.
Finally the quantities that occur in mechanics are of two kinds, known as scalars and vectors. Scalar quantities are measured by the ordinary positive and negative numbers of arithmetic and algebra. Vector quantities, however, involve the idea of direction as well as magnitude; and in order to treat them simply and directly we shall begin with a chapter in the elements of vector algebra.
As very full cross references are given in this book, an article as well as a page number is given at the top of each page. Equations are referred to by article and number; thus ( 17, 2) means equation (2) in 17. Figures are given the number of the article in which they first appear, followed by a letter in case there is more than one in the article in question; Fig. 176, for example, is the second figure in 17. In the text all letters in heavy type denote vector quantities. In a figure, a heavy letter denotes the vector adjacent; a light letter near a vector denotes its length or magnitude.
After reading an article and carefully studying the solved examples, the student should next attack the problems as a test of his mastery of the subject-matter. Nearly all the answers of the problems are given in a list at the end of the book; this should be consulted after the solution is complete.
The following directions should be observed in solving problems.
1. Read the problem through carefully. If no figure is given, draw one, paying strict attention to the wording of the problem statement and lettering the drawing accordingly. (Draw, for example, the figure for Problem 2, 53.) If the lettering is not given, supply letters so that (a) the figure may be briefly referred to, (6) all unknown quantities have designations. It is frequently advisable to denote also the known quantities with letters and to solve the problem in general terms.
2. Plan a method of solution. If there are alternative methods, try to choose the shortest and most direct. Remember, however, that solving a problem in two ways affords a valuable check for correctness.
3. Draw free-body diagrams for all problems in dynamics. All forces acting on the body should be indicated by arrows pointing in the right directions. Do not forget the reactions of other bodies on the body in question. Frictional forces oppose impending slippage. In this connection the words smooth and rough mean respectively that friction is neglected or taken into account.
4. State briefly the principles of mechanics involved in the solution. The examples solved in the text show how this may be simply done. If several equations are involved it is well to number them for convenient reference.
5. If the problem consists of several parts, label each part of the solution.
6. Make a practice of solving problems in general terms. All known and unknown quantities are then designated by letters and the solution is effected algebraically. The solution should then be tested by the check of dimensions. Besides affording this valuable check, this method shows the structure of the final results and facilitates the numerical calculations effected by substituting numerical values in the formulas of the general solution. The arithmetic is thus postponed to the end of the problem, cancellations are readily made, and the final calculations carried out by slide-rule or four-place logarithms. The student may thus concentrate on mechanical principles in solving the problem, unhampered by the details of arithmetic. Although the student will not believe this at the outset, a general problem is often more readily solved than a specific one. To quote a saying of a great American scientist, J. Willard Gibbs The whole is simpler than its parts.
7. Carry out the solution neatly and systematically to the end. Careless or badly arranged work frequently leads to serious errors and is difficult to retrace in checking.
CONTENTS
CHAPTER I - ARTICLE VECTOR ALGEBRA
1. Scalars and Vectors
2. Equality of Vectors
3. Addition of Vectors
4. Negative of a Vector
5. Subtraction of Vectors
6. Multiplication of Vectors by Real Numbers
7. Point of Division
8. Vectors in a Plane
9. Vectors in Space
10. Component of a Vector on an Axis
11. Rectangular Axes
12. Centroid
13. Products of Two Vectors
14. Scalar Product of Two Vectors
15. Distributive Law for Scalar Products
16. Vector Product of Two Vectors
17. Distributive Law for Vector Products
18. Scalar Triple Product
19. Vector Triple Product
20. Summary, Chapter I
CHAPTER II - STATICS. FUNDAMENTAL PRINCIPLES
21. Force
22. Local and Standard Weight
23. Particles and Rigid Bodies
24. The Fundamental Principles of Statics
25. Principle A: Vector Addition of Forces
26. Computation of Resultant
27. Principle B : Transmissibility of a Force
28. Equivalent Systems of Forces
29. Resultant of Parallel Forces
30. Principle C: Static Equilibrium
31. Principle D: Action and Reaction
32. Contact Forces. Friction
33. Axial Stress
34. Summary, Chapter II
CHAPTER III - STATICS OF A PARTICLE
35. Equilibrium of a Particle
36. Free-Body Diagram
37. Scalar Conditions of Equilibrium
38. Equilibrium of Concurrent, Coplanar Forces
39. Equilibrium of Concurrent Forces in Space
40. Systems of Particles
41. Summary, Chapter III
CHAPTER IV - PLANE STATICS
42. Law of the Lever
43. Moment of a Force about an Axis
44. Computation of Moments
45. Moments about the Coordinate Axes
46. Theorems of Moments
47. Couples
48. Reduction of Coplanar Forces
49. Funicular Polygon
50. Resultant of Coplanar Forces
51. Center of Gravity
52. Equilibrium of a Rigid Body: Coplanar Forces
53. Three Forces in Equilibrium
54. Problem of Three Forces
55. Trusses
56. Statically Determinate Trusses
57. Stresses in Simple Structures
58. Index Stresses
59. Maxwell Diagrams
60. Method of Sections
61. Members Subject to Non-Axial Stress
62. Systems of Rigid Bodies
63. Cone of Friction
64. Journal Friction
65. Summary, Chapter IV
CHAPTER V - STATICS IN THREE DIMENSIONS
66. Moment of a Force about a Point
67. Theorem of Moments
68. Force-sum and Moment-sum
69. Moment of a Couple
70. Reduction of Forces Acting on a Rigid Body
71. Reduction in Special Cases
72. Equilibrium of a Rigid Body
73. Body with a Fixed Axis
74. Equivalent Systems
75. Resultant of Parallel Forces
76. Center of Gravity
77. Center of Gravity: Continuation
78. Centroids
79. Square-threaded Screw
80. Pivot Friction
81. Friction Clutches
82. Summary, Chapter V
CHAPTER VI - VECTOR CALCULUS
83. Derivative of a Vector
84. Derivatives of Sums and Products
85. Unit Tangent Vector
80. Curvature
87. Plane Curves
88. Integral of a Vector
89. Definite Integral
90. Summary, Chapter VI
CHAPTER VII - FLEXIBLE CABLES
91. Principle E: Rigidification
92. Flexible Cables
93. Scalar Equations of Equilibrium
94. String Stretched over a Smooth Surface
95. Rope or Belt Friction
96. Parabolic Cable
97. The Catenary
98. Cable with Supports on the Same Level
99. Cable with Supports on Different Levels
100. Concentrated Load on Cable
101. Summary, Chapter VII
CHAPTER VIII - KINEMATICS OF A PARTICLE
102. Speed
103. Space-Time Curve
104. Velocity
105. Acceleration
106. Rectangular Components of Velocity and Acceleration
107. Tangential and Normal Components of Acceleration
108. Rotation about a Fixed Axis
109. Circular Motion
110. Relative Motion
111. Summary, Chapter VIII
CHAPTER IX - KINEMATICS OF RECTILINEAR MOTION
112. Rectilinear Motion
113. Velocity-Time Curve
114. Velocity-Space Curve
115. Equation of Motion
116. Uniformly Accelerated Motion
117. Uniformly Accelerated Rotation
118. The Acceleration of Gravity
119. Simple Harmonic Motion
120. Summary, Chapter IX
CHAPTER X - KINEMATICS OF PLANE MOTION
121. Plane Motion of a Rigid Body
122. Translation and Rotation
123. Fundamental Kinematic Equations
124. Velocities in Plane Motion
125. Instantaneous Center
126. Centrodes
127. Accelerations in Plane Motion
128. Center of Acceleration
129. Velocity and Acceleration Images
130. Resume
131. Construction of Polar Velocity Image
132. Construction of Polar Acceleration Image
133. Relative Time Rates
134. Theorem of Coriolis
135. Instantaneous Center in Relative Motion
136. Kinematic Chains
137. The Criterion of Constraint
138. Inversion
139. The Theorem of Three Centers
140. Relative Angular Velocity
141. Theorem on Angular Velocity Ratios
142. Translatory Motions
143. Speed Ratios
144. Polar Diagrams of Velocity and Acceleration
145. Angular Velocity Ratio in Higher Pairing
146. Spur Gears
147. Involute Teeth
148. Summary, Chapter X
CHAPTER XI - DYNAMICS. FUNDAMENTAL PRINCIPLES
149. Orientation
150. Principle I: Force and Acceleration
151. Mass
152. Principle II: Vector Addition of Forces
153. Principle III: Action and Reaction
154. Calculation of Mass
155. Law of Inertia
156. Summary, Chapter XI
CHAPTER XII - DYNAMICS OF A PARTICLE
157. Scalar Equations of Motion
158. Sliding Friction
159. Two Particles
160. Differential Equations of Motion
161. Momentum and Impulse
162. Kinetic Energy and Power
163. Work
164. Graphical Representation of Work
165. Principle of Work and Energy
166. Conservation of Energy
167. Units and Dimensions
168. Free Motion under Gravity
169. Central Forces
170. Harmonic Motion
171. Simple Pendulum
172. Rectilinear Motion in a Resisting Medium
173. Damped Oscillations
174. Non-periodic Motion under Central Attraction
175. Forced Oscillations
176. Universal Gravitation
177. The Solar System
178. Summary, Chapter XII
CHAPTER XIII - ARTICLE DYNAMICS OF A SYSTEM OF PARTICLES PAGE
179. Two Basic Theorems
180. Conservation of Momentum
181. Center of Mass
182. Problem of Two Bodies
183. Moment of Momentum
184. Moment of Relative Momentum
185. Kinetic Energy
180, Work and Energy
187. Summary, Chapter XIII
CHAPTER XIV - DYNAMICS OF RIGID BODIES
188. Bo dies as Continuous Mass Distributions
189. Principle of Work and Energy for a Rigid Body
190. Kinetics of Translation
191. Rotation about a Fixed Axis
192. Comparison of Translation and Rotation
193. Moment of Inertia of Solids of Revolution
194. Transfer Theorem
195. Moment of Inertia of Thin Flat Plates
196. Application of Transfer Theorem
197. Physical Pendulum
198. Kinetics of Rotation
199. Center of Percussion
200. Torsion Pendulum
201. Uniform Rotation
202. Balance of Revolving Masses
203. Balancing by Two Masses in Given Axial Planes
204. Balance of Masses in S.H.M
205. Balance of Reciprocating Masses
206. Governors
207. Pendulum Governors
208. Characteristic Curve of a Governor
209. Shaft Governors
210. Kinetics of Plane Motion
211. Energy Equation in Plane Motion
212. Rolling Resistance
213. Kinematics of a Rigid Body
214. Kinetics of a Rigid Body with One Point Fixed
215. Equation of Energy
216. Composition of Angular Velocities
217. Gyroscope
218. Steady Precession
220. Statics of a. Rigid Body
221. Summary, Chapter XIV
CHAPTER XV - IMPACT
222. Fundamental Equations of Impact
223. Direct Impact of Spheres
224. The Restitution Equation
225. Impact in Cases of Plane Motion
226. Reduced Masses
227. Summary, Chapter XV
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