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Theoretical mechanics - Smith, Longley
THEORETICAL MECHANICS
BY PEECEY E. SMITH, Ph.D. AND WILLIAM EAYMOND LONGLEY, Ph.D.
GINN AND COMPANY, 1910
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PREFACE
The study of Mechanics as presented in this volume is founded upon a course in mathematics extending through the Calculus. It is assumed, moreover, that the student has already become familiar with the fundamental ideas of force, energy, and work through such preliminary courses as are included in textbooks on General Physics. In short, this volume presents the subject of Mechanics in that relation to other mathematical subjects which has become established in the curricula of the technical schools of this country. It should be emphasized, however, that the volume includes, for purposes of review, a discussion of the fundamental notions and many simple exercises involving these notions.
Attention may be called to the arrangement in the text. This arrangement is founded upon experience in teaching the subject for many years in the Sheffield Scientific School of Yale University. In 1903 Professor E. R. Hedrick prepared a mimeographed text which followed the conventional arrangement of treating statics first. This text was used for one year. It then developed that an obvious disadvantage existed in not taking up directly upon the conclusion of the study of the Integral Calculus the calculation of the integrals of Mechanics involving centers of gravity and moments of inertia. The point was that this formal integration out of the way, the continuous study of Mechanics proper need not afterwards be interrupted. Acting upon this conviction, the present text was prepared essentially as here published in 1907, and has since that time been used in mimeographed form. The general plan of the arrangement is that a single problem may at any one time be under discussion. Thus, when the question of energy of rotation is solved, the appearance of the moment of inertia integral presents no complication. This has been disposed of already. Similarly, the equations of motion presenting themselves as solutions of the force equations have been previously discussed. Another feature is the departure from convention by arranging types of motion under the corresponding fields of force. In this way it is made clear that the emphasis is to be laid upon the force and velocity of projection.
In the case of a book which, like the present volume, has been long in the making, it is difficult to record definite acknowledgments of aid and indebtedness. There are included in the text many problems suggested by past and present members of the mathematical department of the Sheffield Scientific School. Further, the text has been the subject of discussion at frequent departmental conferences, and for all suggestions received on these occasions the authors gratefully here record their thanks. The diagrams were skillfully prepared by Mr. S. J. Berard of the department of mechanical engineering.
Attention may be called to the arrangement in the text. This arrangement is founded upon experience in teaching the subject for many years in the Sheffield Scientific School of Yale University. In 1903 Professor E. R. Hedrick prepared a mimeographed text which followed the conventional arrangement of treating statics first. This text was used for one year. It then developed that an obvious disadvantage existed in not taking up directly upon the conclusion of the study of the Integral Calculus the calculation of the integrals of Mechanics involving centers of gravity and moments of inertia. The point was that this formal integration out of the way, the continuous study of Mechanics proper need not afterwards be interrupted. Acting upon this conviction, the present text was prepared essentially as here published in 1907, and has since that time been used in mimeographed form. The general plan of the arrangement is that a single problem may at any one time be under discussion. Thus, when the question of energy of rotation is solved, the appearance of the moment of inertia integral presents no complication. This has been disposed of already. Similarly, the equations of motion presenting themselves as solutions of the force equations have been previously discussed. Another feature is the departure from convention by arranging types of motion under the corresponding fields of force. In this way it is made clear that the emphasis is to be laid upon the force and velocity of projection.
In the case of a book which, like the present volume, has been long in the making, it is difficult to record definite acknowledgments of aid and indebtedness. There are included in the text many problems suggested by past and present members of the mathematical department of the Sheffield Scientific School. Further, the text has been the subject of discussion at frequent departmental conferences, and for all suggestions received on these occasions the authors gratefully here record their thanks. The diagrams were skillfully prepared by Mr. S. J. Berard of the department of mechanical engineering.
CONTENTS
CHAPTER I. MOMENTS OF MASS AND INERTLA
1. Center of gravity
2. Moment of area
3. Symmetry
4. Theorem on the center of gravity
5. Moment of mass (solids of revolution)
6. Moment of mass (particular solids)
7. Moment of mass. Any solid
8. Principle of combination
9. Center of gravity of an arc
10. Theorems of Pappus
11. Moment of inertia. Plane areas
12. Theorems on moments of inertia
13. Further theorems
14. Polar moment of inertia
15. Flat thin plates or laminae
16. Solids of revolution
17. Moments of inertia of solids in general
18. Parallel axes
19. Relation between moment of inertia of a beam and polar moment of a right section
20. Combined solids and areas
21. Routh's rules
22. System of material particles
23. Ellipse of inertia
CHAPTER II. KINEMATICS OF A POINT. RECTILINEAR MOTION
24. Motion on a straight line
25. Velocity
26. Acceleration
27. Distance-time diagram. Discussion
CHAPTER III. KINEMATICS OF A POINT. CURVILINEAR MOTION
28. Position in a plane or in space. Vectors
29. Addition of vectors
30. Subtraction of vectors
31. Multiplication of a vector by a scalar
32. Resolution of plane vectors
33. Vectors in space
34. Displacement in a plane. Path
35. Velocity in the plane. Velocity curve
36. Acceleration in a plane
37. Motion in space
38. Discussion of any motion
39. Motion in a prescribed path
40. Tangential and normal accelerations
41. Equations in polar coordinates
42. Rotation
CHAPTER IV. KINETICS OF A MATERIAL PARTICLE
43. Momentum
44. Force
45. Units of force
46. Rectilinear motion
47. Resultant force in rectilinear motion
48. Curvilinear motion. Axioms on force action. Concurrent forces
49. Curvilinear motion
50. Intrinsic force equations
51. Polar equations
CHAPTER V. WORK, ENERGY, IMPULSE
52. Work
53. Kinetic energy
54. Constrained motion. Dynamic pressure
55. Units of work and energy. Power
56. Impulse
57. Impact
58. Force-moments in a plane
59. Moment of momentum
60. Angular momentum
61. Fundamental equations
62. Formulas in dynamics of a particle
CHAPTER VL MOTION OF A PARTICLE IN A CONSTANT FIELD
63. Field of force
64. Rectilinear motion under a constant force
65. Curvilinear free motion
66. Constrained motion
67. Inclined plane
68. Motion on a smooth circle. Simple pendulum
69. Motion on a smooth cycloid
70. Seconds pendulum
CHAPTER VII. CENTRAL FORCES
71. Central field of force
72. Areal velocity
73. Law of areas for central forces
74. Converse of the theorem of areas
75. The energy equation
76. Circular orbits
77. Differential equation of the orbit
78. Determination of the orbit when the law of force is known
79. Position in the orbit
80. Complete solution of a problem in central motion
81. Planetary motion. The law of gravitation
CHAPTER VIII. HARMONIC FIELD
82. Harmonic central field
83. Energy equation
84. Simple harmonic motion
85. Composition of simple harmonic motions in a given field
86. Composition with different periods. Forced vibrations
87. General harmonic field
CHAPTER IX. MOTION IN A RESISTING MEDIUM
88. Law of resistance
89. Constant field. Resistance proportional to the square of the velocity
90. Damped harmonic motion. Resistance varying as velocity
CHAPTER X. POTENTIAL AND POTENTIAL ENERGY
91. Potential
92. Conservative field
93. Potential energy. Conservation of energy
94. Equipotential lines and lines of force
95. Non-conservative forces. Friction
96. Newtonian potential
CHAPTER XI. SYSTEM OF MATERIAL PARTICLES
97. System in a plane
98. System in space
99. Moment equation for a system of particles
100. Work and energy of the system
101. Rigid system of particles
CHAPTER XII. DYNAMICS OF A RIGID BODY
102. Rigid body
103. Translation
104. Rotation
105. Uniplanar motion
106. Centrodes
107. Screw motion
108. Force equations. Work and energy
109. Kinetic energy
110. Moment equation in rotation
111. Comparison of formulas in translation and rotation
112. Fundamental equations for uniplanar motion
CHAPTER XIII. EQUILIBRIUM OF COPLANAE FORCES
113. Equilibrium of forces
114. Analytic conditions for equilibrium of coplanar forces
116. General method of solving problems in equilibrium
116. Friction
117. Equilibrium of flexible cords
118. The common catenary
119. Load distributed uniformly along the horizontal
120. Stability
CHAPTER XIV. COLLECTION OF FORMULAS
Formulas from algebra
Formulas from geometry
Formulas from trigonometry
Formulas from analytic geometry
Formulas from calculus
Differential equations
Formulas for differentiation
Table of integrals
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