Augustus W. (Augustus William) Smith.

An elementary treatise on mechanics, embracing the theory of statics and dynamics, and its applications to solids and fluids online

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Online LibraryAugustus W. (Augustus William) SmithAn elementary treatise on mechanics, embracing the theory of statics and dynamics, and its applications to solids and fluids → online text (page 1 of 20)
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AN



ELEMENTARY TREATISE



MECHANICS,



EMBRACING THE



THEORY OF STATICS AND DYNAMICS,



AND ITS APPLICATION TO



SOLIDS AND FLUIDS.



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BY AUGUSTUS W. SMITH, LL.D.,



NEW YORK:

HARPER & BROTHERS, PUBLISHERS.

18 68.



1°[



*3l I
S



Entered, according to Act of Congress, in the year one thousand
eight hundred and forty-nine, by

Harper & Brothers,

ia the Clerk's Office of the District Court of the Southern District
of New York.



PREFACE,



The preparation of the present treatise was undertaken un-
der the impression that an elementary work on analytical Me-
chanics, suited to the purposes and exigencies of the course of
study in colleges, was needed. This impression is the result
of long experience in teaching, and a fair trial of all known
American works and reprints designed for such use.

It can scarcely be necessary, at this period, to assign at
length the reasons for adopting the analytical methods of in-
vestigation. Whether the object be intellectual discipline, or
a knowledge of facts and principles, or both, the preference
must be given to the modern analysis. It affords a wider field
for the exercise of judgment, calls more fully into exercise the
inventive powers, and taxes the memory less with unimportant
particulars, thus developing and strengthening more of the
mental faculties, and more equably by far than the geometric-
al methods. It is more universal in its application, shorter in
practice, and far more fruitful in results. It is, indeed, the only
method by which the student can advance beyond the merest
rudiments of the science, without an expense of time and en-
ergy wholly disproportioned to the ends accomplished — the
only method by which he can acquire a self-sustaining and a
progressive power.

As the hope of furnishing to the student some additional
facilities for a pleasant and profitable prosecution of this
branch of study was the motive for undertaking the prepara-
tion of this manual, it will be proper,to refer to some of the

2



PREFACE.



specific objects had in view. The most formidable obstacles
to the acquisition of any branch of science are generally found
at the very outset. It has therefore been a specific object to
introduce the subject by giving distinctness to the elementary
truths, dwelling upon them till they are rendered familiar,
adopting the simplest mode of investigation and proof consist-
ent with rigor of demonstration, and avoiding all reference to
the metaphysics of the science as out of place in a work de-
signed for beginners. At every successive stage of advance-
ment the student is required to review the ground passed over
by the use of the principles learned, in the solution of exam-
ples which will require their application, and test at once his
knowledge of them and his ability to apply them. Having
examined in this way each division of forces as classified,
more general methods are introduced, and their application
illustrated by numerous problems. These methods are often
employed in particular cases when others less general would
be shorter ; for it is a readiness in their use and a familiarity
with their application that gives to the student his power over
more difficult and complicated questions.

To adapt the work to the exigencies of the recitation-room,
the whole is divided, at the risk, perhaps, of too much apparent
formality, into distinct portions or propositions, suitable for an
individual exercise. In each case the object to be accomplish-
ed is distinctly proposed, or the truth to be established is
briefly and clearly enunciated. The student has thus a defin-
ite object before him when called upon to recite, acquires a
convenient formulary of words by which to quote and apply
his arguments, and the more clearly conceives and marks his
own progress.

In this institution the mathematical and experimental courses
are assigned to different departments. It is not, however, for
this reason alone that I.have purposely abstained from swell-



PREFACE. ill

ing the volume with diffuse verbal explanations, and the intro-
duction of experimental illustrations. The experienced teach-
er will always have at command an abundance of matter of
this kind to meet every emergency, and can adapt the kind
and mode of illustration to the specific difficulty that arises in
the mind of his pupil. There is an objection to the introduc
tion of such matter, growing out of a tendency, in some at
least, to become the passive recipients of their mental aliment.
These mental dyspeptics loathe that which costs them labor,
and, satisfied with the stimulus of inflation, seize upon the
lighter portions, and neglect that which alone can impart
vigor to their mental constitution. Whoever caters for this
class of students must share in the responsibility of raising up
a race effeminate in mind, if not in body.

The investigations are limited to forces in the same plane,
except in the case of parallel forces. This limit, while it is
sufficiently ample to include the most important topics of ter-
restrial mechanics, and to embrace an interesting field of celes-
tial mechanics, is fully sufficient to occupy the time which the
crowded course of study in our colleges will admit of being
appropriated to it. It may, without marring the integrity of
the work, be very much reduced by the omission of all those
portions in which the integral calculus is employed ; and still
further, if desired, by omitting all that relates to the principle
of virtual velocities, and its application to the mechanical
powers. A copious analysis is given in the contents, conven-
ient for frequent and rapid reviews, and suggestive of ques-
tions for examination.

In this work no claim is advanced to originality. The ma-
terials have been sought and freely taken from all available
sources. Nearly all the matter, in some form, is found in al-
most every author consulted, and credit could not be given in
every case, if, in an elementary work designed as a text-book,



IV PREFACE.

it were desirable to do so. In the portion on Statics, I am
most indebted to the excellent introductory work of Profess-
or Potter, of University College, London. The chapter on
Couples is substantially taken from Poinsot's Elemens de Sta-
tique. The questions, generally simplified in their, character,
are mostly taken from Walton's Mechanical and Hydrostatical
Problems, and Wrigley and Johnston's Examples. The works
more especially consulted are those of Poisson, Francaeur,
Gregory, Whewell, Walker, Moseley, and Jamieson. Some-
thing has been taken from each, but modified to suit the spe-
cific object kept constantly in view — the preparation of a
manual which should be simple in its character, would most
naturally, easily, and successfully induct the student into the
elementary principles of the science, and prepare him, if so
disposed, to prosecute the study further, without the necessity
of beginning again and studying entirely new methods. How
far I have succeeded must be left to the decision of others,
especially of my co-laborers in this department of instruction.

Wesleyan University,
Middletown, Conn., Jan., 1849.



I^H1




CONTENTS.



Irt. t*&

INTRODUCTION 1

1. Definition of Mechanics and its Subdivisions.

2. Definition of Force and its Mechanical Effects.

3. Definition of its Intensity and Measure, its Direction and Point of Application.

4. Definition of Analytical Mechanics.

5. Definition of Concurring and Conspiring Forces.

6. Definition of Body, Rigid, Flexible, and Elastic.



STATICS.

CHAPTER I.

COMPOSITION AND EQUILIBRIUM OF CONCURRING FORCES IN THE SAME PLANE. 3

7. Two equal and opposite Forces in Equilibrium.

8. Two Forces inclined to each other can not Equilibrate.

9. Definition of Resultant and Components.

10. Resultant of several Conspiring Forces.

11. Resultant of two unequal opposite Forces.
12 Resultant of any number of opposite Forces.

13. Point of Application at any Point in its Direction.

14. Direction of the Resultant of several Forces.

15. Direction of the Resultant of two equal Forces.
1C. Direction of the Resultant of two unequal Forces.

17. Variation of the Magnitude of the Resultant and its Components.

18. Equilibrium of three equal Forces.

19. Resultant of two equal Forces at an Angle of 120°.

20. Each of three equilibrating Forces equal to the Resultant of the other two in

Magnitude.

21. Parallelogram of Forces.

22. Triangle of Forces.

23. Resolution of Forces.

24. Polygon of Forces.

25. Representation of equilibrated Forces.

26. Graphical determination of Resultant.

27. Parallelopiped of Forces.

28. Ratios of three equilibrated Forces.

29. Expression for the Resultant of two Forces.

30. Definition of Moment of a Force — Origin of Moments.

31. Equality of the Moment of the Resultant with the Sum of the Moments of two

Components.
3-2. Equality of the Moment of the Resultant with the Sum of the Momenta ol any
member of Components.

33. When the Origin of Moments is fixed.

34. When the Forces are in Equilibrium.

35. Examples.



VI CONTENTS.

Art. Pag,

CHAPTER II.

PARALLEL FORCES 15

'J6. Resultant of two Parallel Forces.

37. Definition of Amis of Forces.

38. Equilibrium of two Parallel Forces by a third Force.

39. Point of Application of the Resultant.

40. When the Forces act in opposite Directions.

41. When the Forces are Equal and Opposite.

42. Such Forces constitute a Statical Couple.

43. Resultant of any Number of Parallel Forces.

44. Definition of Center of Parallel Forces.

45. Equality of the Moment of the Resultant with the Sum of the Moments of the

Components.
40. Definition of Moment of a Force- in reference to a Plane.

47. Conditions of Equilibrium of any Number of Parallel Forces.

48. Condition of Rotation.

49. When in Equilibrium, each Force equal in Magnitude to the Resultant of all the

others.

50. Equilibrium independent of their Direction.

51. Examples.

CHAPTER III.

THEORY OF COUPLES 24

52. Definition of a Statical Couple.

53. Definition of the Arms of a Couple.

54. Definition of the Moment of a Couple.

55. A Couple may be turned round in its own Plane.

56. A Couple may be removed parallel to itself in its own Plane.

57. A Couple may be removed to a Parallel Plane.

58. Couples are equivalent when their Planes are Parallel and Moments are EquaL

59. Couples may be changed into others having Amis of a given Length.
GO. Definition of the Axis of a Couple.

61. Properties of an Axis.

62. Definition of the Resultant of two or more Couples.

63. Equality of the Moment of the Resultant with the Sum of the Moments of the

Components.

64. Equality of the Axis of the Resultant with the Sum of the Axes of the Com-

ponents.

65. The Resultant of two Couples inclined to each other.

66. Representative of the Axis of the Resultant of two Couples.

67. Parallelogram of Couples.

CHAPTER IV.

ANALYTICAL STATICS IN TWO DIMENSIONS 31

68. Resultant of any Number of Concurring Forces.

69. Directions of the Rectangular Components involved in their Trigonometrical Values

70. Conditions of Equilibrium of Concurring Forces.

71. Resultant Force and Resultant Couple when the Forces do not concur.

72. Construction of the Results.

73. Equation of the Resultant.

74. Equilibrium of non-concurring Forces.

75. Equilibrium when there is a fixed Point in the System,

76. Equilibrium of a Point on a Plane Curve.



CONTENTS. Vll

trt Page

77. Conditions of Equilibrium.

78. Definition of Virtual Velocitius.

79. Principle of Virtual Velocities.

80. Principle of Virtual Velocities obtains in Concurring Forces in the same Plane.

, 81. Principle of Virtual Velocities obtains in non-concurring Forces in the same Plane.
L»2. The Converse.

CHAPTER V.

CENTER OF GRAVITY 43

83. Definition of Gravity.

84. Laws of Gravity.

85. Definition of a Heavy Body.

66. Definition of the Weight of a Body.

67. Definition of its Mass.

88. Expression for Weight.

89. Definition of Density.

00. Another Expression for Weight.

91. Relations of Masses to Volumes of the same Density.

92. Relations of Densities to Volumes of the same Mass.

93. Relations of Densities to Masses of the same Volume.

94. Definition of Center of Gravity.

95. Connection of the Center of Gravity with the Doctrine of Parallel Forces.

96. Definition of a Body symmetrical with respect to a Plane.

97. Position of its Center of Gravity.

98. Definition of a Body symmetrical with respect to an Axis.

99. Position of its Center of Gravity.

100. Center of Gravity of a Body symmetrical with respect to two Axes.

101. Definition of Center of Figure.

102. Center of Gravity of any Number of heavy Particles.

103. Their Center of Gravity when their Positions are given by their Co-ordinates

104. Their Center of Gravity when they are all in the same Line.

105. Their Center of Gravity when they are Homogeneous.

106. The Center of Gravity of the Whole and a Part given to find that of the other Part.

107. Examples — 1. Of a Straight Line — 2. Triangle— 3. Parallelogram — 4. Polygon —

5. Triangular Pyramid — 6. Any Pyramid — 7. Frustum of a Cone — 8. Perimeter
of a Triangle — 9. Of a Triangle in Terms of its Co-ordinates.

CONDITIONS OF EQUILIBRIUM OF BODIES FROM THE ACTION OF GRAVITY... 54

108. When the Body has a Fixed Point in it.

109. Definition of Stable, Unstable, and Neutral Equilibrium.

110. Position of the Centerof Gravity when the Equilibrium is Stable aud when Unstable.

111. Pressure on the Fixed Point.

112. Position and Pressure when there are two Fixed Points.

113. Position and Pressure when there are three Fixed Points.

114. Position and Pressure when a Body touches a Horizontal Plane in one Point.

115. Position and Pressure when a Body touches a Horizontal Plane in two Points.

116. Position and Pressure when a Body touches a Horizontal Plane in three Points

117. Position and Pressure when a Body touches a Horizontal Plane in any Number of

Points.

118. Measure of the Stability on a Horizontal Plane.

119. Case of a Body on an Inclined Plane.

120. Examples.

APPLICATION OF THE INTEGRAL CALCULUS TO THE DETERMINATION OF THE CEN.
TER OF GRAVITY 53



Vlil CONTENTS.

Art P'S*

121. General differential Expressions for the Co-ordimites of the Center of Gravity.

122. General differential Expressions for the Center of Gravity of a Plane Curve.

123. General differential Expressions for the Center of Gravity of a Plane Area.

124. General differential Expressions for the Co-ordinates of a Surface of Revolution.

125. General differential Expressions for the Co-ordinates of a Solid of Revolution.

126. Determination of a Surface of Revolution.

127. Determination of a Solid of Revolution.

128. Examples— 1. Of a Circular Arc— 2. Circular Segment— 3. Surface of a Spherical

Segment — 4. Spherical Segment.

129. Examples on the preceding Chapters.

CHAPTER VI.

THE MECHANICAL POWERS 7fe

130. Classification of the Mechanical Powers.

§ I. THE LEVER.

131. Definition of a Lever.

132. Kinds of Lever.

133. Conditions of Equilibrium when the Forces are Parallel

134. Conditions of Equilibrium when the Forces are Inclined.

135. Conditions of Equilibrium when the Lever is Bent or Curved.

136. Conditions of Equilibrium when any Number of Forces in the same Plane act on

a Lever of any Form — Examples.

§11. WHEEL AND AXLE 86

137. Definition of Wheel and Axle.

138. Conditions of Equilibrium when two Forces act Tangentially to the Surface of the

Wheel and Axle.

139. Perpetual Lever.

140. Pressure on the Axis.

141. Conditions of Equilibrium of any Number of Forces.

142. Definition of Cogged Wheels, Crown Wheels, Beveled Wheels, Piuions, Leaver.

143. Conditions of Equilibrium in Cogged Wheels.

144. Conditions of Equilibrium when the Cogs are of equal Breadth.

145. Conditions of Equilibrium in Cogged Wheels and Pinions.

§ III. the cord 91

146. Definition of the Cord— Of Tension.

147. Conditions of Equilibrium when there are three Forces.

148. Conditions of Equilibrium when the Ends are fixed.

149. Conditions of Equilibrium when the Ends are fixed and a third Force applied to

a Running Knot.

150. Conditions of Equilibrium when there is any number of Forces.

151. Relations of the Forces when in Equilibrium.

152. Relations of the Forces when the Ends are Fixed and the Forces Parallel.

153. Relations of the Forces when the Ends are Fixed and the Forces are Weights.

154. Point of Application of the Resultant.

155. Catenary — Examples.

§ IV. the pulley •-■• 88

156. Definition of the Pulley— Fixed and Movable.

157. Use of Fixed Pulley.

158. Equilibrium in single Movable Pulley.

159. Systems of Pulleys.

160. Equilibrium in first System.
161 Equilibrium in the second.

If2. Equilibrium in the third — Examples.



CONTENTS. IX

Art P"gl

§ V. THE INCLINED PLANE 108

163. Definition of Inclined Plane.

164. Equilibrium when the Body is sustained by a Force acting in any Direction.

165. Equilibrium when the Body is sustained by a Force parallel to the Plane.

166. Equilibrium when the Body is sustained by a Force parallel to the Base of the Plane.

167. Equilibrium when two Bodies rest on two Inclined Planes.

§ VI. THE WEDGE 103

168. Definition of the Wedge — Faces, Angle Back.

169. Conditions of Equilibrium in the Wedge.

170. Defect in the Theory.

171. Illustrative Problem.

§ VII. THE screw 103

172. Definition of the Screw.

173. Conditions of Equilibrium in the Screw.

$ VIII. BALANCES AND COMBINATIONS OF THE MECHANICAL POWERS.. 107

174. The common Balance.

175. Requisites for good Balance.

176. Conditions of Horizontality of the Beam.

177. Conditions of Sensibility.

178. Conditions of Stability.

179. Relations of the Requisites.

180. Steelyard Balance.

181. Law of Graduation.

182. Bent Lever Balance.

183. Law of Graduation.

184. Roberval's Balance.

185. Condition of Equilibrium.

186. Conditions of Equilibrium in a Combination of Levers.

187. Conditions of Equilibrium in the Endless Screw.

188. Conditions of Equilibrium in any Combination of the Mechanical Powers-

189. Conditions of Equilibrium in the Knee.

*— CHAPTER VII.

APPLICATION OF THE PRINCIPLE OF VIRTUAL VELOCITIES TO THE MECHANICAL
POWERS 116

190. Preliminary Considerations.

191. Application to the Wheel and Axle.

192. Application to Toothed Wheels.

193. Application to Movable Pulley with Parallel Cords.

194. Application to the first System of Pulleys.

195. Application to the second System.

196. Application to the third System.

197. Application to the Inclined Plane.

198. Application to the Wedge.

199. Application to the Lever of any Form.

200. Application to the single Movable Pulley with inclined Cords.

CHAPTER VIII.

FRICTION 134

201. Definition of Friction— Kinds.

202. Measurement of Friction.

203. Laws of Friction.

S04. Value of the Coefficient of Friction.



* CONTENTS.

Art. Pif.

205. Limits of the Ratio of the Power to the Weight ou the Inclined Plane.

206. Limits of the Ratio of the Power to the Weight in the Screw.

207. One Limit obtained directly from the other. — Examples.
203. Examples on Chapters VI., VII., and VIII.



DYNAMICS.

INTRODUCTION *.. 181

209. In Dynamics, Time an Element.

210. Definition of Motion.

211. Definition of Absolute Motion.

212. Definition of Relative Motion.

213. Definition of Velocity — Its Measure.

214. Definition of Variable Velocity — Its Measure.

215. Definition of Relative Velocity.

216. Definition of Inertia — First Law of Motion.

217. Definition of Center of Inertia.

218. Definition of the Path of a Body.

219. Definition of Free and Constrained Motion.

220. Definition of an Impulsive Force.

221. Definition of an Incessant Force.

222. Definition of a Constant Force — Its Measure.

223. Definition of a Variable Force — Its Measure.

224. Definition of Momentum — Its Measure — Of Living Force.

225. Definition of a Moving Force — Its Measure.

226. The second Law of Motion.

227. The third Law of Motion.

CHAPTER I.

UNIFORM MOTION j 3fi

228. Point to which the Force must be applied.

229. General Equation of Uniform Motion.

230. Relation of Spaces to Velocities when the Times are Equal.

231. Relation of Spaces to the Times when the Velocities are Equal.

232. Relation of Velocities to the Times when the Spaces are EquaL

233. Measure of an Impulsive Force.

234. The Velocity resulting from the Action of several Forces.

235. Parallelogram of Velocities.

236. Rectangular Composition and Resolution of Velocities.

237. Relations of Space, Time, and Velocity of two Bodies moving in the same Straight

Line.

238. Relations of Space, Time, and Velocity of two Bodies moving in the Circumference

of a Circle.

239. Examples.

CHAPTER II.

IMPACT OF BODIES 143

210. Definition of Direct, Central, and Oblique Impact.

241. Definition of Elasticity — Perfect — Imperfect — Its Modulus.

242. Definition of Hard and Soft.

243. Velocity of two Inelastic Bodies after Impact.

244. Loss of Living Force in the Impact of Inelastic Bodies.

245. Velocities of imperfectly Elastic Bodies after Impact.



CONTENTS. X

Art. Pa^t

246. Velocity of the nth Body in a Series of perfectly Elastic Bodies.

247. Velocity of the common Center of Gravity before and after Impact.

248. Conservation of the Motion of the Center of Gravity.

249. Definition of Angles of Incidence and Reflection.

250. Motion of an Inelastic Body after Oblique Impact on a Hard Plane.

251. Motion of an Elastic Body after Oblique Impact on a Hard Plane.

252. Direction of Motion before Impact, that a Body after Impact may pass through a

given Point.

253. Measure of the Modulus of Elasticity.

254. Mode of determining it — Table of Moduli.

255. Examples.

CHAPTER III.

MOTION FROM THE ACTION OF A CONSTANT FORCE 153

256. Uniformly accelerated Motion — Acquired Velocity.

257. Space in Terms of the Force and Time.

258. Space in Terms of the Force and Velocity.

259. Space described in the last n Seconds.

260. The Velocity and Space from the joint Action of a Projectile and Constant Force.

261. The Velocity when the Space is given.

262. Velocity lost and gained by the Action of a Constant Force when the Space is

the same.
{(63. Scholium on Universal Gravitation.

264. Scholium on the Numerical Value of the Force of Gravity.
°.65. Examples.

' CHAPTER IV.

P ROJ E CT I L E 3 161

266. The Path of a Projectile is a Parabola.

267. Equation of the Path when referred to Horizontal and Vertical Axes.

268. Definition of Horizontal Range — Time of Flight — Impetus.

269. Time of Flight on a Horizontal Plane.

270. Range on a Horizontal Plane — The same for two Angles of Elevation.

271. Greatest Height.

272. Range and Time of Flight on an Inclined Plane, and Co-ordinates of Point of Impact

273. Formula for Velocity of a Ball or Shell.

274. Examples.

CHAPTER V.

CONSTRAINED MOTION.
$ I. MOTION ON INCLINED PLANES 169

275. Relations of Space, Time, and Velocity.

276. Velocity down the Plane and its Height.

277. Times down Inclined Planes of the same Height.

278. Relations of Space, Time, and Velocity when projected up or down the Plane

279. Time of Descent down the Chords of a Circle.

280. Straight Line of quickest Descent from a Point within a Circle to its Circumference.

281. Straight Line of quickest Descent from a given Point to an Inclined Plane.

282. Motion of two Bodies suspended by a Cord over a Fixed Pulley.

283. Motion of two Bodies when the Inertia of the Pulley is considered.

§ II. MOTION IN CIRCULAR ARCS 173

284. Velocity acquired down the Arc of a Circle.

285. Velocity lost in passing from one Side of a Polygon to the next



\



XU CONTENTS.

Art. Page

286. Velocity lost when the Sides are Infinite in Numler.

267. Direction and Intensity of an Impulse at each Angle, to make a Body describe a
Polygon with a uniform Velocity.

288. Direction and Intensity when the Polygon becomes a Circle.

289. Definition of Centrifugal and Centripetal Force.

290. Discussion of the Motion of a Body in a Circle by the Action of a Central Force.


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Online LibraryAugustus W. (Augustus William) SmithAn elementary treatise on mechanics, embracing the theory of statics and dynamics, and its applications to solids and fluids → online text (page 1 of 20)