An Elementary Treatise on Mechanics: Embracing the Theory of Statics and Dynamics, and Its Applications to Solids and Fluids

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Harper, 1855 - 307 pages
 

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Contents

Resultant of any number of opposite Forces
12
Point of Application at any Point in its Direction
13
Direction of the Resultant of several Forces
14
Direction of the Resultant of two equal Forces
15
Direction of the Resultant of two unequal Forces
16
Variation of the Magnitude of the Resultant and its Components
17
Equilibrium of three equal Forces
18
Resultant of two equal Forces at an Angle of 120
20
Parallelogram of Forces
21
Triangle of Forces
22
Resolution of Forces
23
Polygon of Forces
24
Representation of equilibrated Forces
25
Graphical determination of Resultant
26
Parallelopiped of Forces
27
Ratios of three equilibrated Forces
28
Expression for the Resultant of two Forces
29
Definition of Moment of a ForceOrigin of Moments
30
Equality of the Moment of the Resultant with the Sum of the Moments of
31
Equality of the Moment of the Resultant with the Sum of the Moments of
32
number of Components 33 When the Origin of Moments is fixed 34 When the Forces are in Equilibrium
33
Examples
35
Definition of Arms of Forces
37
Equilibrium of two Parallel Forces by a third Force
38
Point of Application of the Resultant
39
When the Forces act in opposite Directions i
40
When the Forces are Equal and Opposite
41
Such Forces constitute a Statical Couple
42
Resultant of any Number of Parallel Forces
43
Definition of Center of Parallel Forces Page
44
Equality of the Moment of the Resultant with the Sum of the Moments of the Components
45
Definition of Moment of a Force in reference to a Plane
46
Conditions of Equilibrium of any Number of Parallel Forces
47
Condition of Rotation
48
When in Equilibrium each Force equal in Magnitude to the Resultant of all the others
49
Equilibrium independent of their Direction
50
Examples CHAPTER III
51
Definition of a Statical Couple
52
Definition of the Arms of a Couple
53
Definition of the Moment of a Couple
54
A Couple may be turned round in its own Plane
55
A Couple may be removed parallel to itself in its own Plane
56
A Couple may be removed to a Parallel Plane
57
Couples are equivalent when their Planes are Parallel and Moments are Equal
58
Couples may be changed into others having Arms of a given Length
59
Definition of the Axis of a Couple
60
Properties of an Axis
61
Definition of the Resultant of two or more Couples 24
62
Equality of the Moment of the Resultant with the Sum of the Moments of the Components
63
Equality of the Axis of the Resultant with the Sum of the Axes of the Com ponents
64
The Resultant of two Couples inclined to each other
65
Representative of the Axis of the Resultant of two Couples
66
Parallelogram of Couples CHAPTER IV
67
Resultant of any Number of Concurring Forces
68
Directions of the Rectangular Components involved in their Trigonometrical Values
69
Conditions of Equilibrium of Concurring Forces
70
Resultant Force and Resultant Couple when the Forces do not concur
71
Construction of the Results
72
Equation of the Resultant
73
Equilibrium of nonconcurring Forces
74
Equilibrium when there is a fixed Point in the System
75
Definition of Virtual Velocities
78
Principle of Virtual Velocities Page
79
Principle of Virtual Velocities obtains in Concurring Forces in the same Plane
80
Principle of Virtual Velocities obtains in nonconcurring Forces in the same Plane
81
The Converse
82
Definition of Gravity
83
Laws of Gravity CHAPTER V
84
Definition of a Heavy Body
85
Definition of the Weight of a Body
86
Definition of its Mass
87
Expression for Weight
88
Definition of Density
89
Another Expression for Weight
90
Relations of Masses to Volumes of the same Density
91
Relations of Densities to Volumes of the same Mass
92
Relations of Densities to Masses of the same Volume
93
Definition of Center of Gravity
94
Connection of the Center of Gravity with the Doctrine of Parallel Forces
95
Definition of a Body symmetrical with respect to a Plane
96
Position of its Center of Gravity
97
Definition of a Body symmetrical with respect to an Axis
98
Position of its Center of Gravity
99
Center of Gravity of a Body symmetrical with respect to two Axes
100
Definition of Center of Figure
101
Center of Gravity of any Number of heavy Particles
102
Their Center of Gravity when their Positions are given by their Coordinates
103
Their Center of Gravity when they are all in the same Line
104
Their Center of Gravity when they are Homogeneous 43
105
The Center of Gravity of the Whole and a Part given to find that of the other Part
106
Examples1 Of a Straight Line2 Triangle3 Parallelogram4 Polygon 5 Triangular Pyramid6 Any Pyramid7 Frustum of a Cone8 Perimeter of a Triangle...
107
When the Body has a Fixed Point in
108
Definition of Stable Unstable and Neutral Equilibrium
109
Position of the Center of Gravity when the Equilibrium is Stable and when Unstable
110
Pressure on the Fixed Point
111
Position and Pressure when there are two Fixed Points
112
Position and Pressure when there are three Fixed Points
113
Position and Pressure when a Body touches a Horizontal Plane in one Point
114
Position and Pressure when a Body touches a Horizontal Plane in two Points
115
Position and Pressure when a Body touches a Horizontal Plane in three Points
116
Position and Pressure when a Body touches a Horizontal Plane in any Number of Points
117
Measure of the Stability on a Horizontal Plane
118
Case of a Body on an Inclined Plane
119
Examples
120
General differential Expressions for the Center of Gravity of a Plane Curve
122
General differential Expressions for the Center of Gravity of a Plane Area
123
General differential Expressions for the Coordinates of a Surface of Revolution
124
General differential Expressions for the Coordinates of a Solid of Revolution
125
Determination of a Surface of Revolution
126
Determination of a Solid of Revolution
127
Examples1 Of a Circular Arc2 Circular Segment3 Surface of a Spherical Segment4 Spherical Segment 1
128
Examples on the preceding Chapters CHAPTER VI
129
Classification of the Mechanical Powers I THE LEVER
130
Definition of a Lever
131
Kinds of Lever
132
Conditions of Equilibrium when the Forces are Parallel
133
Conditions of Equilibrium when the Forces are Inclined
134
Conditions of Equilibrium when the Lever is Bent or Curved
135
Conditions of Equilibrium when any Number of Forces in the same Plane act on a Lever of any FormExamples II WHEEL AND AXLE
136
Definition of Wheel and Axle 86
137
Conditions of Equilibrium when two Forces act Tangentially to the Surface of the Wheel and Axle
138
Perpetual Lever
139
Pressure on the Axis
140
Conditions of Equilibrium of any Number of Forces
141
Definition of Cogged Wheels Crown Wheels Beveled Wheels Pinions Leaves
142
Relations of the Forces when in Equilibrium
151
Relations of the Forces when the Ends are Fixed and the Forces Parallel
152
Relations of the Forces when the Ends are Fixed and the Forces are Weights
153
Point of Application of the Resultant
154
CatenaryExamples IV THE PULLEY
155
Definition of the PulleyFixed and Movable
156
Use of Fixed Pulley
157
Equilibrium in single Movable Pulley
158
Systems of Pulleys
159
Equilibrium in first System
160
Equilibrium in the second
161
Equilibrium in the thirdExamples
162
Conditions of Equilibrium in a Combination of Levers
186
Conditions of Equilibrium in the Endless Screw
187
Conditions of Equilibrium in any Combination of the Mechanical Powers
188
Conditions of Equilibrium in the Knee CHAPTER VII
189
Preliminary Considerations
190
Application to the Wheel and Axle
191
Application to Toothed Wheels
192
Application to Movable Pulley with Parallel Cords
193
Application to the first System of Pulleys
194
Application to the second System 116
195
Application to the third System
196
Application to the Inclined Plane
197
Application to the Wedge
198
Application to the Lever of any Form
199
Application to the single Movable Pulley with inclined Cords
200
Definition of FrictionKinds CHAPTER VIII
201
Measurement of Friction
202
Laws of Friction
203
Value of the Coefficient of Friction
204
Limits of the Ratio of the Power to the Weight in the Screw
206
One Limit obtained directly from the other Examples
207
DYNAMICS INTRODUCTION
208
In Dynamics Time an Element
209
Definition of Motion
210
Definition of Absolute Motion
211
Definition of Relative Motion
212
Definition of VelocityIts Measure
213
Definition of Variable VelocityIts Measure
214
Definition of Relative Velocity
215
Definition of InertiaFirst Law of Motion
216
Definition of Center of Inertia
217
Definition of the Path of a Body
218
Definition of Free and Constrained Motion
219
Definition of an Impulsive Force
220
Definition of an Incessant Force
221
Definition of a Constant ForceIts Measure
222
Definition of a Variable ForceIts Measure
223
124
224
Definition of a Moving ForceIts Measure
225
The second Law of Motion
226
The third Law of Motion A Page
227
Point to which the Force must be applied
228
General Equation of Uniform Motion
229
Relation of Spaces to Velocities when the Times are Equal
230
Relation of Spaces to the Times when the Velocities are Equal
231
Relation of Velocities to the Times when the Spaces are Equal
232
Measure of an Impulsive Force
233
The Velocity resulting from the Action of several Forces
234
Parallelogram of Velocities
235
Rectangular Composition and Resolution of Velocities 136
236
Relations of Space Time and Velocity of two Bodies moving in the same Straight Line
237
Relations of Space Time and Velocity of two Bodies moving in the Circumference of a Circle
238
Examples CHAPTER II
239
Definition of Direct Central and Oblique Impact
240
Definition of ElasticityPerfectImperfectIts Modulus
241
Definition of Hard and Soft
242
Velocity of two Inelastic Bodies after Impact
243
Loss of Living Force in the Impact of Inelastic Bodies
244
Velocities of imperfectly Elastic Bodies after Impact
245
APPLICATION OF THE INTEGRAL CALCULUS TO THE DETERMINATION OF THE CEN TER OF GRAVITY
247
Definition of Angles of Incidence and Reflection
249
Motion of an Inelastic Body after Oblique Impact on a Hard Plane
250
Motion of an Elastic Body after Oblique Impact on a Hard Plane Page
251
Direction of Motion before Impact that a Body after Impact may pass through given Point
252
Measure of the Modulus of Elasticity
253
143
254
Examples CHAPTER III
255
Uniformly accelerated MotionAcquired Velocity
256
Space in Terms of the Force and Time
257
Space in Terms of the Force and Velocity
258
Space described in the last n Seconds 153
259
The Velocity and Space from the joint Action of a Projectile and Constant Force
260
The Velocity when the Space is given
261
Velocity lost and gained by the Action of a Constant Force when the Space
262
CHAPTER II
263
the same 263 Scholium on Universal Gravitation 264 Scholium on the Numerical Value of the Force of Gravity
264
Examples
265
The Path of a Projectile is a Parabola
266
Equation of the Path when referred to Horizontal and Vertical Axes
267
Definition of Horizontal RangeTime of FlightImpetus
268
Time of Flight on a Horizontal Plane
269
Range on a Horizontal PlaneThe same for two Angles of Elevation
270
Greatest Height
271
Range and Time of Flight on an Inclined Plane and Coordinates of Point of Impact
272
Formula for Velocity of a Ball or Shell
273
Examples
274
Relations of Space Time and Velocity
275
Velocity down the Plane and its Height
276
Times down Inclined Planes of the same Height
277
Relations of Space Time and Velocity when projected up or down the Plane
278
Time of Descent down the Chords of a Circle 169
279
Straight Line of quickest Descent from a Point within a Circle to its Circumference
280
Straight Line of quickest Descent from a given Point to an Inclined Plane
281
Motion of two Bodies suspended by a Cord over a Fixed Pulley
282
Motion of two Bodies when the Inertia of the Pulley is considered II MOTION IN CIRCULAR ARCS
283
Velocity acquired down the Arc of a Circle
284
Velocity lost in passing from one Side of a Polygon to the next
285
Velocity lost when the Sides are Infinite in Number
286
Discharge from a Triangular Aperture
287
Velocity of Efflux of an Elastic Fluid 428 Motion of Fluids in Long Pipes
289
173
292
General Method of determining the Discharge from small Orifices 430 General Method of determining the Time for a Vessel to empty itself 431 Gen...
293
HYDROSTATIC AND HYDRAULIC INSTRUMENTS 433 Mariottes Flask
295
Bramahs Press
296
Moon retained in its Orbit by Gravitation
298
Diving Bell
299
Sea Gage 438 Siphon
301
Common Pump
302
Pendulum used to determine the Figure of the Earth
304
Common Pump Conditions of Failure 441 Air Pump
305
Condenser
306
Clepsydra

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Common terms and phrases

accelerating force action angle applied apsis axes axis axle beam body moves center of gravity centrifugal force circle co-ordinates components concurring forces conditions of equilibrium constant force cord couple curve cycloid density descend determine displaced fluid distance dt² earth elastic equal and opposite equation equilibrium feet fixed point fluid force of gravity force varies forces acting friction fulcrum given Hence horizontal impact inclined plane inertia Integrating length lever magnitude mass moment of inertia motion orbit P₁ P₂ parallel forces parallelogram particle pendulum perpendicular point of application polygon position pressure PROP pulley radius of gyration ratio represent resolved resultant SCHOL seconds sides space described substituted surface tension tion triangle v₁ versin vertical line vessel vibration virtual velocities weight wheel νμ

Popular passages

Page 6 - If two forces be represented in magnitude and direction by the two adjacent sides of a parallelogram, the diagonal will represent their resultant in magnitude and direction.‎
Appears in 38 books from 1855-2008
Page 141 - If an iron rod have one end against the sun and the other resting on the earth, the distance of the sun from the earth being 95,125,000 miles, in what time will a blow applied to the end on the earth be felt by the sun, the velocity of an impulse in iron being 11,865 feet per second ? Ans. 490 days. Ex. 2. When the earth is in that part of its orbit nearest to Jupiter, an eclipse of one of Jupiter's satellites is seen 16 minutes 36 seconds sooner than it would be if the earth were in that part of...‎
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Page 129 - A beam, 30 feet long, balances itself on a point at one-third of its length from the thicker end ; but when a weight of 10 Ibs. is suspended from the smaller end, the prop must be moved 2 feet towards it, in order to maintain the equilibrium. Find the weight of the beam.‎
Appears in 25 books from 1830-1917
Page 13 - ... components. By compounding the resultant R with a third force, we should obtain a like result. In the same manner, the proposition may be extended to any number of forces. 33. COR. 2. If the origin of moments be a fixed point, and taken in the direction of the resultant, On will become zero, and...‎
Appears in 10 books from 1855-1956
Page 171 - Find the straight line of quickest descent from a given point to a given straight line, the point and the line being in the same vertical plane.‎
Appears in 20 books from 1845-1904
Page 9 - If any number of forces acting at a point can be represented in magnitude and direction by the sides of a POLYGON taken in order, they are in equilibrium.‎
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Page 248 - V:V^p:o; or the whole volume of the solid is to the part immersed as the density of the fluid is to the density of the solid. COR. 2. If the centers of gravity of the solid and displaced fluid be not in the same vertical line, the body will be acted upon by two parallel forces in opposite directions, and will cause the body to turn round. The point of application of the resultant of these forces may be found by Art. 29. 391. DBF. The section of a floating body made by a plane coincident with the...‎
Appears in 8 books from 1855-1995
Page 46 - STATICS. same distance from the plane, and their moments will be equal and have contrary signs. But all the particles, taken two and two, are thus placed (Art. 96). Therefore the resultant of the system of forces will be in that plane, and, consequently, the center of gravity also. 98. DBF. A body is said to be symmetrical with respect to an axis when it is symmetrical with respect to two planes passing through that axis. 99. PROP. The center of gravity of a homogeneous body, symmetrical with respect...‎
Appears in 9 books from 1855-1902
Page 1 - Hydrodynamics investigates the effects of forces on fluids when motion results.* 2. Force is that which produces or tends to produce motion or change of motion.‎
Appears in 34 books from 1830-2003
Page 164 - For in this case 2a=90°, and sin. 2a=l. .-. x=2h. or the greatest horizontal range is equal to twice the height due to the velocity of projection, or twice the distance from the point of projection to the focus of the trajectory. COR. 2. The range is the same for any two angles of elevation, the difference between which and 45° is the same, or for (45±0). For sin. (90°+20)=sin. (90°-20), or sin.‎
Appears in 9 books from 1847-1866

Bibliographic information

TitleAn Elementary Treatise on Mechanics: Embracing the Theory of Statics and Dynamics, and Its Applications to Solids and Fluids
AuthorAugustus William Smith
PublisherHarper, 1855
Original fromthe University of Michigan
Digitized23 Nov 2005
Length307 pages
  
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