An Elementary Treatise on Analytic Mechanics: With Numerous Examples

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D. Van Nostrand, 1884 - Mechanics, Analytic - 511 pages
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Contents

Action and Reaction
9
Representation of Forces
10
Kinetic Measure of Force
11
Absolute or Kinetic Unit of Force
13
Three Ways of Measuring Force
14
AST PAGE 24 Gravitation Units of Force and Mass
16
Gravitation and Absolute Measure
17
Examples
18
STATICS REST CHAPTER II
21
Composition of Conspiring Forces
22
Composition of Velocities
23
Composition of Forces
24
Triangle of Forces
25
Three Concurring Forces in Equilibrium
26
The Polygon of Forces
27
Parallelopiped of Forces
28
Resolution of Forces
30
Magnitude and Direction of Resultant 81
31
3i Conditions of Equilibrium
33
Resultant of Concurring Forces in Space
34
Equilibrium of Concurring Forces in Space
35
Equilibrium of Concurring Forces on a Smooth Plane
39
Examples
45
CHAPTER III
57
Resultant of Two Parallel Forces
58
Moment of a Force
60
ART PiOI 47 Signs of Moments
61
Moment of a Couple
62
Effect of a Couple on a Rigid Body
63
Effect of Transferring Couple to Parallel Plane not altered
64
5i5 A Couple replaced by another Couple
65
Resultant of any number of Couples
66
Resultant of Two Couples
67
Varignons Theorem of Moments
69
Varignons Theorem for Parallel Forces
71
Equilibrium of a Rigid Body under Parallel Forces
74
Equilibrium of a Rigid Body under Forces in any Direction
75
Equilibrium under Three Forces
77
Centre of Parallel Forces in Different Planes
85
Equilibrium of Parallel Forces in Space
87
Equilibrium of Forces acting in any Direction in Space
88
Examples
90
T V CHAPTER IV
100
Planes of SymmetryAxes of Symmetry
101
Body Supported on a Surface
102
Centre of Gravity of Two Masses
103
Centre of Gravity of a Triangular Pyramid
105
Centre of Gravity of a Cone
106
Centre of Gravity of Frustum of Pyramid
107
Investigations involving Integration
109
Centre of Gravity of the Arc of a Curve
110
Centre of Gravity of a Plane Area
115
Polar Elements of a IMuue Area
118
ABT PAGE 81 Double IntegrationPolar Formulae
120
Double IntegrationRectangular Formulae
122
Centre of Gravity of a Surface of Revolution
123
Centre of Gravity of any Curved Surface
126
Centre of Gravity of a Solid of Revolution
127
Polar Formulas
130
Centre of Gravity of any Solid 181
131
Polar Elements of Mass
133
Special Methods
136
Theorems of Pappus
138
Examples
140
CHAPTEK V
149
Laws of Friction
150
Magnitudes of Coefficients of Friction
152
Reaction of a Rough Curve or Surface
154
Friction on a Double Inclined Plane
156
Friction on Two Inclined Planes
159
Friction of a Pivot
160
Examples
162
THE PRINCIPLE OF VIRTUAL VELOCITIES 101 Virtual Velocity
166
Principle of Virtual Velocities
167
Nature of the Displacement
169
System of Particles Rigidly Connected
170
Examples
172
CHAPTER VII
177
Mechanical Advantage
178
Simple Machines
180
The Lever
181
The Common Balance
184
Chief Requisites of a Good Balance
185
The Steelyard
188
The Wheel and Axle
190
Toothed Wheels
192
Relation of Power and Weight in Toothed Wheels
193
Relation of Power to Weight in a Train of Wheels
194
The Inclined Plane
196
The Pulley
197
The Simple Movable Pulley
198
Second System of Pulleys
200
Third System of Pulleys
201
The Wedge
202
The Screw
204
129a Pronys Differential Screw
206
Examples
207
Relation when the Acceleration is Constant
234
Relation when Acceleration varies as the Time
235
Equations of Motion for Falling Bodies
237
Particle Projected Vertically Upwards
239
Compositions of Velocities
242
Resolution of Velocities
243
Motion on an Inclined Plane
245
Times of Descent down Chords of a Circle
247
The Straight Line of Quickest Descent
248
Examples
249
CHAPTER II
258
Composition and Resolution of Acceleration
259
Examples
261
Motion of Projectiles in Vacuo
266
The ParameterRangeGreatest HeightHeight of Direc trix
267
Velocity of a Particle at any point of its Path
269
Point at which a Projectile will Strike an Inclined Plane
270
The Elevation that the Particle may pass u Given Point
271
Second Method of Finding Equation of Trajectory
272
Velocity of Discharge of Balls and Shells
274
Angular Velocity and Angular Acceleration
278
Accelerations Along and Perpendicular to Tangent
279
When Acceleration Perpendicular to Radius Vector is zero
281
When Angular Velocity is Constant
282
Examples
284
PART III
289
Remarks on Law I
290
Remarks on Law II
291
Remarks on Law III
294
Two Laws of Motion in the French Treatises
295
Motion under the action of a Variable Repulsive Force
298
Motion under the action of an Attractive Force 2i9
299
Velocity acquired in Falling through a Great Height
300
Motion in a Resisting Medium
302
Motion in the Air against the Action of Gravity
304
Motion of a Projectile in a Resisting Medium
307
Motion against the Resistance of the Atmosphere
308
Motion in the Atmosphere under a small Angle of Elevation
312
Examples
313
CENTRAL FORCES ABT PAQB 180 Definitions
321
The Sectorial Area Swept over by the Radius Vector
325
Orbit when Attraction as the Inverse Square of Distance
329
Suppose the Orbit to be an Ellipse
333
Keplers Laws
335
Examples
338
CHAPTER III
345
To Find the Reaction of the Constraining Curve
348
Point where Particle will leave Constraining Curve
349
Constrained Motion Under Action of Gravity
350
The Simple Pendulum
352
Relation of Time Length and Force of Gravity
353
Height of Mountain Determined with Pendulum 854
355
Centripetal and Centrifugal Forces
356
The Centrifugal Force at the Equator
358
Centrifugal Force at Different Latitudes
359
The Conical PendulumThe Governor
361
Examples
362
CHAPTER IV
370
Impact or Collision
371
Direct and Central Impact
372
Elasticity of BodiesCoefficient of Restitution
373
Oblique Impact of Bodies
380
CHAPTER V
389
Horse Power 895
395
Kinetic and Potential EnergyStored Work
404
Force of a Blow
411
CHAPTER VI
429
Polar Moment of Inertia
436
AKT PAGE 232 Principal Axes of a Body
442
Products of Inertia
446
Examples
447
CHAPTER VII
451
DAlemberts Principle
452
Rotation of a Rigid Body about a Fixed Axis
454
The Compound Pendulum
457
Length of Seconds Pendulum Determined Experimentally
462
Motion of a Body when Unconstrained
464
Principal Radius of Gyration Determined Practically
467
The Ballistic Pendulum
468
Motion of a Body about a Horizontal Axle through its Centre
470
Motion of a Wheel and Axle
471
Motion of a Rigid Body about a Vertical Axis 472
472
Body Rolling down an Inclined Plane
473
Falling Body under an Impulse not through its Centre
475
Examples
477
CHAPTER VIII
481
Independence of the Motions of Translation and Rotation
482
Principle of the Conservation of the Centre of Gravity
485
Principle of the Conservation of Areas
486
Conservation of Vis Viva or Energy
488
Composition of Rotations
493
Motion of a Rigid Body referred to Fixed Axes
494
Axis of Instantaneous Rotation
495
Angular Velocity about Axis of Instantaneous Rotation 49
497
The Integral of Eulers Equations
502
Examples
505
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Page 290 - Change of motion is proportional to the force applied and takes place in the direction of the straight line in which the force acts.
Page 483 - The attraction of a uniform spherical surface on an external point is the same as if the whole mass were collected at the centre.
Page 245 - Let P be the position of the particle at any time t, /the resultant acceleration acting always along OP, O being a fixed point in the line of motion.
Page 290 - To every action there is always an equal and contrary reaction ; or, the mutual actions of any two bodies are always equal and oppositely directed in the same straight line.
Page 24 - The Parallelogram of Forces. — If two forces acting at a point be represented in magnitude and direction by the adjacent sides of a parallelogram, the resultant...
Page 329 - A particle is projected from a given point in a given direction and with...
Page 292 - Since forces are measured by the changes of motion they produce, and their directions assigned by the directions in which these changes are produced ; and since the changes of motion of one and the same body are in the directions of, and proportional to, the changes of velocity — a single force, measured by the resultant change of velocity, and in its direction, will be the equivalent of any number of simultaneously acting forces.
Page 181 - The parts of the lever, into which the fulcrum divides it, are called the arms of the lever. When the arms are in the same straight line, it is called a straight lever, otherwise a bended, or more commonly, a bent lever.
Page 27 - If a number of forces acting at a point can be represented in magnitude and direction by the sides of a closed polygon taken...
Page 365 - A railway train is moving smoothly along a curve at the rate of 60 miles an hour, and in one of the carriages a pendulum, which would ordinarily oscillate seconds, is observed to oscillate 121 times in 2 min.

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