Vectors and Rotors: With Applications

Front Cover
E. Arnold, 1903 - Electronic book - 204 pages
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Contents

Steps in a line
9
Subtraction of steps
10
Areas
11
Aspect of planes
12
Turning about an axis
13
CHAPTER I
15
Notation
16
Addition of two Vectors
17
Vectorpolygon
18
Like Vectors
19
Subtraction of Vectors
20
Algebraic rules hold for addition
21
Two nonparallel vectors are independent
22
Three space vectors independent
23
Position vector to midpoint of a line
24
EXERCISES I
27
Vectors and Coordinates
28
Equal Vectors equal components
31
Distance between two points
32
Vectors in Space
33
Any vector expressed in terms of three independent vectors
35
Rectangular coordinates and direction cosines
37
Distance between two points
38
SECTION 111 APPLICATIONS TO CONCURRENT FORCES 55 Definition of Resultant Force
40
Vector Polygon closed
41
Friction
43
Laws of Friction
44
Examples
45
Three Forces in Equilibrium must pass through a point
46
EXERCISES IV
47
CHAPTER II
49
Centre of Mean Position
50
Centre of Mean Position independent of origin
51
Mass
52
Definition of masspoint masscentre resultant mass
53
Calculation
54
Geometrical Theorems
55
Masspoints in a plane
56
W z of masscentre
58
EXERCISES V
59
The Linkpolygon
60
EXERCISES VI
63
Masscentre of sides of a Triangle
64
Masscentre of Circular Arc
65
Graphical construction
66
Masscentre of Areas
68
Masscentre of Quadrilateral
69
Masscentre of Sector of Circle
70
Negative mass
72
Masscentre of Segment of Circle
73
Masscentre of Irregular Area
74
Masscentre of Areas by Linkpolygon
76
Masscentre of Volumes planes of Symmetry
77
Masscentre of Prisms
78
Masscentre of Tetrahedron
79
Masscentre of Pyramids and Cones
80
Masscentre of Surfaces of Cones and Cylinders
81
Masscentre of Cap of sphere
82
Projection of Vectors on a line
83
Deduction of the coordinates of Masscentre
84
Projection of Vectors on a plane
85
EXERCISES VII
86
PRODUCTS OF TWO VECTORS PAGE 107 Notation
90
Summary in tabular form
104
a+BaB
105
Orthocentre of Triangle
106
The straight line equation
107
Mechanisms for drawing a straight line
108
The Hart Contraparallelogram
110
The Circle equation
111
Area of a triangle in coordinates
113
Angle between two straight lines
114
Coordinates of Masscentre from scalar product
115
Application to work done
116
The Peaucellier Cell
118
CHAPTER IV
119
Equality of two Rotors
120
Momental Areas
121
If the momental area of a Rotor is zero for three non collinear points the rotor itself vanishes
122
If two rotors have equal momental areas for all poles the rotors are equal If a system of Rotors has a resultant there is only one
123
Resultant of two intersecting Rotors Resultant of any concurrent system of Rotors
124
Coplanar Rotors first method of finding Resultant
125
EXERCISES IX
126
Couples definitions of a Couple and its Momental Area
127
Equivalence of two Couples Resultant Couple
128
Vectorpolygon closed
130
Vector and Linkpolygons closed
131
Summary
132
EXERCISES X
134
Measurement of Momental Areas of Parallel Rotors
135
1 two lines intersecting on rotor 2 two parallel rotors 3 three nonparallel lines 4 a parallel and equal one and a couple
137
Any system of rotors equivalent to a single rotor and a couple 176 The Equilibrium of Rotors M0 for three noncol
141
Equilibrium of a two rotors b three rotors
142
new notation
144
Determination of Reactions again considered
145
Bending Moment and Shearing Force
146
Bending Moment due to a Travelling Load
148
Bending Moment when forces are not parallel
149
ExampleCrane
150
EXERCISES XI
151
STRESSES IN FRAMES PAGE 186 Rigid Bodies
155
Stress
156
Hookes Law
157
determination of Stresses
158
Stress diagram
159
Another Cantilever
160
Warren Girder
161
Web and Booms of Girder
163
Suspension Bridges
164
Funicular Polygon
166
Roofprincipal
170
EXERCISES XII
172
Examples
173
Reciprocal Figures
174
Theorem
176
Stress diagram by method of trial
178
Stress diagram by resolution of forces
179
Stress diagram by Moments
180
Rigid nonrigid and overrigid frames
181
Simple examples of overrigid frames
182
Degrees of freedom
183
Single and double joints in a frame
184
Redundancy of lines in some stress diagrams
186
EXERCISES XIII
193
Relations between the sides and angles of a triangle
201

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Page 21 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
Page 29 - ... line and the extremities of the base have the same ratio which the other sides of the triangle have to one...
Page 8 - If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by the two parts.
Page 112 - ... is equal to the rectangle contained by the segments of the other.
Page 17 - ... from the beginning of the first to the end of the last...

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