Analytical Elements of Mechanics

Preface1 Differentiation Of Vectors 1.1 Vector Functions Of A Scalar Variable 1 1.1.1 Definition Of A Vector Function Of A Scalar Variable In A Reference Frame; Independence Of A Variable In A Reference Frame; Vectors Fixed In A Reference Frame 1.1.2 Dependence On A Variable In One Reference Frame, Independence Of The Same Variable In Another Reference Frame 1.1.3 Measure Numbers Characterize The Behavior Of A Vector Function 1.1.4 Constant Measure Numbers 1.1.5 Values Of A Vector Function 1.1.6 Equality 1.1.7 Dependence Of Results On The Reference Frame In Which An Operation Is Performed 1.1.8 Independence Of Results Of The Reference Frame In Which An Operation Is Performed 1.1.9 Notation 1.1.10 Expression Of Results In Terms Of Unit Vectors Fixed In Any Reference Frame 1.1.11 Functional Character Of Results Of Operations Involving Vector Functions 1.2 The First Derivative Of A Vector Function 1.2.1 Definition Of The First Derivative 1.2.2 First Derivatives Equal To Zero 1.2.3 Dimensions Of The First Derivative 1.2.4 Expression Of The First Derivative In Terms Of Unit Vectors Fixed In Any Reference Frame 1.2.5 Equality Of First Derivatives Of Equal Vector Functions 1.2.6 Notation 1.2.7 The First Derivative As A Limit 1.2.8 Properties Of The Derivative 1.3 The Second And Higher Derivatives Of Vector Functions 12 1.3.1 Definition Of The Second Derivative Of A Vector Function; Definition Of Higher Derivatives Of A Vector Function 1.4 Derivatives Of Sums 1.4.1 Equality Of The First Derivative Of A Sum And The Sum Of The First Derivatives 1.4.2 Applicability Of 1.4.1 To Second And Higher Derivatives 1.5 Derivatives Of Products 1.5.1 First Derivative Of The Product Of A Scalar And A Vector Function 1.5.2 First Derivative Of The Scalar Product Of Two Vectors 1.5.3 First Derivative Of The Vector Product Of Two Vectors 1.5.4 First Derivative Of The Continued Product Of Any Number Of Vector And Scalar Functions. 1.6 Derivatives Of Implicit Functions 1.6.1 First Derivative Of An Implicit Function Of A Scalar Variable 1.7 The First Derivative Of A Unit Vector Which Remains Perpendicular To A Line Fixed In A Reference Frame 1.7.1 Expression For The First Derivative 1.7.2 Interpretation Of One Of The Terms Appearing In 1.7.1, As A Rate Of Rotation. 1.7.3 Convenience Of 1.7.1 When Limited Information Available 1.8 Taylor's Theorem For Vector Functions 1.8.1 Statement Of The Theorem 1.8.2 The Use Of Taylor's Theorem For Purposes Of Computation And In Connection With Functions Not Specified Explicitly 1.9 Vector Tangents Of A Space Curve 1.9.1 Vector Tangents Expressed In Terms Of The First Derivative Of A Position Vector 1.9.2 Sense Of The Vector Tangents Obtained By Using Various Scalar Variables 1.9.3 Expression For The Vector Tangent In Terms Of The Derivative Of The Position Vector With Respect To Arc-Length Displacement 1.9.4 Definition Of The Normal Plane At A Point Of A Space Curve 1.10 Vector Binormals Of A Space Curve 1.10.1 Vector Binormals Expressed In Terms Of Derivatives Of A Position Vector 1.10.2 Perpendicularity Of Vector Tangents And Vector Binormals 1.10.3 Sense Of Vector Binormals Obtained By Using Various Scalar Variables 1.10.4 Simplification Introduced By The Use Of Arc-Length Displacement As Independent Variable 1.10.5 Definition Of The Plane Of Curvature Or Osculating Plane At A Point Of A Space Curve 1.11 The Vector Principal Normal Of A Space Curve 1.11.1 Definition Of The Vector Principal Normal 1.11.2 Expression For The Vector Principal Normal In Terms Of Derivatives Of A Position Vector 1.11.3 Expression For The Vector Principal Normal In Terms Of The Second Derivative With Respect To Arc-Length Displacement 1.11.4 Definition Of The Rectifying Plane At A Point Of A Space Curve 1.12 The Vector Radius Of Curvature Of A Space Curve 1.12.1 The Vector Radius Of Curvature Expressed In Terms Of Derivatives Of A Position Vector 1.12.2 The Vector Radius Of Curvature As The Product Of A Scalar And The Vector Principal Normal 1.12.3 Expressions In Terms Of Derivatives With Respect To Arc-Length Displacement 1.13 The Serret-Frenet Formulas 1.13.1 Derivatives Of Vector Tangents, Binormals, And Principal Normal With Respect To Arc-Length Displacement 1.13.2 The Torsion Of A Space Curve, Expressed In Terms Of Derivatives Of A Position Vector.2.Kinematics 2.1 Rates Of Change Of Orientation Of A Rigid Body 2.1.1 Definition Of The Rate Of Change Of Orientation Of A Rigid Body In A Reference Frame With Respect To A Scalar Variable 2.1.2 Importance Of Rates Of Change Of Orientation As Analytical Tools 2.1.3 Symmetry Of The Expression For Rates Of Change Of Orientation 2.1.4 The Relationship Between The First Derivatives Of A Vector Function In Two Reference Frames 2.1.5 The Derivative In Two Reference Frames Of The Rate Of Change Of Orientation 2.1.6 Interchange Of Reference Frames 2.2 Angular Velocity 2.2.1 Definition Of The Angular Velocity Of A Rigid Body In A Reference Frame. 2.2.2 Expression For The Angular Velocity As A Product Of An Angular Speed Ancl A Unit Vector 2.2.3 Pictorial Representation Of Angular Velocity 2.2.4 Angular Velocity Of Fixed Orientation 2.2.5 Omission Of Qualifying Phrases In The Description Of Frequently Encountered Systems 2.2.6 Application Of 2.2.4 To The Motion Of Bodies Possessing No Fixed Point 2.2.7 Addition Of Angular Velocities 2.2.8 Resolution Of Angular Velocities Into Components 2.2.9 Kinematic Chains 2.2.10 Reference Frames Having No Physical Counterparts 2.3 Angular Acceleration 2.3.1 Definition Of The Angular Acceleration Of A Rigid Body In A Reference Frame 2.3.2 The Relationship Between Measure Numbers Of Components Of Angular Velocity And Angular Acceleration Vectors 2.3.3 Interchange Of Reference Frames 2.3.4 Expression For The Angular Acceleration As The Product Of A Scalar Angular Acceleration And A Unit Vector 2.3.5 The Relationship Between Angular Speed And Scalar Angular Acceleration 2.3.6 Pictorial Representation Of Angular Acceleration 2.3.7 Angular Acceleration Of A Body Having An Angular Velocity Of Fixed Orientation 2.3.8 Plane Linkages 2.3.9 Graphical Method For The Determination Of The Scalar Product Of Unit Vectors 2.3.10 Applicability Of 2.3.8 To Linkages Containing Sliding Pairs 2.3.11 Addition Of Angular Accelerations 2.4 Relative Velocity And Acceleration 2.4.1 Definitions Of Velocity And Acceleration Of One Point Relative To Another 2.4.2 The Relationship Between The Velocity Of P Relative To Q And The Velocity Of Q Relative To P 2.4.3 Relative Velocity And Acceleration Of Two Points Fixed In A Reference Frame 2.4.4 Addition Of Relative Velocities; Addition Of Relative Accelerations 2.4.5 Relative Velocity And Relative Acceleration Of Two Points Fixed On A Rigid Body 2.5 Absolute Velocity And Acceleration 2.5.1 Definitions Of Absolute Velocity And Absolute Acceleration Of A Point 2.5.2 Velocity And Acceleration Of A Point Fixed In A Reference Frame 2.5.3 Expression For The Velocity As The Product Of A Speed And A Vector Tangent 2.5.4 Tangential And Normal Accelerations; Scalar Tangential And Scalar Normal Accelerations 2.5.5 Convenience Of Normal And Tangential Accelerations 2.5.6 Velocity And Acceleration Of A Point Fixed On A Rigid Body Which Is Rotating About A Fixed Axis 2.5.7 Rectilinear Motion 2.5.8 Rectilinear Motion As A Limiting Case Of Curvilinear Motion 2.5.9 Velocity And Acceleration Of Points Fixed On A Rigid Body Possessing No Fixed Point 2.5.10 Rolling; Pure Rolling Contact; Pivoting 2.5.11 The Instantaneous Axis Of A Rigid Body; Minimum Velocity 2.5.12 Plane Motion Of A Rigid Body; Instantaneous Centers 2.5.13 The Relationship Between The Velocities Of A Point In Two Reference Frames; The Relationship Between The Accelerations Of A Point In Two Reference Frames; Coriolis Acceleration 2.5.14 Equality Of The Lengths Of Contact Arcs During Rolling 2.5.15 Expressions For Relative Velocities And Relative Accelerations In Terms Of Absolute Velocities And Absolute Accelerations3. Second Moments 3.1 Second Moments Of A Point 3.1.1 Definition Of The Second Moment Of One Point With Respect To Another 3.1.2 Expression Of The Second Moment Of A Point For One Direction In Terms Of The Second Moments Of The Point For Three Mutually Perpendicular Directions 3.1.3 Parallelism Of The Second Moment For A Direction With That Direction 3.1.4 Definition Of The Second Moment Of One Point With Respect To Another For A Pair Of Directions 3.1.5 Alternative Expression For The Second Moment Of One Point With Respect To Another For A Pair Of Directions 3.1.6 Symmetry Of Second Moments For A Pair Of Directions 3.1.7 Definition Of The Second Moment Of A Point With Respect To A Line 3.1.8 Expression For The Second Moment Of One Point With Respect To Another In Terms Of Second Moments For Three Pairs Of Directions 3.1.9 Expression For The Second Moment Of One Point With Respect To Another For A Pair Of Directions In Terms Of Second Moments For Three Pairs Of Mutually Perpendicular Directions 3.1.10 Expression Of The Second Moment Of A Point With Respect To A Line In Terms Of Second Moments For Three Pairs Of Directions 3.2 Second Moments Of A Set Of Points 3.2.1 Definition Of The Second Moment Of A Set Of Points With Respect To A Point 3.2.2 Definition Of The Second Moment Of A Set Of Points With Respect To A Point For A Pair Of Directions 3.2.3 Definition Of The Second Moment Of A Set Of Points With Respect To A Line 3.2.4 Applicability To Second Moments Of Sets Of Points Of Relationships Discussed In Secs. 3.1.2-3.1.10 3.2.5 Definition Of The Radius Of Gyration Of A Set Of Points With Respect To A Line 3.2.6 Parallel Axes Theorems 3.2.7 Parallel Axes Theorem For Second Moments With Respect To A Line 3.2.8 Parallel Axes Theorem For Radii Of Gyration. 3.2.9 Determination Of All Second Moments Of A Set Of Points By Successive Use Of Sees. 3.2.4, 3.2.6, And 3.2.7. 3.3 Principal Directions, Axes, Planes, Second Moments, And Radii Of Gyration Of A Set Of Points 3.3.1 Definition Of Principal Directions, Principal Axes, Principal Planes, Principal Second Moments, Principal Radii Of Gyration 3.3.2 Necessary And Sufficient Condition That A Unit Vector Define A Principal Direction Of A Set Of Points 3.3.3 Zero Second Moment For A Pair Of Perpendicular Directions When One Is A Principal Direction 3.3.4 Location Of A Principal Direction By Consideration Of Zero Second Moments For Two Pairs Of Directions 3.3.5 Nonzero Second Moments 3.3.6 Planes Of Symmetry 3.3.7 Coplanar Points 3.3.8 Location Of Two Principal Axes In A Principal Plane 3.3.9 Location Of Three Principal Axes For Any Set Of Points 3.3.10 Use Of Principal Axes And Principal Second Moments In The Determination Of Second Moments For Arbitrary Directions 3.3.11 Determination Of Second Moments By Successive Use Of Sees. 3.2.6, 3.2.7, 3.3.10 3.3.12 Centroidal Principal Directions 3.3.13 The Momental Ellipsoid Of A Set Of Points 3.3.14 Lines Of Maximum Or Minimum Second Moment 3.3.15 Minimum Second Moment Of A Set Of Points 3.4 Second Moments Of Curves, Surfaces, And Solids 3.4.1 Definition Of The Second Moment Of A Figure With Respect To A Point 3.4.2 Integral Expression For The Second Moment Of A Figure With Respect To A Point 3.4.3 Definition Of The Second Moment Of A Figure With Respect To A Point For A Pair Of Directions 3.4.4 Definition Of The Second Moment Of A Figure With Respect To A Line 3.4.5 Definition Of The Radius Of Gyration Of A Figure With Respect To A Line 3.4.6 Principal Directions, Axes, Planes, Second Moments, Radii Of Gyration Of A Figure 3.4.7 Use Of Relationships Discussed In Parts 3.2 And 3.3 For The Solution Of Problems Involving Curves, Surfaces, And Solids. 3.4.8 Polar Second Moments 3.4.9 Explanation Of The Appendix. 3.4.10 Decomposition Of Complex Figures 3.4.11 Figures Obtained By Subtraction 3.4.12 Radii Of Gyration Obtained By Regarding A Surface As A Limiting Case Of A Solid 3.5 Second Moments Of Sets Of Particles And Continuous Bodies 3.5.1 Definitions Of The Second Moment Of A Set Of Particles With Respect To A Point, Products Of Inertia, And Moments Of Inertia 3.5.2 Definition Of The Second Moment Of A Continuous Body With Respect To A Point 3.5.3 Definition Of The Second Moment Of A Continuous Body With Respect To A Point For A Pair Of Directions 3.5.4 Moment Of Inertia Of A Continuous Body About A Line 3.5.5 Definition Of The Radius Of Gyration Of A Continuous Body With Respect To A Line 3.5.6 Principal Directions, Axes, Planes, Second Moments, Moments Of Inertia, And Radii Of Gyration 3.5.7 Applicability Of Relationships Discussed In Parts 3.2, 3.3, 3.4. 3.5.8 The Relationship Between Second Moments Of A Uniform Body And Second Moments Of The Figure Occupied By The Body. 3.5.9 Coincidence Of Principal Direction, Axes, And Planes Of A Uniform Body And Of The Figure Occupied By The Body4. Laws Of Motion 4.1 Inertia Forces And Force Systems 4.1.1 Definition Of The Inertia Force Acting On A Particle In A Reference Frame 4.1.2 Definition Of The Inertia Force And The Inertia Couple Acting On A Continuous Body In A Reference Frame 4.1.3 Expressions For The Inertia Force And Inertia Torque Acting On A Rigid Body 4.1.4 Resolution Of The Inertia Torque Acting On A Rigid Body Into Any Mutually Perpendicular Components 4.1.5 Resolution Of The Inertia Torque Acting On A Rigid Body Into Components Parallel To A Right-Handed Set Of Mutually Perpendicular Principal Directions. 4.1.6 Resolution Of The Inertia Torque Acting On A Rigid Body Into Components Parallel To A Right-Handed Set Of Mutually Perpendicular Principal Directions For The Mass Center 4.1.7 Expression For The Inertia Torque Acting On A Rigid Body Which Has An Angular Velocity Of Fixed Orientation 4.2 D'Alembert's Principle 4.2.1 Statement Of D'Alembert's Principle. 4.2.2 Necessary And Sufficient Condition That A Reference Frame Be A Newtonian Reference Frame 4.2.3 Equations Of Motion And Free-Body Diagrams 4.2.4 Evaluation Of The Earth As A Newtonian Reference Frame 4.2.5 Foucault's Pendulum 4.2.6 Motion Of A Particle In The Neighborhood Of The Earth's Surface 4.2.7 Motion Of Ballistic Missiles And Earth Satellites 4.2.8 Approximately Newtonian Reference Frames 4.3 Motions Of Rigid Bodies 4.3.1 Equations Of Motions For A Rigid Body 4.3.2 Evaluation Of Contact Forces Acting On A Rigid Body Whose Motion Is Specified 4.3.3 Plane Free-Body Diagrams 4.3.4 Several Body Problems 4.3.5 The Law Of Action And Reaction 4.4 Linear And Angular Momentum 4.4.1 Definition Of The Linear Momentum Of A Set Of Particles Relative To A Point In A Reference Frame 4.4.2 Linear Momentum Of A Continuous Body Relative To A Point In A Reference Frame 4.4.3 Linear Momentum Of A Body Relative To A Point In A Reference Frame 4.4.4 The Linear Momentum Of A Body Relative To The Mass Center Of The Body 4.4.5 The Absolute Linear Momentum Of A Body In A Reference Frame 4.4.6 The Linear Momentum Principle 4.4.7 The Principle Of Conservation Of Linear Momentum 4.4.8 Definition Of The Angular Momentum Of A Set Of Particles Relative To A Point In A Reference Frame 4.4.9 Angular Momentum Of A Continuous Body Relative To A Point In A Reference Frame 4.4.10 Angular Momentum Of A Body Relative To A Point In A Reference Frame 4.4.11 The Angular Momentum Of A Rigid Body Relative To A Point Fixed On The Body 4.4.12 Absolute Angular Momentum Of A Body In A Reference Frame 4.4.13 The Angular Momentum Principle 4.4.14 Modified Form Of The Angular Momentum Principle 4.4.15 The Principle Of Conservation Of Angular Momentum 4.4.16 Relative Advantages And Disadvantages Of The Angular Momentum Principle 4.5 Activity And Kinetic Energy 4.5.1 Definition Of The Kinetic Energy Of A Set Of Particles Relative To A Point In A Reference Frame 4.5.2 Kinetic Energy Of A Continuous Body Relative To A Point In A Reference Frame 4.5.3 Kinetic Energy Of A Body Relative To A Point In A Reference Frame 4.5.4 Kinetic Energy Of A Rigid Body Relative To A Point In A Reference Frame 4.5.5 Absolute Kinetic Energy Of A Body In A Reference Frame 4.5.6 The Activity-Energy Principle For A Particle 4.5.7 Relative Advantages And Disadvantages Of The Activity-Energy Principle For A Particle 4.5.8 The Activity-Energy Principle For A Rigid Body 4.5.9 Elimination Of Contact Forces By Means Of The Activity-Energy Principle For A Rigid Body 4.5.10 The Activity Energy Principle For A Set Of Rigid Bodies 4.5.11 Rigid Bodies Connected To Each Other By Light, Helical Springs 4.5.12 The Use Of The Activity-Energy Principle For A Set Of Rigid Bodies When The Number Of Bodies Is Large.Problem Sets Problem Set 1 (Sections 1.1.1-1.7.3) Problem Set 2 (Sections 1.8.1-1.13.3) Problem Set 3 (Sections 2.1.1-2.3.11) Problem Set 4 (Sections 2.4.1-2.5.8) Problem Set 5 (Sections 2.5.9-2.5.15) Problem Set 6 (Sections 3.1.1-3.2.9) Problem Set 7 (Sections 3.3.1-3.5.9) Problem Set 8 (Sections 4.1.1-4.1.7) Problem Set 9 (Sections 4.2.1-4.2.8) Problem Set 10 (Sections 4.3.1-4.3.5) Problem Set 11 (Sections 4.4.1-4.4.16) Problem Set 12 (Sections 4.5.1-4.5.12)Appendix Curves Surfaces SolidsIndex