Online electronic version May not be emailed or posted ANYWHERE May not be copied, or printed without express written permission of the authors. Introduction to STATICS and DYNAMICS Filename:Saskyalaunch3517 Andy Ruina and Rudra Pratap Oxford University Press (Preprint) Most recent modifications on January 20, 2015. Reference Tables: The front and back tables concisely summarize much of the text material. Summary of Mechanics 0) The laws of mechanics apply to any collection of material or ‘body.’ This body could be the overall system of study or any part of it.
In the equations below, the forces and moments are those that show on a free body diagram. Interacting bodies cause equal and opposite forces and moments on each other. I) Linear Momentum Balance (LMB)/Force Balance Equation of Motion Fi L The total force on a body is equal (I) to its rate of change of linear momentum. t2 Impulse-momentum F i ·dt L Net impulse is equal to the change in (Ia) (integrating in time) t1 momentum.
Conservation of momentum L=0 When there is no net force the linear (Ib) (if F i 0) L = L2 L1 0 momentum does not change. Statics Fi 0 If the inertial terms are zero the (Ic) (if L is negligible) net force on system is zero. II) Angular Momentum Balance (AMB)/Moment Balance Equation of motion MC H C The sum of moments is equal to the (II) rate of change of angular momentum. t2 Impulse-momentum (angular) MC dt HC The net angular impulse is equal to (IIa) (integrating in time) t1 the change in angular momentum.
Conservation of angular momentum HC 0 If there is no net moment about point (IIb) (if MC 0) H C H C2 H C1 0 C then the angular momentum about point C does not change. Statics MC 0 If the inertial terms are zero then the (IIc) (if H C is negligible) total moment on the system is zero. III) Power Balance (1st law of thermodynamics) Equation of motion Q P EK EP E int Heat flow plus mechanical power (III) E into a system is equal to its change in energy (kinetic + potential + internal). t2 t2 for finite time Qdt Pdt E The net energy flow going in is equal (IIIa) t1 t1 to the net change in energy.
Conservation of Energy E 0 If no energy flows into a system, (IIIb) (if Q P 0) E E2 E1 0 then its energy does not change. Statics Q P EP E int If there is no change of kinetic energy (IIIc) (if E K is negligible) then the change of potential and internal energy is due to mechanical work and heat flow. Pure Mechanics (if heat flow and dissipation P EK EP In a system well modeled as purely (IIId) are negligible) mechanical the change of kinetic and potential energy is due to mechanical Filename:Summaryofmechanics work on the system. Some definitions (Also see the index and back tables) r * or x * Position r r e., *i *i=O is the position of a point i relative to the origin, O., *i *i=O is the velocity of a point i relative to O, measured in a non-rotating ref- erence frame.
ddtv D ddt 2r * 2* a * Acceleration a a e., *i *i=O is the acceleration of a point i relative to O, measured in a Newtonian frame., the force on A from B is FA from B. M* or M D M=C * C * Moment or Torque e., the moment of a collection of forces about point C. ! * Angular velocity A measure of rotational velocity of a rigid ob- ! ject. *B = angular velocity of rigid object B.
* !P * Angular acceleration A measure of rotational acceleration of a rigid object. 8 L * < P m* i vi discrete Linear momentum A measure of a system’s net translational rate : vdm continuous R* (weighted by mass). D m *v 8tot cm L*P < P m* i ai discrete Rate of change of linear momen- The aspect of motion that balances the net : R* adm continuous tum force on a system. D m a * 8tot cm H*=C < P ri=C mi*vi discrete * Angular momentum about point C A measure of the rotational rate of a system : R* r=C *vdm continuous about a point C (weighted by mass and dis- tance from C).
8 HP =C * < P*ri=C mi*ai discrete Rate of change of angular momen- The aspect of motion that balances the net : r=C *adm continuous R* tum about point C torque on a system about a point C. 8 < 1 P mi v 2 EK 2R i discrete : 1 v 2 dm Kinetic energy A scalar measure of net system motion. 2 continuous Eint D (heat-like terms) Internal energy The non-kinetic non-potential part of a sys- tem’s total energy. P P** Fi vi C P M*i *!i Power of forces and torques The mechanical energy flow into a system.
Also, P WP , rate of work. 2 3 I cm Ixy 6 xx cm Ixz cm 7 I 64 Ixy Iyy cm 6 cm cm Iyz cm 7 7 5 Moment of inertia matrix about A measure of the mass distribution in a rigid cm Iyz Ixz cm Izz cm center of mass (cm) object. c Rudra Pratap and Andy Ruina, 1994-2014. All rights reserved.
No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, or otherwise, without prior written permission of the authors. This book is a pre-release version of a book in progress for Oxford University Press. The following are amongst those who have helped with this book as editors, artists, tex programmers, advisors, critics or suggesters and creators of content: William Adams, Alexa Barnes, Pranav Bhounsule, Joseph Burns, Hye Yeon Choe, Jason Cortell, Gabor Domokos, Max Donelan, Thu Dong, Gail Fish, Mike Fox, John Gibson, Robert Ghrist, Vivek Gupta, Saptarsi Haldar, Dave Heimstra, Theresa Howley, Herbert Hui, Dirk Martin (Mark) Luchtenburg, Michael Marder, Elaina McCartney, Saskya van Nouhuys, Horst Nowacki, Jim Papadopoulos, Kalpana Pratap, Dane Quinn, Richard Rand, C. Radakrishnan, Nidhi Rathi, Phoebus Rosakis, Les Schaffer, Ishan Sharma, David Shipman, Jill Startzell, Brett Tallman, Tian Tang, Kim Turner and Bill Zobrist.
Our on-again off-again editor Peter Gordon has been supportive throughout. Many other friends, colleagues, relatives, students, and anonymous reviewers have also made helpful suggestions. We certify Arthur Ogawa, Ivan Dobrianov, and Stephen Hicks as TeX geniuses. Mike Coleman worked extensively on the text, wrote many of the examples and homework problems and made many figures.
Manjula, Abhay and Mieke Ruina drew or improved most of the drawings. Credit for some of the homework problems retrieved from Cornell archives is due to various Theoretical and Applied Mechanics faculty. Harry Soodak and Martin Tiersten provided some problems from their incomplete book. Software we have used to prepare this book includes TEXshop (for LATEX) with many custom features implemented by Stephen Hicks, Adobe Illustrator, GraphicsConverter and MATLAB.
Introduction to Statics and Dynamics, c Andy Ruina and Rudra Pratap 1994-2014. Introduction to Statics and Dynamics, c Andy Ruina and Rudra Pratap 1994-2014. Brief Contents Front tables. 12 To the student.
16 Part I: Basics for Mechanics 26 1 What is mechanics?. 26 2 Vectors: position, force and moment. 126 Part II: Statics 190 4 Statics of one object. 190 5 Trusses and frames.
264 6 Transmissions and mechanisms. 330 7 Tension, shear and bending moment. 408 Part III: Dynamics 426 9 Dynamics in 1D. 506 11 Particles in space.
552 12 Many particles in space. 600 13 Straight line motion. 626 14 Circular motion of a particle. 666 15 Circular motion of a rigid object.
698 16 Planar motion of an object. 778 17 Time-varying basis vectors. 864 18 Constrained particles and rigid objects. 934 Appendices 1004 A Units and dimensions.
1004 B Friction: perspectives on friction laws. 1016 C The simplest ODEs and their solutions. 1026 D Theorems for Systems. 1030 Answers to some homework problems.
1049 Introduction to Statics and Dynamics, c Andy Ruina and Rudra Pratap 1994-2014. Detailed Contents Front tables i Summary of mechanics. i Some basic definitions. ii Brief Contents 2 Detailed Contents 3 Preface 12 General issues about content, level, organization, style and motivation.
Study advice starts on page ??. To the student 16 How to study. The use of computers.1 A note on computation. 21 Box: Informal computer commands.
24 Part I: Basics for Mechanics 26 1 What is mechanics? 26 Mechanics can predict forces and motions by using the three pillars of the subject: I. models of physical behavior, II. geometry, and III. the basic mechanics balance laws.
The laws of mechanics are informally summa- rized in this introductory chapter. The extreme accuracy of Newtonian mechanics is emphasized, despite relativity and quantum mechanics sup- posedly having ‘overthrown’ seventeenth-century physics. Various uses of the word ‘model’ are described.1 The three pillars .2 Mechanics is wrong, why study it? .3 The hierarchy of models. 35 2 Vectors: position, force and moment 42 The key vectors for statics, namely relative position, force, and mo- ment, are used to develop vector skills.
Notational clarity is empha- sized because good vector calculation demands distinguishing vectors from scalars. Vector addition is motivated by the need to add forces and relative positions. Dot products are motivated as the tool which reduces vector equations to scalar equations. And cross products are motivated as Introduction to Statics and Dynamics, c Andy Ruina and Rudra Pratap 1994-2014.
Detailed Contents Detailed Contents the formula which correctly calculates the heuristically motivated quan- tities of moment and moment about an axis.1 Notation and addition .1 The scalars in mechanics .2 The Vectors in Mechanics .2 The dot product of two vectors .3 Basic features of the vector dot product.4 ab cos ) ax bx C ay by C az bz .3 Vector cross product .5 Uses of the cross product .6 Cross product as a matrix multiply .7 The cross product: from geometry to components .5 Solving vector equations .8 The rules of vector algebra.9 Vector triangles and the laws of sines and cosines .10 Existence, uniqueness, and geometry. 112 Problems for Chapter 2. 117 3 FBDs 126 A free-body diagram is a sketch of the system to which you will apply the laws of mechanics. The diagram shows all of the non-negligible external forces and couples which act on the system.
The diagram tells what ma- terial is in the system and also what is known, and what is not known, about the forces. Mechanics reasoning depends on free-body diagrams so we give tips about how to avoid common mistakes. On a free-body diagram systems of forces are often replaced with ‘equivalent’ forces, a special case of which is a weight force at the center of gravity.1 EquivalentP force systems .2 Equivalent at one point ) equivalent at all points 132 Box 3.3 A “wrench” can represent any force system .2 Center of mass and P gravity .4 Like , the symbol also means add .5 Each subsystem is like a particle .6 The COM of a triangle is at h=3 .3 Interactions, forces & partial FBDs. 153 Vector notation for FBDs .7 Free-body diagram first, mechanics reasoning after 163 Box 3.8 Action and reaction on partial FBD’s .4 Contact: Sliding, friction, and rolling.
168 Problems for Chapter 3. 182 Introduction to Statics and Dynamics, c Andy Ruina and Rudra Pratap 1994-2014.
