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University Physics I: Classical Mechanics University of Arkansas, Fayetteville ScholarWorks@UARK Open Educational Resources 2-8-2019 University Physics I: Classical Mechanics Julio Gea-Banacloche University of Arkansas, Fayetteville Follow this and additional works at: https://scholarworks.uark.edu/oer Part of the Atomic, Molecular and Optical Physics Commons, Elementary Particles and Fields and String Theory Commons, Engineering Physics Commons, and the Other Physics Commons Citation Gea-Banacloche, J. (2019). University Physics I: Classical Mechanics. Open Educational Resources. Retrieved from https://scholarworks.uark.edu/oer/3 This Textbook is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Open Educational Resources by an authorized administrator of ScholarWorks@UARK. For more information, please contact [email protected]. University Physics I: Classical Mechanics Julio Gea-Banacloche Cover image from NASA, https://www.nasa.gov/image-feature/jpl/not-really-starless-at-saturn University Physics I: Classical Mechanics Julio Gea-Banacloche First revision, Fall 2019 This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. Developed thanks to a grant from the University of Arkansas Libraries ii Contents Preface i 1 Reference frames, displacement, and velocity 1 1.1 Introduction ......................................... 1 1.1.1 Particles in classical mechanics .......................... 1 1.1.2 Aside: the atomic perspective ........................... 3 1.2 Position, displacement, velocity .............................. 3 1.2.1 Position ....................................... 4 1.2.2 Displacement .................................... 6 1.2.3 Velocity ....................................... 8 1.3 Reference frame changes and relative motion ...................... 14 1.4 In summary ......................................... 19 1.5 Examples .......................................... 21 1.5.1 Motion with (piecewise) constant velocity .................... 21 1.5.2 Addition of velocities, relative motion ...................... 24 iii iv CONTENTS 1.6 Problems .......................................... 27 2 Acceleration 31 2.1 The law of inertia ...................................... 31 2.1.1 Inertial reference frames .............................. 33 2.2 Acceleration ......................................... 35 2.2.1 Average and instantaneous acceleration ..................... 35 2.2.2 Motion with constant acceleration ........................ 39 2.2.3 Acceleration as a vector .............................. 41 2.2.4 Acceleration in different reference frames .................... 41 2.3 Free fall ........................................... 42 2.4 In summary ......................................... 44 2.5 Examples .......................................... 46 2.5.1 Motion with piecewise constant acceleration ................... 46 2.6 Problems .......................................... 48 3 Momentum and Inertia 51 3.1 Inertia ............................................ 51 3.1.1 Relative inertia and collisions ........................... 51 3.1.2 Inertial mass: definition and properties ..................... 55 3.2 Momentum ......................................... 56 3.2.1 Conservation of momentum; isolated systems .................. 56 CONTENTS v 3.3 Extended systems and center of mass ........................... 58 3.3.1 Center of mass motion for an isolated system .................. 59 3.3.2 Recoil and rocket propulsion ........................... 61 3.4 In summary ......................................... 61 3.5 Examples .......................................... 63 3.5.1 Reading a collision graph ............................. 63 3.5.2 Collision in different reference frames, center of mass,andrecoil ....... 65 3.6 Problems .......................................... 67 4 Kinetic Energy 69 4.1 Kinetic Energy ....................................... 69 4.1.1 Kinetic energy in collisions ............................ 70 4.1.2 Relative velocity and coefficient of restitution .................. 74 4.2 “Convertible” and “translational” kinetic energy .................... 76 4.2.1 Kinetic energy and momentum in different reference frames .......... 79 4.3 In summary ......................................... 80 4.4 Examples .......................................... 82 4.4.1 Collision graph revisited .............................. 82 4.4.2 Inelastic collision and explosive separation .................... 83 4.5 Problems .......................................... 86 5 Interactions and energy 89 vi CONTENTS 5.1 Conservative interactions ................................. 89 5.1.1 Potential energy .................................. 90 5.1.2 Potential energy functions and “energy landscapes” .............. 94 5.2 Dissipation of energy and thermal energy ........................ 96 5.3 Fundamental interactions, and other forms of energy .................. 98 5.4 Conservation of energy ................................... 99 5.5 In summary .........................................101 5.6 Examples ..........................................103 5.6.1 Inelastic collision in the middle of a swing ....................103 5.6.2 Kinetic energy to spring potential energy: block collides with spring .....104 5.7 Advanced Topics ......................................106 5.7.1 Two carts colliding and compressing a spring ..................106 5.7.2 Getting the potential energy function from collision data ............107 5.8 Problems ..........................................109 6 Interactions, part 2: Forces 113 6.1 Force .............................................113 6.1.1 Forces and systems of particles ..........................115 6.2 Forces and potential energy ................................116 6.3 Forces not derived from a potential energy ........................120 6.3.1 Tensions .......................................120 CONTENTS vii 6.3.2 Normal forces ....................................122 6.3.3 Static and kinetic friction forces .........................123 6.3.4 Air resistance ....................................125 6.4 Free-body diagrams ....................................126 6.5 In summary .........................................127 6.6 Examples ..........................................129 6.6.1 Dropping an object on a weighing scale .....................129 6.6.2 Speeding up and slowing down ..........................130 6.7 Problems ..........................................134 7 Impulse, Work and Power 137 7.1 Introduction: work and impulse ..............................137 7.2 Work on a single particle .................................138 7.2.1 Work done by the net force, and the Work-Energy Theorem ..........140 7.3 The “center of mass work” .................................141 7.4 Work done on a system by all the external forces ....................142 7.4.1 The no-dissipation case ..............................143 7.4.2 The general case: systems with dissipation ...................147 7.4.3 Energy dissipated by kinetic friction .......................150 7.5 Power ............................................150 7.6 In summary .........................................151 viii CONTENTS 7.7 Examples ..........................................153 7.7.1 Braking .......................................153 7.7.2 Work, energy and the choice of system: dissipative case ............154 7.7.3 Work, energy and the choice of system: non-dissipativecase..........155 7.7.4 Jumping .......................................156 7.8 Problems ..........................................158 8 Motion in two dimensions 161 8.1 Dealing with forces in two dimensions ..........................161 8.2 Projectile motion ......................................164 8.3 Inclined planes .......................................167 8.4 Motion on a circle (or part of a circle) ..........................169 8.4.1 Centripetal acceleration and centripetal force ..................169 8.4.2 Kinematic angular variables ...........................172 8.5 In summary .........................................175 8.6 Examples ..........................................177 8.6.1 The penny on the turntable ............................177 8.7 Advanced Topics ......................................179 8.7.1 Staying on track ..................................179 8.7.2 Going around a banked curve ...........................181 8.7.3 Rotating frames of reference: Centrifugal force and Coriolis force .......184 CONTENTS ix 8.8 Problems ..........................................186 9 Rotational dynamics 191 9.1 Rotational kinetic energy, and moment of inertia ....................191 9.2 Angular momentum ....................................193 9.3 The cross product and rotational quantities .......................197 9.4 Torque ............................................201 9.5 Statics ............................................204 9.6 Rolling motion .......................................207 9.7 In summary .........................................211 9.8 Examples ..........................................213 9.8.1 Torques and forces on the wheels of an accelerating bicycle ..........213 9.8.2 Blocks connected by rope over a pulley with non-zero mass ..........214 9.9 Problems ..........................................217 10 Gravity 221 10.1 The inverse-square law ...................................221 10.1.1 Gravitational potential energy ..........................224 10.1.2 Types of orbits under an inverse-square force ..................227
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