General Physics I: Classical Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences and Engineering Prince George’s C ommunity College Largo, Maryland Fall 2013 Last updated: December 16, 2013 Contents Acknowledgments 11 1WhatisPhysics? 12 2Units 14 2.1 SystemsofUnits 14 2.2 SIUnits 15 2.3 CGSSystemsofUnits 18 2.4 British Engineering Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5 UnitsasanError-CheckingTechnique 18 2.6 UnitConversions 19 2.7 CurrencyUnits 20 2.8 OddsandEnds 21 3 Problem-Solving Strategies 22 4Density 24 4.1 SpecificGravity 25 4.2 DensityTrivia 25 5 Kinematics in One Dimension 27 5.1 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2 Velocity 27 5.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.4 HigherDerivatives 29 5.5 DotNotation 29 5.6 InverseRelations 29 5.7 Constant Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.8 Summary 32 5.9 GeometricInterpretations 33 6 Vectors 35 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.2 Arithmet i c: Graphical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.3 Arithmetic: Algebraic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.4 Derivatives 40 6.5 Integrals 40 1 Prince George’s Community College General Physics I D.G. Simpson 6.6 OtherVectorOperations 40 7 The Dot Product 42 7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 7.2 Component Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 7.3 Properties 43 7.4 MatrixFormulation 44 8 Kinematics in Two or Three Dimensions 46 8.1 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 8.2 Velocity 46 8.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 8.4 InverseRelations 47 8.5 Constant Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 8.6 Verticalvs.HorizontalMotion 48 8.7 Summary 49 9 Projectile Motion 51 9.1 Range 52 9.2 Maximum Altitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 9.3 Shape of the Projectile Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 9.4 Hitting a Target on the Ground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 9.5 Hitting a Target on a Hill. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 9.6 OtherConsiderations 57 9.7 The Monkey and the Hunter Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 9.8 Summary 59 10 Newton’s Method 60 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 10.2 TheMethod 60 10.3 Example:SquareRoots 60 10.4 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 11 Mass 63 12 Force 64 12.1 TheFourForcesofNature 64 12.2 Hooke’s Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 12.3 Weight 65 12.4 NormalForce 65 12.5 Tension 65 13 Newton’s Laws of Motion 66 13.1 FirstLawofMotion 67 13.2 SecondLawofMotion 67 13.3 ThirdLawofMotion 67 2 Prince George’s Community College General Physics I D.G. Simpson 14 The Inclined Plane 68 15 Atwood’s Machine 69 16 Statics 73 16.1 MassSuspendedbyTwoRopes 73 16.2 ThePulley 76 16.3 TheElevator 76 17 Friction 78 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 17.2 StaticFriction 78 17.3 KineticFriction 79 17.4 Rolling Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 17.5 The CoefficientofFriction 79 18 Resistive Forces in Fluids 81 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 18.2 Model I: F R / v 81 18.3 Model II: F R / v 2 83 19 Circular Motion 86 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 19.2 CentripetalForce 87 19.3 CentrifugalForce 88 19.4 RelationsbetweenCircularandLinearMotion 89 19.5 Examples 89 20 Work 90 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 20.2 Case I: Constant F k r 90 20.3 Case II: Constant F ¬ r 91 20.4 Case III: Variable F k r 91 20.5 Case IV (General Case): Variable F ¬ r 91 20.6 Summary 92 21 Energy 93 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 21.2 KineticEnergy 93 21.3 PotentialEnergy 94 21.4 OtherFormsofEnergy 97 21.5 ConservationofEnergy 97 21.6 TheWork-EnergyTheorem 98 21.7 TheVirialTheorem 98 22 Conservative Forces 100 23 Power 101 23.1 Energy Conversion of a Falling Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3 Prince George’s Community College General Physics I D.G. Simpson 23.2 RateofChangeofPower 102 23.3 VectorEquation 103 24 Linear Momentum 104 24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 24.2 ConservationofMomentum 104 24.3 Newton’sSecondLawofMotion 104 25 Impulse 106 26 Collisions 108 26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 26.2 The Coefficient of Restitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 26.3 Perfectly Inelastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 26.4 Perfectly Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 26.5 Newton’sCradle 111 26.6 Inelastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 26.7 Collisions in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 27 The Ballistic Pendulum 114 28 Rockets 116 28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 28.2 TheRocketEquation 116 28.3 MassFraction 117 28.4 Staging 118 29 Center of Mass 119 29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 29.2 DiscreteMasses 119 29.3 Continuous Bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 30 The Cross Product 123 30.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 30.2 Component Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 30.3 Properties 124 30.4 MatrixFormulation 126 30.5 Inverse 126 31 Rotational Moti on 128 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 31.2 Translationalvs.RotationalMotion 128 31.3 ExampleProblems 130 32 Moment of Inertia 132 32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 32.2 ParallelAxisTheorem 136 32.3 PlaneFigureTheorem 138 32.4 Routh’sRule 138 4 Prince George’s Community College General Physics I D.G. Simpson 32.5 Lees’Rule 138 33 Torque 140 33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 33.2 RotationalVersionsofNewton’sLaws 141 33.3 Rotational Version of Hooke’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 34 The Pendulum 142 34.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 34.2 The Simple Plane Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 34.3 The Spherical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 34.4 The Conical Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 34.5 The Torsional Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 34.6 The Physical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 34.7 Other Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 35 Simple Harm onic Motion 148 35.1 Energy 150 35.2 FrequencyandPeriod 152 35.3 MassonaSpring 152 36 Rolling Bodies 154 36.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 36.2 Velocity 154 36.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 36.4 KineticEnergy 156 36.5 TheWheel 157 36.6 Ball Rolling in a Bowl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 37 Galileo’s Law 160 37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 37.2 ModernTreatment 160 38 The Coriolis Force 162 38.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 38.2 Examples 163 39 Angular Momentum 164 39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 39.2 Conservation of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 40 Conservation Laws 166 41 The Gyroscope 167 41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 41.2 Precession 167 41.3 Nutation 168 5 Prince George’s Community College General Physics I D.G. Simpson 42 Roller Coasters 169 42.1 GeneralPrinciples 169 42.2 LoopShapes 169 43 Elasticity 170 43.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 43.2 Longitudinal(Normal)Stress 170 43.3 Transverse(Shear)Stress—Translational 171 43.4 Transverse(Shear)Stress—Torsional 172 43.5 VolumeStress 172 43.6 ElasticLimit 173 43.7 Summary 173 44 Fluid Mechanics 174 44.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 44.2 Archimedes’Principle 174 44.3 FloatingBodies 174 44.4 Pressure 175 44.5 ChangeinFluidPressurewithDepth 176 44.6 Pascal’sLaw 177 44.7 FluidDynamics 177 44.8 TheContinuityEquation 178 44.9 Bernoulli’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 44.10 Torricelli’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 44.11 The Siphon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 44.12 Viscosity 182 44.13 The Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 44.14 Stokes’sLaw 184 44.15 Fluid Flow through a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 44.16 Superfluids 185 45 Hydraulics 188 45.1 TheHydraulicPress 188 46 Pneumatics 190 47 Gravity 191 47.1 Newton’sLawofGravity 191 47.2 GravitationalPotential 191 47.3 TheCavendishExperiment 192 47.4 Helmert’sEquation 192 47.5 EscapeVelocity 193 47.6 Gauss’sFormulation 193 47.7 GeneralRelativity 197 47.8 BlackHoles 198 6 Prince George’s Community College General Physics I D.G. Simpson 48 Earth Rotation 199 48.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 48.2 Precession 199 48.3 Nutation 199 48.4 PolarMotion 201 48.5 RotationRate 201 49 Geodesy 203 49.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 49.2 TheCosineFormula 203 49.3 Vincenty’s Formulæ: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 49.4 Vincenty’sFormulæ:DirectProblem 204 49.5 Vincenty’sFormulæ:InverseProblem 206 50 Celestial Mechanics 209 50.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 50.2 Kepler’sLaws 209 50.3 Time 210 50.4 OrbitReferenceFrames 210 50.5 OrbitalElements 211 50.6 RightAscensionandDeclination 212 50.7 Computing a Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 50.8 TheInverseProblem 214 50.9 CorrectionstotheTwo-BodyCalculation 214 50.10 Bound and Unbound Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 50.11 The Vis Viva Equation 215 50.12 Bertrand’sTheorem 216 50.13 DifferentialEquationforanOrbit 216 50.14 LagrangePoints 217 50.15 TheRingsofSaturn 218 50.16 HyperbolicOrbits 220 50.17 ParabolicOrbits 221 51 Astrodynamics 222 51.1 CircularOrbits 222 51.2 Geosynchronous Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 51.3 Elliptical Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 51.4 TheHohmannTransfer 228 51.5 GravityAssistManeuvers 229 51.6 TheInternationalCometaryExplorer 231 52 Partial Derivatives 233 52.1 FirstPartialDerivatives 233 52.2 Higher-OrderPartialDerivatives 234 53 Lagrangian Mechanics 235 53.1 Examples 236 7 Prince George’s Community College General Physics I D.G. Simpson 54 Hamiltonian Mechanics 238 54.1 Examples 238 55 Special Relativity 241 55.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 55.2 Postulates 241 55.3 TimeDilation 241 55.4 LengthContraction 242 55.5 AnExample 242 55.6 Momentum 242 55.7 Addition of Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 55.8 Energy 243 56 Quantum Mechanics 245 56.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 56.2 ReviewofNewtonianMechanics 245 56.3 QuantumMechanics 245 56.4 Example: Simple Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 56.5 TheHeisenbergUncertaintyPrinciple 248 AFurtherReading 250 B Greek Alphabet 254 C Trigonometry 255 D Useful Series 258 ESIUnits 259 F Gaussian Units 262 G Units of Physical Quantities 264 H Physical Constants 267 IAstronomicalData 268 J Unit Conversion Tables 269 K Angular Measure 272 K.1 PlaneAngle 272 K.2 SolidAngle 272 L Vector Arithmetic 274 M Matrix Properties 277 N Newton’s Laws of Motion (Original) 279 8 Prince George’s Community College General Physics I D.G. Simpson O The Simple Plane Pendulum: Exact Solution 280 O.1 EquationofMotion 280 O.2 Solution, Â.t/ 281 O.3 Period 281 P Motion of a Falling Body 285 Q Tabl e of Viscosities 287 R TI-83+ / TI-84+ Calculator Programs 289 R.1 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 R.2 Kepler’sEquation 290 R.3 HyperbolicKepler’sEquation 290 R.4 Barker’sEquation 291 R.5 ReductionofanAngle 291 R.6 Helmert’sEquation 292 R.7 Pendulum Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 R.8 1D Perfectly Elastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 S TI-89 / TI-92 / Voyage 200 Calculator Programs 295 S.1 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 S.2 Kepler’sEquation 296 S.3 HyperbolicKepler’sEquation 296 S.4 Barker’sEquation 297 S.5 ReductionofanAngle 298 S.6 Helmert’sEquation 298 S.7 Pendulum Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 S.8 1D Perfectly Elastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 T HP 35s / HP 15C Calculator Programs 301 T.1 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 T.2 Kepler’sEquation 303 T.3 HyperbolicKepler’sEquation 304 T.4 Barker’sEquation 306 T.5 ReductionofanAngle 307 T.6 Helmert’sEquation 308 T.7 Pendulum Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 T.8 1D Perfectly Elastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 U HP 48 / HP 50g Calculator Programs 313 U.1 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 U.2 Kepler’sEquation 314 U.3 HyperbolicKepler’sEquation 314 U.4 Barker’sEquation 315 U.5 ReductionofanAngle 315 U.6 Helmert’sEquation 316 U.7 Pendulum Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 U.8 1D Perfectly Elastic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 9