The Physics GRE Solution Guide RA FT Sample, GR8677, GR9277, GR9677 and GR0177 Tests D http://groups.yahoo.com/group/physicsgre_v2 November 6, 2009 Author: David S Latchman D RA FT David S Latchman ©2009 Preface RA David Latchman FT This solution guide initially started out on the Yahoo Groups web site and was pretty successful at the time Unfortunately, the group was lost and with it, much of the the hard work that was put into it This is my attempt to recreate the solution guide and make it more widely avaialble to everyone If you see any errors, think certain things could be expressed more clearly, or would like to make suggestions, please feel free to so Document Changes 05-11-2009 Added diagrams to GR0177 test questions 1-25 Revised solutions to GR0177 questions 1-25 D 04-15-2009 First Version D RA FT ii David S Latchman ©2009 Preface Classical Mechanics i 1.1 Kinematics 1.2 Newton’s Laws 1.3 Work & Energy 1.4 Oscillatory Motion 1.5 Rotational Motion about a Fixed Axis 1.6 Dynamics of Systems of Particles 10 1.7 Central Forces and Celestial Mechanics 10 1.8 Three Dimensional Particle Dynamics 12 1.9 Fluid Dynamics 12 RA FT Contents D 1.10 Non-inertial Reference Frames 13 1.11 Hamiltonian and Lagrangian Formalism 13 Electromagnetism 15 2.1 Electrostatics 15 2.2 Currents and DC Circuits 20 2.3 Magnetic Fields in Free Space 20 2.4 Lorentz Force 20 2.5 Induction 20 2.6 Maxwell’s Equations and their Applications 20 2.7 Electromagnetic Waves 20 2.8 Contents AC Circuits 20 2.9 Magnetic and Electric Fields in Matter 20 iv 2.10 Capacitance 21 2.11 Energy in a Capacitor 21 2.12 Energy in an Electric Field 21 2.13 Current 21 2.14 Current Destiny 21 2.15 Current Density of Moving Charges 21 2.16 Resistance and Ohm’s Law 21 FT 2.17 Resistivity and Conductivity 22 2.18 Power 22 2.19 Kirchoff’s Loop Rules 22 2.20 Kirchoff’s Junction Rule 22 2.21 RC Circuits 22 RA 2.22 Maxwell’s Equations 22 2.23 Speed of Propagation of a Light Wave 23 2.24 Relationship between E and B Fields 23 2.25 Energy Density of an EM wave 24 2.26 Poynting’s Vector 24 Optics & Wave Phonomena 25 3.1 Wave Properties 25 3.2 Superposition 25 3.3 Interference 25 3.4 Diffraction 25 3.5 Geometrical Optics 25 3.6 Polarization 25 3.7 Doppler Effect 26 3.8 Snell’s Law 26 D Thermodynamics & Statistical Mechanics 27 4.1 Laws of Thermodynamics 27 4.2 Thermodynamic Processes 27 David S Latchman ©2009 Contents v 4.3 Equations of State 27 4.4 Ideal Gases 27 4.5 Kinetic Theory 27 4.6 Ensembles 27 4.7 Statistical Concepts and Calculation of Thermodynamic Properties 28 4.8 Thermal Expansion & Heat Transfer 28 4.9 Heat Capacity 28 4.10 Specific Heat Capacity 28 4.11 Heat and Work 28 FT 4.12 First Law of Thermodynamics 28 4.13 Work done by Ideal Gas at Constant Temperature 29 4.14 Heat Conduction Equation 29 4.15 Ideal Gas Law 30 4.16 Stefan-Boltzmann’s FormulaStefan-Boltzmann’s Equation 30 30 RA 4.17 RMS Speed of an Ideal Gas 4.18 Translational Kinetic Energy 30 4.19 Internal Energy of a Monatomic gas 30 4.20 Molar Specific Heat at Constant Volume 31 4.21 Molar Specific Heat at Constant Pressure 31 4.22 Equipartition of Energy 31 4.23 Adiabatic Expansion of an Ideal Gas 33 D 4.24 Second Law of Thermodynamics 33 Quantum Mechanics 35 5.1 Fundamental Concepts 35 5.2 Schrodinger Equation 35 ¨ 5.3 Spin 40 5.4 Angular Momentum 41 5.5 Wave Funtion Symmetry 41 5.6 Elementary Perturbation Theory 41 Atomic Physics 43 6.1 Properties of Electrons 43 ©2009 David S Latchman 6.3 Energy Quantization 44 6.4 Atomic Structure 44 6.5 Atomic Spectra 45 6.6 Selection Rules 45 6.7 Black Body Radiation 45 6.8 X-Rays 46 6.9 Atoms in Electric and Magnetic Fields 47 Special Relativity 51 7.1 Introductory Concepts 51 7.2 Time Dilation 51 7.3 Length Contraction 51 7.4 Simultaneity 52 7.5 Energy and Momentum 52 7.6 Four-Vectors and Lorentz Transformation 53 7.7 Velocity Addition 54 7.8 Relativistic Doppler Formula 54 7.9 Lorentz Transformations 55 RA FT 6.2 Contents Bohr Model 43 vi 7.10 Space-Time Interval 55 Laboratory Methods 57 8.1 Data and Error Analysis 57 8.2 Instrumentation 59 8.3 Radiation Detection 59 8.4 Counting Statistics 59 8.5 Interaction of Charged Particles with Matter 60 8.6 Lasers and Optical Interferometers 60 8.7 Dimensional Analysis 60 8.8 Fundamental Applications of Probability and Statistics 60 D Sample Test 9.1 61 Period of Pendulum on Moon 61 David S Latchman ©2009 Contents vii 9.2 Work done by springs in series 62 9.3 Central Forces I 63 9.4 Central Forces II 64 9.5 Electric Potential I 65 9.6 Electric Potential II 66 9.7 Faraday’s Law and Electrostatics 66 9.8 AC Circuits: RL Circuits 66 9.9 AC Circuits: Underdamped RLC Circuits 68 FT 9.10 Bohr Model of Hydrogen Atom 70 9.11 Nuclear Sizes 73 9.12 Ionization of Lithium 74 9.13 Electron Diffraction 74 9.14 Effects of Temperature on Speed of Sound 75 RA 9.15 Polarized Waves 75 9.16 Electron in symmetric Potential Wells I 76 9.17 Electron in symmetric Potential Wells II 77 9.18 Relativistic Collisions I 77 9.19 Relativistic Collisions II 77 9.20 Thermodynamic Cycles I 78 9.21 Thermodynamic Cycles II 78 D 9.22 Distribution of Molecular Speeds 79 9.23 Temperature Measurements 79 9.24 Counting Statistics 80 9.25 Thermal & Electrical Conductivity 80 9.26 Nonconservation of Parity in Weak Interactions 81 9.27 Moment of Inertia 82 9.28 Lorentz Force Law I 83 9.29 Lorentz Force Law II 84 9.30 Nuclear Angular Moment 85 9.31 Potential Step Barrier 85 ©2009 David S Latchman Contents 87 viii 10 GR8677 Exam Solutions 10.1 Motion of Rock under Drag Force 87 10.2 Satellite Orbits 88 10.3 Speed of Light in a Dielectric Medium 88 10.4 Wave Equation 88 10.5 Inelastic Collision and Putty Spheres 89 10.6 Motion of a Particle along a Track 90 10.7 Resolving Force Components 90 10.8 Nail being driven into a block of wood 91 FT 10.9 Current Density 91 10.10Charge inside an Isolated Sphere 92 10.11Vector Identities and Maxwell’s Laws 93 10.12Doppler Equation (Non-Relativistic) 93 10.13Vibrating Interference Pattern 93 RA 10.14Specific Heat at Constant Pressure and Volume 93 10.15Helium atoms in a box 94 10.16The Muon 95 10.17Radioactive Decay 95 10.18Schrodinger’s Equation 96 ¨ 10.19Energy Levels of Bohr’s Hydrogen Atom 96 10.20Relativistic Energy 97 D 10.21Space-Time Interval 97 10.22Lorentz Transformation of the EM field 98 10.23Conductivity of a Metal and Semi-Conductor 98 10.24Charging a Battery 99 10.25Lorentz Force on a Charged Particle 99 10.26K-Series X-Rays 99 10.27Electrons and Spin 100 10.28Normalizing a wavefunction 101 10.29Right Hand Rule 102 10.30Electron Configuration of a Potassium atom 102 10.31Photoelectric Effect I 103 David S Latchman ©2009 GR0177 Exam Solutions 298 13.90 Springs in Series and Parallel The Period of a Mass-Spring System is ω= ke m (13.90.1) Now it comes down to solving the effective spring constants for springs in series and parallel The addition of springs in series and parallel are the same as capacitors but if you didn’t recall this you can solve the relationships Springs in Parallel FT 13.90.1 In the parallel case, both springs extend by the same amount, x The Forces on both springs also add up sych that F = F1 + F2 = −k1 x − k2 x (13.90.2) RA This Parallel arrangement is the same as a Mass-Spring system with only one spring of spring constant, ke We have F = −ke x (13.90.3) (eq (13.90.2)) = (eq (13.90.3)), gives ke = k1 + k2 13.90.2 (13.90.4) Springs in Series D Springs in Series is a bit more challenging For this case, we will assume light springs such that the tension throughout the springs is constant So we have F = −k1 x1 = −k2 x2 k1 x2 = ⇒ k2 x1 (13.90.5) This is equivalent to a single Spring system where we again have a single Spring System but x = x1 + x2 (13.90.6) Since the Forces are equal, we can say kx = k2 x2 k (x1 + x2 ) = k2 x2 k1 k2 ⇒k= k1 + k2 David S Latchman (13.90.7) ©2009 Cylinder rolling down an incline We have thus shown that 299 1 = + k k1 k2 (13.90.8) The period of the Mass-Spring Sytem for the Series Arrangement becomes ke = k1 + k2 = 2k Ts = 2π m ke = 2π m 2k (13.90.9) FT The period for the Mass-Spring System in the Parallel arrangement becomes 1 = + k k k = k Tp = 2π RA The ratio between Tp and Ts is 2m k Tp 2π = Ts 2π (13.90.10) 2m k m 2k =2 (13.90.11) Answer: (E)15 Cylinder rolling down an incline D 13.91 As the cylinder rolls down the hill, Gravitational Potential Energy is converted to Translational Kinetic Energy and Rotational Kinectic Energy This can be expressed as PEgravity = KEtranslational + KErotational 1 mgh = mv2 + Iω2 2 (13.91.1) Solving for I, we have I= MR2 2gH − v2 v2 (13.91.2) 15 Springs are like Capacitors; when in parallel, their spring constants add and when in series, the inverse of the total spring constant is the sum of the inverse of the individual ones ©2009 David S Latchman GR0177 Exam Solutions 300 Plugging in for v, we get I = MR2 8gH 2gH − 8gH = MR2 (13.91.3) Answer: (B) 13.92 Hamiltonian of Mass-Spring System The Hamiltonian of a System is H =T+V (13.92.1) FT p2i 2m where Ti = and V = V(q) So the Hamiltonian is the sum of the kinetic energies of the partiles and the energy stroed in the spring Thus H= 2 p1 p2 + +k( − m m 13.93 RA Answer: (E) 0) (13.92.2) Radius of the Hydrogen Atom The radial probability density for the ground state of the Hydrogen atom is found by multiplying the square of the wavefunction by the spherical shell volume element Pr = ψ0 dV (13.93.1) D The Volume of a sphere is V = 43 πr3 , so dV = 4πr2 From the above equation we see that − 2r dPr e a0 = · 4πr2 (13.93.2) dr πa30 We find the maxima and thus the most probable position by determining Differentiating eq (13.93.2) gives Solving for r gives d2 Pr dr2 = d2 P − a2r − a2r 2 − r · = 2r · e · e =0 dr2 a0 a30 (13.93.3) r = a0 (13.93.4) This is Bohr’s Radius which was found using semi-classical methods In this case, Schroedinger’s Equation confirms the first Bohr radius as the most probable radius and more; the semi-classical Bohr’s Theory does not Answer: (C) David S Latchman ©2009 Perturbation Theory 13.94 301 Perturbation Theory Perturbation Theory NOT FINISHED 13.95 Electric Field in a Dielectric The Electric Field has magnitude 4πκ q σ = κ 0r (13.95.1) FT E= In a vacuum, κ = and the strength of the Electric Field is E0 So E= 13.96 (13.95.2) RA Answer: (A) E0 κ EM Radiation Though the size of the sphere oscillates between R1 and R2 , the charge remains the same So the power radiated is zero Answer: (E) Dispersion of a Light Beam D 13.97 The Angular Spread of the light beam can be calculated by using Snell’s Law n1 sin θ1 = n2 sin θ2 (13.97.1) For an Air-Glass system, this becomes sin θ1 = n sin θ (13.97.2) We know that for dispersion to take place then θ = θ (λ) n = n(λ) ©2009 (13.97.3) (13.97.4) David S Latchman 302 Differentiating eq (13.97.2) with respect to λ, we have GR0177 Exam Solutions d d (sin θ) = (n sin θ ) dλ dλ d (n sin θ ) 0= dλ dn dθ = sin θ + n cos θ dλ dλ dθ dn ⇒ = tan θ dλ n dλ dn ∴ δθ = tan θ δλ n dλ 13.98 FT Answer: (E) (13.97.5) Average Energy of a Thermal System RA The thermodynamic total energy is simply the expected value of the energy; this is the sum of the microstate energies weighed by their probabilities This look like Ei e−Ei /kT E = i e−Ei /kT (13.98.1) i D Answer: (A) 13.99 Pair Production in vincinity of an electron The familiar pair production reaction takes place in the Coulomb field of a massive atom As this nucleus is massive, we can ignore any recoil action of this spectator to calculate the minimum energy needed for our photon This time, our pair production process takes place in the neighbourhood of an electron thus forcing us to take the momenta and energies of all participants preset16 Our pair production process is γ + e− −→ e− + e− + e+ 16 This question was covered as an example question here David S Latchman ©2009 Pair Production in vincinity of an electron 13.99.1 303 Solution Momentum and Energy is conserved during the process The energy of our photon is, E Conservation of Momentum shows us E = c 3me v v 1− c (13.99.1) The left hand side of the equation is the momentum of our photon and the right hand side is the momentum of all our electrons17 We assume that their momenta is the same for all Energy conservation gives us 3me c2 E + me c2 = (13.99.2) FT v 1− c Dividing eq (13.99.1) by eq (13.99.2) gives us E v = E + me c2 c (13.99.3) Substituting eq (13.99.3) into eq (13.99.1) yields E E = 3me c c E + me c2 RA E + me c2 2E me c2 + (me c2 )2 (13.99.4) After some very quick simplification, we get E = 4me c2 13.99.2 (13.99.5) Solution D You may find the above a bit calculation intensive; below is a somewhat quicker solution but the principle is exactly the same We use the same equations in a different form The total relativistic energy before our collision is Ei = E + me c2 (13.99.6) After collision, the relativistic energy of one electron is E2e = pe c + me c2 (13.99.7) We have three electrons so the final energy is E f = 3Ee =3 pe c + (me c2 )2 (13.99.8) 17 I am using electrons to indicate both electrons & positrons As they have the same rest mass we can treat them the same We don’t have to pay attention to their charges ©2009 David S Latchman GR0177 Exam Solutions 304 Thus we have E + me c2 = pe c + (me c2 )2 (13.99.9) As momentum is conserved, we can say pe = p (13.99.10) where p is the momentum of the photon, which happens to be E = pc (13.99.11) Substituting eq (13.99.11) and eq (13.99.10) into eq (13.99.9) gives us (13.99.12) FT E = 4me c2 Which is exactly what we got the first time we worked it out18 Answer: (D) 13.100 Michelson Interferometer RA A fringe shift is registered when the movable mirror moves a full wavelength So we can say d = mλ (13.100.1) where m is the number of fringes If m g and mr are the number of fringes for green and red light respectively, the wavelength of green light will be λg = mr λr mg (13.100.2) D This becomes (85865)(632.82) 100000 86000 · 630 ≈ 100000 = 541 · · · λg = (13.100.3) Answer: (B) 18 The maximum wavelength of this works out to be λ= h λc = 4me c (13.99.13) where λc is the Compton Wavelength David S Latchman ©2009 Appendix A A.1 Constants Symbol c G me NA R k e Value 2.99 × 108 m/s 6.67 × 10−11 m3 /kg.s2 9.11 × 10−31 kg 6.02 × 1023 mol-1 8.31 J/mol.K 1.38 × 10−23 J/K 1.60 × 10−9 C 8.85 × 10−12 C2 /N.m2 4π × 10−7 T.m/A 1.0 × 105 M/m2 0.529 × 10−10 m D RA Constant Speed of light in a vacuum Gravitational Constant Rest Mass of the electron Avogadro’s Number Universal Gas Constant Boltzmann’s Constant Electron charge Permitivitty of Free Space Permeability of Free Space Athmospheric Pressure Bohr Radius FT Constants & Important Equations µ0 atm a0 Table A.1.1: Something A.2 Vector Identities A.2.1 Triple Products A · (B × C) = B · (C × A) = C · (A × B) A × (B × C) = B (A · C) − C (A · B) (A.2.1) (A.2.2) Constants & Important Equations 306 A.2.2 Product Rules ∇ f g = f ∇g + g ∇ f ∇ (A · B) = A × (∇ × B) + B × (∇ × A) + (A · ∇) B + (B · ∇) A ∇ · f A = f (∇ · A) + A · ∇ f ∇ · (A × B) = B · (∇ × A) − A · (∇ × B) ∇ × f A = f (∇ × A) − A × ∇ f ∇ × (A × B) = (B · ∇) A − (A · ∇) B + A (∇ · B) − B (∇ · A) Second Derivatives FT A.2.3 ∇ · (∇ × A) = ∇ × ∇f = ∇ × (∇ × A) = ∇ (∇ · A) − ∇2 A Commutators A.3.1 Lie-algebra Relations RA A.3 D [A, A] = [A, B] = −[B, A] [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = A.3.2 A.3.3 (A.2.3) (A.2.4) (A.2.5) (A.2.6) (A.2.7) (A.2.8) (A.2.9) (A.2.10) (A.2.11) (A.3.1) (A.3.2) (A.3.3) Canonical Commutator [x, p] = i (A.3.4) Kronecker Delta Function δmn = if m n; if m = n; For a wave function ψm (x)∗ ψn (x)dx = δmn David S Latchman (A.3.5) ©2009 Linear Algebra 307 A.4 Linear Algebra A.4.1 Vectors Vector Addition The sum of two vectors is another vector Associative (A.4.1) |α + |β = |β + |α (A.4.2) |α + |β + |γ Zero Vector = |α + |β + |γ (A.4.3) |α + |0 = |α (A.4.4) |α + | − α = |0 (A.4.5) D RA Inverse Vector FT Commutative |α + |β = |γ ©2009 David S Latchman Constants & Important Equations D RA FT 308 David S Latchman ©2009 Bibliography FT [1] Stephen Gasiorowicz Paul M Fishbane and Stephen T Thornton Physics for Scientists and Engineers with Modern Physics, chapter 24.2, page 687 Prentice Hall, third edition, 2005 [2] Wikipedia Maxwell’s equations — wikipedia, the free encyclopedia, 2009 [Online; accessed 21-April-2009] D RA [3] David J Griffiths Introduction to Electrodyanmics, chapter 5.3.4, page 232 Prentice Hall, third edition, 1999 Index RLC Circuits Sample Test Q09, 68 RL Circuits Sample Test Q08, 66 D RA FT Commutators, 306 Canonical Commutators, 306 Kronecker Delta Function, 306 Lie-algebra Relations, 306 Compton Effect, 46 Amplifiers Compton Wavelength GR8677 Q39, 105 GR8677 Q45, 108 Angular Momentum, see Rotational Mo- Conductivity tion GR8677 Q23, 98 Counting Statistics, 59 Binding Energy GR8677 Q40, 105 GR8677 Q41, 106 Current Density Bohr Model GR8677 Q09, 91 GR0177 Q18, 257 GR0177 Q93, 300 Dielectrics GR8677 Q19, 96 GR8677 Q03, 88 Hydrogen Model, 43 Digital Circuits GR8677 Q38, 105 Capacitors Doppler Effect, GR0177 Q10, 252 Drag Force Celestial Mechanics, 10 GR8677 Q01, 87 Circular Orbits, 11 Escape Speed, 10 Elastic Colissions Kepler’s Laws, 11 GR8677 Q05, 89 Newton’s Law of Gravitation, 10 Electric Potential Orbits, 11 Sample Test Q05, 65 Potential Energy, 10 Work Central Forces Sample Test Q06, 66 Sample Test Q03, 63 Electricity Sample Test Q04, 64 GR8677 Q24, 99 Cetripetal Motion Electron Spin GR8677 Q06, 90 GR8677 Q27, 100 Circular Motion Electronic Configuration GR0177 Q2, 246 GR8677 Q30, 102 Circular Orbits, see Celestial Mechanics Equipartition Theorem Collisions GR0177 Q4, 248 GR0177 Q5, 249 Index Faraday’s Law Sample Test Q07, 66 Fleming’s Right Hand Rule GR8677 Q29, 102 Fourier Series GR9277 Q39, 159 Franck-Hertz Experiment, 49 GR8677 Q47, 109 311 Nuclear Physics Radioactive Decay GR8677 Q17, 95 Hall Effect GR8677 Q50, 110 Hamiltonian GR8677 Q35, 104 Interference GR8677 Q13, 93 Parallel Axis Theorem, see Rotational Motion Particle Physics Muon GR8677 Q16, 95 Pendulum Simple GR0177 Q1, 245 Sample Test Q01, 61 Photoelectric Effect GR8677 Q31, 103 GR8677 Q32, 103 GR8677 Q33, 103 Potential Energy GR8677 Q34, 103 Principle of Least Action GR8677 Q36, 104 Probability GR8677 Q15, 94 RA Kepler’s Laws, see Celestial Mechanics GR0177 Q3, 247 Kronecker Delta Function, 306 FT Gauss’ Law GR8677 Q10, 92 Gravitation, see Celestial Mechanics Oscillations Underdamped Sample Test Q09, 68 Oscillatory Motion, Coupled Harmonic Oscillators, GR8677 Q43, 106 Damped Motion, Kinetic Energy, Potential Energy, Simple Harmonic Motion Equation, Small Oscillations, GR9677 Q92, 237 Total Energy, D Laboratory Methods GR8677 Q40, 105 Linear Algebra, 307 Vectors, 307 Lorentz Force Law GR8677 Q25, 99 Lorentz Transformation GR8677 Q22, 98 Maximum Power Theorem GR8677 Q64, 117 Maxwell’s Equations Sample Test Q07, 66 Maxwell’s Laws GR8677 Q11, 93 Mechanics GR8677 Q07, 90 GR8677 Q08, 91 GR8677 Q37, 104 Moment of Inertia, see Rotational Motion Rolling Kinetic Energy, see Rotational Motion Rotational Kinetic Energy, see Rotational Motion Rotational Motion, Angular Momentum, Moment of Inertia, Parallel Axis Theorem, Newton’s Law of Gravitation, see Celestial Rolling Kinetic Energy, Mechanics ©2009 David S Latchman Index 312 RA Satellite Orbits GR8677 Q02, 88 Schrodinger’s Equation ¨ GR8677 Q18, 96 Space-Time Interval GR8677 Q21, 97 Special Relativity Doppler Shift GR8677 Q12, 93 Energy GR8677 Q20, 97 Specific Heat GR8677 Q14, 93 Springs Work Sample Test Q02, 62 Stefan-Boltzmann’s Equation GR8677 Q46, 108 Subject, 30 System of Particles, 10 FT Rotational Kinetic Energy, Torque, Thin Film Interference GR8677 Q73, 122 Torque, see Rotational Motion D Vector Identities, 305 Product Rules, 306 Second Derivatives, 306 Triple Products, 305 Wave Equation GR8677 Q04, 88 Wave function GR8677 Q28, 101 X-Rays GR8677 Q26, 99 David S Latchman ©2009 [...]... exactly how the masses move with each mode by substituting ω2 into the equations of motion Where k We see that m k + κ − mω2 = κ (1.4.33) D Substituting this into the equation of motion yields y1 = y2 (1.4.34) We see that the masses move in phase with each other You will also notice the absense of the spring constant term, κ, for the connecting spring As the masses are moving in step, the spring isn’t... assume solutions for the equations of motion to be of the form y1 = cos(ωt + δ1 ) y2 = B cos(ωt + δ2 ) y¨ 1 = −ωy1 y¨ 2 = −ωy2 (1.4.27) Substituting the values for y¨ 1 and y¨ 2 into the equations of motion yields k + κ − mω2 y1 − κy2 = 0 (1.4.28) −κy1 + k + κ − mω2 y2 = 0 (1.4.29) k + κ − mω2 −κ Solving the determinant gives mω2 This yields 2 FT We can get solutions from solving the determinant of the. .. (1.4.35) Substituting this into the equation of motion yields y1 = −y2 (1.4.36) Here the masses move out of phase with each other In this case we see the presence of the spring constant, κ, which is expected as the spring playes a role It is being stretched and compressed as our masses oscillate ©2009 David S Latchman Classical Mechanics 8 1.4.7 Doppler Effect The Doppler Effect is the shift in frequency and... Axis Theorem I = Icm + Md2 (1.5.3) τ=r×F τ = Iα (1.5.4) (1.5.5) Torque 1.5.5 FT where α is the angular acceleration Angular Momentum L = Iω (1.5.6) dL dt (1.5.7) RA we can find the Torque τ= 1.5.6 Kinetic Energy in Rolling D With respect to the point of contact, the motion of the wheel is a rotation about the point of contact Thus 1 (1.5.8) K = Krot = Icontact ω2 2 Icontact can be found from the Parallel... with respect to the medium, a receiver moving with respect to the medium or a moving medium Moving Source If a source is moving towards an observer, then in one period, τ0 , it moves a distance of vs τ0 = vs / f0 The wavelength is decreased by λ =λ− The frequency change is vs v − vs − f0 f0 v v = f0 λ v − vs FT f = (1.4.37) (1.4.38) Moving Observer As the observer moves, he will measure the same wavelength,... Mechanics 4 1.3.5 Hooke’s Law F = −kx (1.3.8) where k is the spring constant 1.3.6 Potential Energy of a Spring 1 U(x) = kx2 2 1.4.1 Oscillatory Motion FT 1.4 (1.3.9) Equation for Simple Harmonic Motion x(t) = A sin (ωt + δ) (1.4.1) 1.4.2 RA where the Amplitude, A, measures the displacement from equilibrium, the phase, δ, is the angle by which the motion is shifted from equilibrium at t = 0 Period of... Oscillations The Energy of a system is 1 E = K + V(x) = mv(x)2 + V(x) 2 (1.4.13) We can solve for v(x), 2 (E − V(x)) (1.4.14) m where E ≥ V(x) Let the particle move in the potential valley, x1 ≤ x ≤ x2 , the potential can be approximated by the Taylor Expansion v(x) = V(x) = V(xe ) + (x − xe ) ©2009 dV(x) dx x=xe d2 V(x) 1 + (x − xe )2 2 dx2 + ··· (1.4.15) x=xe David S Latchman 6 Classical Mechanics 2 At the. .. the case of a simple pendulum of length, , and the mass of the bob is m1 For small displacements, the equation of motion is θ¨ + ω0 θ = 0 (1.4.18) RA We can express this in cartesian coordinates, x and y, where x = cos θ ≈ y = sin θ ≈ θ (1.4.19) (1.4.20) y¨ + ω0 y = 0 (1.4.21) eq (1.4.18) becomes This is the equivalent to the mass-spring system where the spring constant is D k = mω20 = mg (1.4.22)... Third Law When a particle A exerts a force on another particle B, B simultaneously exerts a force on A with the same magnitude in the opposite direction FAB = −FBA David S Latchman (1.2.2) ©2009 Work & Energy 1.2.2 3 Momentum p = mv 1.2.3 Impulse ∆p = J = 1.3.1 Fdt = Favg dt Work & Energy Kinetic Energy 1 K ≡ mv2 2 1.3.2 (1.3.1) The Work-Energy Theorem The net Work done is given by Wnet = K f − Ki RA... Icontact ω2 2 Icontact can be found from the Parallel Axis Theorem Icontact = Icm + MR2 (1.5.9) Substitute eq (1.5.8) and we have 1 Icm + MR2 ω2 2 1 1 = Icm ω2 + mv2 2 2 K= (1.5.10) The kinetic energy of an object rolling without slipping is the sum of hte kinetic energy of rotation about its center of mass and the kinetic energy of the linear motion of the object ©2009 David S Latchman Classical Mechanics ... We see that the masses move in phase with each other You will also notice the absense of the spring constant term, κ, for the connecting spring As the masses are moving in step, the spring isn’t... along the rod’s axis, with the origin at the start of the rod dq 4π x λdx = 4π x dV = This becomes V= (2.1.36) λ x2 ln 4π x1 (2.1.37) where x1 and x2 are the distances from O, the end of the rod... this into the equation of motion yields y1 = −y2 (1.4.36) Here the masses move out of phase with each other In this case we see the presence of the spring constant, κ, which is expected as the spring