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# CLASSICAL DYNAMICS PHYS 5306

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This 10 page Class Notes was uploaded by Harry Jerde on Thursday October 22, 2015. The Class Notes belongs to PHYS 5306 at Texas Tech University taught by Juyang Huang in Fall. Since its upload, it has received 33 views. For similar materials see /class/226443/phys-5306-texas-tech-university in Physics 2 at Texas Tech University.

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Date Created: 10/22/15

Motions Relative to Center of Mass Relative position The position of a particle ri and be expressed by the center of mass R and the relative position of the particle to CM ri ri39 39 Relative velocigg where V a dRdt the velocity of CM and vi39 a dri39dt the relative velocity of the particle to CM Note MR Emlfi Emil MR l I thus 0 i Relative angular mom entuii ff x i 27 x mini 427 R5343 i 5d x 7 x miv li x 7 Rx2mi gj Rx2miv l l mil i l l The 2nd and 3rd terms vanish Thus T Total angular momentum angular momentum of CM angular momentum of motion about CM By Prof Juyang Huang Texas Tech University All rights reserved W12 5 Ef1z 39 i2f1z 239d i Eflz zj39 i iyjij Total work work done by external forces work done by internal forces AlSO W12 Eff 39 dig Eflzmividt 2f12dmivi22f12dTi and Thus Kinetic energy of the system Work done change in total kinetic energy Kinetic Energy T 12ZimiV Vi 3939V W T 12ZimiV2 12ZimiVi392 V39 EmiVi39 Again the last term vanishes So the total KE T 12MV2 12 mivi39z The total Kinetic Energy of a many particle system the Kinetic Energy of the CM the Kinetic Energy of motion about the CM Work and Potential Energy wagEff da2ff ltegtd i Ef emi iyjij By Prof Juyang Huang Texas Tech University All rights reserved a Assume that all external forces are conservative forces and Vi are external potential then the lSt term become 2f12Ee39d i 2f12ViVi 55 2Vi2 l l 1 b Assume that potential functions Vij exist such that for each particle pair ij Fij ViVij then I Fij viVij VIII711 Fji r1 I jf1 ij where fl ij 1s a scalar functlon The 2 term become see book EV llji j l 2 7 1 2 1 Efl Fij 39dgi 5 Viszj drzj iji j iyji J Together for conservative external forces amp conservative central internal forces it is possible to define a potential energy function for the system IV E VG t Va E ZiVi 12ZijltigtVijl and the total work done in a process is W12 VlV2AV In general W12 T2T1AT Combining gt V1 V2 T2 T1 or AT AV 0 or or E T V constant E T V a Total Mechanical Energy Energy Conservation Theorem for a Many Particle System If only conservative external forces amp conservative central By Prof Juyang Huang Texas Tech University All rights reserved internal forces are acting on a system then the total mechanical energy of the system E T V is conserved 13 Constraints To find the equation of state of a system we need to solve a set of simultaneous coupled 2nd order differential equations which come from Newton s 2rld Law applied to each particle individually dpidt mid2ridt2 Fie 3F Given forces and initial conditions problem can be solved However many systems have CONSTRAINTS which limit the motion of particles Rigid body Gas molecules Rolling ball In a container no jumping rij constant 0 s r s R y fx 0 h Types of Constraints Holonomic Constraints expressed by fr1r2r3 rNt 0 By Prof Juyang Huang Texas Tech University All rights reserved Example 1 rigid body r r1Z DuZ 0 2 particle on a surface yfx 0 NonHolonomic Constraints onstrainm cannot be expressed as frt 0 Example Particle confined to surface of rigid sphere with a radius of R r2 2 R2 Time dependent constrains Rh enom1c or rhenomous constrains Time independent constrains Fixed or scleronomic or scleronomous constraints Difficulties constraints introduce in problems 1 Coordinates r are no longer all independent They are connected by constraint equations 2 To apply Newton s 2 01 Law need totalforce acting on each particle Forces of constraint are often unknown an difficult to calculate Don t believe me Ok what are the constraint forces in the rolling ball example above Generalized Coordinates By Prof Juyang Huang Texas Tech University All rights reserved Solving problems with holonomic constraints Generalized Coordinates can be introduced They are alternatives to the usual Cartesian coordinates For anN particle system the degree of freedom 3N without constraint k holonomic constraints 3N k degree of freedom or 3Nk independent coordinates with the constraints Introduce s 3N k independent Generalized Coordinates to completely specify the system q1qz Or q1 l2 s In principle one can always find relations between generalized coordinates and Cartesian vector coordinates r rq1qzqst i l23N These are transformation equation from the set of coordinates r to the set ql In principle can combine with k constraint equations to obtain inverse transformation ql q1r1rzr3t Z 12 s The generalized coordinates need not be Cartesian spherical cylindrical coordinates The generalized coordinates need not have units of length By Prof Juyang Huang Texas Tech University All rights reserved Example Double pendulum A convenient choice of Two coordinates can completely specify the system NonHolonomic constraint uations expressing constraint can t be used to eliminate dependent coordinates Example ofRolling Constraint Disk radius a constrained to be vertical rolling on the horizontal xy plane F1gure 0 FIGURE 15 Vermin disk rolling on ii rim110nm prime Use four generalized coordinates point of contact of disk with plane xy angle between disk axis 9 and xaxis angle of rotation about disk axis b By Piaf Juyang Huang TexasTech Umva slty All ngms reserved Constraint Velocity v of disk center is related to angular velocity det of disk rotation v R Also Cartesian components of v vX dxdt v sine vy dydt v cose Combine the above relations we have a sine db dy a cose db 0 Neither can be integrated without solving the problem That is a function fxyeb 0 cannot be found The number of coordinates cannot be reduced by the constraint 39Non Holonomic constraints can also involve higher order derivatives or inequalities 39Holonomic constraints are preferred since easiest to deal with No general method to treat problems with NonHolonomic constraints Treat on casebycase basis The bottom line Physicists hate constraints Lagrangian and Hamiltonian formulations were introduced to avoid direct dealing with the forces of constraint By Piaf Juyang Huang TexasTech Umva slty All ngms reserved 311 Transformation of the Scattering Problem to Lab Coordinates In the last section we treat the scattering problem as lbody problem Assumed 1 particle scatters off by a stationary Center of Force It means that we are doing problem in the Center of Mass coordinate system for 2 bodies and that we are looking at the behavior of the reduced mass u and relative motion of two particles Actual scattering is of course a 2body problem 2 masses m1 and m 2 scattering off each other In lab coordinate system we need to account for the motions of both bodies The treatment we used so far is valid in the lab frame if m 2 gtgt m1 thus m1 a so that the recoil of m due to m1 can be neglected FIGURE 3 24 Scattering ol lwo particles as viewed in the laboratory system The scattering angle measured in the lab 6 5 angle between final and incident directions of the scattered particle in the lab coordinate system Scattering angle calculated in previous discussion 9 Angle between the initial and nal directions of the relative By Prof Juyang Huang Texas Tech University All ring reserved coordinate r between m1 and m 2 in the CM coordinate system 6 9 only if m is stationary ie with 00 mass throughout the scattering Problem Generally 6 Q Must nd a way to convert between two First study the problem in CM system then convert it back to the lab frame In the CM frame the scattering event looks simple 1 The total linear momentum of the 2 particles is always 0 2 Before scattering the particles move directly towards each other 3 Afterwards they move directly away from each other 4 The scattering angle 9 is the same as scattering angle of either particle FIGURE 325 Scattering of two panicles as izwcd in he ccmcr of mass system Conversion Between CM and Laborato Coordinates 39For a 2body problem Center of Mass vector R s m 1r 1 m2r2MII Total mass Msm1m2 Velocity of CM Vis a constant vector Relative coordinate r Er r2 By Prof Juyang Huang Texas Tech University All rights reserved Reduced mass u s mjm rn rrnz M NH CM CM RED IN m a 1m lbl 7139 7milf and 7239 p n MLZF r1 and r2 be the radii vectors of the two particles relative to CM 0 r 1 and v are the position and velocity after scattering of the incident particle m in the laboratory frame 0 r and v1 are the position and velocity after scattering of the incident particle rn 1 in the CM frame Coordinate Conversion r R r1 R Iamgr Velocity Conversion v V vj V yrn v As shown in the gure v1 and v1 39make angles 6 and 6 respectively with direction of V By Prof Juyahg Huang Texas Tech University All rights reserved Let v0 be the initial velocity of rn 1 in lab system and m2 is initially at rest in the lab system From conservation of momentum m1 m2V mlvo or V llm2quot0 1 0 From the gure vlcos vl39cosG V 2 and vlsinB vl39sinG 3 Divide 3 b 2 amp use 1 tanB sin cos p 4 where p a uv0m2v139 Note ifrn is in nite thenp 0 and 6 9 0 Alternative relation can be derived from the Law of Cosines From the gure v12 vl392 V2 2v139Vcos Also vlsin vl39sinG and V um2v0 gt osB cos p l2pcos p2 2 439 39Consider p a uv0m2v139 From the CM definition vl39 pm1v v W relative speed after collision gt p mjrnz vlv p is depending on the nature of collision scattering 39Elastic KE conserving scattering v0 v p rnjrnz By Prof Juyahg Huang Texas Tech University All rights reserved Inelastic KE nonconserving scattering E0 12uv02 Let 12uv2 12uv02 a Q a Q value of collision Clearly since KE is lost Q lt 0 Algebra gives M m1m2 1 1 V0 m1 p m21l1 inelastic scattering 5 Not only 0 and 9 are differrent also need to transform the cross section 6 itself from a function of 9 to a function of 0 69 gt 69 39Connection conservation of particle number number of particles scattered into a given differential solid angle d9 must be the same Whether measured in the lab or CM frame So 2nlo sin d 21I60sin0d0 gt 60 o sin sin0d d0 o dcos dcos Use kinematic result cos0 cos p12pcos p2 2 Take derivative and get p mlm21 Mm2QE quot2 60 o 12pcos p23 21 pcos 6 By Prof Juyang Huang Texas Tech University All rights reserved 39Note 6 0 and o are both measured in the lab frame They re expressed in terms of different coordinates 39Special Case 1 Elastic scattering With m1 m2 p 1 gt cos0 1 cos 2V2 cos 2 gt 0 12 Since 9 s 1 in this case cannot have 0gt 1211 gt In the lab system all scattering is in forward hemisphere In this case 6 becomes 60 4cos o gt Even in the very special case Where o constant 60 still depends on angle 39Special Case 2 Elastic scattering With m1 ltlt m2 effectively m2 is infinite gt p z 0 gt 60 z o More Details Obviously scattering slows down the incident particle More kinematics We had v12 v1392 V2 2v139Vcos A150 P V0lm2V1rl and V Wm2Vo Combine these to get algebra VIv02 Hzm2P211 2pcos p2 a 39Special case Elastic scattering gt p mlm2 Let E0 2 12m1v02 initial KB of m1 before scattering Let E1 2 12m1v12 final KB of m1 after scattering a becomes ElEO 12 pcos p21 p2 If m1 m2 ElEo 1 cos 2 cos20 By Prof Juyang Huang Texas Tech University All rights reserved For max 9 1 0 152 gt ElEo 0 The incident particle stops in the lab system Principle behind moderator in neutron scattering Classical Mech vs QM Some nal thoughts on classical scattering discussion 39 All we ve used is simple conservation of momentum and energy The cross section results are classical 39 However as long as we know the Q value and momentum is conserved it doesn t really matter if it is QM or classical scattering Why Because we ve analyzed the outgoing particle beam mostly except for Coulomb scattering without caring what the details of the scattering were Details of the scattering of course usually require QM analysis The results ofMOST of Sects 310 and 311 can be used in analyzing experiments for almost any kind of low energy scattering Exception At high enough energies need to do all of this with Relativity See Sect 77 However the authors forgot to mention the wave property of particles Scattering signals are results of superposition of scattering waves By Prof Juyang Huang Texas Tech University All rights reserved Important Send me an email juyanghuangttuedu this week and you will get 10 extra credits for your 1st homework I will send timely announcements to you through out the semester Course website wwwphvsttueduhuang24Teachingths5306 You can download and print my lecture notes before classes Also I will post my latest announcements there Textbook quotClassicalMechanicsquot by H Goldstein C Poole amp J Safko 3rd Ed Addision Wesley 2002 ISBN 0 201657023 39 This is by far the most popular textbook for graduate level classical dynamics course which is one of the core courses in this department and many other physics departments nationwide 39 There are many supporting materials available for this textbook such as solution to problems and corrections Please do a Google search to help yourself d 39 There are many prmtmg errors 1n th1s 3r ed1t10n and previous editions so be careful CLASSIFICATION OF PHYSICS Classical Physics before 1905 Dealing with macroscopic world at low speed Classical physics Newton s mechanics Boltzmann s Statistical Mechanics Thermodynamics Electrodynamics Relativity Physics of highspeed world c 3x108 ms Chapter 7 covers special theory of relativity System with acceleration general relativity Quantum Mechanics Physics of microscopic world Atoms 4 FUNDAMENTAL FORCES OF NATURE 39Strong Nuclear Force Binds nuclei together Still being researched 39Electromagnetic Force EampM phenomena Chemical forces Most everyday forces Maxwell Coulomb Ampere Faraday 39Weak Nuclear Force Nuclear decay Fermi Bethe others Still being researched 39Gravitational Force Newton classical mechanics Einstein general relativity Relative LongDistance Interaction CurremTheory Mediators sumgmm 32mm Rangem Quantum St chrom namics quot5 quotmg m gmquot quot 38 discussion m 0 below Quantum l Electromagnetic Electrodynamics photons 1036 2 in nite QED Eleclroweak W and Z quotquot 12 Weak 1025 7 Io a bosons r General 1 Gravilalion Relativity gravilons 1 7 in nite GR 39 39 ElectroWeak Force Electromagnetic Force amp Weak Nuclear Force combined into one theory S Weinberg amp A Salaam 1972 Nobel Prize in Physics A Nobel Prize project for you Unify all 4 forces in one consistent theory I f successful not only you will receive A for this course but also become the greatest physicist in the world

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