THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions


 Kerry Singleton
 5 years ago
 Views:
Transcription
1 THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN Copyrght George Hardgrove 996. All rghts reserved. You are welcome to use ths document n your own classes but commercal use s not allowed wthout the permsson of the author. Ths document contans the nstructons for the Mathcad fle enttled corun.mcd. The equatons n ths document are btmap mages and not Mathcad actve. The Mathcad actve equatons are found n corun.mcd. Students should frst complete the exercses for the smple harmonc oscllator that are found n horun.mcd and ts accompanyng nstructonal fle hons.mcd. In ths lesson we wll examne the mathematcal bass for representng the vbratons of CO. We wll assume that the force constant for each CO bond s k. The molecule s arranged along the x axs wth coordnates of the three atoms beng x, x, and x 3, the left oxygen, the carbon, and the rght oxygen respectvely. Note that m = m 3. Sketch the molecule. Goal: To explore classcal harmonc oscllator moton n a lnear tratomc molecule. Objectves:. To use matrx algebra to transform between coordnate systems used to represent a molecule. To wrte the Laplacan and equatons of moton for a lnear tratomc molecule. 3. To explan the moton of tratomc molecules n terms of the potental and knetc energy contrbutons from each atom. Prerequstes:. Successful completon of hons.mcd and horun.mcd exercses;. Mathcad 6.0+ s requred to to solve the dfferental equatons; 3. Dfferental calculus s needed n order to take the requred dervatves. Students should understand what "to solve a dfferental equaton" means. 4. Only elementary level skll wth Mathcad s requred to use the corun.mcd document. More advanced skll would be requred to author documents of ths type. Created: Sprng 996 Modfed: June 997 cons.mcd page
2 The knetc energy s the sum of the knetc energy contrbuted by each atom: T = ( mx + mx + mx 3 ) () The extent Q that the left bond s stretched or compressed compared to ts equlbrum bond length of r e s Q = x x r e () Smlarly for Q Q = x3 x r e (3) Where at equlbrum x  x = r e and x 3  x = r e Takng the tme dervatves we get Q = x  x and Q = x  x (4) 3 The center of mass of the molecule s gven by C = m x m The tme dervatve of the center of mass yelds equaton 5. m x + m x + m x M 3 3 = C (5) Created: Sprng 996 Modfed: June 997 cons.mcd page
3 Next we wrte the potental energy n terms of the stretchng of each bond. Notce how Hooke's law s used here. V = (k Q + k Q ) (6) Above we descrbed the knetc energy T n terms of x coordnates, and the potental energy V n terms of Q coordnates. In order to obtan the equatons of moton we must descrbe T and V n the same coordnates. Snce we wll use the Q coordnates below, we must obtan both T and V n terms of the Q's. Ths can be done by usng a matrx transformaton. The matrx equaton to accomplsh ths s Q 0 x Q 0 x C = m m m 3 x 3 M M M (7) To obtan the x coordnates n terms of the Qs we nvert the 3x3 matrx and multply each sde of equaton 7 on the left by ths nverse to get: x x x = 3 m m m M Q m m M Q M m m + m M C (8) Where M = m + m. Recall that m = m 3. Queston. Wrte the exp resson for the x ' s as a functon of Q ' s and C. Created: Sprng 996 Modfed: June 997 cons.mcd page 3
4 The knetc energy as a functon of the Q 's can be obtaned by substtutng the x 's from the above matrx equaton nto equaton. After consderable algebra we get equaton 9. m m T = + M ( m ) ( Q Q ) m + + M Q Q + mc (9) Queston. Does the knetc energy depend on the center of mass? What coordnates are essental for descrbng ths system of three atoms? Complete the algebrac step leadng from equaton to equaton 9. Ths s not a trval algebrac process. Be careful. Collectng the constants nto B and B we can rewrte the knetc energy as T = B (Q +Q ) + B Q Q (0) whch wth the potental energy n terms of the Q's V = k (Q + Q ) () can be substtuted nto Lagrange's dfferental equaton (), frst for Q then for Q and then for Cdot. d dt L Q  L Q = 0 () to gve the followng equatons of moton B Q + B Q + kq = 0 (3) BQ + BQ + kq = 0 (4) mc = 0 (5) Equaton 5 does not contrbute to the equatons of moton descrbng the poston of the oxygens. We wll thus solve equatons 3 and 4 as a coupled par of dfferental equatons Created: Sprng 996 Modfed: June 997 cons.mcd page 4
5 Queston 3. Wrte the Lagrangan for the CO molecule. Queston 4. Complete the detals of obtanng equatons 3 and 4. Equatons 3 and 4 cannot be solved by MATHCAD n ths form shown above. We must frst solve them for each of the second dervatves n order to use the rkfxed Mathcad functon. The rkfxed functon gves us a numercal soluton to the set of dfferental equatons. Q = B k Q  k Q B B B  4 B (6) Q = k B B Q  k Q B B B (7) The values of k and the masses of the atoms are used wth these equatons n the computatonal document where the dfferental equatons are solved. Queston 5: Complete the detals of obtanng equatons 6 and 7 from equatons 3 and 4. EXERCISE The nput to the dfferental equatons routne has some adjustable constants. We must set values of the masses of the atoms m and m whch can be taken as 6 and for O (m and m 3 ) and C (m ) respectvely. The value of the force constant should be set to 00. The column matrx Q n corun.mcd gves the ntal values of the Q's and ther ntal veloctes: Q Q = Q Q Q Created: Sprng 996 Modfed: June 997 cons.mcd page 5
6 Set each ntal velocty equal to 3 and each ntal poston equal to zero. Calculate all four pages of the document. Whch vbratonal mode does the chosen ntal condtons smulate? For the anmaton part place the cursor (red cross) just to the rght of the top of the anmaton plot and clck on Graphcs and Anmaton and create. Then wth the mouse block out the porton of the pcture you wsh to anmate. If necessary press the lghtbulb button to actvate the automatc mode. The anmaton wll now be recorded and when t s complete a wndow wll appear at the left for the playback feature. Clckng on the lower left button n ths wndow wll actvate the playback. Note the Fourer transform of the moton of Q on the last page. Ths s a plot n frequency space of the vbratonal frequences contaned n ths moton. EXERCISE Go to the Q matrx and change one of the dervatves from 3 to 3. Repeat the calculatons. Has the frequency become larger or smaller? Whch vbratonal mode does ths represent? EXERCISE 3 Go agan to the Q matrx and change the ntal POSITION Q to and change the ntal VELOCITY for Q to 3. Repeat the calculaton. Note that the moton becomes a mxture of the two modes and appears to be haphazard. But the Fourer transform can dsplay two frequences (even though the phasng of the maxma s not great). Queston 6: What s the physcal meanng of havng ntal value of Q for partcle equal to 3 and the ntal velocty of partcle equal to zero and at the same tme have the ntal poston value of Q for partcle equal to zero and the correspondng velocty for ths coordnate equal to 3? Created: Sprng 996 Modfed: June 997 cons.mcd page 6
12. The HamiltonJacobi Equation Michael Fowler
1. The HamltonJacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationPHYS 705: Classical Mechanics. HamiltonJacobi Equation
1 PHYS 705: Classcal Mechancs HamltonJacob Equaton HamltonJacob Equaton There s also a very elegant relaton between the Hamltonan Formulaton of Mechancs and Quantum Mechancs. To do that, we need to derve
More informationNMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 48/58 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 48/58 7. Robot Dynamcs 7.5 The Equatons of Moton Gven that we wsh to fnd the path q(t (n jont space) whch mnmzes the energy
More informationIntegrals and Invariants of EulerLagrange Equations
Lecture 16 Integrals and Invarants of EulerLagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
More informationAPPENDIX 2 FITTING A STRAIGHT LINE TO OBSERVATIONS
Unversty of Oulu Student Laboratory n Physcs Laboratory Exercses n Physcs 1 1 APPEDIX FITTIG A STRAIGHT LIE TO OBSERVATIOS In the physcal measurements we often make a seres of measurements of the dependent
More informationLecture 20: Noether s Theorem
Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationMolecular structure: Diatomic molecules in the rigid rotor and harmonic oscillator approximations Notes on Quantum Mechanics
Molecular structure: Datomc molecules n the rgd rotor and harmonc oscllator approxmatons Notes on Quantum Mechancs http://quantum.bu.edu/notes/quantummechancs/molecularstructuredatomc.pdf Last updated
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationSolution of Linear System of Equations and Matrix Inversion Gauss Seidel Iteration Method
Soluton of Lnear System of Equatons and Matr Inverson Gauss Sedel Iteraton Method It s another wellknown teratve method for solvng a system of lnear equatons of the form a + a22 + + ann = b a2 + a222
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More information10. Canonical Transformations Michael Fowler
10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationThree views of mechanics
Three vews of mechancs John Hubbard, n L. Gross s course February 1, 211 1 Introducton A mechancal system s manfold wth a Remannan metrc K : T M R called knetc energy and a functon V : M R called potental
More informationNovember 5, 2002 SE 180: Earthquake Engineering SE 180. Final Project
SE 8 Fnal Project Story Shear Frame u m Gven: u m L L m L L EI ω ω Solve for m Story Bendng Beam u u m L m L Gven: m L L EI ω ω Solve for m 3 3 Story Shear Frame u 3 m 3 Gven: L 3 m m L L L 3 EI ω ω ω
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More information6.3.4 Modified Euler s method of integration
6.3.4 Modfed Euler s method of ntegraton Before dscussng the applcaton of Euler s method for solvng the swng equatons, let us frst revew the basc Euler s method of numercal ntegraton. Let the general from
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationModelli Clamfim Equazione del Calore Lezione ottobre 2014
CLAMFIM Bologna Modell 1 @ Clamfm Equazone del Calore Lezone 17 15 ottobre 2014 professor Danele Rtell danele.rtell@unbo.t 1/24? Convoluton The convoluton of two functons g(t) and f(t) s the functon (g
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationEPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski
EPR Paradox and the Physcal Meanng of an Experment n Quantum Mechancs Vesseln C Nonnsk vesselnnonnsk@verzonnet Abstract It s shown that there s one purely determnstc outcome when measurement s made on
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationBe true to your work, your word, and your friend.
Chemstry 13 NT Be true to your work, your word, and your frend. Henry Davd Thoreau 1 Chem 13 NT Chemcal Equlbrum Module Usng the Equlbrum Constant Interpretng the Equlbrum Constant Predctng the Drecton
More informationChapter 4 The Wave Equation
Chapter 4 The Wave Equaton Another classcal example of a hyperbolc PDE s a wave equaton. The wave equaton s a secondorder lnear hyperbolc PDE that descrbes the propagaton of a varety of waves, such as
More informationLagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013
Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned
More information5.04, Principles of Inorganic Chemistry II MIT Department of Chemistry Lecture 32: Vibrational Spectroscopy and the IR
5.0, Prncples of Inorganc Chemstry II MIT Department of Chemstry Lecture 3: Vbratonal Spectroscopy and the IR Vbratonal spectroscopy s confned to the 005000 cm  spectral regon. The absorpton of a photon
More informationHow Differential Equations Arise. Newton s Second Law of Motion
page 1 CHAPTER 1 FrstOrder Dfferental Equatons Among all of the mathematcal dscplnes the theory of dfferental equatons s the most mportant. It furnshes the explanaton of all those elementary manfestatons
More informationWorkshop: Approximating energies and wave functions Quantum aspects of physical chemistry
Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:55: Copyrght 25 Dan Dll (dan@bu.edu) Department
More informationQuantum Mechanics I Problem set No.1
Quantum Mechancs I Problem set No.1 Septembe0, 2017 1 The Least Acton Prncple The acton reads S = d t L(q, q) (1) accordng to the least (extremal) acton prncple, the varaton of acton s zero 0 = δs = t
More informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: mgk F 1 x O
More informationNewton s Method for One  Dimensional Optimization  Theory
Numercal Methods Newton s Method for One  Dmensonal Optmzaton  Theory For more detals on ths topc Go to Clck on Keyword Clck on Newton s Method for One Dmensonal Optmzaton You are free to Share to copy,
More informationPhysics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2
Physcs 607 Exam 1 Please be wellorganzed, and show all sgnfcant steps clearly n all problems. You are graded on your wor, so please do not just wrte down answers wth no explanaton! Do all your wor on
More information1 Rabi oscillations. Physical Chemistry V Solution II 8 March 2013
Physcal Chemstry V Soluton II 8 March 013 1 Rab oscllatons a The key to ths part of the exercse s correctly substtutng c = b e ωt. You wll need the followng equatons: b = c e ωt 1 db dc = dt dt ωc e ωt.
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationLecture Note 3. Eshelby s Inclusion II
ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte
More informationIntegrals and Invariants of
Lecture 16 Integrals and Invarants of Euler Lagrange Equatons NPTEL Course Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng, Indan Insttute of Scence, Banagalore
More information11. Dynamics in Rotating Frames of Reference
Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons
More informationArmy Ants Tunneling for Classical Simulations
Electronc Supplementary Materal (ESI) for Chemcal Scence. Ths journal s The Royal Socety of Chemstry 2014 electronc supplementary nformaton (ESI) for Chemcal Scence Army Ants Tunnelng for Classcal Smulatons
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationModule 14: THE INTEGRAL Exploring Calculus
Module 14: THE INTEGRAL Explorng Calculus Part I Approxmatons and the Defnte Integral It was known n the 1600s before the calculus was developed that the area of an rregularly shaped regon could be approxmated
More informationcoordinates. Then, the position vectors are described by
Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,
More informationSnce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t
8.5: Manybody phenomena n condensed matter and atomc physcs Last moded: September, 003 Lecture. Squeezed States In ths lecture we shall contnue the dscusson of coherent states, focusng on ther propertes
More information6. Hamilton s Equations
6. Hamlton s Equatons Mchael Fowler A Dynamcal System s Path n Confguraton Sace and n State Sace The story so far: For a mechancal system wth n degrees of freedom, the satal confguraton at some nstant
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06FnteDfference Method (Onedmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the faketest data; fxed
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s nonnegatve Hence the equatons x, x, x + 7 0 etc are not
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationIrregular vibrations in multimass discretecontinuous systems torsionally deformed
(2) 4 48 Irregular vbratons n multmass dscretecontnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscretecontnuous systems consstng of an arbtrary number rgd bodes connected
More information763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.
7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warmup Show that the egenvalues of a Hermtan operator Â are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
More informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
More informationSpinrotation coupling of the angularly accelerated rigid body
Spnrotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 Emal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationPhysics 240: Worksheet 30 Name:
(1) One mole of an deal monatomc gas doubles ts temperature and doubles ts volume. What s the change n entropy of the gas? () 1 kg of ce at 0 0 C melts to become water at 0 0 C. What s the change n entropy
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationPHYS 215C: Quantum Mechanics (Spring 2017) Problem Set 3 Solutions
PHYS 5C: Quantum Mechancs Sprng 07 Problem Set 3 Solutons Prof. Matthew Fsher Solutons prepared by: Chatanya Murthy and James Sully June 4, 07 Please let me know f you encounter any typos n the solutons.
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 97457X (Onlne), ISSN 9745718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 1314 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationLesson 5: Kinematics and Dynamics of Particles
Lesson 5: Knematcs and Dynamcs of Partcles hs set of notes descrbes the basc methodology for formulatng the knematc and knetc equatons for multbody dynamcs. In order to concentrate on the methodology and
More informationEN40: Dynamics and Vibrations. Homework 7: Rigid Body Kinematics
N40: ynamcs and Vbratons Homewor 7: Rgd Body Knematcs School of ngneerng Brown Unversty 1. In the fgure below, bar AB rotates counterclocwse at 4 rad/s. What are the angular veloctes of bars BC and C?.
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fntedmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationIn this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
More informationENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15
NGN 40 ynamcs and Vbratons Homework # 7 ue: Frday, Aprl 15 1. Consder a concal hostng drum used n the mnng ndustry to host a mass up/down. A cable of dameter d has the mass connected at one end and s wound/unwound
More informationMAE140  Linear Circuits  Winter 16 Final, March 16, 2016
ME140  Lnear rcuts  Wnter 16 Fnal, March 16, 2016 Instructons () The exam s open book. You may use your class notes and textbook. You may use a hand calculator wth no communcaton capabltes. () You have
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationSalmon: Lectures on partial differential equations. Consider the general linear, secondorder PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of secondorder equatons There are general methods for classfyng hgherorder partal dfferental equatons. One s very general (applyng even to
More informationThe classical spinrotation coupling
LOUAI H. ELZEIN 2018 All Rghts Reserved The classcal spnrotaton couplng Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 louaelzen@gmal.com Abstract Ths paper s prepared to show that a rgd
More informationDO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.
EE 539 Homeworks Sprng 08 Updated: Tuesday, Aprl 7, 08 DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED. For full credt, show all work. Some problems requre hand calculatons.
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationMathematics Intersection of Lines
a place of mnd F A C U L T Y O F E D U C A T I O N Department of Currculum and Pedagog Mathematcs Intersecton of Lnes Scence and Mathematcs Educaton Research Group Supported b UBC Teachng and Learnng Enhancement
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationy i x P vap 10 A T SOLUTION TO HOMEWORK #7 #Problem
SOLUTION TO HOMEWORK #7 #roblem 1 10.11 a. In order to solve ths problem, we need to know what happens at the bubble pont; at ths pont, the frst bubble s formed, so we can assume that all of the number
More informationClassical Mechanics ( Particles and Biparticles )
Classcal Mechancs ( Partcles and Bpartcles ) Alejandro A. Torassa Creatve Commons Attrbuton 3.0 Lcense (0) Buenos Ares, Argentna atorassa@gmal.com Abstract Ths paper consders the exstence of bpartcles
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationPhysics 2A Chapters 6  Work & Energy Fall 2017
Physcs A Chapters 6  Work & Energy Fall 017 These notes are eght pages. A quck summary: The workenergy theorem s a combnaton o Chap and Chap 4 equatons. Work s dened as the product o the orce actng on
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed multpartcle systems! Internal and external forces! Laws of acton and
More informationSPANC  SPlitpole ANalysis Code User Manual
Functonal Descrpton of Code SPANC  SPltpole ANalyss Code User Manual Author: Dale Vsser Date: 14 January 00 Spanc s a code created by Dale Vsser for easer calbratons of poston spectra from magnetc spectrometer
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
More informationChapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
Chapter. Ordnar Dfferental Equaton Boundar Value (BV) Problems In ths chapter we wll learn how to solve ODE boundar value problem. BV ODE s usuall gven wth x beng the ndependent space varable. p( x) q(
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More informationNotes on Analytical Dynamics
Notes on Analytcal Dynamcs Jan Peters & Mchael Mstry October 7, 004 Newtonan Mechancs Basc Asssumptons and Newtons Laws Lonely pontmasses wth postve mass Newtons st: Constant velocty v n an nertal frame
More informationHomework Notes Week 7
Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we
More informationMAE140  Linear Circuits  Winter 16 Midterm, February 5
Instructons ME140  Lnear Crcuts  Wnter 16 Mdterm, February 5 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More informationQuantum Mechanics I  Session 4
Quantum Mechancs I  Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................
More informationReport on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton
More informationChapter 8. Potential Energy and Conservation of Energy
Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and nonconservatve forces Mechancal Energy Conservaton of Mechancal
More informationLAGRANGIAN MECHANICS
LAGRANGIAN MECHANICS Generalzed Coordnates State of system of N partcles (Newtonan vew): PE, KE, Momentum, L calculated from m, r, ṙ Subscrpt covers: 1) partcles N 2) dmensons 2, 3, etc. PE U r = U x 1,
More informationCALCULUS CLASSROOM CAPSULES
CALCULUS CLASSROOM CAPSULES SESSION S86 Dr. Sham Alfred Rartan Valley Communty College salfred@rartanval.edu 38th AMATYC Annual Conference Jacksonvlle, Florda November 8, 202 2 Calculus Classroom Capsules
More informationFrequency calculations can serve a number of different purposes:
4. Frequency calculatons Frequency calculatons can serve a number of dfferent purposes: To predct the IR and Raman spectra of molecules (frequences and ntenstes. To compute force constants for a geometry
More informationCHARACTERISTICS OF COMPLEX SEPARATION SCHEMES AND AN ERROR OF SEPARATION PRODUCTS OUTPUT DETERMINATION
Górnctwo Geonżynera Rok 0 Zeszyt / 006 Igor Konstantnovch Mladetskj * Petr Ivanovch Plov * Ekaterna Nkolaevna Kobets * Tasya Igorevna Markova * CHARACTERISTICS OF COMPLEX SEPARATION SCHEMES AND AN ERROR
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More information= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the righthand side of the system.
Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and
More information2 Finite difference basics
Numersche Methoden 1, WS 11/12 B.J.P. Kaus 2 Fnte dfference bascs Consder the one The bascs of the fnte dfference method are best understood wth an example. dmensonal transent heat conducton equaton T
More information