21 Vectors: The Cross Product & Torque


 Aubrey Higgins
 2 years ago
 Views:
Transcription
1 21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rghthand rule for the cross product of two vectors dscussed n ths chapter or the rghthand rule for somethng curl somethng straght dscussed n the precedng chapter. There s a relatonal operator 1 for vectors that allows us to bpass the calculaton of the moment arm. The relatonal operator s called the cross product. It s represented b the smbol read cross. The torque τ can be epressed as the cross product of the poston vector r for the pont of applcaton of the force, and the force vector F tself: τ =r F 211 efore we begn our mathematcal dscusson of what we mean b the cross product, a few words about the vector r are n order. It s mportant for ou to be able to dstngush between the poston vector r for the force, and the moment arm, so we present them below n one and the same dagram. We use the same eample that we have used before: s of Rotaton O Poston of the Pont of pplcaton of the Force F n whch we are loong drectl along the as of rotaton so t loos le a dot and the force les n a plane perpendcular to that as of rotaton. We use the dagramatc conventon that, the pont at whch the force s appled to the rgd bod s the pont at whch one end of the arrow n the dagram touches the rgd bod. Now we add the lne of acton of the force and the moment arm r to the dagram, as well as the poston vector r of the pont of applcaton of the force. 1 You are much more famlar wth relatonal operators then ou mght reale. The sgn s a relatonal operator for scalars numbers. The operaton s addton. pplng t to the numbers 2 and 3 elds 23=5. You are also famlar wth the relatonal operators,, and for subtracton, multplcaton, and dvson of scalars respectvel. 132
2 The Moment rm Lne of cton of the Force r O r Poston Vector for the Pont of pplcaton of the Force F The moment arm can actuall be defned n terms of the poston vector for the pont of applcaton of the force. Consder a tlted  coordnate sstem, havng an orgn on the as of rotaton, wth one as parallel to the lne of acton of the force and one as perpendcular to the lne of acton of the force. We label the as for perpendcular and the as for parallel. O r F 133
3 and Chapter 21 Vectors: The Cross Product & Torque Now we brea up the poston vector r nto ts component vectors along the aes. r O r r F From the dagram t s clear that the moment arm r s ust the magntude of the component vector, n the perpendculartotheforce drecton, of the poston vector of the pont of applcaton of the force. 134
4 Now let s dscuss the cross product n general terms. Consder two vectors, and that are nether parallel nor antparallel 2 to each other. Two such vectors defne a plane. Let that plane be the plane of the page and defne θ to be the smaller of the two angles between the two vectors when the vectors are drawn tal to tal. θ The magntude of the cross product vector s gven b = snθ The drecton of the cross product vector s gven b the rghthand rule for the cross product of two vectors 3. To appl ths rghthand rule, etend the fngers of our rght hand so that the are pontng drectl awa from our rght elbow. Etend our thumb so that t s at rght angles to our fngers Two vectors that are antparallel are n eact opposte drectons to each other. The angle between them s 180 degrees. ntparallel vectors le along parallel lnes or along one and the same lne, but pont n opposte drectons. 3 You need to learn two rghthand rules for ths course: the rghthand rule for somethng curl somethng straght, and ths one, the rghthand rule for the cross product of two vectors. 135
5 Keepng our fngers algned wth our forearm, pont our fngers n the drecton of the frst vector the one that appears before the n the mathematcal epresson for the cross product; e.g. the n. Now rotate our hand, as necessar, about an magnar as etendng along our forearm and along our mddle fnger, untl our hand s orented such that, f ou were to close our fngers, the would pont n the drecton of the second vector. Ths thumb s pontng straght out of the page, rght at ou! Your thumb s now pontng n the drecton of the cross product vector. C =. The cross product vector C s alwas perpendcular to both of the vectors that are n the cross product the and the n the case at hand. Hence, f ou draw them so that both of the vectors that are n the cross product are n the plane of the page, the cross product vector wll alwas be perpendcular to the page, ether straght nto the page, or straght out of the page. In the case at hand, t s straght out of the page. 136
6 When we use the cross product to calculate the torque due to a force F whose pont of applcaton has a poston vector r, relatve to the pont about whch we are calculatng the torque, we get an aal torque vector τ. To determne the sense of rotaton that such a torque vector would correspond to, about the as defned b the torque vector tself, we use The Rght Hand Rule For Somethng Curl Somethng Straght. Note that we are calculatng the torque wth respect to a pont rather than an as the as about whch the torque acts, comes out n the answer. Calculatng the Cross Product of Vectors that are Gven n,, Notaton Unt vectors allow for a straghtforward calculaton of the cross product of two vectors under even the most general crcumstances, e.g. crcumstances n whch each of the vectors s pontng n an arbtrar drecton n a threedmensonal space. To tae advantage of the method, we need to now the cross product of the Cartesan coordnate as unt vectors,, and wth each other. Frst off, we should note that an vector crossed nto tself gves ero. Ths s evdent from equaton 212: = snθ, because f and are n the same drecton, then θ = 0, and snce sn 0 = 0, we have = 0. Regardng the unt vectors, ths means that: = 0 = 0 = 0 Net we note that the magntude of the cross product of two vectors that are perpendcular to each other s ust the ordnar product of the magntudes of the vectors. Ths s also evdent from equaton 212: = snθ, because f s perpendcular to then θ = 90 and sn 90 = 1 so = Now f and are unt vectors, then ther magntudes are both 1, so, the product of ther magntudes s also 1. Furthermore, the unt vectors,, and are all perpendcular to each other so the magntude of the cross product of an one of them wth an other one of them s the product of the two magntudes, that s,
7 Now how about the drecton? Let s use the rght hand rule to get the drecton of : Fgure 1 Wth the fngers of the rght hand pontng drectl awa from the rght elbow, and n the same drecton as, the frst vector n to mae t so that f one were to close the fngers, the would pont n the same drecton as, the palm must be facng n the drecton. That beng the case, the etended thumb must be pontng n the drecton. Puttng the magntude the magntude of each unt vector s 1 and drecton nformaton together we see 4 that =. Smlarl: =, =, =, =, and =. One wa of rememberng ths s to wrte,, twce n successon:,,,,,. Then, crossng an one of the frst three vectors nto the vector mmedatel to ts rght elds the net vector to the rght. ut crossng an one of the last three vectors nto the vector 4 You ma have pced up on a bt of crcular reasonng here. Note that n Fgure 1, f we had chosen to have the as pont n the opposte drecton eepng and as shown then would be pontng n the drecton. In fact, havng chosen the and drectons, we defne the drecton as that drecton that maes =. Dong so forms what s referred to as a rghthanded coordnate sstem whch s, b conventon, the nd of coordnate sstem that we use n scence and mathematcs. If = then ou are dealng wth a lefthanded coordnate sstem, somethng to be avoded. 138
8 139 mmedatel to ts left elds the negatve of the net vector to the left lefttorght, but rghttoleft. Now we re read to loo at the general case. n vector can be epressed n terms of unt vectors: = Dong the same for a vector then allows us to wrte the cross product as: = Usng the dstrbutve rule for multplcaton we can wrte ths as: = = Usng, n each term, the commutatve rule and the assocatve rule for multplcaton we can wrte ths as: = Now we evaluate the cross product that appears n each term: = Elmnatng the ero terms and groupng the terms wth together, the terms wth together, and the terms wth together elds:
9 140 = Factorng out the unt vectors elds: = whch can be wrtten on one lne as: = 213 Ths s our end result. We can arrve at ths result much more qucl f we borrow a tool from that branch of mathematcs nown as lnear algebra the mathematcs of matrces. We form the 33 matr b wrtng,, as the frst row, then the components of the frst vector that appears n the cross product as the second row, and fnall the components of the second vector that appears n the cross product as the last row. It turns out that the cross product s equal to the determnant of that matr. We use absolute value sgns on the entre matr to sgnf the determnant of the matr. So we have: = 214 To tae the determnant of a 33 matr ou wor our wa across the top row. For each element n that row ou tae the product of the elements along the dagonal that etends down and to the rght, mnus the product of the elements down and to the left; and ou add the three results one result for each element n the top row together. If there are no elements down and to the approprate sde, ou move over to the other sde of the matr see below to complete the dagonal.
10 For the frst element of the frst row, the, tae the product down and to the rght, ths elds mnus the product down and to the left the product downandtotheleft s. For the frst element n the frst row, we thus have: whch can be wrtten as:. Repeatng the process for the second and thrd elements n the frst row the and the we get and respectvel. ddng the three results, to form the determnant of the matr results n: = as we found before, the hard wa
where the coordinates are related to those in the old frame as follows.
Chapter 2  Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of noncoplanar vectors Scalar product
More informationGoals Rotational quantities as vectors. Math: Cross Product. Angular momentum
Physcs 106 Week 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap 11.2 to 3 Rotatonal quanttes as vectors Cross product Torque expressed as a vector Angular momentum defned Angular momentum as a
More informationSCALAR A physical quantity that is completely characterized by a real number (or by its numerical value) is called a scalar. In other words, a scalar
SCALAR A phscal quantt that s completel charactered b a real number (or b ts numercal value) s called a scalar. In other words, a scalar possesses onl a magntude. Mass, denst, volume, temperature, tme,
More informationQUANTUM MECHANICS, BRAS AND KETS
PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented
More informationThe Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets
. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely
More informationRotation Kinematics, Moment of Inertia, and Torque
Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute
More informationGraph Theory and Cayley s Formula
Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll
More informationExperiment 8 Two Types of Pendulum
Experment 8 Two Types of Pendulum Preparaton For ths week's quz revew past experments and read about pendulums and harmonc moton Prncples Any object that swngs back and forth can be consdered a pendulum
More informationThe Mathematical Derivation of Least Squares
Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the ageold queston: When the hell
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationChapter 3 Group Theory p. 1  Remark: This is only a brief summary of most important results of groups theory with respect
Chapter 3 Group Theory p.  3. Compact Course: Groups Theory emark: Ths s only a bref summary of most mportant results of groups theory wth respect to the applcatons dscussed n the followng chapters. For
More information2. Linear Algebraic Equations
2. Lnear Algebrac Equatons Many physcal systems yeld smultaneous algebrac equatons when mathematcal functons are requred to satsfy several condtons smultaneously. Each condton results n an equaton that
More informationAryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006
Aryabhata s Root Extracton Methods Abhshek Parakh Lousana State Unversty Aug 1 st 1 Introducton Ths artcle presents an analyss of the root extracton algorthms of Aryabhata gven n hs book Āryabhatīya [1,
More informationFaraday's Law of Induction
Introducton Faraday's Law o Inducton In ths lab, you wll study Faraday's Law o nducton usng a wand wth col whch swngs through a magnetc eld. You wll also examne converson o mechanc energy nto electrc energy
More informationTexas Instruments 30Xa Calculator
Teas Instruments 30Xa Calculator Keystrokes for the TI30Xa are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the tet, check
More informationPart IB Paper 1: Mechanics Examples Paper Kinematics
Engneerng Trpos Part B SECOND YEAR Straghtforward questons are marked t Trpos standard questons are marked *. Part B Paper 1: Mechancs Examples Paper Knematcs 1 ~..~~." 'SSUED ON t 7 JAN 20t4
More informationChapter 11 Torque and Angular Momentum
Chapter 11 Torque and Angular Momentum I. Torque II. Angular momentum  Defnton III. Newton s second law n angular form IV. Angular momentum  System of partcles  Rgd body  Conservaton I. Torque  Vector
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationChapter 6 Balancing of Rotating Masses
Chapter 6 Balancng of otatng Masses All rotors have soe eccentrct. Eccentrct s present when geoetrcal center of the rotor and the ass center do not concde along ther length (gure ). Eaples of rotors are
More informationLinear Algebra for Quantum Mechanics
prevous ndex next Lnear Algebra for Quantum Mechancs Mchael Fowler 0/4/08 Introducton We ve seen that n quantum mechancs, the state of an electron n some potental s gven by a ψ x t, and physcal varables
More informationTexas Instruments 30X IIS Calculator
Texas Instruments 30X IIS Calculator Keystrokes for the TI30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationRotation and Conservation of Angular Momentum
Chapter 4. Rotaton and Conservaton of Angular Momentum Notes: Most of the materal n ths chapter s taken from Young and Freedman, Chaps. 9 and 0. 4. Angular Velocty and Acceleraton We have already brefly
More informationHYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION
HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR Emal: ghapor@umedumy Abstract Ths paper
More informationElectric circuit components. Direct Current (DC) circuits
Electrc crcut components Capactor stores charge and potental energy, measured n Farads (F) Battery generates a constant electrcal potental dfference ( ) across t. Measured n olts (). Resstor ressts flow
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationMechanics of Rigid Body
Mechancs of Rgd Body 1. Introducton Knematcs, Knetcs and Statc 2. Knematcs. Types of Rgd Body Moton: Translaton, Rotaton General Plane Moton 3. Knetcs. Forces and Acceleratons. Energy and Momentum Methods.
More informationThinking about Newton's Laws
Newtonan modellng In ths actvty you wll see how Newton s Laws of Moton are used to connect the moton of an object wth the forces actng on t. You wll practse applyng Newton s three laws n some real contexts.
More informationPart 1. Electromagnetic Induction. Faraday s Law. Faraday s observation. Problem. Induced emf (Voltage) from changing Magnetic Flux.
Electromagnetc Inducton Part 1 Faraday s Law Chapter 1 Faraday s obseraton Electrc currents produce magnetc elds. 19 th century puzzle: Can magnetc elds produce currents? A statc magnet wll produce no
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationELE427  Testing Linear Sensors. Linear Regression, Accuracy, and Resolution.
ELE47  Testng Lnear Sensors Lnear Regresson, Accurac, and Resoluton. Introducton: In the frst three la eperents we wll e concerned wth the characterstcs of lnear sensors. The asc functon of these sensors
More informationLecture 2: Single Layer Perceptrons Kevin Swingler
Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCullochPtts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses
More information6. EIGENVALUES AND EIGENVECTORS 3 = 3 2
EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a nonzero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :
More informationQ3.8: A person trying to throw a ball as far as possible will run forward during the throw. Explain why this increases the distance of the throw.
Problem Set 3 Due: 09/3/, Tuesda Chapter 3: Vectors and Moton n Two Dmensons Questons: 7, 8,, 4, 0 Eercses & Problems:, 7, 8, 33, 37, 44, 46, 65, 73 Q3.7: An athlete performn the lon jump tres to acheve
More informationSIGNIFICANT FIGURES (4SF) (4SF) (5SF) (3 SF) 62.4 (3 SF) x 10 4 (5 SF) Figure 0
SIGIFICAT FIGURES Wth any calbrated nstrument, there s a lmt to the number of fgures that can be relably kept n the answer. For dgtal nstruments, the lmtaton s the number of fgures appearng n the dsplay.
More informationChapter 20 Rigid Body: Translation and Rotational Motion Kinematics for Fixed Axis Rotation
Chapter 20 Rgd Body: Translaton and Rotatonal Moton Knematcs for Fxed Axs Rotaton 201 Introducton 1 202 Constraned Moton: Translaton and Rotaton 1 2021 Rollng wthout slppng 5 Example 201 Bcycle Wheel Rollng
More informationJoe Pimbley, unpublished, 2005. Yield Curve Calculations
Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward
More informationHALL EFFECT SENSORS AND COMMUTATION
OEM770 5 Hall Effect ensors H P T E R 5 Hall Effect ensors The OEM770 works wth threephase brushless motors equpped wth Hall effect sensors or equvalent feedback sgnals. In ths chapter we wll explan how
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More informationPortfolio Risk Decomposition (and Risk Budgeting)
ortfolo Rsk Decomposton (and Rsk Budgetng) Jason MacQueen RSquared Rsk Management Introducton to Rsk Decomposton Actve managers take rsk n the expectaton of achevng outperformance of ther benchmark Mandates
More informationAttention: This material is copyright Chris Hecker. All rights reserved.
Attenton: Ths materal s copyrght 19951997 Chrs Hecker. All rghts reserved. You have permsson to read ths artcle for your own educaton. You do not have permsson to put t on your webste (but you may lnk
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationLaws of Electromagnetism
There are four laws of electromagnetsm: Laws of Electromagnetsm The law of BotSavart Ampere's law Force law Faraday's law magnetc feld generated by currents n wres the effect of a current on a loop of
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationChapter 4: Motion in Two Dimensions
Answers and Solutons. (a) Tme s a scalar quantt. (b) Dsplacement s a vector quantt. (c) Veloct s a vector quantt. (d) Speed s a scalar quantt.. An nspecton of the lengths, or magntudes, or the vectors
More informationConversion between the vector and raster data structures using Fuzzy Geographical Entities
Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationThe eigenvalue derivatives of linear damped systems
Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by YeongJeu Sun Department of Electrcal Engneerng IShou Unversty Kaohsung, Tawan 840, R.O.C emal: yjsun@su.edu.tw
More informationLECTURE 1: MOTIVATION
LECTURE 1: MOTIVATION STEVEN SAM AND PETER TINGLEY 1. Towards quantum groups Let us begn by dscussng what quantum groups are, and why we mght want to study them. We wll start wth the related classcal objects.
More informationSolutions to First Midterm
rofessor Chrstano Economcs 3, Wnter 2004 Solutons to Frst Mdterm. Multple Choce. 2. (a) v. (b). (c) v. (d) v. (e). (f). (g) v. (a) The goods market s n equlbrum when total demand equals total producton,.e.
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationIntroduction: Analysis of Electronic Circuits
/30/008 ntroducton / ntroducton: Analyss of Electronc Crcuts Readng Assgnment: KVL and KCL text from EECS Just lke EECS, the majorty of problems (hw and exam) n EECS 3 wll be crcut analyss problems. Thus,
More information( ) Homework Solutions Physics 8B Spring 09 Chpt. 32 5,18,25,27,36,42,51,57,61,76
Homework Solutons Physcs 8B Sprng 09 Chpt. 32 5,8,25,27,3,42,5,57,,7 32.5. Model: Assume deal connectng wres and an deal battery for whch V bat = E. Please refer to Fgure EX32.5. We wll choose a clockwse
More informationHomework Solutions Physics 8B Spring 2012 Chpt. 32 5,18,25,27,36,42,51,57,61,76
Homework Solutons Physcs 8B Sprng 202 Chpt. 32 5,8,25,27,3,42,5,57,,7 32.5. Model: Assume deal connectng wres and an deal battery for whch V bat =. Please refer to Fgure EX32.5. We wll choose a clockwse
More informationSystematic Circuit Analysis (T&R Chap 3)
Systematc Crcut Analyss TR Chap ) Nodeoltage analyss Usng the oltages of the each node relate to a ground node, wrte down a set of consstent lnear equatons for these oltages Sole ths set of equatons usng,
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationState function: eigenfunctions of hermitian operators> normalization, orthogonality completeness
Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators> normalzaton, orthogonalty completeness egenvalues and
More informationPolitecnico di Torino. Porto Institutional Repository
Poltecnco d orno Porto Insttutonal Repostory [Proceedng] rbt dynamcs and knematcs wth full quaternons rgnal Ctaton: Andres D; Canuto E. (5). rbt dynamcs and knematcs wth full quaternons. In: 16th IFAC
More informationGeneral Physics (PHY 2130)
General Physcs (PHY 30) Lecture 8 Moentu Collsons Elastc and nelastc collsons http://www.physcs.wayne.edu/~apetro/phy30/ Lghtnng Reew Last lecture:. Moentu: oentu and pulse oentu conseraton Reew Proble:
More informationGEOLOGIC STRUCTURES. I. Tectonic Forces (Forces that move and deform rocks, especially along plate boundaries.)
GEOLOGIC STRUCTURES Structural geology s the branch of geology that pertans to the shapes, arrangement, and nterrelatonshps of bedrock unts and the forces that cause them. Structural geologsts study the
More informationCHAPTER 8 Potential Energy and Conservation of Energy
CHAPTER 8 Potental Energy and Conservaton o Energy One orm o energy can be converted nto another orm o energy. Conservatve and nonconservatve orces Physcs 1 Knetc energy: Potental energy: Energy assocated
More informationComplex Number Representation in RCBNS Form for Arithmetic Operations and Conversion of the Result into Standard Binary Form
Complex Number epresentaton n CBNS Form for Arthmetc Operatons and Converson of the esult nto Standard Bnary Form Hatm Zan and. G. Deshmukh Florda Insttute of Technology rgd@ee.ft.edu ABSTACT Ths paper
More information1E6 Electrical Engineering AC Circuit Analysis and Power Lecture 12: Parallel Resonant Circuits
E6 Electrcal Engneerng A rcut Analyss and Power ecture : Parallel esonant rcuts. Introducton There are equvalent crcuts to the seres combnatons examned whch exst n parallel confguratons. The ssues surroundng
More informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
More informationPreAlgebra. MixNMatch. LineUps. InsideOutside Circle. A c t. t i e s & v i
IV A c t PreAlgebra v MxNMatch 1. Operatons on Fractons.................15 2. Order of Operatons....................15 3. Exponental Form......................16 4. Scentfc Notaton......................16
More informationHomework: 49, 56, 67, 60, 64, 74 (p. 234237)
Hoework: 49, 56, 67, 60, 64, 74 (p. 3437) 49. bullet o ass 0g strkes a ballstc pendulu o ass kg. The center o ass o the pendulu rses a ertcal dstance o c. ssung that the bullet reans ebedded n the pendulu,
More information5. Simultaneous eigenstates: Consider two operators that commute: Â η = a η (13.29)
5. Smultaneous egenstates: Consder two operators that commute: [ Â, ˆB ] = 0 (13.28) Let Â satsfy the followng egenvalue equaton: Multplyng both sdes by ˆB Â η = a η (13.29) ˆB [ Â η ] = ˆB [a η ] = a
More information4088858 05 AUG 10 Rev L
SL Jack Tool Kt 1725150 [ ] Instructon Sheet 4088858 05 AUG 10 PROPER USE GUIDELINES Cumulatve Trauma Dsorders can result from the prolonged use of manually powered hand tools. Hand tools are ntended
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationIDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS
IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,
More informationThe circuit shown on Figure 1 is called the common emitter amplifier circuit. The important subsystems of this circuit are:
polar Juncton Transstor rcuts Voltage and Power Amplfer rcuts ommon mtter Amplfer The crcut shown on Fgure 1 s called the common emtter amplfer crcut. The mportant subsystems of ths crcut are: 1. The basng
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationLoop Parallelization
  Loop Parallelzaton C52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I,J]+B[I,J] ED FOR ED FOR analyze
More informationReview C: Work and Kinetic Energy
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department o Physcs 8.2 Revew C: Work and Knetc Energy C. Energy... 2 C.. The Concept o Energy... 2 C..2 Knetc Energy... 3 C.2 Work and Power... 4 C.2. Work Done by
More informationRing structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAMReport 201448 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon
More informationGibbs Free Energy and Chemical Equilibrium (or how to predict chemical reactions without doing experiments)
Gbbs Free Energy and Chemcal Equlbrum (or how to predct chemcal reactons wthout dong experments) OCN 623 Chemcal Oceanography Readng: Frst half of Chapter 3, Snoeynk and Jenkns (1980) Introducton We want
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationWe assume your students are learning about selfregulation (how to change how alert they feel) through the Alert Program with its three stages:
Welcome to ALERT BINGO, a funflled and educatonal way to learn the fve ways to change engnes levels (Put somethng n your Mouth, Move, Touch, Look, and Lsten) as descrbed n the How Does Your Engne Run?
More informationSIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA
SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA E. LAGENDIJK Department of Appled Physcs, Delft Unversty of Technology Lorentzweg 1, 68 CJ, The Netherlands Emal: e.lagendjk@tnw.tudelft.nl
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More informationCHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES
CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 19982016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More information9.1 The Cumulative Sum Control Chart
Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s
More informationToday in Physics 217: the divergence and curl theorems
Today n Physcs 217: the dvergence and curl theorems Flux and dvergence: proof of the dvergence theorem, à lá Purcell. rculaton and curl: proof of tokes theorem, also followng Purcell. ee Purcell, chapter
More informationSemiconductor sensors of temperature
Semconductor sensors of temperature he measurement objectve 1. Identfy the unknown bead type thermstor. Desgn the crcutry for lnearzaton of ts transfer curve.. Fnd the dependence of forward voltage drop
More informationPassive Filters. References: Barbow (pp 265275), Hayes & Horowitz (pp 3260), Rizzoni (Chap. 6)
Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called
More informationEE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN
EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson  3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson  6 Hrs.) Voltage
More informationIntroduction to the Least Squares Fit. Table of Contents. Least Squares Fitting with Excel.1
Least Squares Fttng wth Ecel. Introducton to the Least Squares Ft Tale of Contents. Uses for a Least Squares Ft: Lnear Dependence. Methods of Fndng the Best Ft Lne: Estmatng, Usng Ecel, and Calculatng
More information1. Select the extent of the freebody and detach it from the ground and all other bodies.
Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes 4/1 Rgd bod n equlbum gd bod s sad to be n equlbum the esultant of all foces actng on t s zeo. hus, the esultant foce R and the esultant couple ae both
More informationVision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION
Vson Mouse Saurabh Sarkar a* a Unversty of Cncnnat, Cncnnat, USA ABSTRACT The report dscusses a vson based approach towards trackng of eyes and fngers. The report descrbes the process of locatng the possble
More informationIMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1
Nov Sad J. Math. Vol. 36, No. 2, 2006, 009 IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS Modrag S. Petkovć 2, Dušan M. Mloševć 3 Abstract. A new theorem concerned
More informationNordea G10 Alpha Carry Index
Nordea G10 Alpha Carry Index Index Rules v1.1 Verson as of 10/10/2013 1 (6) Page 1 Index Descrpton The G10 Alpha Carry Index, the Index, follows the development of a rule based strategy whch nvests and
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationSection B9: Zener Diodes
Secton B9: Zener Dodes When we frst talked about practcal dodes, t was mentoned that a parameter assocated wth the dode n the reverse bas regon was the breakdown voltage, BR, also known as the peaknverse
More information1. Measuring association using correlation and regression
How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a
More information