Test 1 phy What mass of a material with density ρ is required to make a hollow spherical shell having inner radius r i and outer radius r o?

Size: px
Start display at page:

Download "Test 1 phy What mass of a material with density ρ is required to make a hollow spherical shell having inner radius r i and outer radius r o?"

Transcription

1 Test 1 phy 0 1. a) What s the pupose of measuement? b) Wte all fou condtons, whch must be satsfed by a scala poduct. (Use dffeent symbols to dstngush opeatons on ectos fom opeatons on numbes.) c) What s a pojecton of a ecto? d) What does t mean that multplcaton of a ecto by a numbe s dstbute oe addton of numbes? (Use unambguous symbols to dstngush opeatons on ectos fom opeatons on numbes.) I am on the ege of a majo beakthough, but I m also at that pont whee chemsty leaes off and physcs begns, so I ll hae to dop the whole thng. In a Catesan coodnate system, ectos and B ae gen by the followng lnea combnatons of the base ectos: + j and B - + j. a) ssgn (somophcally) a pa of scala components to each ecto. b) Fnd the scala poduct of the two ectos. c) Detemne the angle between the ectos.. What mass of a mateal wth densty ρ s equed to make a hollow sphecal shell hang nne adus and oute adus o?. Two ectos and B hae pecsely equal magntudes. Fo the magntude of +B to be 100 tmes geate than the magntude of -B what must the angle 1 between them be? (Hnt: sn 1 ( 1 ; cos ( 1+ )

2 - 1 - a) In a measuement, a mathematcal quantty (numbe, ecto, ) s assgned to a physcal quantty. b) The followng condtons ae satsfed by scala poduct: 1. o B B o (commutate). ( α ) o B α( o B) (mxed assocate) B o C o C + Bo C (dstbute oe addton of ectos). ( ) ( ) ( ). 0 0 f and only f 0 c) The pojecton, of ecto, n the decton of a unt ectoe s defned as o ˆ ( e ) eˆ d) If α andβ ae two abtay numbes, and s an abtay ecto then ( α + β) α β

3 - - B j a) The coeffcents n the lnea combnaton of the base ectos consttute the scala components of a ecto. s both ectos ae gen n the fom of the lnea combnaton, the scala components ae pactcally gen [,] and B [ 1,] b) The scala poduct of ectos and B s equal to the scala poduct of the R n ectos assgned n a Catesan coodnate system o B o B o [,] [ 1,] + 6 c) Followng the defnton of the angle between the ectos, the angle between ectos and B s o B ϕ accos accos accos 60 B 1 5 [,] o [ 1,] [,] [ 1,]

4 - - (Fom the defnton of densty, we can elate the mass dm of a small (nfntesnal) segment to the appopate (dffeental) olume of the segment: o C (1) dm ρdv To fnd the mass of the ente object, we hae to ntegate ( add ) the mass of all the segments. ddng the mass s based on the scala popety of ths quantty.) Snce the densty s unfom oe the olume of the body the ntegaton s sgnfcantly smplfed. () m ρdv ρ dv ρv shell shell Knowng the ad, we can fnd the olume of the shell dectly fom the defntton of olume, caclulatng the appopate ntegal. It s ease though to use the scala popety of olume and the geneally known fomula fo the olume of a sphee (theoem). The olume of the shell s equal to the dffeence n olume of the oute and the nne sphee: () V π o π Equatons () and () fom a set wth two unknowns {m,v}. Solng (by substtuton) we fnd the mass of the shell (the answe): m ρv π( o )ρ

5 - - +B B -B -B Soluton 1 Let epesent the angle between the dectons of the two ectos. Fom the defnton of magntude ( ) + B + B + ( 1+ + B o B ( ) B + B B ( 1 cos B o sn The ato gen n the poblem yelds the followng tgonometc equaton cos 1 B tan B sn fom whch 1 actan

6 Soluton (fo oented segments) Snce and B hae the same magntudes (lengths),, B, and +B fom an sosceles tangle n whch the angles ae (π-), / and /. Smlaly, B, and -B fom an sosceles tangle n whch the angles ae, (π-)/ and (π-)/. Fom the law of cosnes, the magntude of both +B and -B can be expessed n tems of the magntude of ecto (and B ) + B B + B B cos ( π ) ( 1+ + B ( 1 cos B cos sn

Set of square-integrable function 2 L : function space F

Set of square-integrable function 2 L : function space F Set of squae-ntegable functon L : functon space F Motvaton: In ou pevous dscussons we have seen that fo fee patcles wave equatons (Helmholt o Schödnge) can be expessed n tems of egenvalue equatons. H E,

More information

24-2: Electric Potential Energy. 24-1: What is physics

24-2: Electric Potential Energy. 24-1: What is physics D. Iyad SAADEDDIN Chapte 4: Electc Potental Electc potental Enegy and Electc potental Calculatng the E-potental fom E-feld fo dffeent chage dstbutons Calculatng the E-feld fom E-potental Potental of a

More information

Chapter I Matrices, Vectors, & Vector Calculus 1-1, 1-9, 1-10, 1-11, 1-17, 1-18, 1-25, 1-27, 1-36, 1-37, 1-41.

Chapter I Matrices, Vectors, & Vector Calculus 1-1, 1-9, 1-10, 1-11, 1-17, 1-18, 1-25, 1-27, 1-36, 1-37, 1-41. Chapte I Matces, Vectos, & Vecto Calculus -, -9, -0, -, -7, -8, -5, -7, -36, -37, -4. . Concept of a Scala Consde the aa of patcles shown n the fgue. he mass of the patcle at (,) can be epessed as. M (,

More information

Integral Vector Operations and Related Theorems Applications in Mechanics and E&M

Integral Vector Operations and Related Theorems Applications in Mechanics and E&M Dola Bagayoko (0) Integal Vecto Opeatons and elated Theoems Applcatons n Mechancs and E&M Ι Basc Defnton Please efe to you calculus evewed below. Ι, ΙΙ, andιιι notes and textbooks fo detals on the concepts

More information

PHY126 Summer Session I, 2008

PHY126 Summer Session I, 2008 PHY6 Summe Sesson I, 8 Most of nfomaton s avalable at: http://nngoup.phscs.sunsb.edu/~chak/phy6-8 ncludng the sllabus and lectue sldes. Read sllabus and watch fo mpotant announcements. Homewok assgnment

More information

Dynamics of Rigid Bodies

Dynamics of Rigid Bodies Dynamcs of Rgd Bodes A gd body s one n whch the dstances between consttuent patcles s constant thoughout the moton of the body,.e. t keeps ts shape. Thee ae two knds of gd body moton: 1. Tanslatonal Rectlnea

More information

VEKTORANALYS FLUX INTEGRAL LINE INTEGRAL. and. Kursvecka 2. Kapitel 4 5. Sidor 29 50

VEKTORANALYS FLUX INTEGRAL LINE INTEGRAL. and. Kursvecka 2. Kapitel 4 5. Sidor 29 50 VEKTORANAYS Ksecka INE INTEGRA and UX INTEGRA Kaptel 4 5 Sdo 9 5 A wnd TARGET PROBEM We want to psh a mne cat along a path fom A to B. Bt the wnd s blowng. How mch enegy s needed? (.e. how mch s the wok?

More information

CSJM University Class: B.Sc.-II Sub:Physics Paper-II Title: Electromagnetics Unit-1: Electrostatics Lecture: 1 to 4

CSJM University Class: B.Sc.-II Sub:Physics Paper-II Title: Electromagnetics Unit-1: Electrostatics Lecture: 1 to 4 CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 Electostatcs: It deals the study of behavo of statc o statonay Chages. Electc Chage: It s popety by

More information

Scalars and Vectors Scalar

Scalars and Vectors Scalar Scalas and ectos Scala A phscal quantt that s completel chaacteed b a eal numbe (o b ts numecal value) s called a scala. In othe wods a scala possesses onl a magntude. Mass denst volume tempeatue tme eneg

More information

Physics 11b Lecture #2. Electric Field Electric Flux Gauss s Law

Physics 11b Lecture #2. Electric Field Electric Flux Gauss s Law Physcs 11b Lectue # Electc Feld Electc Flux Gauss s Law What We Dd Last Tme Electc chage = How object esponds to electc foce Comes n postve and negatve flavos Conseved Electc foce Coulomb s Law F Same

More information

Review of Vector Algebra and Vector Calculus Operations

Review of Vector Algebra and Vector Calculus Operations Revew of Vecto Algeba and Vecto Calculus Opeatons Tpes of vaables n Flud Mechancs Repesentaton of vectos Dffeent coodnate sstems Base vecto elatons Scala and vecto poducts Stess Newton s law of vscost

More information

UNIVERSITÀ DI PISA. Math thbackground

UNIVERSITÀ DI PISA. Math thbackground UNIVERSITÀ DI ISA Electomagnetc Radatons and Bologcal l Inteactons Lauea Magstale n Bomedcal Engneeng Fst semeste (6 cedts), academc ea 2011/12 of. aolo Nepa p.nepa@et.unp.t Math thbackgound Edted b D.

More information

Rotational Kinematics. Rigid Object about a Fixed Axis Western HS AP Physics 1

Rotational Kinematics. Rigid Object about a Fixed Axis Western HS AP Physics 1 Rotatonal Knematcs Rgd Object about a Fxed Axs Westen HS AP Physcs 1 Leanng Objectes What we know Unfom Ccula Moton q s Centpetal Acceleaton : Centpetal Foce: Non-unfom a F c c m F F F t m ma t What we

More information

Chapter 23: Electric Potential

Chapter 23: Electric Potential Chapte 23: Electc Potental Electc Potental Enegy It tuns out (won t show ths) that the tostatc foce, qq 1 2 F ˆ = k, s consevatve. 2 Recall, fo any consevatve foce, t s always possble to wte the wok done

More information

1. A body will remain in a state of rest, or of uniform motion in a straight line unless it

1. A body will remain in a state of rest, or of uniform motion in a straight line unless it Pncples of Dnamcs: Newton's Laws of moton. : Foce Analss 1. A bod wll eman n a state of est, o of unfom moton n a staght lne unless t s acted b etenal foces to change ts state.. The ate of change of momentum

More information

Engineering Mechanics. Force resultants, Torques, Scalar Products, Equivalent Force systems

Engineering Mechanics. Force resultants, Torques, Scalar Products, Equivalent Force systems Engneeng echancs oce esultants, Toques, Scala oducts, Equvalent oce sstems Tata cgaw-hll Companes, 008 Resultant of Two oces foce: acton of one bod on anothe; chaacteed b ts pont of applcaton, magntude,

More information

Physics 202, Lecture 2. Announcements

Physics 202, Lecture 2. Announcements Physcs 202, Lectue 2 Today s Topcs Announcements Electc Felds Moe on the Electc Foce (Coulomb s Law The Electc Feld Moton of Chaged Patcles n an Electc Feld Announcements Homewok Assgnment #1: WebAssgn

More information

Physics 1501 Lecture 19

Physics 1501 Lecture 19 Physcs 1501 ectue 19 Physcs 1501: ectue 19 Today s Agenda Announceents HW#7: due Oct. 1 Mdte 1: aveage 45 % Topcs otatonal Kneatcs otatonal Enegy Moents of Ineta Physcs 1501: ectue 19, Pg 1 Suay (wth copason

More information

V. Principles of Irreversible Thermodynamics. s = S - S 0 (7.3) s = = - g i, k. "Flux": = da i. "Force": = -Â g a ik k = X i. Â J i X i (7.

V. Principles of Irreversible Thermodynamics. s = S - S 0 (7.3) s = = - g i, k. Flux: = da i. Force: = -Â g a ik k = X i. Â J i X i (7. Themodynamcs and Knetcs of Solds 71 V. Pncples of Ievesble Themodynamcs 5. Onsage s Teatment s = S - S 0 = s( a 1, a 2,...) a n = A g - A n (7.6) Equlbum themodynamcs detemnes the paametes of an equlbum

More information

ALL QUESTIONS ARE WORTH 20 POINTS. WORK OUT FIVE PROBLEMS.

ALL QUESTIONS ARE WORTH 20 POINTS. WORK OUT FIVE PROBLEMS. GNRAL PHYSICS PH -3A (D. S. Mov) Test (/3/) key STUDNT NAM: STUDNT d #: -------------------------------------------------------------------------------------------------------------------------------------------

More information

PHYS 705: Classical Mechanics. Derivation of Lagrange Equations from D Alembert s Principle

PHYS 705: Classical Mechanics. Derivation of Lagrange Equations from D Alembert s Principle 1 PHYS 705: Classcal Mechancs Devaton of Lagange Equatons fom D Alembet s Pncple 2 D Alembet s Pncple Followng a smla agument fo the vtual dsplacement to be consstent wth constants,.e, (no vtual wok fo

More information

19 The Born-Oppenheimer Approximation

19 The Born-Oppenheimer Approximation 9 The Bon-Oppenheme Appoxmaton The full nonelatvstc Hamltonan fo a molecule s gven by (n a.u.) Ĥ = A M A A A, Z A + A + >j j (883) Lets ewte the Hamltonan to emphasze the goal as Ĥ = + A A A, >j j M A

More information

COMPUTATIONAL METHODS AND ALGORITHMS Vol. I - Methods of Potential Theory - V.I. Agoshkov, P.B. Dubovski

COMPUTATIONAL METHODS AND ALGORITHMS Vol. I - Methods of Potential Theory - V.I. Agoshkov, P.B. Dubovski METHODS OF POTENTIAL THEORY.I. Agoshkov and P.B. Dubovsk Insttute of Numecal Mathematcs, Russan Academy of Scences, Moscow, Russa Keywods: Potental, volume potental, Newton s potental, smple laye potental,

More information

CSU ATS601 Fall Other reading: Vallis 2.1, 2.2; Marshall and Plumb Ch. 6; Holton Ch. 2; Schubert Ch r or v i = v r + r (3.

CSU ATS601 Fall Other reading: Vallis 2.1, 2.2; Marshall and Plumb Ch. 6; Holton Ch. 2; Schubert Ch r or v i = v r + r (3. 3 Eath s Rotaton 3.1 Rotatng Famewok Othe eadng: Valls 2.1, 2.2; Mashall and Plumb Ch. 6; Holton Ch. 2; Schubet Ch. 3 Consde the poston vecto (the same as C n the fgue above) otatng at angula velocty.

More information

Chapter 13 - Universal Gravitation

Chapter 13 - Universal Gravitation Chapte 3 - Unesal Gataton In Chapte 5 we studed Newton s thee laws of moton. In addton to these laws, Newton fomulated the law of unesal gataton. Ths law states that two masses ae attacted by a foce gen

More information

Remember: When an object falls due to gravity its potential energy decreases.

Remember: When an object falls due to gravity its potential energy decreases. Chapte 5: lectc Potental As mentoned seveal tmes dung the uate Newton s law o gavty and Coulomb s law ae dentcal n the mathematcal om. So, most thngs that ae tue o gavty ae also tue o electostatcs! Hee

More information

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 29,

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 29, hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe 9, 08 47 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe 9, 08 48 I. ETOR AND TENOR ANALYI I... Tenso functon th Let A

More information

Physics 201 Lecture 4

Physics 201 Lecture 4 Phscs 1 Lectue 4 ltoda: hapte 3 Lectue 4 v Intoduce scalas and vectos v Peom basc vecto aleba (addton and subtacton) v Inteconvet between atesan & Pola coodnates Stat n nteestn 1D moton poblem: ace 9.8

More information

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23,

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 47 hapte I ecto and Tenso Analyss I. ecto and Tenso Analyss eptembe, 07 48 I. ETOR AND TENOR ANALYI I... Tenso functon th Let A n n

More information

A. Thicknesses and Densities

A. Thicknesses and Densities 10 Lab0 The Eath s Shells A. Thcknesses and Denstes Any theoy of the nteo of the Eath must be consstent wth the fact that ts aggegate densty s 5.5 g/cm (ecall we calculated ths densty last tme). In othe

More information

Rigid Bodies: Equivalent Systems of Forces

Rigid Bodies: Equivalent Systems of Forces Engneeng Statcs, ENGR 2301 Chapte 3 Rgd Bodes: Equvalent Sstems of oces Intoducton Teatment of a bod as a sngle patcle s not alwas possble. In geneal, the se of the bod and the specfc ponts of applcaton

More information

Potential Theory. Copyright 2004

Potential Theory. Copyright 2004 Copyght 004 4 Potental Theoy We have seen how the soluton of any classcal echancs poble s fst one of detenng the equatons of oton. These then ust be solved n ode to fnd the oton of the patcles that copse

More information

Some Approximate Analytical Steady-State Solutions for Cylindrical Fin

Some Approximate Analytical Steady-State Solutions for Cylindrical Fin Some Appoxmate Analytcal Steady-State Solutons fo Cylndcal Fn ANITA BRUVERE ANDRIS BUIIS Insttute of Mathematcs and Compute Scence Unvesty of Latva Rana ulv 9 Rga LV459 LATVIA Astact: - In ths pape we

More information

Chapter Fifiteen. Surfaces Revisited

Chapter Fifiteen. Surfaces Revisited Chapte Ffteen ufaces Revsted 15.1 Vecto Descpton of ufaces We look now at the vey specal case of functons : D R 3, whee D R s a nce subset of the plane. We suppose s a nce functon. As the pont ( s, t)

More information

THE EQUIVALENCE OF GRAM-SCHMIDT AND QR FACTORIZATION (page 227) Gram-Schmidt provides another way to compute a QR decomposition: n

THE EQUIVALENCE OF GRAM-SCHMIDT AND QR FACTORIZATION (page 227) Gram-Schmidt provides another way to compute a QR decomposition: n HE EQUIVAENCE OF GRA-SCHID AND QR FACORIZAION (page 7 Ga-Schdt podes anothe way to copute a QR decoposton: n gen ectos,, K, R, Ga-Schdt detenes scalas j such that o + + + [ ] [ ] hs s a QR factozaton of

More information

UNIT10 PLANE OF REGRESSION

UNIT10 PLANE OF REGRESSION UIT0 PLAE OF REGRESSIO Plane of Regesson Stuctue 0. Intoducton Ojectves 0. Yule s otaton 0. Plane of Regesson fo thee Vaales 0.4 Popetes of Resduals 0.5 Vaance of the Resduals 0.6 Summay 0.7 Solutons /

More information

Physics 2A Chapter 11 - Universal Gravitation Fall 2017

Physics 2A Chapter 11 - Universal Gravitation Fall 2017 Physcs A Chapte - Unvesal Gavtaton Fall 07 hese notes ae ve pages. A quck summay: he text boxes n the notes contan the esults that wll compse the toolbox o Chapte. hee ae thee sectons: the law o gavtaton,

More information

Physics Exam II Chapters 25-29

Physics Exam II Chapters 25-29 Physcs 114 1 Exam II Chaptes 5-9 Answe 8 of the followng 9 questons o poblems. Each one s weghted equally. Clealy mak on you blue book whch numbe you do not want gaded. If you ae not sue whch one you do

More information

COMPLEMENTARY ENERGY METHOD FOR CURVED COMPOSITE BEAMS

COMPLEMENTARY ENERGY METHOD FOR CURVED COMPOSITE BEAMS ultscence - XXX. mcocd Intenatonal ultdscplnay Scentfc Confeence Unvesty of skolc Hungay - pl 06 ISBN 978-963-358-3- COPLEENTRY ENERGY ETHOD FOR CURVED COPOSITE BES Ákos József Lengyel István Ecsed ssstant

More information

Chapter 8. Linear Momentum, Impulse, and Collisions

Chapter 8. Linear Momentum, Impulse, and Collisions Chapte 8 Lnea oentu, Ipulse, and Collsons 8. Lnea oentu and Ipulse The lnea oentu p of a patcle of ass ovng wth velocty v s defned as: p " v ote that p s a vecto that ponts n the sae decton as the velocty

More information

ON THE FRESNEL SINE INTEGRAL AND THE CONVOLUTION

ON THE FRESNEL SINE INTEGRAL AND THE CONVOLUTION IJMMS 3:37, 37 333 PII. S16117131151 http://jmms.hndaw.com Hndaw Publshng Cop. ON THE FRESNEL SINE INTEGRAL AND THE CONVOLUTION ADEM KILIÇMAN Receved 19 Novembe and n evsed fom 7 Mach 3 The Fesnel sne

More information

Optimization Methods: Linear Programming- Revised Simplex Method. Module 3 Lecture Notes 5. Revised Simplex Method, Duality and Sensitivity analysis

Optimization Methods: Linear Programming- Revised Simplex Method. Module 3 Lecture Notes 5. Revised Simplex Method, Duality and Sensitivity analysis Optmzaton Meods: Lnea Pogammng- Revsed Smple Meod Module Lectue Notes Revsed Smple Meod, Dualty and Senstvty analyss Intoducton In e pevous class, e smple meod was dscussed whee e smple tableau at each

More information

Smoothed Particle Hydrodynamics

Smoothed Particle Hydrodynamics Smooted Patcle Hydodynamcs Applcaton Example Alan Hec Noembe 9, 010 Fluds Nae-Stokes equatons Smooted Patcle Hydodynamcs Smooted Patcle Hydodynamcs Noembe 010 Fluds Lquds, e.g. wate Gasses, e.g. a Plasmas

More information

Stellar Astrophysics. dt dr. GM r. The current model for treating convection in stellar interiors is called mixing length theory:

Stellar Astrophysics. dt dr. GM r. The current model for treating convection in stellar interiors is called mixing length theory: Stella Astophyscs Ovevew of last lectue: We connected the mean molecula weght to the mass factons X, Y and Z: 1 1 1 = X + Y + μ 1 4 n 1 (1 + 1) = X μ 1 1 A n Z (1 + ) + Y + 4 1+ z A Z We ntoduced the pessue

More information

(8) Gain Stage and Simple Output Stage

(8) Gain Stage and Simple Output Stage EEEB23 Electoncs Analyss & Desgn (8) Gan Stage and Smple Output Stage Leanng Outcome Able to: Analyze an example of a gan stage and output stage of a multstage amplfe. efeence: Neamen, Chapte 11 8.0) ntoducton

More information

PHYS Week 5. Reading Journals today from tables. WebAssign due Wed nite

PHYS Week 5. Reading Journals today from tables. WebAssign due Wed nite PHYS 015 -- Week 5 Readng Jounals today fom tables WebAssgn due Wed nte Fo exclusve use n PHYS 015. Not fo e-dstbuton. Some mateals Copyght Unvesty of Coloado, Cengage,, Peason J. Maps. Fundamental Tools

More information

4 SingularValue Decomposition (SVD)

4 SingularValue Decomposition (SVD) /6/00 Z:\ jeh\self\boo Kannan\Jan-5-00\4 SVD 4 SngulaValue Decomposton (SVD) Chapte 4 Pat SVD he sngula value decomposton of a matx s the factozaton of nto the poduct of thee matces = UDV whee the columns

More information

iclicker Quiz a) True b) False Theoretical physics: the eternal quest for a missing minus sign and/or a factor of two. Which will be an issue today?

iclicker Quiz a) True b) False Theoretical physics: the eternal quest for a missing minus sign and/or a factor of two. Which will be an issue today? Clce Quz I egsteed my quz tansmtte va the couse webste (not on the clce.com webste. I ealze that untl I do so, my quz scoes wll not be ecoded. a Tue b False Theoetcal hyscs: the etenal quest fo a mssng

More information

4.4 Continuum Thermomechanics

4.4 Continuum Thermomechanics 4.4 Contnuum Themomechancs The classcal themodynamcs s now extended to the themomechancs of a contnuum. The state aables ae allowed to ay thoughout a mateal and pocesses ae allowed to be eesble and moe

More information

Units, Physical Quantities and Vectors

Units, Physical Quantities and Vectors What s Phscs? Unts, Phscal Quanttes and Vectos Natual Phlosoph scence of matte and eneg fundamental pncples of engneeng and technolog an epemental scence: theo epement smplfed (dealed) models ange of valdt

More information

COORDINATE SYSTEMS, COORDINATE TRANSFORMS, AND APPLICATIONS

COORDINATE SYSTEMS, COORDINATE TRANSFORMS, AND APPLICATIONS Dola Bagaoo 0 COORDINTE SYSTEMS COORDINTE TRNSFORMS ND PPLICTIONS I. INTRODUCTION Smmet coce of coodnate sstem. In solvng Pscs poblems one cooses a coodnate sstem tat fts te poblem at and.e. a coodnate

More information

3.1 Electrostatic Potential Energy and Potential Difference

3.1 Electrostatic Potential Energy and Potential Difference 3. lectostatc Potental negy and Potental Dffeence RMMR fom mechancs: - The potental enegy can be defned fo a system only f consevatve foces act between ts consttuents. - Consevatve foces may depend only

More information

Contact, information, consultations

Contact, information, consultations ontact, nfomaton, consultatons hemsty A Bldg; oom 07 phone: 058-347-769 cellula: 664 66 97 E-mal: wojtek_c@pg.gda.pl Offce hous: Fday, 9-0 a.m. A quote of the week (o camel of the week): hee s no expedence

More information

Tensor. Syllabus: x x

Tensor. Syllabus: x x Tenso Sllabus: Tenso Calculus : Catesan tensos. Smmetc and antsmmetc tensos. Lev Vvta tenso denst. Pseudo tensos. Dual tensos. Dect poduct and contacton. Dads and dadc. Covaant, Contavaant and med tensos.

More information

If there are k binding constraints at x then re-label these constraints so that they are the first k constraints.

If there are k binding constraints at x then re-label these constraints so that they are the first k constraints. Mathematcal Foundatons -1- Constaned Optmzaton Constaned Optmzaton Ma{ f ( ) X} whee X {, h ( ), 1,, m} Necessay condtons fo to be a soluton to ths mamzaton poblem Mathematcally, f ag Ma{ f ( ) X}, then

More information

COLLEGE OF FOUNDATION AND GENERAL STUDIES PUTRAJAYA CAMPUS FINAL EXAMINATION TRIMESTER /2017

COLLEGE OF FOUNDATION AND GENERAL STUDIES PUTRAJAYA CAMPUS FINAL EXAMINATION TRIMESTER /2017 COLLEGE OF FOUNDATION AND GENERAL STUDIES PUTRAJAYA CAMPUS FINAL EXAMINATION TRIMESTER 1 016/017 PROGRAMME SUBJECT CODE : Foundaton n Engneeng : PHYF115 SUBJECT : Phscs 1 DATE : Septembe 016 DURATION :

More information

gravity r2,1 r2 r1 by m 2,1

gravity r2,1 r2 r1 by m 2,1 Gavtaton Many of the foundatons of classcal echancs wee fst dscoveed when phlosophes (ealy scentsts and atheatcans) ted to explan the oton of planets and stas. Newton s ost faous fo unfyng the oton of

More information

Physics 207 Lecture 16

Physics 207 Lecture 16 Physcs 07 Lectue 6 Goals: Lectue 6 Chapte Extend the patcle odel to gd-bodes Undestand the equlbu of an extended object. Analyze ollng oton Undestand otaton about a fxed axs. Eploy consevaton of angula

More information

Review. Physics 231 fall 2007

Review. Physics 231 fall 2007 Reew Physcs 3 all 7 Man ssues Knematcs - moton wth constant acceleaton D moton, D pojectle moton, otatonal moton Dynamcs (oces) Enegy (knetc and potental) (tanslatonal o otatonal moton when detals ae not

More information

3. A Review of Some Existing AW (BT, CT) Algorithms

3. A Review of Some Existing AW (BT, CT) Algorithms 3. A Revew of Some Exstng AW (BT, CT) Algothms In ths secton, some typcal ant-wndp algothms wll be descbed. As the soltons fo bmpless and condtoned tansfe ae smla to those fo ant-wndp, the pesented algothms

More information

Groupoid and Topological Quotient Group

Groupoid and Topological Quotient Group lobal Jounal of Pue and Appled Mathematcs SSN 0973-768 Volume 3 Numbe 7 07 pp 373-39 Reseach nda Publcatons http://wwwpublcatoncom oupod and Topolocal Quotent oup Mohammad Qasm Manna Depatment of Mathematcs

More information

LINEAR MOMENTUM. product of the mass m and the velocity v r of an object r r

LINEAR MOMENTUM. product of the mass m and the velocity v r of an object r r LINEAR MOMENTUM Imagne beng on a skateboad, at est that can move wthout cton on a smooth suace You catch a heavy, slow-movng ball that has been thown to you you begn to move Altenatvely you catch a lght,

More information

1. Starting with the local version of the first law of thermodynamics q. derive the statement of the first law of thermodynamics for a control volume

1. Starting with the local version of the first law of thermodynamics q. derive the statement of the first law of thermodynamics for a control volume EN10: Contnuum Mechancs Homewok 5: Alcaton of contnuum mechancs to fluds Due 1:00 noon Fda Febua 4th chool of Engneeng Bown Unvest 1. tatng wth the local veson of the fst law of themodnamcs q jdj q t and

More information

In statistical computations it is desirable to have a simplified system of notation to avoid complicated formulas describing mathematical operations.

In statistical computations it is desirable to have a simplified system of notation to avoid complicated formulas describing mathematical operations. Chapte 1 STATISTICAL NOTATION AND ORGANIZATION 11 Summation Notation fo a One-Way Classification In statistical computations it is desiable to have a simplified system of notation to avoid complicated

More information

Energy in Closed Systems

Energy in Closed Systems Enegy n Closed Systems Anamta Palt palt.anamta@gmal.com Abstact The wtng ndcates a beakdown of the classcal laws. We consde consevaton of enegy wth a many body system n elaton to the nvese squae law and

More information

8 Baire Category Theorem and Uniform Boundedness

8 Baire Category Theorem and Uniform Boundedness 8 Bae Categoy Theoem and Unfom Boundedness Pncple 8.1 Bae s Categoy Theoem Valdty of many esults n analyss depends on the completeness popety. Ths popety addesses the nadequacy of the system of atonal

More information

( ) α is determined to be a solution of the one-dimensional minimization problem: = 2. min = 2

( ) α is determined to be a solution of the one-dimensional minimization problem: = 2. min = 2 Homewo (Patal Solton) Posted on Mach, 999 MEAM 5 Deental Eqaton Methods n Mechancs. Sole the ollowng mat eqaton A b by () Steepest Descent Method and/o Pecondtoned SD Method Snce the coecent mat A s symmetc,

More information

TEST-03 TOPIC: MAGNETISM AND MAGNETIC EFFECT OF CURRENT Q.1 Find the magnetic field intensity due to a thin wire carrying current I in the Fig.

TEST-03 TOPIC: MAGNETISM AND MAGNETIC EFFECT OF CURRENT Q.1 Find the magnetic field intensity due to a thin wire carrying current I in the Fig. TEST-03 TPC: MAGNETSM AND MAGNETC EFFECT F CURRENT Q. Fnd the magnetc feld ntensty due to a thn we cayng cuent n the Fg. - R 0 ( + tan) R () 0 ( ) R 0 ( + ) R 0 ( + tan ) R Q. Electons emtted wth neglgble

More information

Part V: Velocity and Acceleration Analysis of Mechanisms

Part V: Velocity and Acceleration Analysis of Mechanisms Pat V: Velocty an Acceleaton Analyss of Mechansms Ths secton wll evew the most common an cuently pactce methos fo completng the knematcs analyss of mechansms; escbng moton though velocty an acceleaton.

More information

Summer Workshop on the Reaction Theory Exercise sheet 8. Classwork

Summer Workshop on the Reaction Theory Exercise sheet 8. Classwork Joned Physcs Analyss Cente Summe Wokshop on the Reacton Theoy Execse sheet 8 Vncent Matheu Contact: http://www.ndana.edu/~sst/ndex.html June June To be dscussed on Tuesday of Week-II. Classwok. Deve all

More information

Supersymmetry in Disorder and Chaos (Random matrices, physics of compound nuclei, mathematics of random processes)

Supersymmetry in Disorder and Chaos (Random matrices, physics of compound nuclei, mathematics of random processes) Supesymmety n Dsoe an Chaos Ranom matces physcs of compoun nucle mathematcs of anom pocesses Lteatue: K.B. Efetov Supesymmety n Dsoe an Chaos Cambge Unvesty Pess 997999 Supesymmety an Tace Fomulae I.V.

More information

P 365. r r r )...(1 365

P 365. r r r )...(1 365 SCIENCE WORLD JOURNAL VOL (NO4) 008 www.scecncewoldounal.og ISSN 597-64 SHORT COMMUNICATION ANALYSING THE APPROXIMATION MODEL TO BIRTHDAY PROBLEM *CHOJI, D.N. & DEME, A.C. Depatment of Mathematcs Unvesty

More information

9/12/2013. Microelectronics Circuit Analysis and Design. Modes of Operation. Cross Section of Integrated Circuit npn Transistor

9/12/2013. Microelectronics Circuit Analysis and Design. Modes of Operation. Cross Section of Integrated Circuit npn Transistor Mcoelectoncs Ccut Analyss and Desgn Donald A. Neamen Chapte 5 The pola Juncton Tanssto In ths chapte, we wll: Dscuss the physcal stuctue and opeaton of the bpola juncton tanssto. Undestand the dc analyss

More information

LASER ABLATION ICP-MS: DATA REDUCTION

LASER ABLATION ICP-MS: DATA REDUCTION Lee, C-T A Lase Ablaton Data educton 2006 LASE ABLATON CP-MS: DATA EDUCTON Cn-Ty A. Lee 24 Septembe 2006 Analyss and calculaton of concentatons Lase ablaton analyses ae done n tme-esolved mode. A ~30 s

More information

Generating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences

Generating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences Geneatng Functons, Weghted and Non-Weghted Sums fo Powes of Second-Ode Recuence Sequences Pantelmon Stăncă Aubun Unvesty Montgomey, Depatment of Mathematcs Montgomey, AL 3614-403, USA e-mal: stanca@studel.aum.edu

More information

Description Linear Angular position x displacement x rate of change of position v x x v average rate of change of position

Description Linear Angular position x displacement x rate of change of position v x x v average rate of change of position Chapte 5 Ccula Moton The language used to descbe otatonal moton s ey smla to the language used to descbe lnea moton. The symbols ae deent. Descpton Lnea Angula poston dsplacement ate o change o poston

More information

One-dimensional kinematics

One-dimensional kinematics Phscs 45 Fomula Sheet Eam 3 One-dmensonal knematcs Vectos dsplacement: Δ total dstance taveled aveage speed total tme Δ aveage veloct: vav t t Δ nstantaneous veloct: v lm Δ t v aveage acceleaton: aav t

More information

Asymptotic Solutions of the Kinetic Boltzmann Equation and Multicomponent Non-Equilibrium Gas Dynamics

Asymptotic Solutions of the Kinetic Boltzmann Equation and Multicomponent Non-Equilibrium Gas Dynamics Jounal of Appled Mathematcs and Physcs 6 4 687-697 Publshed Onlne August 6 n ScRes http://wwwscpog/jounal/jamp http://dxdoog/436/jamp64877 Asymptotc Solutons of the Knetc Boltzmann Equaton and Multcomponent

More information

Class #16 Monday, March 20, 2017

Class #16 Monday, March 20, 2017 D. Pogo Class #16 Monday, Mach 0, 017 D Non-Catesian Coodinate Systems A point in space can be specified by thee numbes:, y, and z. O, it can be specified by 3 diffeent numbes:,, and z, whee = cos, y =

More information

CIRCUITS 1 DEVELOP TOOLS FOR THE ANALYSIS AND DESIGN OF BASIC LINEAR ELECTRIC CIRCUITS

CIRCUITS 1 DEVELOP TOOLS FOR THE ANALYSIS AND DESIGN OF BASIC LINEAR ELECTRIC CIRCUITS CCUTS DEELOP TOOLS FO THE ANALYSS AND DESGN OF BASC LNEA ELECTC CCUTS ESSTE CCUTS Here we introduce the basic concepts and laws that are fundamental to circuit analysis LEANNG GOALS OHM S LAW - DEFNES

More information

Electron density: Properties of electron density (non-negative): => exchange-correlation functionals should respect these conditions.

Electron density: Properties of electron density (non-negative): => exchange-correlation functionals should respect these conditions. lecton densty: ρ ( =... Ψ(,,..., ds d... d Pobablty of fndng one electon of abtay spn wthn a volume element d (othe electons may be anywhee. s Popetes of electon densty (non-negatve:.. 3. ρ ( d = ρ( =

More information

AE301 Aerodynamics I UNIT B: Theory of Aerodynamics

AE301 Aerodynamics I UNIT B: Theory of Aerodynamics AE301 Aeodynamics I UNIT B: Theoy of Aeodynamics ROAD MAP... B-1: Mathematics fo Aeodynamics B-2: Flow Field Repesentations B-3: Potential Flow Analysis B-4: Applications of Potential Flow Analysis AE301

More information

Capítulo. Three Dimensions

Capítulo. Three Dimensions Capítulo Knematcs of Rgd Bodes n Thee Dmensons Mecánca Contents ntoducton Rgd Bod Angula Momentum n Thee Dmensons Pncple of mpulse and Momentum Knetc Eneg Sample Poblem 8. Sample Poblem 8. Moton of a Rgd

More information

Multipole Radiation. March 17, 2014

Multipole Radiation. March 17, 2014 Multpole Radaton Mach 7, 04 Zones We wll see that the poblem of hamonc adaton dvdes nto thee appoxmate egons, dependng on the elatve magntudes of the dstance of the obsevaton pont,, and the wavelength,

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 151 Lectue 18 Hamltonan Equatons of Moton (Chapte 8) What s Ahead We ae statng Hamltonan fomalsm Hamltonan equaton Today and 11/6 Canoncal tansfomaton 1/3, 1/5, 1/10 Close lnk to non-elatvstc

More information

Khintchine-Type Inequalities and Their Applications in Optimization

Khintchine-Type Inequalities and Their Applications in Optimization Khntchne-Type Inequaltes and The Applcatons n Optmzaton Anthony Man-Cho So Depatment of Systems Engneeng & Engneeng Management The Chnese Unvesty of Hong Kong ISDS-Kolloquum Unvestaet Wen 29 June 2009

More information

MULTIPOLE FIELDS. Multipoles, 2 l poles. Monopoles, dipoles, quadrupoles, octupoles... Electric Dipole R 1 R 2. P(r,θ,φ) e r

MULTIPOLE FIELDS. Multipoles, 2 l poles. Monopoles, dipoles, quadrupoles, octupoles... Electric Dipole R 1 R 2. P(r,θ,φ) e r MULTIPOLE FIELDS Mutpoes poes. Monopoes dpoes quadupoes octupoes... 4 8 6 Eectc Dpoe +q O θ e R R P(θφ) -q e The potenta at the fed pont P(θφ) s ( θϕ )= q R R Bo E. Seneus : Now R = ( e) = + cosθ R = (

More information

7/1/2008. Adhi Harmoko S. a c = v 2 /r. F c = m x a c = m x v 2 /r. Ontang Anting Moment of Inertia. Energy

7/1/2008. Adhi Harmoko S. a c = v 2 /r. F c = m x a c = m x v 2 /r. Ontang Anting Moment of Inertia. Energy 7//008 Adh Haoko S Ontang Antng Moent of neta Enegy Passenge undego unfo ccula oton (ccula path at constant speed) Theefoe, thee ust be a: centpetal acceleaton, a c. Theefoe thee ust be a centpetal foce,

More information

Chapter 3 Vector Integral Calculus

Chapter 3 Vector Integral Calculus hapte Vecto Integal alculus I. Lne ntegals. Defnton A lne ntegal of a vecto functon F ove a cuve s F In tems of components F F F F If,, an ae functon of t, we have F F F F t t t t E.. Fn the value of the

More information

Rotating Disk Electrode -a hydrodynamic method

Rotating Disk Electrode -a hydrodynamic method Rotatng Dsk Electode -a hdodnamc method Fe Lu Ma 3, 0 ente fo Electochemcal Engneeng Reseach Depatment of hemcal and Bomolecula Engneeng Rotatng Dsk Electode A otatng dsk electode RDE s a hdodnamc wokng

More information

The intrinsic sense of stochastic differential equations

The intrinsic sense of stochastic differential equations The ntnsc sense of stochastc dffeental equatons Detch Ryte RyteDM@gawnet.ch Mdatweg 3 CH-4500 Solothun Swtzeland Phone +413 61 13 07 A change of the vaables (establshng a nomal fom of the fowad and bacwad

More information

Hamiltonian multivector fields and Poisson forms in multisymplectic field theory

Hamiltonian multivector fields and Poisson forms in multisymplectic field theory JOURNAL OF MATHEMATICAL PHYSICS 46, 12005 Hamltonan multvecto felds and Posson foms n multsymplectc feld theoy Mchael Foge a Depatamento de Matemátca Aplcada, Insttuto de Matemátca e Estatístca, Unvesdade

More information

Chapter I Vector Analysis

Chapter I Vector Analysis . Chpte I Vecto nlss . Vecto lgeb j It s well-nown tht n vecto cn be wtten s Vectos obe the followng lgebc ules: scl s ) ( j v v cos ) ( e Commuttv ) ( ssoctve C C ) ( ) ( v j ) ( ) ( ) ( ) ( (v) he lw

More information

Analytical and Numerical Solutions for a Rotating Annular Disk of Variable Thickness

Analytical and Numerical Solutions for a Rotating Annular Disk of Variable Thickness Appled Mathematcs 00 43-438 do:0.436/am.00.5057 Publshed Onlne Novembe 00 (http://www.scrp.og/jounal/am) Analytcal and Numecal Solutons fo a Rotatng Annula Ds of Vaable Thcness Abstact Ashaf M. Zenou Daoud

More information

Budding yeast colony growth study based on circular granular cell

Budding yeast colony growth study based on circular granular cell Jounal of Physcs: Confeence Sees PAPER OPEN ACCESS Buddng yeast colony gowth study based on ccula ganula cell To cte ths atcle: Dev Apant et al 2016 J. Phys.: Conf. Se. 739 012026 Vew the atcle onlne fo

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

Open Systems: Chemical Potential and Partial Molar Quantities Chemical Potential

Open 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 information

E For K > 0. s s s s Fall Physical Chemistry (II) by M. Lim. singlet. triplet

E For K > 0. s s s s Fall Physical Chemistry (II) by M. Lim. singlet. triplet Eneges of He electonc ψ E Fo K > 0 ψ = snglet ( )( ) s s+ ss αβ E βα snglet = ε + ε + J s + Ks Etplet = ε + ε + J s Ks αα ψ tplet = ( s s ss ) ββ ( αβ + βα ) s s s s s s s s ψ G = ss( αβ βα ) E = ε + ε

More information

N = N t ; t 0. N is the number of claims paid by the

N = N t ; t 0. N is the number of claims paid by the Iulan MICEA, Ph Mhaela COVIG, Ph Canddate epatment of Mathematcs The Buchaest Academy of Economc Studes an CECHIN-CISTA Uncedt Tac Bank, Lugoj SOME APPOXIMATIONS USE IN THE ISK POCESS OF INSUANCE COMPANY

More information

A Brief Guide to Recognizing and Coping With Failures of the Classical Regression Assumptions

A Brief Guide to Recognizing and Coping With Failures of the Classical Regression Assumptions A Bef Gude to Recognzng and Copng Wth Falues of the Classcal Regesson Assumptons Model: Y 1 k X 1 X fxed n epeated samples IID 0, I. Specfcaton Poblems A. Unnecessay explanatoy vaables 1. OLS s no longe

More information