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

Size: px
Start display at page:

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

Transcription

1 VEKTORANAYS Ksecka INE INTEGRA and UX INTEGRA Kaptel 4 5 Sdo 9 5

2 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? B We wll ae to the fnal answe n thee steps: - Path: staght lne Decton: the path has the same decton of the wnd - Path: staght lne Decton: the path has not the same decton of the wnd 3- Path: ced lne Wnd not constant B path A wnd path B B wnd A α A path wnd

3 Step : B path A wnd W Step : α // W cos α // Step 3: z B ( W A ( y W W W ( W ( o ey small segments: ( ( W lm d s the lne ntegal of along the path d We need to: - ntodce a VECTOR IED, ( - Defne the nfntesmal dsplacement d along the path 3

4 VECTOR IED A ecto feld assocates a ecto A(,y,z to each pont (,y,z of the space. Eamples: elocty dstbton n a fld magnetc feld aond a magnet electc feld aond an electc chage Two typcal ways to epesent a ecto feld: Aow feld The aow length s popotonal to the feld ampltde The aow decton shows the feld decton ne feld The tangent to the ces s paallel to the ecto feld n all ponts. The densty of the lnes s popotonal to the stength of the feld. 4

5 VECTOR IED The aplane wng eample (elocty feld of a aond a wng ow elocty (hozontal Hgh elocty (hozontal ow elocty (hozontal Hgh elocty (hozontal ( (, EXERCISE: - plot the ecto feld - wte on the ce defned by: ( ( (, 5

6 z B W d ( AND THE INE INTEGRA z + A y y s descbed by ( + + ( ( + + d lm lm [ ( ( ] + lm [ ( ( ] + So, the lne ntegal can be calclated as: b d ( d ( ( d a d EXERCISE: Calclate ( d wth ( (, and defned by ( (, : d d d 6

7 A INE INTEGRA (some sefl popetes THEOREM (4. n the tetbook B - A B ( d ( d PROO If all lne elements change sgn then also the ntegal wll change sgn. DEINITION The lne of ntegal of A along a closed ce C s called cclaton of a A along C: A ( d C (3 (4 7

8 INE INTEGRA DEINITION: A ecto feld A s called conseate f: A ( C d THEOREM (4.3 n the tetbook The cclaton of A along a close ce s zeo f and only f fo all ponts P and Q the lne ntegal of A fom P to Q s ndependent fom the ntegaton path between P and Q. PROO Assme that and ae two ces fom P to Q. P Then - s a closed ce. ( The cclaton s zeo the lne ntegal fom P to Q s ndependent fom the path A ( d A ( d A ( d A ( d A ( d Q A ( d A( d The lne ntegal s ndependent fom the ntegaton path! ( The lne ntegal fom P to Q s ndependent fom the path the cclaton s zeo. A ( d A( d A ( d A ( d A ( d A ( d A ( d The cclaton s zeo 8

9 INE INTEGRA THEOREM 3 (4.4 n the tetbook If Agadφ : Q A ( d φ ( Q φ ( P P So the lne ntegal s ndependent fom the ntegaton path and depends only on the statng pont and on the endng pont (5 PROO If ( s a ce fom P to Q then, sng the chan le fo the patal deate: q d q φ φ φ d dy dz A ( d gad φ d,,,, d d y z d d d ( p p q φ d φ dy φ dz q d d φ( ( d φ( Q φ( P p + + d yd zd p d O, ease: A ( d gad φ d d φ φ( B φ( A 9

10 OTHER KINDS O INE INTEGRAS It s possble to combne scala and ecto lne elements n many dffeent ways along a ce and ths get dffeent knds of lne ntegals φ Some eamples: ( ds φ ( d whee dêds To calclate the ntegals: A ( d ( [, ] : a b whee a b d d d d o ds d d d (6 (7 (8

11 EXAMPE z The foce s: (-yz,z,- Q The path s : (cos, sn, wth : 4π Calclate the wok fom P(,, tll Q(,,4π y fom defnton ( ( b 4 π (( d W d d a d P ( ( ( (sn,(cos, d d ( sn,cos, d ( ( ( sn + cos d + (sn cos 4 π 4 π ( ( 8 4 W d d π π

12 WHICH STATEMENT IS WRONG? WHICH STATEMENT IS WRONG? - The mage aea of a ecto feld A s composed of ectos (yellow - The lne ntegal d s a scala (ed 3- The sgn of the lne ntegal ddepends on the ntegaton path (geen gad A 4- The gadent of a ecto feld can be wtten as: (ble

13 TARGET PROBEM We ae makng canbey jce. Afte canbees ae sqeezed, It s bette to flte the jce! How mch jce flows togh the cloth each second? We need : ( to ndestand how to calclate the fl of a VECTOR IED (, y, z ( a method to ntegate the fl oe the whole sface. 3

14 THE UX In the jce eample, the fl s the olme of the fld V that flows thogh the sface n the tme t. t V STEP : - the fld elocty s pependcla to the sface - the sface s not ced V V t STEP : - the fld elocty s NOT pependcla to the sface - the sface s not ced S S n ˆ S // S t S S S S t STEP 3: - the sface s ced nˆ S d S S S lm ds s the fl ntegal of on the sface S S S S nˆ 4

15 ds AND UX INTEGRA Assme that the sface S s paametezed by (, z S et s consde two dsplacements, - de a change n : ( +, - de a change n : (, + y α snα S The aea S s S sn α nˆ s pependcla to S. Bt also s pependcla to S S n ˆ S 5

16 dd ds d d d d S ds + +, (, (, (, ( lm, (, ( lm lm lm lm AND UX INTEGRA ds ( S dd ds, ( So, the fl ntegal of the ecto feld on the sface S can be calclated as: n the same way: 6

17 EXAMPE Calclate the fl of the ecto feld A(y,,z thogh the sface S: z +y +y n ê z < SOUTION: - fge - Paametezaton of S 3- l calclaton sng eqaton z +y z +y z y y Ths defnes n whch decton we wll calclate the fl. n s chosen so that z-component s negate. z z +y +y n y 7

18 EXAMPE Paametezaton of S z +y +y ϕ (ρ,ϕ y ρ cos ϕ ρsn ϕ z + y ( ρ sn ϕ + ( ρsn ϕ ρ ρ ϕ π ρ and ϕ ae pola coodnates z π (ρ,ϕ y ρ Paametezaton of the ecto feld: A(y,,z (ρ snϕ cosϕ,, ρ 4 8

19 EXAMPE l calclaton sng eqaton A ds A (( ρ, ϕ d ρd ϕ ρ ϕ S S ρ ϕ ( cos ϕ,sn ϕ, ρ ( ρsn ϕ, ρcos ϕ, eˆ ˆ ˆ ey ez cos ϕ sn ϕ ρ ρ ϕ ρsn ϕ ρcos ϕ ( ρ cos ϕ, ρ sn ϕ, ρcos ϕ+ ρsn ϕ ρ ϕ ρ ϕ ρ ( cos, sn, Note that ρ ϕ has a poste z-component, whle the fl was n the othe decton. So we odnay sole the ntegal bt then we change the sgn n the answe! 9

20 EXAMPE AdS A ( ( ρϕ, d ρdϕ ρ ϕ S S π π ( ( 4 sn cos + 5 sn cos,, ( cos, sn, d d 4 ρ ϕ ϕ ρ ρ ϕ ρ ϕ ρ ρ ϕ ρ ϕ ϕ+ ρ dρd ϕ π π 5 6 ρ sn ϕ cos ϕ+ ρ dϕ sn ϕ cos ϕ+ dϕ cos ϕ π + ϕ π Bt we mst change sgn! The answe s 3 π

21 WHICH STATEMENT IS WRONG? WHICH STATEMENT IS WRONG? - The fl ntegal s a scala (yellow - l ntegals can be calclated also on a closed sface. (ed 3- The pependcla to the ntegaton sface ponts ot fom an abtay sde. (geen 4- The fl thogh a membane can be calclated wth fl ntegals. (ble

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

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

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?

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? 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

VEKTORANALYS GAUSS THEOREM STOKES THEOREM. and. Kursvecka 3. Kapitel 6 7 Sidor 51 82

VEKTORANALYS GAUSS THEOREM STOKES THEOREM. and. Kursvecka 3. Kapitel 6 7 Sidor 51 82 VEKTORANAY Kursvecka 3 GAU THEOREM and TOKE THEOREM Kaptel 6 7 dor 51 82 TARGET PROBEM Do magnetc monopoles est? EECTRIC FIED MAGNETIC FIED N +? 1 TARGET PROBEM et s consder some EECTRIC CHARGE 2 - + +

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

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

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

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

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

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

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

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

Lesson 8: Work, Energy, Power (Sections ) Chapter 6 Conservation of Energy

Lesson 8: Work, Energy, Power (Sections ) Chapter 6 Conservation of Energy Lesson 8: Wok, negy, Powe (Sectons 6.-6.8) Chapte 6 Conseaton o negy Today we begn wth a ey useul concept negy. We wll encounte many amla tems that now hae ey specc dentons n physcs. Conseaton o enegy

More information

The Forming Theory and the NC Machining for The Rotary Burs with the Spectral Edge Distribution

The Forming Theory and the NC Machining for The Rotary Burs with the Spectral Edge Distribution oden Appled Scence The Fomn Theoy and the NC achnn fo The Rotay us wth the Spectal Ede Dstbuton Huan Lu Depatment of echancal Enneen, Zhejan Unvesty of Scence and Technoloy Hanzhou, c.y. chan, 310023,

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

3.3 Properties of Vortex Structures

3.3 Properties of Vortex Structures .0 - Maine Hydodynamics, Sping 005 Lecte 8.0 - Maine Hydodynamics Lecte 8 In Lecte 8, paagaph 3.3 we discss some popeties of otex stctes. In paagaph 3.4 we dedce the Benolli eqation fo ideal, steady flow.

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

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

VEKTORANALYS. GAUSS s THEOREM and STOKES s THEOREM. Kursvecka 3. Kapitel 6-7 Sidor 51-82

VEKTORANALYS. GAUSS s THEOREM and STOKES s THEOREM. Kursvecka 3. Kapitel 6-7 Sidor 51-82 VEKTORANAY Kursvecka 3 GAU s THEOREM and TOKE s THEOREM Kaptel 6-7 dor 51-82 TARGET PROBEM EECTRIC FIED MAGNETIC FIED N + Magnetc monopoles do not est n nature. How can we epress ths nformaton for E and

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

Unit_III Complex Numbers: Some Basic Results: 1. If z = x +iy is a complex number, then the complex number z = x iy is

Unit_III Complex Numbers: Some Basic Results: 1. If z = x +iy is a complex number, then the complex number z = x iy is Unt_III Comple Nmbes: In the sstem o eal nmbes R we can sole all qadatc eqatons o the om a b c, a, and the dscmnant b 4ac. When the dscmnant b 4ac

More information

PHYS 2421 Fields and Waves

PHYS 2421 Fields and Waves PHYS 242 Felds nd Wves Instucto: Joge A. López Offce: PSCI 29 A, Phone: 747-7528 Textook: Unvesty Physcs e, Young nd Feedmn 23. Electc potentl enegy 23.2 Electc potentl 23.3 Clcultng electc potentl 23.4

More information

A Tutorial on Multiple Integrals (for Natural Sciences / Computer Sciences Tripos Part IA Maths)

A Tutorial on Multiple Integrals (for Natural Sciences / Computer Sciences Tripos Part IA Maths) A Tutoial on Multiple Integals (fo Natual Sciences / Compute Sciences Tipos Pat IA Maths) Coections to D Ian Rud (http://people.ds.cam.ac.uk/ia/contact.html) please. This tutoial gives some bief eamples

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

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

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

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

Synopsis : 8. ELECTROMAGNETISM

Synopsis : 8. ELECTROMAGNETISM Synopss : 8. ELECTROMAGNETISM MAGNETIC EFFECTS OF CURRENT: 1. Electomagnetsm s the banch of physcs whch deals wth elaton between electcty and magnetsm.. A statc chage poduces only electc feld but movng

More information

Instantaneous velocity field of a round jet

Instantaneous velocity field of a round jet Fee shea flows Instantaneos velocty feld of a ond et 3 Aveage velocty feld of a ond et 4 Vtal ogn nozzle coe Developng egon elf smla egon 5 elf smlaty caled vaables: ~ Q ξ ( ξ, ) y δ ( ) Q Q (, y) ( )

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

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

VECTOR MECHANICS FOR ENGINEERS: Vector Mechanics for Engineers: Dynamics. In the current chapter, you will study the motion of systems of particles.

VECTOR MECHANICS FOR ENGINEERS: Vector Mechanics for Engineers: Dynamics. In the current chapter, you will study the motion of systems of particles. Seeth Edto CHPTER 4 VECTOR MECHNICS FOR ENINEERS: DYNMICS Fedad P. ee E. Russell Johsto, J. Systems of Patcles Lectue Notes: J. Walt Ole Texas Tech Uesty 003 The Mcaw-Hll Compaes, Ic. ll ghts eseed. Seeth

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

Lab #0. Tutorial Exercises on Work and Fields

Lab #0. Tutorial Exercises on Work and Fields Lab #0 Tutoial Execises on Wok and Fields This is not a typical lab, and no pe-lab o lab epot is equied. The following execises will emind you about the concept of wok (fom 1130 o anothe intoductoy mechanics

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

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

Objectives. Chapter 6. Learning Outcome. Newton's Laws in Action. Reflection: Reflection: 6.2 Gravitational Field

Objectives. Chapter 6. Learning Outcome. Newton's Laws in Action. Reflection: Reflection: 6.2 Gravitational Field Chapte 6 Gataton Objectes 6. Newton's Law o nesal Gataton 6. Gatatonal Feld 6. Gatatonal Potental 6. Satellte oton n Ccula Obts 6.5 scape Velocty Leanng Outcoe (a and use the oula / (b explan the eanng

More information

Physics Tutorial V1 2D Vectors

Physics Tutorial V1 2D Vectors Physics Tutoial V1 2D Vectos 1 Resolving Vectos & Addition of Vectos A vecto quantity has both magnitude and diection. Thee ae two ways commonly used to mathematically descibe a vecto. y (a) The pola fom:,

More information

2/24/2014. The point mass. Impulse for a single collision The impulse of a force is a vector. The Center of Mass. System of particles

2/24/2014. The point mass. Impulse for a single collision The impulse of a force is a vector. The Center of Mass. System of particles /4/04 Chapte 7 Lnea oentu Lnea oentu of a Sngle Patcle Lnea oentu: p υ It s a easue of the patcle s oton It s a vecto, sla to the veloct p υ p υ p υ z z p It also depends on the ass of the object, sla

More information

Electric potential energy Electrostatic force does work on a particle : Potential energy (: i initial state f : final state):

Electric potential energy Electrostatic force does work on a particle : Potential energy (: i initial state f : final state): Electc ptental enegy Electstatc fce des wk n a patcle : v v v v W = F s = E s. Ptental enegy (: ntal state f : fnal state): Δ U = U U = W. f ΔU Electc ptental : Δ : ptental enegy pe unt chag e. J ( Jule)

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

Physics Exam 3

Physics Exam 3 Physcs 114 1 Exam 3 The numbe of ponts fo each secton s noted n backets, []. Choose a total of 35 ponts that wll be gaded that s you may dop (not answe) a total of 5 ponts. Clealy mak on the cove of you

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

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

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

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

CHAPTER 3 SYSTEMS OF PARTICLES

CHAPTER 3 SYSTEMS OF PARTICLES HAPTER 3 SYSTEMS O PARTILES 3. Intoducton By systes of patcles I ean such thngs as a swa of bees, a sta cluste, a cloud of gas, an ato, a bck. A bck s ndeed coposed of a syste of patcles atos whch ae constaned

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

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

An Analytical Approach to Turbulent Flow Through Smooth Pipes

An Analytical Approach to Turbulent Flow Through Smooth Pipes An Analytcal Appoach to Tblent Flow Thogh Smooth Ppes AMIN MOOSAIE and GHOLAMALI ATEFI Depatment of Mechancal Engneeng Ian nesty of Scence and Technology P.O. Box 6844, Namak, Tehan IAN amnmoosae@mal.st.ac.

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

CHAPTER 4 EVALUATION OF FORCE-CONSTANT MATRIX

CHAPTER 4 EVALUATION OF FORCE-CONSTANT MATRIX CHAPTER 4 EVALUATION OF FORCE-CONSTANT MATRIX 4.- AIM OF THE WORK As antcpated n the ntodcton the am of the pesent ok s to obtan the nmecal vale of the foce-constant matx fo tantalm. In a fst step expesson

More information

Electrostatic Potential

Electrostatic Potential Chapte 23 Electostatic Potential PowePoint Lectues fo Univesity Physics, Twelfth Edition Hugh D. Young and Roge A. Feedman Lectues by James Pazun Copyight 2008 Peason Education Inc., publishing as Peason

More information

FI 2201 Electromagnetism

FI 2201 Electromagnetism F Eectomagnetism exane. skana, Ph.D. Physics of Magnetism an Photonics Reseach Goup Magnetostatics MGNET VETOR POTENTL, MULTPOLE EXPNSON Vecto Potentia Just as E pemitte us to intouce a scaa potentia V

More information

2 dependence in the electrostatic force means that it is also

2 dependence in the electrostatic force means that it is also lectc Potental negy an lectc Potental A scala el, nvolvng magntues only, s oten ease to wo wth when compae to a vecto el. Fo electc els not havng to begn wth vecto ssues woul be nce. To aange ths a scala

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

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

Chapter 12 Equilibrium and Elasticity

Chapter 12 Equilibrium and Elasticity Chapte 12 Equlbum and Elastcty In ths chapte we wll defne equlbum and fnd the condtons needed so that an object s at equlbum. We wll then apply these condtons to a vaety of pactcal engneeng poblems of

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

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

( ) 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

Review for Midterm-1

Review for Midterm-1 Review fo Midtem-1 Midtem-1! Wednesday Sept. 24th at 6pm Section 1 (the 4:10pm class) exam in BCC N130 (Business College) Section 2 (the 6:00pm class) exam in NR 158 (Natual Resouces) Allowed one sheet

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

Solving the Dirac Equation: Using Fourier Transform

Solving the Dirac Equation: Using Fourier Transform McNa Schola Reeach Jounal Volume Atcle Solvng the ac quaton: Ung oue Tanfom Vncent P. Bell mby-rddle Aeonautcal Unvety, Vncent.Bell@my.eau.edu ollow th and addtonal wok at: http://common.eau.edu/na Recommended

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

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

Vectors Serway and Jewett Chapter 3

Vectors Serway and Jewett Chapter 3 Vectos Sewa and Jewett Chapte 3 Scalas and Vectos Vecto Components and Aithmetic Vectos in 3 Dimensions Unit vectos i, j, k Pactice Poblems: Chapte 3, poblems 9, 19, 31, 45, 55, 61 Phsical quantities ae

More information

Math 209 Assignment 9 Solutions

Math 209 Assignment 9 Solutions Math 9 Assignment 9 olutions 1. Evaluate 4y + 1 d whee is the fist octant pat of y x cut out by x + y + z 1. olution We need a paametic epesentation of the suface. (x, z). Now detemine the nomal vecto:

More information

Sources of Magnetic Fields (chap 28)

Sources of Magnetic Fields (chap 28) Souces of Magnetic Fields (chap 8) In chapte 7, we consideed the magnetic field effects on a moving chage, a line cuent and a cuent loop. Now in Chap 8, we conside the magnetic fields that ae ceated by

More information

Uniform Circular Motion

Uniform Circular Motion Unfom Ccul Moton Unfom ccul Moton An object mong t constnt sped n ccle The ntude of the eloct emns constnt The decton of the eloct chnges contnuousl!!!! Snce cceleton s te of chnge of eloct:!! Δ Δt The

More information

Large scale magnetic field generation by accelerated particles in galactic medium

Large scale magnetic field generation by accelerated particles in galactic medium Lage scale magnetc feld geneaton by acceleated patcles n galactc medum I.N.Toptygn Sant Petesbug State Polytechncal Unvesty, depatment of Theoetcal Physcs, Sant Petesbug, Russa 2.Reason explonatons The

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

10/15/2013. PHY 113 C General Physics I 11 AM-12:15 PM MWF Olin 101

10/15/2013. PHY 113 C General Physics I 11 AM-12:15 PM MWF Olin 101 10/15/01 PHY 11 C Geneal Physcs I 11 AM-1:15 PM MWF Oln 101 Plan fo Lectue 14: Chapte 1 Statc equlbu 1. Balancng foces and toques; stablty. Cente of gavty. Wll dscuss elastcty n Lectue 15 (Chapte 15) 10/14/01

More information

Module 18: Outline. Magnetic Dipoles Magnetic Torques

Module 18: Outline. Magnetic Dipoles Magnetic Torques Module 18: Magnetic Dipoles 1 Module 18: Outline Magnetic Dipoles Magnetic Toques 2 IA nˆ I A Magnetic Dipole Moment μ 3 Toque on a Cuent Loop in a Unifom Magnetic Field 4 Poblem: Cuent Loop Place ectangula

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

The Law of Biot-Savart & RHR P θ

The Law of Biot-Savart & RHR P θ The Law of iot-savat & RHR P R dx x Jean-aptiste iot élix Savat Phys 122 Lectue 19 G. Rybka Recall: Potential Enegy of Dipole Wok equied to otate a cuentcaying loop in a magnetic field Potential enegy

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

Thermodynamics of solids 4. Statistical thermodynamics and the 3 rd law. Kwangheon Park Kyung Hee University Department of Nuclear Engineering

Thermodynamics of solids 4. Statistical thermodynamics and the 3 rd law. Kwangheon Park Kyung Hee University Department of Nuclear Engineering Themodynamcs of solds 4. Statstcal themodynamcs and the 3 d law Kwangheon Pak Kyung Hee Unvesty Depatment of Nuclea Engneeng 4.1. Intoducton to statstcal themodynamcs Classcal themodynamcs Statstcal themodynamcs

More information

MAGNETIC FIELD INTRODUCTION

MAGNETIC FIELD INTRODUCTION MAGNETIC FIELD INTRODUCTION It was found when a magnet suspended fom its cente, it tends to line itself up in a noth-south diection (the compass needle). The noth end is called the Noth Pole (N-pole),

More information

4. Compare the electric force holding the electron in orbit ( r = 0.53

4. Compare the electric force holding the electron in orbit ( r = 0.53 Electostatics WS Electic Foce an Fiel. Calculate the magnitue of the foce between two 3.60-µ C point chages 9.3 cm apat.. How many electons make up a chage of 30.0 µ C? 3. Two chage ust paticles exet a

More information

CS348B Lecture 10 Pat Hanrahan, Spring 2004

CS348B Lecture 10 Pat Hanrahan, Spring 2004 Page 1 Reflecton Models I Today Types of eflecton models The BRDF and eflectance The eflecton equaton Ideal eflecton and efacton Fesnel effect Ideal dffuse Next lectue Glossy and specula eflecton models

More information

ECE 3318 Applied Electricity and Magnetism. Spring Prof. David R. Jackson ECE Dept. Notes 13

ECE 3318 Applied Electricity and Magnetism. Spring Prof. David R. Jackson ECE Dept. Notes 13 ECE 338 Applied Electicity and Magnetism ping 07 Pof. David R. Jackson ECE Dept. Notes 3 Divegence The Physical Concept Find the flux going outwad though a sphee of adius. x ρ v0 z a y ψ = D nˆ d = D ˆ

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