# Chapter 3: Vectors and Two-Dimensional Motion

Size: px
Start display at page:

Transcription

1 Chape 3: Vecos and Two-Dmensonal Moon Vecos: magnude and decon Negae o a eco: eese s decon Mulplng o ddng a eco b a scala Vecos n he same decon (eaed lke numbes) Geneal Veco Addon: Tangle mehod o addon Componens addon Noe ha a eco has a magnude and a decon, bu doesn hae a xed locaon n space. I can be edawn anwhee n space and sll emans he same eco. Veco Addon angula mehod eco subacon

2 Componens o a Veco A wo dmensonal eco can be expessed as he sum o s componens along wo pependcula axes. A A x + A A x A cos θ A A sn θ A ( A x + A ) 1/ Veco Addon b Componens When wo o moe ecos ae boken up no componens n he same coodnae ssem (pependcula axes), he esulan eco om he addon o hese ecos has componens whch ae he smple sums o he nddual eco componens. A A x + A A + B Ax A + A + Bx + B + B ) + ( A + B ) ( x x B B x + B x-componen -componen A+B [ (A x + B x ) + ( A + B ) ] 1/ θ θ an A + B 1 Ax + B x

3 Example: Veco Addon A hke begns a p b s walkng 5. km 45. souh o eas om he base camp. On he second da she walks 4. km n a decon 6. noh o eas, a whch pon she dscoes a oes ange s owe. (a) Deemne he componens o he hke s oal dsplacemen o he p. (b) Fnd he magnude and decon o he dsplacemen om he base camp. dsplacemen aeage eloc Dsplacemen, Vel., and Accel. n D Δ Δ nsananeous eloc Δ lm aeage acceleaon a nsananeous acceleaon Δ Δ a lm

4 Poson, Veloc, and Acceleaon: 1D o D o 3D Fo 1-D moon, we used o know ha.. he slope a a specc me o a plo o he poson as a uncon o me s he nsananeous eloc a ha specc me; he slope a a specc me o a plo o he eloc as a uncon o me s he nsananeous acceleaon a ha specc me, ec. Ae hese sll coec n D? Δ x + ( x x + + ) ( x x ) + ( x ) + Δ + x Δ Poson, Veloc, and Acceleaon: 1D o D o 3D Fo 1-D moon, we used o know ha.. he slope a a specc me o a plo o he poson as a uncon o me s he nsananeous eloc a ha specc me; he slope a a specc me o a plo o he eloc as a uncon o me s he nsananeous acceleaon a ha specc me, ec. Ae hese sll coec n D? The ae sll ald o moons n D o 3D, as a as he componens (o poson, eloc, acceleaon) along a specc decon ae concened. Fo example: The slope o he -componen o he eloc o an objec, ploed agans me, s he -componen o he nsananeous acceleaon. The slope o x-componen o he dsplacemen o an objec, ploed agans me, s he x-componen o he nsananeous eloc, ec.

5 Two-Dmensonal Moon The x-componen o he anslaonal (non-oaonal) moon o an objec s ndependen o s moon n he -decon. Equaons deed o moon n one dmenson can be used o descbe he x- o -componen o he moon n wo dmensons. x x + ax 1 ( x + x ) 1 x + ax x Moe x + aoen x han no: a x x cons. x x + a Δ 1 + ) ( Δ a a Δ Pojecle Moon In he absence o a essance, he hozonal o x componen o he acceleaon s zeo, and he ecal o componen o he acceleaon s he acceleaon due o ga. These wo moons ae ndependen o each ohe. x 1 Δ + ( g) g9.8 m/s

6 How a (hgh, much me) s he kcko wh hs nal eloc? ox / Δ ( 4.9m / s ) How much me s n he a? / g o snθ / g How hgh n a does go? H ( / ) o sn g θ How a does go? R Wha s he ahes a ball wll go when hown wh he same o? x o snθ cos / g θ Range Relae Veloc The eloc o A elae o B, V AB s he eloc o A as ewed om he anage pon o B. (B s unconcened abou possble moon o sel.) To nd V AB : (1) Fnd a coodnae (C) common o boh A and B. The eloces o A and B on hs common coodnae ae V AC and V BC, especel. Then, V AB V AC V BC () In cases whee C s known o moe wh espec o A whle B s known o moe espec o C, an addon o ecos ma be moe anspaen: V AB V AC + V CB Mnemonc dece: Thnk o A, B, C as hee pons on a pece o pape and V AB as he eco om A o B, and ce esa.

7 Chape 3 Example Poblems Ca A s mong n he noheas decon wh a speed o 4. m/s. Ca B s mong due eas wh a speed o 5. m/s. Wha s he eloc o ca A as measued b an obsee on ca B? 55. A home un s h n such a wa ha he baseball jus cleas a wall 1 m hgh, locaed 13 m om home plae. The ball s h a an angle o 35 o he hozonal, and a essance s neglgble. Fnd (a) he nal speed o he ball, (b) he me akes he ball o each he wall, and (c) he eloc componens and he speed o he ball when eaches he wall. (Assume ha he ball s h a a hegh o 1. m aboe he gound.) Reew o Chape 3 Veco componens and addon. Two dmensonal knemacs can be analzed b 1D mehods usng he x and componens o s dsplacemen, eloc, and acceleaon. The x and analses ae done ndependen o each ohe. A an specc me, he x- and -componens can be pu ogehe o eld he eloc (o dsplacemen, o acceleaon) eco. Pojecle moon noles consan acceleaon n decon and no acceleaon n x decon. Smme n pojecle moon: same speed a same hegh, he - componen o he eloc s eesed on wa down om on wa up. Relae eloc.

8 Equaon Shee

### Chapters 2 Kinematics. Position, Distance, Displacement

Chapers Knemacs Poson, Dsance, Dsplacemen Mechancs: Knemacs and Dynamcs. Knemacs deals wh moon, bu s no concerned wh he cause o moon. Dynamcs deals wh he relaonshp beween orce and moon. The word dsplacemen

### PHYS 1443 Section 001 Lecture #4

PHYS 1443 Secon 001 Lecure #4 Monda, June 5, 006 Moon n Two Dmensons Moon under consan acceleraon Projecle Moon Mamum ranges and heghs Reerence Frames and relae moon Newon s Laws o Moon Force Newon s Law

### 5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( )

5-1. We apply Newon s second law (specfcally, Eq. 5-). (a) We fnd he componen of he foce s ( ) ( ) F = ma = ma cos 0.0 = 1.00kg.00m/s cos 0.0 = 1.88N. (b) The y componen of he foce s ( ) ( ) F = ma = ma

### L4:4. motion from the accelerometer. to recover the simple flutter. Later, we will work out how. readings L4:3

elave moon L4:1 To appl Newon's laws we need measuemens made fom a 'fed,' neal efeence fame (unacceleaed, non-oang) n man applcaons, measuemens ae made moe smpl fom movng efeence fames We hen need a wa

### s = rθ Chapter 10: Rotation 10.1: What is physics?

Chape : oaon Angula poson, velocy, acceleaon Consan angula acceleaon Angula and lnea quanes oaonal knec enegy oaonal nea Toque Newon s nd law o oaon Wok and oaonal knec enegy.: Wha s physcs? In pevous

### Motion in Two Dimensions

Phys 1 Chaper 4 Moon n Two Dmensons adzyubenko@csub.edu hp://www.csub.edu/~adzyubenko 005, 014 A. Dzyubenko 004 Brooks/Cole 1 Dsplacemen as a Vecor The poson of an objec s descrbed by s poson ecor, r The

### Go over vector and vector algebra Displacement and position in 2-D Average and instantaneous velocity in 2-D Average and instantaneous acceleration

Mh Csquee Go oe eco nd eco lgeb Dsplcemen nd poson n -D Aege nd nsnneous eloc n -D Aege nd nsnneous cceleon n -D Poecle moon Unfom ccle moon Rele eloc* The componens e he legs of he gh ngle whose hpoenuse

### Displacement, Velocity, and Acceleration. (WHERE and WHEN?)

Dsplacemen, Velocy, and Acceleraon (WHERE and WHEN?) Mah resources Append A n your book! Symbols and meanng Algebra Geomery (olumes, ec.) Trgonomery Append A Logarhms Remnder You wll do well n hs class

### Field due to a collection of N discrete point charges: r is in the direction from

Physcs 46 Fomula Shee Exam Coulomb s Law qq Felec = k ˆ (Fo example, f F s he elecc foce ha q exes on q, hen ˆ s a un veco n he decon fom q o q.) Elecc Feld elaed o he elecc foce by: Felec = qe (elecc

### 2-d Motion: Constant Acceleration

-d Moion: Consan Acceleaion Kinemaic Equaions o Moion (eco Fom Acceleaion eco (consan eloci eco (uncion o Posiion eco (uncion o The eloci eco and posiion eco ae a uncion o he ime. eloci eco a ime. Posiion

### Lecture 5. Plane Wave Reflection and Transmission

Lecue 5 Plane Wave Reflecon and Tansmsson Incden wave: 1z E ( z) xˆ E (0) e 1 H ( z) yˆ E (0) e 1 Nomal Incdence (Revew) z 1 (,, ) E H S y (,, ) 1 1 1 Refleced wave: 1z E ( z) xˆ E E (0) e S H 1 1z H (

### UNIT 1 ONE-DIMENSIONAL MOTION GRAPHING AND MATHEMATICAL MODELING. Objectives

UNIT 1 ONE-DIMENSIONAL MOTION GRAPHING AND MATHEMATICAL MODELING Objeces To learn abou hree ways ha a physcs can descrbe moon along a sragh lne words, graphs, and mahemacal modelng. To acqure an nue undersandng

### WebAssign HW Due 11:59PM Tuesday Clicker Information

WebAssgn HW Due 11:59PM Tuesday Clcker Inormaon Remnder: 90% aemp, 10% correc answer Clcker answers wll be a end o class sldes (onlne). Some days we wll do a lo o quesons, and ew ohers Each day o clcker

### 2 shear strain / L for small angle

Sac quaons F F M al Sess omal sess foce coss-seconal aea eage Shea Sess shea sess shea foce coss-seconal aea llowable Sess Faco of Safe F. S San falue Shea San falue san change n lengh ognal lengh Hooke

### CHAPTER 10: LINEAR DISCRIMINATION

HAPER : LINEAR DISRIMINAION Dscmnan-based lassfcaon 3 In classfcaon h K classes ( k ) We defned dsmnan funcon g () = K hen gven an es eample e chose (pedced) s class label as f g () as he mamum among g

### Course Outline. 1. MATLAB tutorial 2. Motion of systems that can be idealized as particles

Couse Oulne. MATLAB uoal. Moon of syses ha can be dealzed as pacles Descpon of oon, coodnae syses; Newon s laws; Calculang foces equed o nduce pescbed oon; Deng and solng equaons of oon 3. Conseaon laws

### Today - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations

Today - Lecue 13 Today s lecue coninue wih oaions, oque, Noe ha chapes 11, 1, 13 all inole oaions slide 1 eiew Roaions Chapes 11 & 1 Viewed fom aboe (+z) Roaional, o angula elociy, gies angenial elociy

### CHAPTER 2 Quick Quizzes

CHAPTER Quck Quzzes (a) 00 yd (b) 0 (c) 0 (a) False The car may be slowng down, so ha he drecon o s acceleraon s oppose he drecon o s elocy (b) True I he elocy s n he drecon chosen as negae, a pose acceleraon

### Mechanics Physics 151

Mechancs Physcs 5 Lecure 0 Canoncal Transformaons (Chaper 9) Wha We Dd Las Tme Hamlon s Prncple n he Hamlonan formalsm Dervaon was smple δi δ Addonal end-pon consrans pq H( q, p, ) d 0 δ q ( ) δq ( ) δ

### Chapter 6 Plane Motion of Rigid Bodies

Chpe 6 Pne oon of Rd ode 6. Equon of oon fo Rd bod. 6., 6., 6.3 Conde d bod ced upon b ee een foce,, 3,. We cn ume h he bod mde of e numbe n of pce of m Δm (,,, n). Conden f he moon of he m cene of he

### ÖRNEK 1: THE LINEAR IMPULSE-MOMENTUM RELATION Calculate the linear momentum of a particle of mass m=10 kg which has a. kg m s

MÜHENDİSLİK MEKANİĞİ. HAFTA İMPULS- MMENTUM-ÇARPIŞMA Linea oenu of a paicle: The sybol L denoes he linea oenu and is defined as he ass ies he elociy of a paicle. L ÖRNEK : THE LINEAR IMPULSE-MMENTUM RELATIN

### Physics 120 Spring 2007 Exam #1 April 20, Name

Phc 0 Spng 007 E # pl 0, 007 Ne P Mulple Choce / 0 Poble # / 0 Poble # / 0 Poble # / 0 ol / 00 In eepng wh he Unon College polc on cdec hone, ued h ou wll nehe ccep no pode unuhozed nce n he copleon o

### Chapter 4: Motion in Two Dimensions Part-1

Lecue 4: Moon n Two Dmensons Chpe 4: Moon n Two Dmensons P- In hs lesson we wll dscuss moon n wo dmensons. In wo dmensons, s necess o use eco noon o descbe phscl qunes wh boh mnude nd decon. In hs chpe,

### Physics 201 Lecture 15

Phscs 0 Lecue 5 l Goals Lecue 5 v Elo consevaon of oenu n D & D v Inouce oenu an Iulse Coens on oenu Consevaon l oe geneal han consevaon of echancal eneg l oenu Consevaon occus n sses wh no ne eenal foces

### PHY2053 Summer C 2013 Exam 1 Solutions

PHY053 Sue C 03 E Soluon. The foce G on o G G The onl cobnon h e '/ = doubln.. The peed of lh le 8fulon c 86,8 le 60 n 60n h 4h d 4d fonh.80 fulon/ fonh 3. The dnce eled fo he ene p,, 36 (75n h 45 The

### Mechanics Physics 151

Mechancs Physcs 5 Lecure 9 Hamlonan Equaons of Moon (Chaper 8) Wha We Dd Las Tme Consruced Hamlonan formalsm H ( q, p, ) = q p L( q, q, ) H p = q H q = p H = L Equvalen o Lagrangan formalsm Smpler, bu

### Rotations.

oons j.lbb@phscs.o.c.uk To s summ Fmes of efeence Invnce une nsfomons oon of wve funcon: -funcons Eule s ngles Emple: e e - - Angul momenum s oon geneo Genec nslons n Noehe s heoem Fmes of efeence Conse

### Mechanics Physics 151

Mechancs Physcs 5 Lecure 9 Hamlonan Equaons of Moon (Chaper 8) Wha We Dd Las Tme Consruced Hamlonan formalsm Hqp (,,) = qp Lqq (,,) H p = q H q = p H L = Equvalen o Lagrangan formalsm Smpler, bu wce as

### Solution in semi infinite diffusion couples (error function analysis)

Soluon n sem nfne dffuson couples (error funcon analyss) Le us consder now he sem nfne dffuson couple of wo blocks wh concenraon of and I means ha, n a A- bnary sysem, s bondng beween wo blocks made of

### . The geometric multiplicity is dim[ker( λi. number of linearly independent eigenvectors associated with this eigenvalue.

Lnear Algebra Lecure # Noes We connue wh he dscusson of egenvalues, egenvecors, and dagonalzably of marces We wan o know, n parcular wha condons wll assure ha a marx can be dagonalzed and wha he obsrucons

### Notes on the stability of dynamic systems and the use of Eigen Values.

Noes on he sabl of dnamc ssems and he use of Egen Values. Source: Macro II course noes, Dr. Davd Bessler s Tme Seres course noes, zarads (999) Ineremporal Macroeconomcs chaper 4 & Techncal ppend, and Hamlon

### Unit 1 Test Review Physics Basics, Movement, and Vectors Chapters 1-3

A.P. Physics B Uni 1 Tes Reiew Physics Basics, Moemen, and Vecors Chapers 1-3 * In sudying for your es, make sure o sudy his reiew shee along wih your quizzes and homework assignmens. Muliple Choice Reiew:

### Relative and Circular Motion

Relaie and Cicula Moion a) Relaie moion b) Cenipeal acceleaion Mechanics Lecue 3 Slide 1 Mechanics Lecue 3 Slide 2 Time on Video Pelecue Looks like mosly eeyone hee has iewed enie pelecue GOOD! Thank you

### Chapter Lagrangian Interpolation

Chaper 5.4 agrangan Inerpolaon Afer readng hs chaper you should be able o:. dere agrangan mehod of nerpolaon. sole problems usng agrangan mehod of nerpolaon and. use agrangan nerpolans o fnd deraes and

### ( ) () we define the interaction representation by the unitary transformation () = ()

Hgher Order Perurbaon Theory Mchael Fowler 3/7/6 The neracon Represenaon Recall ha n he frs par of hs course sequence, we dscussed he chrödnger and Hesenberg represenaons of quanum mechancs here n he chrödnger

### Physics 15 Second Hour Exam

hc 5 Second Hou e nwe e Mulle hoce / ole / ole /6 ole / ------------------------------- ol / I ee eone ole lee how ll wo n ode o ecee l ced. I ou oluon e llegle no ced wll e gen.. onde he collon o wo 7.

### Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation

Lecue 8: Kineics of Phase Gowh in a Two-componen Sysem: geneal kineics analysis based on he dilue-soluion appoximaion Today s opics: In he las Lecues, we leaned hee diffeen ways o descibe he diffusion

### ESS 265 Spring Quarter 2005 Kinetic Simulations

SS 65 Spng Quae 5 Knec Sulaon Lecue une 9 5 An aple of an lecoagnec Pacle Code A an eaple of a knec ulaon we wll ue a one denonal elecoagnec ulaon code called KMPO deeloped b Yohhau Oua and Hoh Mauoo.

### Sections 3.1 and 3.4 Exponential Functions (Growth and Decay)

Secions 3.1 and 3.4 Eponenial Funcions (Gowh and Decay) Chape 3. Secions 1 and 4 Page 1 of 5 Wha Would You Rahe Have... \$1million, o double you money evey day fo 31 days saing wih 1cen? Day Cens Day Cens

### THE PHYSICS BEHIND THE SODACONSTRUCTOR. by Jeckyll

THE PHYSICS BEHIND THE SODACONSTRUCTOR b Jeckll THE PHYSICS BEHIND THE SODACONSTRUCTOR - b Jeckll /3 CONTENTS. INTRODUCTION 5. UNITS OF MEASUREMENT 7 3. DETERMINATION OF THE PHYSICAL CONSTANTS ADOPTED

### 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)

### 1 Constant Real Rate C 1

Consan Real Rae. Real Rae of Inees Suppose you ae equally happy wh uns of he consumpon good oday o 5 uns of he consumpon good n peod s me. C 5 Tha means you ll be pepaed o gve up uns oday n eun fo 5 uns

### Including the ordinary differential of distance with time as velocity makes a system of ordinary differential equations.

Soluons o Ordnary Derenal Equaons An ordnary derenal equaon has only one ndependen varable. A sysem o ordnary derenal equaons consss o several derenal equaons each wh he same ndependen varable. An eample

### Chapter Finite Difference Method for Ordinary Differential Equations

Chape 8.7 Fne Dffeence Mehod fo Odnay Dffeenal Eqaons Afe eadng hs chape, yo shold be able o. Undesand wha he fne dffeence mehod s and how o se o solve poblems. Wha s he fne dffeence mehod? The fne dffeence

### In the complete model, these slopes are ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL. (! i+1 -! i ) + [(!") i+1,q - [(!

ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL The frs hng o es n wo-way ANOVA: Is here neracon? "No neracon" means: The man effecs model would f. Ths n urn means: In he neracon plo (wh A on he horzonal

### Suppose we have observed values t 1, t 2, t n of a random variable T.

Sppose we have obseved vales, 2, of a adom vaable T. The dsbo of T s ow o belog o a cea ype (e.g., expoeal, omal, ec.) b he veco θ ( θ, θ2, θp ) of ow paamees assocaed wh s ow (whee p s he mbe of ow paamees).

### Ordinary Differential Equations in Neuroscience with Matlab examples. Aim 1- Gain understanding of how to set up and solve ODE s

Ordnary Dfferenal Equaons n Neuroscence wh Malab eamples. Am - Gan undersandng of how o se up and solve ODE s Am Undersand how o se up an solve a smple eample of he Hebb rule n D Our goal a end of class

### . The geometric multiplicity is dim[ker( λi. A )], i.e. the number of linearly independent eigenvectors associated with this eigenvalue.

Mah E-b Lecure #0 Noes We connue wh he dscusson of egenvalues, egenvecors, and dagonalzably of marces We wan o know, n parcular wha condons wll assure ha a marx can be dagonalzed and wha he obsrucons are

### 10. A.C CIRCUITS. Theoretically current grows to maximum value after infinite time. But practically it grows to maximum after 5τ. Decay of current :

. A. IUITS Synopss : GOWTH OF UNT IN IUIT : d. When swch S s closed a =; = d. A me, curren = e 3. The consan / has dmensons of me and s called he nducve me consan ( τ ) of he crcu. 4. = τ; =.63, n one

### EGN 3321 Final Exam Review Spring 2017

EN 33 l Em Reew Spg 7 *T fshg ech poblem 5 mues o less o pcce es-lke me coss. The opcs o he pcce em e wh feel he bee sessed clss, bu hee m be poblems o he es o lke oes hs pcce es. Use ohe esouces lke he

### Science Advertisement Intergovernmental Panel on Climate Change: The Physical Science Basis 2/3/2007 Physics 253

Science Adeisemen Inegoenmenl Pnel on Clime Chnge: The Phsicl Science Bsis hp://www.ipcc.ch/spmfeb7.pdf /3/7 Phsics 53 hp://www.fonews.com/pojecs/pdf/spmfeb7.pdf /3/7 Phsics 53 3 Sus: Uni, Chpe 3 Vecos

### Example: MOSFET Amplifier Distortion

4/25/2011 Example MSFET Amplfer Dsoron 1/9 Example: MSFET Amplfer Dsoron Recall hs crcu from a prevous handou: ( ) = I ( ) D D d 15.0 V RD = 5K v ( ) = V v ( ) D o v( ) - K = 2 0.25 ma/v V = 2.0 V 40V.

### s in boxe wers ans Put

Pu answers in boxes Main Ideas in Class Toda Inroducion o Falling Appl Old Equaions Graphing Free Fall Sole Free Fall Problems Pracice:.45,.47,.53,.59,.61,.63,.69, Muliple Choice.1 Freel Falling Objecs

### CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 4

CS434a/54a: Paern Recognon Prof. Olga Veksler Lecure 4 Oulne Normal Random Varable Properes Dscrmnan funcons Why Normal Random Varables? Analycally racable Works well when observaon comes form a corruped

### 7 Wave Equation in Higher Dimensions

7 Wave Equaion in Highe Dimensions We now conside he iniial-value poblem fo he wave equaion in n dimensions, u c u x R n u(x, φ(x u (x, ψ(x whee u n i u x i x i. (7. 7. Mehod of Spheical Means Ref: Evans,

### Calculus 241, section 12.2 Limits/Continuity & 12.3 Derivatives/Integrals notes by Tim Pilachowski r r r =, with a domain of real ( )

Clculu 4, econ Lm/Connuy & Devve/Inel noe y Tm Plchow, wh domn o el Wh we hve o : veco-vlued uncon, ( ) ( ) ( ) j ( ) nume nd ne o veco The uncon, nd A w done wh eul uncon ( x) nd connuy e he componen

### Nanoparticles. Educts. Nucleus formation. Nucleus. Growth. Primary particle. Agglomeration Deagglomeration. Agglomerate

ucs Nucleus Nucleus omaon cal supesauaon Mng o eucs, empeaue, ec. Pmay pacle Gowh Inegaon o uson-lme pacle gowh Nanopacles Agglomeaon eagglomeaon Agglomeae Sablsaon o he nanopacles agans agglomeaon! anspo

### Problem Set #10 Math 471 Real Analysis Assignment: Chapter 8 #2, 3, 6, 8

Poblem Set #0 Math 47 Real Analysis Assignment: Chate 8 #2, 3, 6, 8 Clayton J. Lungstum Decembe, 202 xecise 8.2 Pove the convese of Hölde s inequality fo = and =. Show also that fo eal-valued f / L ),

### Slide. King Saud University College of Science Physics & Astronomy Dept. PHYS 103 (GENERAL PHYSICS) CHAPTER 5: MOTION IN 1-D (PART 2) LECTURE NO.

Slde Kng Saud Unersty College of Scence Physcs & Astronomy Dept. PHYS 103 (GENERAL PHYSICS) CHAPTER 5: MOTION IN 1-D (PART ) LECTURE NO. 6 THIS PRESENTATION HAS BEEN PREPARED BY: DR. NASSR S. ALZAYED Lecture

### The ray paths and travel times for multiple layers can be computed using ray-tracing, as demonstrated in Lab 3.

C. Trael me cures for mulple reflecors The ray pahs ad rael mes for mulple layers ca be compued usg ray-racg, as demosraed Lab. MATLAB scrp reflec_layers_.m performs smple ray racg. (m) ref(ms) ref(ms)

### Summary:Linear Motion

Summary:Linear Moion D Saionary objec V Consan velociy D Disance increase uniformly wih ime D = v. a Consan acceleraion V D V = a. D = ½ a 2 Velociy increases uniformly wih ime Disance increases rapidly

### Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai Pupose: o exen he Enegy appoach in eiving euaions of oion i.e. Lagange s Meho fo Mechanical Syses. opics: Genealize Cooinaes Lagangian Euaion

### J i-1 i. J i i+1. Numerical integration of the diffusion equation (I) Finite difference method. Spatial Discretization. Internal nodes.

umercal negraon of he dffuson equaon (I) Fne dfference mehod. Spaal screaon. Inernal nodes. R L V For hermal conducon le s dscree he spaal doman no small fne spans, =,,: Balance of parcles for an nernal

### Review Equations. Announcements 9/8/09. Table Tennis

Announcemens 9/8/09 1. Course homepage ia: phsics.bu.edu Class web pages Phsics 105 (Colon J). (Class-wide email sen) Iclicker problem from las ime scores didn ge recorded. Clicker quizzes from lecures

### Guest Lecturer Friday! Symbolic reasoning. Symbolic reasoning. Practice Problem day A. 2 B. 3 C. 4 D. 8 E. 16 Q25. Will Armentrout.

Pracice Problem day Gues Lecurer Friday! Will Armenrou. He d welcome your feedback! Anonymously: wrie somehing and pu i in my mailbox a 111 Whie Hall. Email me: sarah.spolaor@mail.wvu.edu Symbolic reasoning

### Outline. GW approximation. Electrons in solids. The Green Function. Total energy---well solved Single particle excitation---under developing

Peenaon fo Theoecal Condened Mae Phyc n TU Beln Geen-Funcon and GW appoxmaon Xnzheng L Theoy Depamen FHI May.8h 2005 Elecon n old Oulne Toal enegy---well olved Sngle pacle excaon---unde developng The Geen

### WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x

WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile

### Physics 101: Lecture 03 Kinematics Today s lecture will cover Textbook Sections (and some Ch. 4)

Physics 101: Lecure 03 Kinemaics Today s lecure will coer Texbook Secions 3.1-3.3 (and some Ch. 4) Physics 101: Lecure 3, Pg 1 A Refresher: Deermine he force exered by he hand o suspend he 45 kg mass as

### One-Dimensional Kinematics

One-Dimensional Kinemaics One dimensional kinemaics refers o moion along a sraigh line. Een hough we lie in a 3-dimension world, moion can ofen be absraced o a single dimension. We can also describe moion

### Density Matrix Description of NMR BCMB/CHEM 8190

Densy Marx Descrpon of NMR BCMBCHEM 89 Operaors n Marx Noaon If we say wh one bass se, properes vary only because of changes n he coeffcens weghng each bass se funcon x = h< Ix > - hs s how we calculae

### Modern Energy Functional for Nuclei and Nuclear Matter. By: Alberto Hinojosa, Texas A&M University REU Cyclotron 2008 Mentor: Dr.

Moden Enegy Funconal fo Nucle and Nuclea Mae By: lbeo noosa Teas &M Unvesy REU Cycloon 008 Meno: D. Shalom Shlomo Oulne. Inoducon.. The many-body poblem and he aee-fock mehod. 3. Skyme neacon. 4. aee-fock

### Energy Storage Devices

Energy Sorage Deces Objece of Lecure Descrbe he consrucon of a capacor and how charge s sored. Inroduce seeral ypes of capacors Dscuss he elecrcal properes of a capacor The relaonshp beween charge, olage,

### N 1. Time points are determined by the

upplemena Mehods Geneaon of scan sgnals In hs secon we descbe n deal how scan sgnals fo 3D scannng wee geneaed. can geneaon was done n hee seps: Fs, he dve sgnal fo he peo-focusng elemen was geneaed o

### THERMODYNAMICS 1. The First Law and Other Basic Concepts (part 2)

Company LOGO THERMODYNAMICS The Frs Law and Oher Basc Conceps (par ) Deparmen of Chemcal Engneerng, Semarang Sae Unversy Dhon Harano S.T., M.T., M.Sc. Have you ever cooked? Equlbrum Equlbrum (con.) Equlbrum

### Physics 201, Lecture 6

Physics 201, Lectue 6 Today s Topics q Unifom Cicula Motion (Section 4.4, 4.5) n Cicula Motion n Centipetal Acceleation n Tangential and Centipetal Acceleation q Relatie Motion and Refeence Fame (Sec.

### calculating electromagnetic

Theoeal mehods fo alulang eleomagne felds fom lghnng dshage ajeev Thoapplll oyal Insue of Tehnology KTH Sweden ajeev.thoapplll@ee.kh.se Oulne Despon of he poblem Thee dffeen mehods fo feld alulaons - Dpole

### 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

### Ch04: Motion in two and three dimensions (2D and 3D)

Ch4: Motion in two and thee dimensions (D and 3D) Displacement, elocity and acceleation ectos Pojectile motion Cicula motion Relatie motion 4.: Position and displacement Position of an object in D o 3D

### Main Ideas in Class Today

Main Ideas in Class Toda Inroducion o Falling Appl Consan a Equaions Graphing Free Fall Sole Free Fall Problems Pracice:.45,.47,.53,.59,.61,.63,.69, Muliple Choice.1 Freel Falling Objecs Refers o objecs

### MEEN 617 Handout #11 MODAL ANALYSIS OF MDOF Systems with VISCOUS DAMPING

MEEN 67 Handou # MODAL ANALYSIS OF MDOF Sysems wih VISCOS DAMPING ^ Symmeic Moion of a n-dof linea sysem is descibed by he second ode diffeenial equaions M+C+K=F whee () and F () ae n ows vecos of displacemens

### A. Inventory model. Why are we interested in it? What do we really study in such cases.

Some general yem model.. Inenory model. Why are we nereed n? Wha do we really udy n uch cae. General raegy of machng wo dmlar procee, ay, machng a fa proce wh a low one. We need an nenory or a buffer or

### 4.5 Constant Acceleration

4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),

### Chapter 6: AC Circuits

Chaper 6: AC Crcus Chaper 6: Oulne Phasors and he AC Seady Sae AC Crcus A sable, lnear crcu operang n he seady sae wh snusodal excaon (.e., snusodal seady sae. Complee response forced response naural response.

### Reflection and Refraction

Chape 1 Reflecon and Refacon We ae now neesed n eplong wha happens when a plane wave avelng n one medum encounes an neface (bounday) wh anohe medum. Undesandng hs phenomenon allows us o undesand hngs lke:

### β A Constant-G m Biasing

p 2002 EE 532 Anal IC Des II Pae 73 Cnsan-G Bas ecall ha us a PTAT cuen efeence (see p f p. 66 he nes) bas a bpla anss pes cnsan anscnucance e epeaue (an als epenen f supply lae an pcess). Hw h we achee

### Kinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8.

Kinemaics Vocabulary Kinemaics and One Dimensional Moion 8.1 WD1 Kinema means movemen Mahemaical descripion of moion Posiion Time Inerval Displacemen Velociy; absolue value: speed Acceleraion Averages

### Physics 101 Fall 2006: Exam #1- PROBLEM #1

Physics 101 Fall 2006: Exam #1- PROBLEM #1 1. Problem 1. (+20 ps) (a) (+10 ps) i. +5 ps graph for x of he rain vs. ime. The graph needs o be parabolic and concave upward. ii. +3 ps graph for x of he person

WORK POWER AND ENERGY Consevaive foce a) A foce is said o be consevaive if he wok done by i is independen of pah followed by he body b) Wok done by a consevaive foce fo a closed pah is zeo c) Wok done

### Density Matrix Description of NMR BCMB/CHEM 8190

Densy Marx Descrpon of NMR BCMBCHEM 89 Operaors n Marx Noaon Alernae approach o second order specra: ask abou x magnezaon nsead of energes and ranson probables. If we say wh one bass se, properes vary

### Chapter 3 Kinematics in Two Dimensions

Chaper 3 KINEMATICS IN TWO DIMENSIONS PREVIEW Two-dimensional moion includes objecs which are moing in wo direcions a he same ime, such as a projecile, which has boh horizonal and erical moion. These wo

### 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

### by Lauren DeDieu Advisor: George Chen

b Laren DeDe Advsor: George Chen Are one of he mos powerfl mehods o nmercall solve me dependen paral dfferenal eqaons PDE wh some knd of snglar shock waves & blow-p problems. Fed nmber of mesh pons Moves

### MAE 210B. Homework Solution #6 Winter Quarter, U 2 =r U=r 2 << 1; ) r << U : (1) The boundary conditions written in polar coordinates,

MAE B Homewok Solution #6 Winte Quate, 7 Poblem a Expecting a elocity change of oe oe a aial istance, the conition necessay fo the ow to be ominate by iscous foces oe inetial foces is O( y ) O( ) = =

### Linear Motion I Physics

Linear Moion I Physics Objecives Describe he ifference beween isplacemen an isance Unersan he relaionship beween isance, velociy, an ime Describe he ifference beween velociy an spee Be able o inerpre a

### Department of Chemical Engineering University of Tennessee Prof. David Keffer. Course Lecture Notes SIXTEEN

D. Keffe - ChE 40: Hea Tansfe and Fluid Flow Deamen of Chemical Enee Uniesi of Tennessee Pof. Daid Keffe Couse Lecue Noes SIXTEEN SECTION.6 DIFFERENTIL EQUTIONS OF CONTINUITY SECTION.7 DIFFERENTIL EQUTIONS

### rt () is constant. We know how to find the length of the radius vector by r( t) r( t) r( t)

Cicula Motion Fom ancient times cicula tajectoies hae occupied a special place in ou model of the Uniese. Although these obits hae been eplaced by the moe geneal elliptical geomety, cicula motion is still

### Physics Unit Workbook Two Dimensional Kinematics

Name: Per: L o s A l o s H i g h S c h o o l Phsics Uni Workbook Two Dimensional Kinemaics Mr. Randall 1968 - Presen adam.randall@mla.ne www.laphsics.com a o 1 a o o ) ( o o a o o ) ( 1 1 a o g o 1 g o

### I-POLYA PROCESS AND APPLICATIONS Leda D. Minkova

The XIII Inenaonal Confeence Appled Sochasc Models and Daa Analyss (ASMDA-009) Jne 30-Jly 3, 009, Vlns, LITHUANIA ISBN 978-9955-8-463-5 L Sakalaskas, C Skadas and E K Zavadskas (Eds): ASMDA-009 Seleced