ENGI 1313 Mechanics I
|
|
- Asher McBride
- 5 years ago
- Views:
Transcription
1 ENGI 1313 Mechanics I Lectue 05: Catesian Vects Shawn Kenny, Ph.D., P.Eng. ssistant Pfess Faculty f Engineeing and pplied Science Memial Univesity f Newfundland spkenny@eng.mun.ca
2 Chapte Objectives t eview cncepts fm linea algeba t sum fces, detemine fce esultants and eslve fce cmpnents f D vects using Paallelgam Law t expess fce and psitin in Catesian vect fm t intduce the cncept f dt pduct 007 S. Kenny, Ph.D., P.Eng.
3 Lectue 05 Objectives t futhe examine Catesian vect ntatin and extend t epesentatin f a 3D vect t sum 3D cncuent fce systems S. Kenny, Ph.D., P.Eng.
4 Recall D Catesian Vect Cplana Fce Vect Summatin Unit vect dimensinless F F R R F 1 F + F 3 ( F F + F ) i + ( F + F F ) j 1x x 3 x ) 1y y 3 y ) Unit Vect; ^ i F X F X ^ Unit Vect; j F y F y Fce Vects Cmpnent Vects Resultant Fce Vect S. Kenny, Ph.D., P.Eng.
5 Extend t 3D Why Use Vects Simplifies mathematical peatins Rectangula Cdinate System Right-hand ule S. Kenny, Ph.D., P.Eng.
6 Catesian Vect in 3D Thee Unit Vects Cmpnent magnitude x, y, z scala Cmpnent sense +, - Catesian quadant Cmpnent diectin i, j, k unit vects S. Kenny, Ph.D., P.Eng.
7 7 007 S. Kenny, Ph.D., P.Eng. Summatin Catesian Vects in 3D Use D Pinciples Vect Vect B ( ) ( ) ( )kˆ B ĵ B î B B z z y y x x kˆ ĵ î z y x + + kˆ B ĵ B î B B z y x + + ( ) ( ) ( )kˆ B ĵ B î B B z z y y x x + +
8 Catesian Vect Magnitude Successive pplicatin f D Pinciple Pythagean theem Find cmpnents x and y + x y Cmbine with z cmpnent + z x + y + z Magnitude f vect S. Kenny, Ph.D., P.Eng.
9 Catesian Vect Diectin Cdinate Diectin ngle Vect tail with cdinate axis α, β, and γ Range fm 0 t 180 Visualizatin aid using ight ectangula pism S. Kenny, Ph.D., P.Eng.
10 Catesian Vect Diectin (cnt.) Cdinate Diectin ngle, α Measued +x-axis t tail f vect csα x S. Kenny, Ph.D., P.Eng.
11 Catesian Vect Diectin (cnt.) Cdinate Diectin ngle, β Measued +y-axis t tail f vect csα cs β x y S. Kenny, Ph.D., P.Eng.
12 Catesian Vect Diectin (cnt.) Cdinate Diectin ngle, γ Measued +z-axis t tail f vect csα cs β cs γ x y z S. Kenny, Ph.D., P.Eng.
13 S. Kenny, Ph.D., P.Eng. Catesian Vect Diectin (cnt.) Diectin Csines Cdinate diectin angles (α, β, & γ) detemined by cs -1 cs x α cs y β cs z γ cs x 1 α cs y 1 β cs z 1 γ
14 Catesian Vect Diectin (cnt.) Expess as Catesian Vect, x î + y ĵ + z kˆ Recall Unit Vect Fm Unit Vect, u^ û x y z î + ĵ + kˆ + + x y ^ Unit Vect; u z u S. Kenny, Ph.D., P.Eng.
15 Catesian Vect Diectin (cnt.) Unit Vect, ^u x y û î + ĵ + Relate t Diectin Csines csα x Theefe û cs β z y kˆ csα î + cs β ĵ + cs γ csγ kˆ z u S. Kenny, Ph.D., P.Eng.
16 Catesian Vect Diectin (cnt.) Find Unit Vect mplitude, u ^ Recall geneal case f vect and magnitude x î + y ĵ + z kˆ + + Whee the unit vect and magnitude is x y z û csα î + cs β ĵ + cs γ kˆ Theefe cs û cs α + cs β + cs γ 1 α + cs β + cs γ S. Kenny, Ph.D., P.Eng.
17 Catesian Vect Oientatin (cnt.) Typical Catesian Vect Pblems Magnitude and cdinate angles Example.8 Magnitude and pjectin angles Example.10 Only need t knw angles S. Kenny, Ph.D., P.Eng.
18 Cmpehensin Quiz 5-01 If yu nly knw u (unit vect) yu can detemine the f uniquely. ) magnitude () B) angles (α, β, and γ) C) cmpnents ( x, y, & z ) D) ll f the abve nswe B Unit vect (u ) defines diectin defines magnitude S. Kenny, Ph.D., P.Eng.
19 Example Pblem 5-01 Fces F and G ae applied t a hk. Fce F makes 60 angle with the X-Y plane. Fce G has a magnitude f 80 lb with α 111 and β G 80lb α 111 β 69.3 Find the esultant fce in Catesian vect fm S. Kenny, Ph.D., P.Eng.
20 Example Pblem 5-01 (cnt.) Reslve Fce F cmpnents F z 100 lb sin lb F 100 lb cs lb F z G 80lb α 111 β 69.3 F x 50 lb cs lb F y 50 lb sin lb F y F x Catesian Vect Ntatin F { î ĵ kˆ }lb F S. Kenny, Ph.D., P.Eng.
21 Example Pblem 5-01 (cnt.) Detemine γ f Vect G cs α + cs β + cs γ 1 G 80lb α 111 β 69.3 cs cs cs γ 1 γ csγ 1 ( ) + ( ) ± γ S. Kenny, Ph.D., P.Eng.
22 Example Pblem 5-01 (cnt.) Cdinate Diectin ngles α111 z α 111 -x G 80lb G csα x G G csα G x x y Unit cicle G x G x 80 cs cs G x 8.67lb lb ( ) lb 007 S. Kenny, Ph.D., P.Eng.
23 Example Pblem 5-01 (cnt.) Cdinate Diectin ngles β and γ z -x G 80lb γ 30. β 69.3 y x S. Kenny, Ph.D., P.Eng.
24 Example Pblem 5-01 (cnt.) Catesian Vect G -x z G 80lb α 111 γ 30. x β 69.3 y { G 80 cs111 î + 80 cs69.3 ĵ + 80 cs 30. kˆ }lb G { 8.67 î ĵ kˆ }lb S. Kenny, Ph.D., P.Eng.
25 Example Pblem 5-01 (cnt.) Cmbine Fce Vects R F + G F { î ĵ kˆ }lb G { 8.67 î ĵ kˆ }lb G 80lb α 111 β 69.3 Resultant Vect R + R { 6.69 î 7.08 ĵ kˆ }lb {( ) î + ( ) ĵ + ( ) kˆ }lb S. Kenny, Ph.D., P.Eng.
26 Gup Pblem 5-01 Pblem -57 (Hibbele, 007) Detemine the magnitude and cdinate diectin angles f F and F 1. Sketch each fce n an x, y, z efeence. F { 1 60 î 50 ĵ + 40 kˆ }N F { 40 î 85 ĵ + 30 kˆ }N S. Kenny, Ph.D., P.Eng.
27 Gup Pblem 5-01 (cnt.) Fce F 1 F { 1 60 î 50 ĵ + 40 kˆ }N F ( 60N) + ( 50N) + ( 40N) 87.7 N 1 F1x 60N csα1 α F 87.75N 1 F1y 50N cs β 1 β1 15 F 87.75N F1z 40N cs γ1 γ F 87.75N S. Kenny, Ph.D., P.Eng.
28 Gup Pblem 5-01 (cnt.) Fce F F { 40 î 85 ĵ + 30 kˆ }N F ( 40N) + ( 85N) + ( 30N) 98.6 N F x 40N csα α 114 F 98.6N F y 85N cs β β 150 F 98.6N Fz 30N cs γ γ 7. 3 F 98.6N S. Kenny, Ph.D., P.Eng.
29 Gup Pblem 5-0 Pblem -59 (Hibbele, 007) Detemine the magnitude and cdinate angles f F acting n the stake S. Kenny, Ph.D., P.Eng.
30 Gup Pblem 5-0 Detemine F and cmpnents 5 F 40N 50N 4 3 F z 40N 30N 4 F x 40N cs N F x F F y F z F y 40N sin N S. Kenny, Ph.D., P.Eng.
31 Gup Pblem 5-0 Detemine Cdinate Diectin ngles Fx 13.68N csα α F 50N Fy 37.58N cs β β F 50N Fz 30N cs γ γ F 50N S. Kenny, Ph.D., P.Eng.
32 Classificatin f Textbk Pblems Hibbele (007) Pblem Set Cncept Degee f Difficulty Estimated Time -57 Catesian: Magnitude & diectin Medium 10-15min -58 t -60 Catesian: Fces Easy 5-10min -61 t -69 Catesian; Fce cmpnents & esultant Easy 5-10min -70 t -71 Catesian; Fce cmpnents & esultant Had 0-5min -7 t -78 Catesian; Fce cmpnents & esultant Medium 15-0min S. Kenny, Ph.D., P.Eng.
33 Refeences Hibbele (007) mech_1 en.wikipedia.g S. Kenny, Ph.D., P.Eng.
Example 11: The man shown in Figure (a) pulls on the cord with a force of 70
Chapte Tw ce System 35.4 α α 100 Rx cs 0.354 R 69.3 35.4 β β 100 Ry cs 0.354 R 111 Example 11: The man shwn in igue (a) pulls n the cd with a fce f 70 lb. Repesent this fce actin n the suppt A as Catesian
More informationENGI 1313 Mechanics I
ENGI 1313 Mechanics I Lectue 04: oce Vectos and System of Coplana oces Shawn Kenny, Ph.D., P.Eng. Assistant Pofesso aculty of Engineeing and Applied Science Memoial Univesity of Newfoundland spkenny@eng.mun.ca
More informationENGI 1313 Mechanics I
ENGI 1313 Mechanics I Lecture 11: 2D and 3D Particle Equilibrium Shawn Kenny, Ph.D., P.Eng. Assistant Prfessr aculty f Engineering and Applied Science Memrial University f Newfundland spkenny@engr.mun.ca
More information5.1 Moment of a Force Scalar Formation
Outline ment f a Cuple Equivalent System Resultants f a Fce and Cuple System ment f a fce abut a pint axis a measue f the tendency f the fce t cause a bdy t tate abut the pint axis Case 1 Cnside hizntal
More informationENGI 1313 Mechanics I
ENGI 1313 Mechanics I Lectue 03: Foce Vectos and Paallelogam Law Shawn Kenny, Ph.D., P.Eng. Assistant Pofesso Faculty of Engineeing and Applied Science Memoial Univesity of Newfoundland spkenny@eng.mun.ca
More informationMEM202 Engineering Mechanics Statics Course Web site:
0 Engineeing Mechanics - Statics 0 Engineeing Mechanics Statics Cuse Web site: www.pages.dexel.edu/~cac54 COUSE DESCIPTION This cuse cves intemediate static mechanics, an extensin f the fundamental cncepts
More informationWork, Energy, and Power. AP Physics C
k, Eneg, and Pwe AP Phsics C Thee ae man diffeent TYPES f Eneg. Eneg is expessed in JOULES (J) 4.19 J = 1 calie Eneg can be expessed me specificall b using the tem ORK() k = The Scala Dt Pduct between
More informationIntroduction. Electrostatics
UNIVESITY OF TECHNOLOGY, SYDNEY FACULTY OF ENGINEEING 4853 Electmechanical Systems Electstatics Tpics t cve:. Culmb's Law 5. Mateial Ppeties. Electic Field Stength 6. Gauss' Theem 3. Electic Ptential 7.
More informationSummary chapter 4. Electric field s can distort charge distributions in atoms and molecules by stretching and rotating:
Summa chapte 4. In chapte 4 dielectics ae discussed. In thse mateials the electns ae nded t the atms mlecules and cannt am fee thugh the mateial: the electns in insulats ae n a tight leash and all the
More informationElectric Charge. Electric charge is quantized. Electric charge is conserved
lectstatics lectic Chage lectic chage is uantized Chage cmes in incements f the elementay chage e = ne, whee n is an intege, and e =.6 x 0-9 C lectic chage is cnseved Chage (electns) can be mved fm ne
More informationAnalytical Solution to Diffusion-Advection Equation in Spherical Coordinate Based on the Fundamental Bloch NMR Flow Equations
Intenatinal Junal f heetical and athematical Phsics 5, 5(5: 4-44 OI:.593/j.ijtmp.555.7 Analtical Slutin t iffusin-advectin Equatin in Spheical Cdinate Based n the Fundamental Blch N Flw Equatins anladi
More informationThe Gradient and Applications This unit is based on Sections 9.5 and 9.6, Chapter 9. All assigned readings and exercises are from the textbook
The Gadient and Applicatins This unit is based n Sectins 9.5 and 9.6 Chapte 9. All assigned eadings and eecises ae fm the tetbk Objectives: Make cetain that u can define and use in cntet the tems cncepts
More informationChapter 4 Motion in Two and Three Dimensions
Chapte 4 Mtin in Tw and Thee Dimensins In this chapte we will cntinue t stud the mtin f bjects withut the estictin we put in chapte t me aln a staiht line. Instead we will cnside mtin in a plane (tw dimensinal
More informationSection 4.2 Radians, Arc Length, and Area of a Sector
Sectin 4.2 Radian, Ac Length, and Aea f a Sect An angle i fmed by tw ay that have a cmmn endpint (vetex). One ay i the initial ide and the the i the teminal ide. We typically will daw angle in the cdinate
More informationDonnishJournals
DonnishJounals 041-1189 Donnish Jounal of Educational Reseach and Reviews. Vol 1(1) pp. 01-017 Novembe, 014. http:///dje Copyight 014 Donnish Jounals Oiginal Reseach Pape Vecto Analysis Using MAXIMA Savaş
More informationWYSE Academic Challenge Sectional Mathematics 2006 Solution Set
WYSE Academic Challenge Sectinal 006 Slutin Set. Cect answe: e. mph is 76 feet pe minute, and 4 mph is 35 feet pe minute. The tip up the hill takes 600/76, 3.4 minutes, and the tip dwn takes 600/35,.70
More informationENGI 1313 Mechanics I
ENGI 1313 echnics I Lecte 18: oment of oce Abot Specified Ais Shwn Kenn, Ph.D., P.Eng. Assistnt Pofesso clt of Engineeing nd Applied Science emoil Univesit of Newfondlnd spkenn@eng.mn.c id-tem o This Clss
More informationChapter 5 Trigonometric Functions
Chapte 5 Tignmetic Functins Sectin 5.2 Tignmetic Functins 5-5. Angles Basic Teminlgy Degee Measue Standad Psitin Cteminal Angles Key Tems: vetex f an angle, initial side, teminal side, psitive angle, negative
More informationPolar Coordinates. a) (2; 30 ) b) (5; 120 ) c) (6; 270 ) d) (9; 330 ) e) (4; 45 )
Pola Coodinates We now intoduce anothe method of labelling oints in a lane. We stat by xing a oint in the lane. It is called the ole. A standad choice fo the ole is the oigin (0; 0) fo the Catezian coodinate
More informationMODULE 5a and 5b (Stewart, Sections 12.2, 12.3) INTRO: In MATH 1114 vectors were written either as rows (a1, a2,..., an) or as columns a 1 a. ...
MODULE 5a and 5b (Stewat, Sections 2.2, 2.3) INTRO: In MATH 4 vectos wee witten eithe as ows (a, a2,..., an) o as columns a a 2... a n and the set of all such vectos of fixed length n was called the vecto
More informationENGI 1313 Mechanics I
ENGI 1313 Mechanics I Lecture 43: Course Material Review Shawn Kenny, Ph.D., P.Eng. ssistant Professor aculty of Engineering and pplied Science Memorial University of Newfoundland spkenny@engr.mun.ca inal
More informationELECTRIC & MAGNETIC FIELDS I (STATIC FIELDS) ELC 205A
LCTRIC & MAGNTIC FILDS I (STATIC FILDS) LC 05A D. Hanna A. Kils Assciate Pfess lectnics & Cmmnicatins ngineeing Depatment Faclty f ngineeing Cai Univesity Fall 0 f Static lecticity lectic & Magnetic Fields
More informationPart V: Closed-form solutions to Loop Closure Equations
Pat V: Closed-fom solutions to Loop Closue Equations This section will eview the closed-fom solutions techniques fo loop closue equations. The following thee cases will be consideed. ) Two unknown angles
More informationElectromagnetic Waves
Chapte 3 lectmagnetic Waves 3.1 Maxwell s quatins and ectmagnetic Waves A. Gauss s Law: # clsed suface aea " da Q enc lectic fields may be geneated by electic chages. lectic field lines stat at psitive
More informationSection 8.2 Polar Coordinates
Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal
More informationENGI 1313 Mechanics I
ENGI 1313 Mechanics I Lecture 25: Equilibrium of a Rigid Body Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@engr.mun.ca
More informationStrees Analysis in Elastic Half Space Due To a Thermoelastic Strain
IOSR Junal f Mathematics (IOSRJM) ISSN: 78-578 Vlume, Issue (July-Aug 0), PP 46-54 Stees Analysis in Elastic Half Space Due T a Themelastic Stain Aya Ahmad Depatment f Mathematics NIT Patna Biha India
More informationENGI 4430 Non-Cartesian Coordinates Page xi Fy j Fzk from Cartesian coordinates z to another orthonormal coordinate system u, v, ˆ i ˆ ˆi
ENGI 44 Non-Catesian Coodinates Page 7-7. Conesions between Coodinate Systems In geneal, the conesion of a ecto F F xi Fy j Fzk fom Catesian coodinates x, y, z to anothe othonomal coodinate system u,,
More informationName Date. Trigonometric Functions of Any Angle For use with Exploration 5.3
5.3 Tigonometic Functions of An Angle Fo use with Eploation 5.3 Essential Question How can ou use the unit cicle to define the tigonometic functions of an angle? Let be an angle in standad position with,
More informationChapter 2: Introduction to Implicit Equations
Habeman MTH 11 Section V: Paametic and Implicit Equations Chapte : Intoduction to Implicit Equations When we descibe cuves on the coodinate plane with algebaic equations, we can define the elationship
More informationOutline. Steady Heat Transfer with Conduction and Convection. Review Steady, 1-D, Review Heat Generation. Review Heat Generation II
Steady Heat ansfe ebuay, 7 Steady Heat ansfe wit Cnductin and Cnvectin ay Caett Mecanical Engineeing 375 Heat ansfe ebuay, 7 Outline eview last lectue Equivalent cicuit analyses eview basic cncept pplicatin
More informationA) N B) 0.0 N C) N D) N E) N
Cdinat: H Bahluli Sunday, Nvembe, 015 Page: 1 Q1. Five identical pint chages each with chage =10 nc ae lcated at the cnes f a egula hexagn, as shwn in Figue 1. Find the magnitude f the net electic fce
More informationREVIEW Polar Coordinates and Equations
REVIEW 9.1-9.4 Pola Coodinates and Equations You ae familia with plotting with a ectangula coodinate system. We ae going to look at a new coodinate system called the pola coodinate system. The cente of
More informationn Power transmission, X rays, lightning protection n Solid-state Electronics: resistors, capacitors, FET n Computer peripherals: touch pads, LCD, CRT
.. Cu-Pl, INE 45- Electmagnetics I Electstatic fields anda Cu-Pl, Ph.. INE 45 ch 4 ECE UPM Maagüe, P me applicatins n Pwe tansmissin, X as, lightning ptectin n lid-state Electnics: esists, capacits, FET
More information3D-Central Force Problems I
5.73 Lectue #1 1-1 Roadmap 1. define adial momentum 3D-Cental Foce Poblems I Read: C-TDL, pages 643-660 fo next lectue. All -Body, 3-D poblems can be educed to * a -D angula pat that is exactly and univesally
More informationPhysics 111 Lecture 5 (Walker: 3.3-6) Vectors & Vector Math Motion Vectors Sept. 11, 2009
Physics 111 Lectue 5 (Walke: 3.3-6) Vectos & Vecto Math Motion Vectos Sept. 11, 2009 Quiz Monday - Chap. 2 1 Resolving a vecto into x-component & y- component: Pola Coodinates Catesian Coodinates x y =
More informationPHYS 110B - HW #7 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased
PHYS 0B - HW #7 Sping 2004, Solutions by David Pace Any efeenced euations ae fom Giffiths Poblem statements ae paaphased. Poblem 0.3 fom Giffiths A point chage,, moves in a loop of adius a. At time t 0
More informationRotational Motion: Statics and Dynamics
Physics 07 Lectue 17 Goals: Lectue 17 Chapte 1 Define cente of mass Analyze olling motion Intoduce and analyze toque Undestand the equilibium dynamics of an extended object in esponse to foces Employ consevation
More informationPhysics for Scientists and Engineers
Phsics 111 Sections 003 and 005 Instucto: Pof. Haimin Wang E-mail: haimin@flae.njit.edu Phone: 973-596-5781 Office: 460 Tienan Hall Homepage: http://sola.njit.edu/~haimin Office Hou: 2:30 to 3:50 Monda
More informationLecture 4. Electric Potential
Lectue 4 Electic Ptentil In this lectue yu will len: Electic Scl Ptentil Lplce s n Pissn s Eutin Ptentil f Sme Simple Chge Distibutins ECE 0 Fll 006 Fhn Rn Cnell Univesity Cnsevtive Ittinl Fiels Ittinl
More informationAnnouncements Candidates Visiting Next Monday 11 12:20 Class 4pm Research Talk Opportunity to learn a little about what physicists do
Wed., /11 Thus., /1 Fi., /13 Mn., /16 Tues., /17 Wed., /18 Thus., /19 Fi., / 17.7-9 Magnetic Field F Distibutins Lab 5: Bit-Savat B fields f mving chages (n quiz) 17.1-11 Pemanent Magnets 18.1-3 Mic. View
More information2 Governing Equations
2 Govening Equations This chapte develops the govening equations of motion fo a homogeneous isotopic elastic solid, using the linea thee-dimensional theoy of elasticity in cylindical coodinates. At fist,
More informationHotelling s Rule. Therefore arbitrage forces P(t) = P o e rt.
Htelling s Rule In what fllws I will use the tem pice t dente unit pfit. hat is, the nminal mney pice minus the aveage cst f pductin. We begin with cmpetitin. Suppse that a fim wns a small pa, a, f the
More informationTHE LAPLACE EQUATION. The Laplace (or potential) equation is the equation. u = 0. = 2 x 2. x y 2 in R 2
THE LAPLACE EQUATION The Laplace (o potential) equation is the equation whee is the Laplace opeato = 2 x 2 u = 0. in R = 2 x 2 + 2 y 2 in R 2 = 2 x 2 + 2 y 2 + 2 z 2 in R 3 The solutions u of the Laplace
More informationChapter 1: Introduction to Polar Coordinates
Habeman MTH Section III: ola Coodinates and Comple Numbes Chapte : Intoduction to ola Coodinates We ae all comfotable using ectangula (i.e., Catesian coodinates to descibe points on the plane. Fo eample,
More informationPhysics 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 informationVectors. Chapter. Introduction of Vector. Types of Vector. Vectors 1
Vect 1 Chapte 0 Vect Intductin f Vect Phical quantitie haing magnitude, diectin and being la f ect algeba ae called ect. Eample : Diplacement, elcit, acceleatin, mmentum, fce, impule, eight, thut, tque,
More informationClass #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 informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationCHAPTER 24 GAUSS LAW
CHAPTR 4 GAUSS LAW LCTRIC FLUX lectic flux is a measue f the numbe f electic filed lines penetating sme suface in a diectin pependicula t that suface. Φ = A = A csθ with θ is the angle between the and
More information5.8 Trigonometric Equations
5.8 Tigonometic Equations To calculate the angle at which a cuved section of highwa should be banked, an enginee uses the equation tan =, whee is the angle of the 224 000 bank and v is the speed limit
More informationMicroelectronics Circuit Analysis and Design. ac Equivalent Circuit for Common Emitter. Common Emitter with Time-Varying Input
Micelectnics Cicuit Analysis and Design Dnald A. Neamen Chapte 6 Basic BJT Amplifies In this chapte, we will: Undestand the pinciple f a linea amplifie. Discuss and cmpae the thee basic tansist amplifie
More informationCHAPTER GAUSS'S LAW
lutins--ch 14 (Gauss's Law CHAPTE 14 -- GAU' LAW 141 This pblem is ticky An electic field line that flws int, then ut f the cap (see Figue I pduces a negative flux when enteing and an equal psitive flux
More informationTopic/Objective: Essential Question: How do solve problems involving radian and/or degree measure?
Topic/Objective: 4- RADIAN AND DEGREE MEASURE Name: Class/Peiod: Date: Essential Question: How do solve poblems involving adian and/o degee measue? Questions: TRIGONOMETRY. Tigonomety, as deived fom the
More informationIntroduction and Vectors
SOLUTIONS TO PROBLEMS Intoduction and Vectos Section 1.1 Standads of Length, Mass, and Time *P1.4 Fo eithe sphee the volume is V = 4! and the mass is m =!V =! 4. We divide this equation fo the lage sphee
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Test # Review Math (Pe -calculus) Name MULTIPLE CHOICE. Choose the one altenative that best completes the statement o answes the question. Use an identit to find the value of the epession. Do not use a
More informationVector d is a linear vector function of vector d when the following relationships hold:
Appendix 4 Dyadic Analysis DEFINITION ecto d is a linea vecto function of vecto d when the following elationships hold: d x = a xxd x + a xy d y + a xz d z d y = a yxd x + a yy d y + a yz d z d z = a zxd
More informationChapter 15. ELECTRIC POTENTIALS and ENERGY CONSIDERATIONS
Ch. 15--Elect. Pt. and Enegy Cns. Chapte 15 ELECTRIC POTENTIALS and ENERGY CONSIDERATIONS A.) Enegy Cnsideatins and the Abslute Electical Ptential: 1.) Cnside the fllwing scenai: A single, fixed, pint
More informationVectors and 2D Motion. Vectors and Scalars
Vectos and 2D Motion Vectos and Scalas Vecto aithmetic Vecto desciption of 2D motion Pojectile Motion Relative Motion -- Refeence Fames Vectos and Scalas Scala quantities: equie magnitude & unit fo complete
More informationFREE Download Study Package from website: &
.. Linea Combinations: (a) (b) (c) (d) Given a finite set of vectos a b c,,,... then the vecto xa + yb + zc +... is called a linea combination of a, b, c,... fo any x, y, z... R. We have the following
More information1. Show that if the angular momentum of a boby is determined with respect to an arbitrary point A, then. r r r. H r A can be expressed by H r r r r
1. Shw that if the angula entu f a bb is deteined with espect t an abita pint, then H can be epessed b H = ρ / v + H. This equies substituting ρ = ρ + ρ / int H = ρ d v + ρ ( ω ρ ) d and epanding, nte
More informationradians). Figure 2.1 Figure 2.2 (a) quadrant I angle (b) quadrant II angle is in standard position Terminal side Terminal side Terminal side
. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an
More informationSolution: (a) C 4 1 AI IC 4. (b) IBC 4
C A C C R A C R C R C sin 9 sin. A cuent f is maintaine in a single cicula lp f cicumfeence C. A magnetic fiel f is iecte paallel t the plane f the lp. (a) Calculate the magnetic mment f the lp. (b) What
More informationPROBLEM (page 126, 12 th edition)
PROBLEM 13-27 (page 126, 12 th edition) The mass of block A is 100 kg. The mass of block B is 60 kg. The coefficient of kinetic fiction between block B and the inclined plane is 0.4. A and B ae eleased
More informationPHYS 1114, Lecture 21, March 6 Contents:
PHYS 1114, Lectue 21, Mach 6 Contents: 1 This class is o cially cancelled, being eplaced by the common exam Tuesday, Mach 7, 5:30 PM. A eview and Q&A session is scheduled instead duing class time. 2 Exam
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More informationEPr over F(X} AA+ A+A. For AeF, a generalized inverse. ON POLYNOMIAL EPr MATRICES
Intenat. J. Hath. & Math. S. VOL. 15 NO. 2 (1992) 261-266 ON POLYNOMIAL EP MATRICES 261 AR. MEENAKSHI and N. ANANOAM Depatment f Mathematics, Annamalai Univeslty, Annamalainaga- 68 2, Tamll Nadu, INDIA.
More informationPhysics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism
Physics 2020, Sping 2005 Lab 5 page 1 of 8 Lab 5. Magnetism PART I: INTRODUCTION TO MAGNETS This week we will begin wok with magnets and the foces that they poduce. By now you ae an expet on setting up
More informationThe Derivative of the Sine and Cosine. A New Derivation Approach
The Deivative of the Sine and Cosine. A New Deivation Appoach By John T. Katsikadelis Scool of Civil Engineeing, National Technical Univesity of Athens, Athens 15773, Geece. e-mail: jkats@cental.ntua.g
More informationPhysics 161 Fall 2011 Extra Credit 2 Investigating Black Holes - Solutions The Following is Worth 50 Points!!!
Physics 161 Fall 011 Exta Cedit Investigating Black Holes - olutions The Following is Woth 50 Points!!! This exta cedit assignment will investigate vaious popeties of black holes that we didn t have time
More informationSolving Problems of Advance of Mercury s Perihelion and Deflection of. Photon Around the Sun with New Newton s Formula of Gravity
Solving Poblems of Advance of Mecuy s Peihelion and Deflection of Photon Aound the Sun with New Newton s Fomula of Gavity Fu Yuhua (CNOOC Reseach Institute, E-mail:fuyh945@sina.com) Abstact: Accoding to
More informationForce between two parallel current wires and Newton s. third law
Foce between two paallel cuent wies and Newton s thid law Yannan Yang (Shanghai Jinjuan Infomation Science and Technology Co., Ltd.) Abstact: In this pape, the essence of the inteaction between two paallel
More informationVECTOR MECHANICS FOR ENGINEERS: STATICS
4 Equilibium CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Fedinand P. Bee E. Russell Johnston, J. of Rigid Bodies Lectue Notes: J. Walt Ole Texas Tech Univesity Contents Intoduction Fee-Body Diagam
More informationContinuous Charge Distributions: Electric Field and Electric Flux
8/30/16 Quiz 2 8/25/16 A positive test chage qo is eleased fom est at a distance away fom a chage of Q and a distance 2 away fom a chage of 2Q. How will the test chage move immediately afte being eleased?
More informationTrigonometry Standard Position and Radians
MHF 4UI Unit 6 Day 1 Tigonomety Standad Position and Radians A. Standad Position of an Angle teminal am initial am Angle is in standad position when the initial am is the positive x-axis and the vetex
More informationMagnetism. Chapter 21
1.1 Magnetic Fields Chapte 1 Magnetism The needle f a cmpass is pemanent magnet that has a nth magnetic ple (N) at ne end and a suth magnetic ple (S) at the the. 1.1 Magnetic Fields 1.1 Magnetic Fields
More information2. Plane Elasticity Problems
S0 Solid Mechanics Fall 009. Plane lasticity Poblems Main Refeence: Theoy of lasticity by S.P. Timoshenko and J.N. Goodie McGaw-Hill New Yok. Chaptes 3..1 The plane-stess poblem A thin sheet of an isotopic
More information=0, (x, y) Ω (10.1) Depending on the nature of these boundary conditions, forced, natural or mixed type, the elliptic problems are classified as
Chapte 1 Elliptic Equations 1.1 Intoduction The mathematical modeling of steady state o equilibium phenomena geneally esult in to elliptic equations. The best example is the steady diffusion of heat in
More informationENGI 5708 Design of Civil Engineering Systems
ENGI 5708 Design of Civil Engineering Systems Lecture 09: Characteristics of Simplex Algorithm Solutions Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial
More informationJournal of Theoretics
Junal f Theetics Junal Hme Page The Classical Pblem f a Bdy Falling in a Tube Thugh the Cente f the Eath in the Dynamic They f Gavity Iannis Iaklis Haanas Yk Univesity Depatment f Physics and Astnmy A
More informationStress, Cauchy s equation and the Navier-Stokes equations
Chapte 3 Stess, Cauchy s equation and the Navie-Stokes equations 3. The concept of taction/stess Conside the volume of fluid shown in the left half of Fig. 3.. The volume of fluid is subjected to distibuted
More informationChapter 12. Kinetics of Particles: Newton s Second Law
Chapte 1. Kinetics of Paticles: Newton s Second Law Intoduction Newton s Second Law of Motion Linea Momentum of a Paticle Systems of Units Equations of Motion Dynamic Equilibium Angula Momentum of a Paticle
More informationCartesian Coordinate System and Vectors
Catesian Coodinate System and Vectos Coodinate System Coodinate system: used to descibe the position of a point in space and consists of 1. An oigin as the efeence point 2. A set of coodinate axes with
More informationEFFECTS OF FRINGING FIELDS ON SINGLE PARTICLE DYNAMICS. M. Bassetti and C. Biscari INFN-LNF, CP 13, Frascati (RM), Italy
Fascati Physics Seies Vol. X (998), pp. 47-54 4 th Advanced ICFA Beam Dynamics Wokshop, Fascati, Oct. -5, 997 EFFECTS OF FRININ FIELDS ON SINLE PARTICLE DYNAMICS M. Bassetti and C. Biscai INFN-LNF, CP
More informationAE301 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 information3D-Central Force Problems II
5.73 ectue # - 1 3D-Cental Foce Poblems II ast time: [x,p] vecto commutation ules: genealize fom 1-D to 3-D conugate position and momentum components in Catesian coodinates Coespondence Pinciple Recipe
More informationa Particle Forces the force. of action its sense is of application. Experimen demonstra forces ( P Resultant of Two Note: a) b) momentum)
Chapter 2 : Statics of a Particle 2.2 Force on a Particle: Resultant of Two Forces Recall, force is a vector quantity whichh has magnitude and direction. The direction of the the force. force is defined
More informationRotational Motion. Lecture 6. Chapter 4. Physics I. Course website:
Lectue 6 Chapte 4 Physics I Rotational Motion Couse website: http://faculty.uml.edu/andiy_danylov/teaching/physicsi Today we ae going to discuss: Chapte 4: Unifom Cicula Motion: Section 4.4 Nonunifom Cicula
More information1.6. Trigonometric Functions. 48 Chapter 1: Preliminaries. Radian Measure
48 Chapte : Peliminaies.6 Tigonometic Functions Cicle B' B θ C A Unit of cicle adius FIGURE.63 The adian measue of angle ACB is the length u of ac AB on the unit cicle centeed at C. The value of u can
More informationQuestion Bank. Section A. is skew-hermitian matrix. is diagonalizable. (, ) , Evaluate (, ) 12 about = 1 and = Find, if
Subject: Mathematics-I Question Bank Section A T T. Find the value of fo which the matix A = T T has ank one. T T i. Is the matix A = i is skew-hemitian matix. i. alculate the invese of the matix = 5 7
More informationPhysics 111. Exam #1. January 26, 2018
Physics xam # Januay 6, 08 ame Please ead and fllw these instuctins caefully: Read all pblems caefully befe attempting t slve them. Yu wk must be legible, and the ganizatin clea. Yu must shw all wk, including
More informationΔt The textbook chooses to say that the average velocity is
1-D Motion Basic I Definitions: One dimensional motion (staight line) is a special case of motion whee all but one vecto component is zeo We will aange ou coodinate axis so that the x-axis lies along the
More informationTEAL Physics and Mathematics Documentation
Vesin. 7/7/008 TAL Phsics and Mathematics cumentatin Jhn Belche, Stanislaw Olbet, and Nman eb IN PF FORMAT THIS OCUMNT HAS BOOKMARKS FOR NAVIGATION CLICK ON TH LFT BOOKMARK TAB IN TH PF RAR Vesin., Jul
More informationis the instantaneous position vector of any grid point or fluid
Absolute inetial, elative inetial and non-inetial coodinates fo a moving but non-defoming contol volume Tao Xing, Pablo Caica, and Fed Sten bjective Deive and coelate the govening equations of motion in
More informationA) 100 K B) 150 K C) 200 K D) 250 K E) 350 K
Phys10 Secnd Maj-09 Ze Vesin Cdinat: k Wednesday, May 05, 010 Page: 1 Q1. A ht bject and a cld bject ae placed in themal cntact and the cmbinatin is islated. They tansfe enegy until they each a final equilibium
More informationTrigonometric Functions of Any Angle 9.3 (, 3. Essential Question How can you use the unit circle to define the trigonometric functions of any angle?
9. Tigonometic Functions of An Angle Essential Question How can ou use the unit cicle to define the tigonometic functions of an angle? Let be an angle in standad position with, ) a point on the teminal
More informationA) (0.46 î ) N B) (0.17 î ) N
Phys10 Secnd Maj-14 Ze Vesin Cdinat: xyz Thusday, Apil 3, 015 Page: 1 Q1. Thee chages, 1 = =.0 μc and Q = 4.0 μc, ae fixed in thei places as shwn in Figue 1. Find the net electstatic fce n Q due t 1 and.
More informationPhysics 107 TUTORIAL ASSIGNMENT #8
Physics 07 TUTORIAL ASSIGNMENT #8 Cutnell & Johnson, 7 th edition Chapte 8: Poblems 5,, 3, 39, 76 Chapte 9: Poblems 9, 0, 4, 5, 6 Chapte 8 5 Inteactive Solution 8.5 povides a model fo solving this type
More informationPhysics 1502: Lecture 4 Today s Agenda
1 Physics 1502: Today s genda nnouncements: Lectues posted on: www.phys.uconn.edu/~cote/ HW assignments, solutions etc. Homewok #1: On Mastephysics today: due next Fiday Go to masteingphysics.com and egiste
More informationChapter 4 Motion in Two and Three Dimensions
Chpte 4 Mtin in Tw nd Thee Dimensins In this chpte we will cntinue t stud the mtin f bjects withut the estictin we put in chpte t me ln stiht line. Insted we will cnside mtin in plne (tw dimensinl mtin)
More information