V V The circumflex (^) tells us this is a unit vector

Size: px
Start display at page:

Download "V V The circumflex (^) tells us this is a unit vector"

Transcription

1 Vecto Vecto have Diection and Magnitude Mike ailey Magnitude: V V V V x y z vecto.pptx Vecto Can lo e Defined a the oitional Diffeence etween Two oint 3 Unit Vecto have a Magnitude =.0 4 ( x, y, z ( x, y, z ( V, V, V (,, x y z x x y y z z V V V V V Vˆ V x y z The cicumflex (^ tell u thi i a unit vecto Dot oduct 5 hyical Intepetation of the Dot oduct 6 ( x, y, z ( x, y, z ( co x x y y z z ecaue it poduce a cala eult (i.e., a ingle numbe, thi i alo called the cala oduct The amount of the foce acceleating the ca along the i how much of i in the hoizontal diection? co Thi i eay to ee in D, but a 3D veion of the ame poblem i tickie.

2 hyical Intepetation of the Dot oduct 7 hyical Intepetation of the Dot oduct 8 The amount of the foce acceleating the ca along the i how much of i in the diection? co ˆ co ˆ Genealizing How Much of Live in the Diection 9 Genealizing How Much of Live ependicula to the Diection 0 ˆ coθ coθ ˆ which i the length of the pojection of onto the line ^ o, how much of live in the diection i that magnitude time the unit vecto: om the peviou lide, how much of live in the diection i : ^ That, plu the pependicula vecto eual, o that how much of i pependicula to the ^ diection i: Dot oduct ae Commutative The ependicula to a D Vecto If V (x,y then V ( y,x Dot oduct ae Ditibutive ( C ( ( C You can tell that thi i tue becaue VV (x,y ( y,x xyxy 0 co90

3 Co oduct 3 The ependicula opety of the Co oduct 4 ( x, y, z The vecto i both pependicula to and pependicula to ( x, y, z x. The ight-hand-ule opety of the Co oduct Cul the finge of you ight hand in the diection that tat at and head towad. You thumb point in the diection of x. (,, y z z y z x x z x y y x in ecaue it poduce a vecto eult (i.e., thee numbe, thi i alo called the Vecto oduct Co oduct ae Not Commutative 5 Ue fo the Co oduct : inding a Vecto ependicula to a lane (= the uface Nomal 6 x. x. n n( ( Co oduct ae Ditibutive ( C ( ( C Ue fo the Co oduct : inding a Vecto ependicula to a lane (= the uface Nomal Thi i ued in CG Lighting 7 n Ue fo the Co and Dot oduct : I a oint Inide a Tiangle? 3D (X-Y-Z Veion 8 Let: n( ( n ( ( n ( ( n ( ( If ( nn,( nn, and( nn ae all poitive, then i inide the tiangle 3

4 n If I a oint Inide a Tiangle? Thi can be implified if you ae in D (X-Y E, E, E E ( ( whee: and: (, imilaly, ae all poitive, then i inide the tiangle x x y y (, E ( ( E ( ( y y x x 9 height Ue fo the Co oduct : inding the ea of a 3D Tiangle ea aeheight ae Height in ea in ( ( 0 Deivation of the Law of Coine Deivation of the Law of ine ( ( * ea( ( ( in ut, the aea i the ame egadle of which two ide we ue to compute it, o: [( ( ] [( ( ] [( ( ] [( ( ] ( ( co Dividing by ( give: in in in in in in d Ditance fom a oint to a lane nˆ In high chool, you defined a plane by: x + y + Cz + D = 0 3 Whee doe a line egment inteect an infinite plane? nˆ 4 It i moe ueful to define it by a point on the plane combined with the plane nomal vecto The euation of the line egment i: ( t t 0 If you want the familia euation of the plane, it i: x,y,z,, (n,n,n 0 x y z x y z which expand out to become the moe familia x + y + Cz + D = 0 The pependicula ditance fom the point to the plane i baed on the plane euation: d nˆ The dot poduct i anweing the uetion How much of (- i in the diection?. Note that thi give a igned ditance. If d > 0., then i on the ame ide of the plane a the nomal point. Thi i vey ueful. ˆn If point i in the plane, then:,,,, (n,n,n 0 x y z x y z x y z If we ubtitute the paametic expeion fo into the plane euation, then the only thing we don t know in that euation i t. olve it fo t*. Knowing t* will let u compute the (x,y,z of the actual inteection uing the line euation. If t* ha a zeo in the denominato, then that tell u that t*=, and the line mut be paallel to the plane. Thi give u the point of inteection with the infinite plane. We could now ue the method coveed a few lide ago to ee if lie inide a paticula tiangle. 0 4

5 Minimal Ditance etween Two 3D Line 5 nothe ue fo Dot oduct : oce One Vecto to be ependicula to nothe Vecto 6 v p d v Hee, we want to foce to become pependicula to 0 0 The euation of the line ae : 0 t v p 0 t The minimal ditance vecto between the two line mut be pependicula to both v ˆ ut, The tategy i to get id of the paallel component, leaving jut the pependicula ( ˆ ˆ vecto between them that i pependicula to both i: We need to anwe the uetion How much of ( 0-0 i in the v diection?. To do thi, we once again ue the dot poduct: d 0 0 vˆ v v p v o that ( ˆ ˆ Thi i known a Gam-chmidt othogonalization 5

V V The circumflex (^) tells us this is a unit vector

V V The circumflex (^) tells us this is a unit vector Vector 1 Vector have Direction and Magnitude Mike ailey mjb@c.oregontate.edu Magnitude: V V V V x y z vector.pptx Vector Can lo e Defined a the oitional Difference etween Two oint 3 Unit Vector have a

More information

V V V V. Vectors. Mike Bailey. Vectors have Direction and Magnitude. Magnitude: x y z. Computer Graphics.

V V V V. Vectors. Mike Bailey. Vectors have Direction and Magnitude. Magnitude: x y z. Computer Graphics. 1 Vector Mike Bailey mjb@c.oregontate.edu vector.pptx Vector have Direction and Magnitude Magnitude: V V V V x y z 1 Vector Can lo Be Defined a the Poitional Difference Between Two Point 3 ( x, y, z )

More information

Chapter 19 Webassign Help Problems

Chapter 19 Webassign Help Problems Chapte 9 Webaign Help Poblem 4 5 6 7 8 9 0 Poblem 4: The pictue fo thi poblem i a bit mileading. They eally jut give you the pictue fo Pat b. So let fix that. Hee i the pictue fo Pat (a): Pat (a) imply

More information

Vectors Mike Bailey Oregon State University Oregon State University Computer Graphics

Vectors Mike Bailey Oregon State University Oregon State University Computer Graphics 1 Vectors Mike Bailey mjb@cs.oregonstate.edu vectors.pptx Vectors have Direction and Magnitude 2 Magnitude: V V V V 2 2 2 x y z A Vector Can Also Be Defined as the Positional Difference Between Two Points

More information

Physics Spring 2012 Announcements: Mar 07, 2012

Physics Spring 2012 Announcements: Mar 07, 2012 Physics 00 - Sping 01 Announcements: Ma 07, 01 HW#6 due date has been extended to the moning of Wed. Ma 1. Test # (i. Ma ) will cove only chaptes 0 and 1 All of chapte will be coveed in Test #4!!! Test

More information

Solutions Practice Test PHYS 211 Exam 2

Solutions Practice Test PHYS 211 Exam 2 Solution Pactice Tet PHYS 11 Exam 1A We can plit thi poblem up into two pat, each one dealing with a epaate axi. Fo both the x- and y- axe, we have two foce (one given, one unknown) and we get the following

More information

. Using our polar coordinate conversions, we could write a

. Using our polar coordinate conversions, we could write a 504 Chapte 8 Section 8.4.5 Dot Poduct Now that we can add, sutact, and scale vectos, you might e wondeing whethe we can multiply vectos. It tuns out thee ae two diffeent ways to multiply vectos, one which

More information

SPH3UW/SPH4U Unit 3.2 Forces in Cetripetal Motion Page 1 of 6. Notes Physics Tool Box

SPH3UW/SPH4U Unit 3.2 Forces in Cetripetal Motion Page 1 of 6. Notes Physics Tool Box SPH3UW/SPH4U Unit 3. Foce in Cetipetal Motion Page 1 o 6 Note Phyic Tool Box Net Foce: acting on an object in uniom cicula motion act towad the cente o the cicle. Magnitude o Net Foce: combine Newton Second

More information

PHYSICS 151 Notes for Online Lecture 2.6

PHYSICS 151 Notes for Online Lecture 2.6 PHYSICS 151 Note fo Online Lectue.6 Toque: The whole eaon that we want to woy about cente of ma i that we ae limited to lookin at point mae unle we know how to deal with otation. Let eviit the metetick.

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

Static Electric Fields. Coulomb s Law Ε = 4πε. Gauss s Law. Electric Potential. Electrical Properties of Materials. Dielectrics. Capacitance E.

Static Electric Fields. Coulomb s Law Ε = 4πε. Gauss s Law. Electric Potential. Electrical Properties of Materials. Dielectrics. Capacitance E. Coulomb Law Ε Gau Law Electic Potential E Electical Popetie of Mateial Conducto J σe ielectic Capacitance Rˆ V q 4πε R ρ v 2 Static Electic Field εe E.1 Intoduction Example: Electic field due to a chage

More information

Ch 30 - Sources of Magnetic Field! The Biot-Savart Law! = k m. r 2. Example 1! Example 2!

Ch 30 - Sources of Magnetic Field! The Biot-Savart Law! = k m. r 2. Example 1! Example 2! Ch 30 - Souces of Magnetic Field 1.) Example 1 Detemine the magnitude and diection of the magnetic field at the point O in the diagam. (Cuent flows fom top to bottom, adius of cuvatue.) Fo staight segments,

More information

Physics 2112 Unit 14

Physics 2112 Unit 14 Physics 2112 Unit 14 Today s Concept: What Causes Magnetic Fields d 0I ds ˆ 2 4 Unit 14, Slide 1 You Comments Can you give a summay fo eveything we use the ight hand ule fo? Wasn't too clea on this topic.

More information

Physics 121: Electricity & Magnetism Lecture 1

Physics 121: Electricity & Magnetism Lecture 1 Phsics 121: Electicit & Magnetism Lectue 1 Dale E. Ga Wenda Cao NJIT Phsics Depatment Intoduction to Clices 1. What ea ae ou?. Feshman. Sophomoe C. Junio D. Senio E. Othe Intoduction to Clices 2. How man

More information

Then the number of elements of S of weight n is exactly the number of compositions of n into k parts.

Then the number of elements of S of weight n is exactly the number of compositions of n into k parts. Geneating Function In a geneal combinatoial poblem, we have a univee S of object, and we want to count the numbe of object with a cetain popety. Fo example, if S i the et of all gaph, we might want to

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

Cartesian Coordinate System and Vectors

Cartesian 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 information

Hopefully Helpful Hints for Gauss s Law

Hopefully Helpful Hints for Gauss s Law Hopefully Helpful Hints fo Gauss s Law As befoe, thee ae things you need to know about Gauss s Law. In no paticula ode, they ae: a.) In the context of Gauss s Law, at a diffeential level, the electic flux

More information

constant t [rad.s -1 ] v / r r [m.s -2 ] (direction: towards centre of circle / perpendicular to circle)

constant t [rad.s -1 ] v / r r [m.s -2 ] (direction: towards centre of circle / perpendicular to circle) VISUAL PHYSICS ONLINE MODULE 5 ADVANCED MECHANICS NON-UNIFORM CIRCULAR MOTION Equation of a cicle x y Angula displacement [ad] Angula speed d constant t [ad.s -1 ] dt Tangential velocity v v [m.s -1 ]

More information

This lecture. Transformations in 2D. Where are we at? Why do we need transformations?

This lecture. Transformations in 2D. Where are we at? Why do we need transformations? Thi lectue Tanfomation in 2D Thoma Sheme Richa (Hao) Zhang Geomet baic Affine pace an affine tanfomation Ue of homogeneou cooinate Concatenation of tanfomation Intouction to Compute Gaphic CMT 36 Lectue

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

MODULE 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 (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 information

FI 2201 Electromagnetism

FI 2201 Electromagnetism FI Electomagnetim Aleande A. Ikanda, Ph.D. Phyic of Magnetim and Photonic Reeach Goup ecto Analyi CURILINEAR COORDINAES, DIRAC DELA FUNCION AND HEORY OF ECOR FIELDS Cuvilinea Coodinate Sytem Cateian coodinate:

More information

Physics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism

Physics 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 information

MCV4U Final Exam Review. 1. Consider the function f (x) Find: f) lim. a) lim. c) lim. d) lim. 3. Consider the function: 4. Evaluate. lim. 5. Evaluate.

MCV4U Final Exam Review. 1. Consider the function f (x) Find: f) lim. a) lim. c) lim. d) lim. 3. Consider the function: 4. Evaluate. lim. 5. Evaluate. MCVU Final Eam Review Answe (o Solution) Pactice Questions Conside the function f () defined b the following gaph Find a) f ( ) c) f ( ) f ( ) d) f ( ) Evaluate the following its a) ( ) c) sin d) π / π

More information

Gravity. David Barwacz 7778 Thornapple Bayou SE, Grand Rapids, MI David Barwacz 12/03/2003

Gravity. David Barwacz 7778 Thornapple Bayou SE, Grand Rapids, MI David Barwacz 12/03/2003 avity David Bawacz 7778 Thonapple Bayou, and Rapid, MI 495 David Bawacz /3/3 http://membe.titon.net/daveb Uing the concept dicued in the peceding pape ( http://membe.titon.net/daveb ), I will now deive

More information

Estimation and Confidence Intervals: Additional Topics

Estimation and Confidence Intervals: Additional Topics Chapte 8 Etimation and Confidence Inteval: Additional Topic Thi chapte imply follow the method in Chapte 7 fo foming confidence inteval The text i a bit dioganized hee o hopefully we can implify Etimation:

More information

Basic propositional and. The fundamentals of deduction

Basic propositional and. The fundamentals of deduction Baic ooitional and edicate logic The fundamental of deduction 1 Logic and it alication Logic i the tudy of the atten of deduction Logic lay two main ole in comutation: Modeling : logical entence ae the

More information

CALCULUS II Vectors. Paul Dawkins

CALCULUS II Vectors. Paul Dawkins CALCULUS II Vectos Paul Dawkins Table of Contents Peface... ii Vectos... 3 Intoduction... 3 Vectos The Basics... 4 Vecto Aithmetic... 8 Dot Poduct... 13 Coss Poduct... 21 2007 Paul Dawkins i http://tutoial.math.lama.edu/tems.aspx

More information

To Feel a Force Chapter 7 Static equilibrium - torque and friction

To Feel a Force Chapter 7 Static equilibrium - torque and friction To eel a oce Chapte 7 Chapte 7: Static fiction, toque and static equilibium A. Review of foce vectos Between the eath and a small mass, gavitational foces of equal magnitude and opposite diection act on

More information

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50

working 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 information

Physics NYB problem set 5 solution

Physics NYB problem set 5 solution Physics NY poblem set 5 solutions 1 Physics NY poblem set 5 solution Hello eveybody, this is ED. Hi ED! ED is useful fo dawing the ight hand ule when you don t know how to daw. When you have a coss poduct

More information

Implementation of RCWA

Implementation of RCWA Instucto D. Ramond Rumpf (915) 747 6958 cumpf@utep.edu EE 5337 Computational Electomagnetics Lectue # Implementation of RCWA Lectue These notes ma contain copighted mateial obtained unde fai use ules.

More information

Geometry Contest 2013

Geometry Contest 2013 eomety ontet 013 1. One pizza ha a diamete twice the diamete of a malle pizza. What i the atio of the aea of the lage pizza to the aea of the malle pizza? ) to 1 ) to 1 ) to 1 ) 1 to ) to 1. In ectangle

More information

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018

Physics 2B Chapter 22 Notes - Magnetic Field Spring 2018 Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field

More information

Recall from last week:

Recall from last week: Recall fom last week: Length of a cuve '( t) dt b Ac length s( t) a a Ac length paametization ( s) with '( s) 1 '( t) Unit tangent vecto T '(s) '( t) dt Cuvatue: s ds T t t t t t 3 t ds u du '( t) dt Pincipal

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

KEPLER S LAWS AND PLANETARY ORBITS

KEPLER S LAWS AND PLANETARY ORBITS KEPE S AWS AND PANETAY OBITS 1. Selected popeties of pola coodinates and ellipses Pola coodinates: I take a some what extended view of pola coodinates in that I allow fo a z diection (cylindical coodinates

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

Σr2=0. Σ Br. Σ br. Σ r=0. br = Σ. Σa r-s b s (1.2) s=0. Σa r-s b s-t c t (1.3) t=0. cr = Σ. dr = Σ. Σa r-s b s-t c t-u d u (1.4) u =0.

Σr2=0. Σ Br. Σ br. Σ r=0. br = Σ. Σa r-s b s (1.2) s=0. Σa r-s b s-t c t (1.3) t=0. cr = Σ. dr = Σ. Σa r-s b s-t c t-u d u (1.4) u =0. 0 Powe of Infinite Seie. Multiple Cauchy Poduct The multinomial theoem i uele fo the powe calculation of infinite eie. Thi i becaue the polynomial theoem depend on the numbe of tem, o it can not be applied

More information

Force & Motion: Newton s Laws

Force & Motion: Newton s Laws oce & otion: Newton Law ( t Law) If no net foce act on a body then the body velocity cannot change. Zeo net foce implie zeo acceleation. The ma of an object detemine how difficult it i to change the object

More information

Auchmuty High School Mathematics Department Advanced Higher Notes Teacher Version

Auchmuty High School Mathematics Department Advanced Higher Notes Teacher Version The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!

More information

21 MAGNETIC FORCES AND MAGNETIC FIELDS

21 MAGNETIC FORCES AND MAGNETIC FIELDS CHAPTER 1 MAGNETIC ORCES AND MAGNETIC IELDS ANSWERS TO OCUS ON CONCEPTS QUESTIONS 1. (d) Right-Hand Rule No. 1 gives the diection of the magnetic foce as x fo both dawings A and. In dawing C, the velocity

More information

PHYSICS 151 Notes for Online Lecture #20

PHYSICS 151 Notes for Online Lecture #20 PHYSICS 151 Notes fo Online Lectue #20 Toque: The whole eason that we want to woy about centes of mass is that we ae limited to looking at point masses unless we know how to deal with otations. Let s evisit

More information

A moving charged particle creates a magnetic field vector at every point in space except at its position.

A moving charged particle creates a magnetic field vector at every point in space except at its position. 1 Pat 3: Magnetic Foce 3.1: Magnetic Foce & Field A. Chaged Paticles A moving chaged paticle ceates a magnetic field vecto at evey point in space ecept at its position. Symbol fo Magnetic Field mks units

More information

(read nabla or del) is defined by, k. (9.7.1*)

(read nabla or del) is defined by, k. (9.7.1*) 9.7 Gadient of a scala field. Diectional deivative Some of the vecto fields in applications can be obtained fom scala fields. This is vey advantageous because scala fields can be handled moe easily. The

More information

Gauss Law. Physics 231 Lecture 2-1

Gauss Law. Physics 231 Lecture 2-1 Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

More information

Physics 11 Chapter 20: Electric Fields and Forces

Physics 11 Chapter 20: Electric Fields and Forces Physics Chapte 0: Electic Fields and Foces Yesteday is not ous to ecove, but tomoow is ous to win o lose. Lyndon B. Johnson When I am anxious it is because I am living in the futue. When I am depessed

More information

Electrostatics (Electric Charges and Field) #2 2010

Electrostatics (Electric Charges and Field) #2 2010 Electic Field: The concept of electic field explains the action at a distance foce between two chaged paticles. Evey chage poduces a field aound it so that any othe chaged paticle expeiences a foce when

More information

Two figures are similar fi gures when they have the same shape but not necessarily the same size.

Two figures are similar fi gures when they have the same shape but not necessarily the same size. NDIN O PIION. o be poficient in math, ou need to ue clea definition in dicuion with othe and in ou own eaoning. imilait and anfomation ential uetion When a figue i tanlated, eflected, otated, o dilated

More information

Calculate the electric potential at B d2=4 m Calculate the electric potential at A d1=3 m 3 m 3 m

Calculate the electric potential at B d2=4 m Calculate the electric potential at A d1=3 m 3 m 3 m MTE : Ch 13 5:3-7pm on Oct 31 ltenate Exams: Wed Ch 13 6:3pm-8:pm (people attending the altenate exam will not be allowed to go out of the oom while othes fom pevious exam ae still aound) Thu @ 9:-1:3

More information

CHAPTER 25 ELECTRIC POTENTIAL

CHAPTER 25 ELECTRIC POTENTIAL CHPTE 5 ELECTIC POTENTIL Potential Diffeence and Electic Potential Conside a chaged paticle of chage in a egion of an electic field E. This filed exets an electic foce on the paticle given by F=E. When

More information

ASTR 3740 Relativity & Cosmology Spring Answers to Problem Set 4.

ASTR 3740 Relativity & Cosmology Spring Answers to Problem Set 4. ASTR 3740 Relativity & Comology Sping 019. Anwe to Poblem Set 4. 1. Tajectoie of paticle in the Schwazchild geomety The equation of motion fo a maive paticle feely falling in the Schwazchild geomety ae

More information

Math Notes on Kepler s first law 1. r(t) kp(t)

Math Notes on Kepler s first law 1. r(t) kp(t) Math 7 - Notes on Keple s fist law Planetay motion and Keple s Laws We conside the motion of a single planet about the sun; fo simplicity, we assign coodinates in R 3 so that the position of the sun is

More information

LECTURE 14. m 1 m 2 b) Based on the second law of Newton Figure 1 similarly F21 m2 c) Based on the third law of Newton F 12

LECTURE 14. m 1 m 2 b) Based on the second law of Newton Figure 1 similarly F21 m2 c) Based on the third law of Newton F 12 CTU 4 ] NWTON W O GVITY -The gavity law i foulated fo two point paticle with ae and at a ditance between the. Hee ae the fou tep that bing to univeal law of gavitation dicoveed by NWTON. a Baed on expeiental

More information

Module 05: Gauss s s Law a

Module 05: Gauss s s Law a Module 05: Gauss s s Law a 1 Gauss s Law The fist Maxwell Equation! And a vey useful computational technique to find the electic field E when the souce has enough symmety. 2 Gauss s Law The Idea The total

More information

Inference for A One Way Factorial Experiment. By Ed Stanek and Elaine Puleo

Inference for A One Way Factorial Experiment. By Ed Stanek and Elaine Puleo Infeence fo A One Way Factoial Expeiment By Ed Stanek and Elaine Puleo. Intoduction We develop etimating equation fo Facto Level mean in a completely andomized one way factoial expeiment. Thi development

More information

Magnetic fields (origins) CHAPTER 27 SOURCES OF MAGNETIC FIELD. Permanent magnets. Electric currents. Magnetic field due to a moving charge.

Magnetic fields (origins) CHAPTER 27 SOURCES OF MAGNETIC FIELD. Permanent magnets. Electric currents. Magnetic field due to a moving charge. Magnetic fields (oigins) CHAPTER 27 SOURCES OF MAGNETC FELD Magnetic field due to a moving chage. Electic cuents Pemanent magnets Magnetic field due to electic cuents Staight wies Cicula coil Solenoid

More information

Rotational Kinetic Energy

Rotational Kinetic Energy Add Impotant Rotational Kinetic Enegy Page: 353 NGSS Standad: N/A Rotational Kinetic Enegy MA Cuiculum Famewok (006):.1,.,.3 AP Phyic 1 Leaning Objective: N/A, but olling poblem have appeaed on peviou

More information

BASIC ALGEBRA OF VECTORS

BASIC ALGEBRA OF VECTORS Fomulae Fo u Vecto Algeba By Mi Mohammed Abbas II PCMB 'A' Impotant Tems, Definitions & Fomulae 01 Vecto - Basic Intoduction: A quantity having magnitude as well as the diection is called vecto It is denoted

More information

PHY2061 Enriched Physics 2 Lecture Notes. Gauss Law

PHY2061 Enriched Physics 2 Lecture Notes. Gauss Law PHY61 Eniched Physics Lectue Notes Law Disclaime: These lectue notes ae not meant to eplace the couse textbook. The content may be incomplete. ome topics may be unclea. These notes ae only meant to be

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Math Pecalculus Ch. 6 Review Name SHORT ANSWER. Wite the wod o phase that best completes each statement o answes the question. Solve the tiangle. ) ) 6 7 0 Two sides and an angle (SSA) of a tiangle ae

More information

University Physics (PHY 2326)

University Physics (PHY 2326) Chapte Univesity Physics (PHY 6) Lectue lectostatics lectic field (cont.) Conductos in electostatic euilibium The oscilloscope lectic flux and Gauss s law /6/5 Discuss a techniue intoduced by Kal F. Gauss

More information

Impulse and Momentum

Impulse and Momentum Impule and Momentum 1. A ca poee 20,000 unit of momentum. What would be the ca' new momentum if... A. it elocity wee doubled. B. it elocity wee tipled. C. it ma wee doubled (by adding moe paenge and a

More information

Math 259 Winter Handout 6: In-class Review for the Cumulative Final Exam

Math 259 Winter Handout 6: In-class Review for the Cumulative Final Exam Math 259 Winte 2009 Handout 6: In-class Review fo the Cumulative Final Exam The topics coveed by the cumulative final exam include the following: Paametic cuves. Finding fomulas fo paametic cuves. Dawing

More information

e.g: If A = i 2 j + k then find A. A = Ax 2 + Ay 2 + Az 2 = ( 2) = 6

e.g: If A = i 2 j + k then find A. A = Ax 2 + Ay 2 + Az 2 = ( 2) = 6 MOTION IN A PLANE 1. Scala Quantities Physical quantities that have only magnitude and no diection ae called scala quantities o scalas. e.g. Mass, time, speed etc. 2. Vecto Quantities Physical quantities

More information

AST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1

AST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1 Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be

More information

Flux. Area Vector. Flux of Electric Field. Gauss s Law

Flux. Area Vector. Flux of Electric Field. Gauss s Law Gauss s Law Flux Flux in Physics is used to two distinct ways. The fist meaning is the ate of flow, such as the amount of wate flowing in a ive, i.e. volume pe unit aea pe unit time. O, fo light, it is

More information

Physics 107 HOMEWORK ASSIGNMENT #15

Physics 107 HOMEWORK ASSIGNMENT #15 Physics 7 HOMEWORK SSIGNMENT #5 Cutnell & Johnson, 7 th eition Chapte 8: Poblem 4 Chapte 9: Poblems,, 5, 54 **4 small plastic with a mass of 6.5 x - kg an with a chage of.5 µc is suspene fom an insulating

More information

P-2: The screw eye is subjected to two forces, ԦF 1 and ԦF 2. Determine the magnitude and direction of the resultant force.

P-2: The screw eye is subjected to two forces, ԦF 1 and ԦF 2. Determine the magnitude and direction of the resultant force. P-1: ԦA=Ԧi +Ԧj -5k and B =Ԧi - 7Ԧj -6k. Detemine;?????? - A B B A A B B A B A B A 7 P-: The scew ee is subjected to two foces, Ԧ 1 and Ԧ. Detemine the magnitude and diection of the esultant foce. P-: The

More information

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

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

More information

Chapter 5 Applications of Newton s Laws

Chapter 5 Applications of Newton s Laws Chapte 5 Application of Newton Law Conceptual Poblem Detemine the Concept Becaue the object ae peeding up (acceleating), thee mut be a net foce acting on them. The foce acting on an object ae the nomal

More information

As is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3.

As is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3. Appendix A Vecto Algeba As is natual, ou Aeospace Stuctues will be descibed in a Euclidean thee-dimensional space R 3. A.1 Vectos A vecto is used to epesent quantities that have both magnitude and diection.

More information

Physics 11 Chapter 3: Vectors and Motion in Two Dimensions. Problem Solving

Physics 11 Chapter 3: Vectors and Motion in Two Dimensions. Problem Solving Physics 11 Chapte 3: Vectos and Motion in Two Dimensions The only thing in life that is achieved without effot is failue. Souce unknown "We ae what we epeatedly do. Excellence, theefoe, is not an act,

More information

Physics 2212 GH Quiz #2 Solutions Spring 2016

Physics 2212 GH Quiz #2 Solutions Spring 2016 Physics 2212 GH Quiz #2 Solutions Sping 216 I. 17 points) Thee point chages, each caying a chage Q = +6. nc, ae placed on an equilateal tiangle of side length = 3. mm. An additional point chage, caying

More information

Look over Chapter 22 sections 1-8 Examples 2, 4, 5, Look over Chapter 16 sections 7-9 examples 6, 7, 8, 9. Things To Know 1/22/2008 PHYS 2212

Look over Chapter 22 sections 1-8 Examples 2, 4, 5, Look over Chapter 16 sections 7-9 examples 6, 7, 8, 9. Things To Know 1/22/2008 PHYS 2212 PHYS 1 Look ove Chapte sections 1-8 xamples, 4, 5, PHYS 111 Look ove Chapte 16 sections 7-9 examples 6, 7, 8, 9 Things To Know 1) What is an lectic field. ) How to calculate the electic field fo a point

More information

Physics 111 Lecture 5 Circular Motion

Physics 111 Lecture 5 Circular Motion Physics 111 Lectue 5 Cicula Motion D. Ali ÖVGÜN EMU Physics Depatment www.aovgun.com Multiple Objects q A block of mass m1 on a ough, hoizontal suface is connected to a ball of mass m by a lightweight

More information

7.2. Coulomb s Law. The Electric Force

7.2. Coulomb s Law. The Electric Force Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat

More information

Gauss s Law Simulation Activities

Gauss s Law Simulation Activities Gauss s Law Simulation Activities Name: Backgound: The electic field aound a point chage is found by: = kq/ 2 If thee ae multiple chages, the net field at any point is the vecto sum of the fields. Fo a

More information

Force and Work: Reminder

Force and Work: Reminder Electic Potential Foce and Wok: Reminde Displacement d a: initial point b: final point Reminde fom Mechanics: Foce F if thee is a foce acting on an object (e.g. electic foce), this foce may do some wok

More information

Phys 201A. Homework 5 Solutions

Phys 201A. Homework 5 Solutions Phys 201A Homewok 5 Solutions 3. In each of the thee cases, you can find the changes in the velocity vectos by adding the second vecto to the additive invese of the fist and dawing the esultant, and by

More information

How can you find the dimensions of a square or a circle when you are given its area? When you multiply a number by itself, you square the number.

How can you find the dimensions of a square or a circle when you are given its area? When you multiply a number by itself, you square the number. 7. Finding Squae Root How can you find the dimenion of a quae o a cicle when you ae given it aea? When you multiply a numbe by itelf, you quae the numbe. Symbol fo quaing i the exponent. = = 6 quaed i

More information

Welcome to Physics 272

Welcome to Physics 272 Welcome to Physics 7 Bob Mose mose@phys.hawaii.edu http://www.phys.hawaii.edu/~mose/physics7.html To do: Sign into Masteing Physics phys-7 webpage Registe i-clickes (you i-clicke ID to you name on class-list)

More information

SUPPLEMENTARY MATERIAL CHAPTER 7 A (2 ) B. a x + bx + c dx

SUPPLEMENTARY MATERIAL CHAPTER 7 A (2 ) B. a x + bx + c dx SUPPLEMENTARY MATERIAL 613 7.6.3 CHAPTER 7 ( px + q) a x + bx + c dx. We choose constants A and B such that d px + q A ( ax + bx + c) + B dx A(ax + b) + B Compaing the coefficients of x and the constant

More information

AP Physics - Coulomb's Law

AP Physics - Coulomb's Law AP Physics - oulomb's Law We ve leaned that electons have a minus one chage and potons have a positive one chage. This plus and minus one business doesn t wok vey well when we go in and ty to do the old

More information

Δt The textbook chooses to say that the average velocity is

Δt 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 information

Module 9: Electromagnetic Waves-I Lecture 9: Electromagnetic Waves-I

Module 9: Electromagnetic Waves-I Lecture 9: Electromagnetic Waves-I Module 9: Electomagnetic Waves-I Lectue 9: Electomagnetic Waves-I What is light, paticle o wave? Much of ou daily expeience with light, paticulaly the fact that light ays move in staight lines tells us

More information

PHYS Summer Professor Caillault Homework Solutions

PHYS Summer Professor Caillault Homework Solutions PHYS 1111 - Summe 2007 - Pofesso Caillault Homewok Solutions Chapte 3 13. Pictue the Poblem: The whale dives along a staight line tilted 20.0 below hoizontal fo 150 m as shown in the figue. Stategy: Resolve

More information

to point uphill and to be equal to its maximum value, in which case f s, max = μsfn

to point uphill and to be equal to its maximum value, in which case f s, max = μsfn Chapte 6 16. (a) In this situation, we take f s to point uphill and to be equal to its maximum value, in which case f s, max = μsf applies, whee μ s = 0.5. pplying ewton s second law to the block of mass

More information

AE301 Aerodynamics I UNIT B: Theory of Aerodynamics

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

More information

Fri Angular Momentum Quiz 10 RE 11.a; HW10: 13*, 21, 30, 39 Mon , (.12) Rotational + Translational RE 11.b Tues.

Fri Angular Momentum Quiz 10 RE 11.a; HW10: 13*, 21, 30, 39 Mon , (.12) Rotational + Translational RE 11.b Tues. Fi. 11.1 Angula Momentum Quiz 10 R 11.a; HW10: 13*, 1, 30, 39 Mon. 11.-.3, (.1) Rotational + Tanslational R 11.b Tues. P10 Mon. 11.4-.6, (.13) Angula Momentum & Toque Tues. Wed. 11.7 -.9, (.11) Toque R

More information

ENGI 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 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 information

Sides and Angles of Right Triangles 6. Find the indicated side length in each triangle. Round your answers to one decimal place.

Sides and Angles of Right Triangles 6. Find the indicated side length in each triangle. Round your answers to one decimal place. Chapte 7 Peequisite Skills BLM 7-1.. Convet a Beaing to an Angle in Standad Position 1. Convet each beaing to an angle in standad position on the Catesian gaph. a) 68 127 c) 215 d) 295 e) N40 W f) S65

More information

Physics 122, Fall October 2012

Physics 122, Fall October 2012 hsics 1, Fall 1 3 Octobe 1 Toda in hsics 1: finding Foce between paallel cuents Eample calculations of fom the iot- Savat field law Ampèe s Law Eample calculations of fom Ampèe s law Unifom cuents in conductos?

More information

PHZ 3113 Fall 2017 Homework #5, Due Friday, October 13

PHZ 3113 Fall 2017 Homework #5, Due Friday, October 13 PHZ 3113 Fall 2017 Homewok #5, Due Fiday, Octobe 13 1. Genealize the poduct ule (fg) = f g +f g to wite the divegence Ö (Ù Ú) of the coss poduct of the vecto fields Ù and Ú in tems of the cul of Ù and

More information

Physics 201 Lecture 18

Physics 201 Lecture 18 Phsics 0 ectue 8 ectue 8 Goals: Define and anale toque ntoduce the coss poduct Relate otational dnamics to toque Discuss wok and wok eneg theoem with espect to otational motion Specif olling motion (cente

More information

( ) Make-up Tests. From Last Time. Electric Field Flux. o The Electric Field Flux through a bit of area is

( ) Make-up Tests. From Last Time. Electric Field Flux. o The Electric Field Flux through a bit of area is Mon., 3/23 Wed., 3/25 Thus., 3/26 Fi., 3/27 Mon., 3/30 Tues., 3/31 21.4-6 Using Gauss s & nto to Ampee s 21.7-9 Maxwell s, Gauss s, and Ampee s Quiz Ch 21, Lab 9 Ampee s Law (wite up) 22.1-2,10 nto to

More information

Review. Electrostatic. Dr. Ray Kwok SJSU

Review. Electrostatic. Dr. Ray Kwok SJSU Review Electostatic D. Ray Kwok SJSU Paty Balloons Coulomb s Law F e q q k 1 Coulomb foce o electical foce. (vecto) Be caeful on detemining the sign & diection. k 9 10 9 (N m / C ) k 1 4πε o k is the Coulomb

More information

The Laws of Motion ( ) N SOLUTIONS TO PROBLEMS ! F = ( 6.00) 2 + ( 15.0) 2 N = 16.2 N. Section 4.4. Newton s Second Law The Particle Under a Net Force

The Laws of Motion ( ) N SOLUTIONS TO PROBLEMS ! F = ( 6.00) 2 + ( 15.0) 2 N = 16.2 N. Section 4.4. Newton s Second Law The Particle Under a Net Force SOLUTIONS TO PROBLEMS The Laws of Motion Section 4.3 Mass P4. Since the ca is moving with constant speed and in a staight line, the esultant foce on it must be zeo egadless of whethe it is moving (a) towad

More information

Kinematics in 2-D (II)

Kinematics in 2-D (II) Kinematics in 2-D (II) Unifom cicula motion Tangential and adial components of Relative velocity and acceleation a Seway and Jewett 4.4 to 4.6 Pactice Poblems: Chapte 4, Objective Questions 5, 11 Chapte

More information