Chapter 3: Vectors and Coordinate Systems


 Bertram Cross
 2 years ago
 Views:
Transcription
1 Coodinate Systems Chapte 3: Vectos and Coodinate Systems Used to descibe the position of a point in space Coodinate system consists of a fied efeence point called the oigin specific aes with scales and labels instuctions on how to label a point elative to the oigin and the aes Catesian Coodinate System Pola Coodinate System Also called ectangula coodinate system  and y aes intesect at the oigin Points ae labeled (,y) Oigin and efeence line ae noted Point is distance fom the oigin in the diection of angle θ, ccw fom efeence line Points ae labeled (,θ) 3 4
2 Pola to Catesian Coodinates Based on foming a ight tiangle fom and θ cos θ y sin θ Catesian to Pola Coodinates is the hypotenuse and θ an angle tanθ y + y θ must be ccw fom positive ais fo these equations to be valid 5 6 Vectos and Scalas Vecto Notation A scala quantity is completely specified by a single value with an appopiate unit and has no diection. A vecto quantity is completely descibed by a numbe and appopiate units plus a diection. When handwitten, use an aow: When pinted, will be in bold pint: A When dealing with just the magnitude of a vecto in pint, an italic lette will be used: A o A The magnitude of the vecto has physical units The magnitude of a vecto is always a positive numbe A
3 Adding Vectos Gaphically Subtacting Vectos Continue dawing the vectos tiptotail The esultant is dawn fom the oigin of A to the end of the last vecto Measue the length of R and its angle Use the scale facto to convet length to actual magnitude Special case of vecto addition If A B, then use A+(B) Continue with standad vecto addition pocedue 9 10 Multiplying o Dividing a Vecto by a Scala The esult of the multiplication o division is a vecto The magnitude of the vecto is multiplied o divided by the scala If the scala is positive, the diection of the esult is the same as of the oiginal vecto If the scala is negative, the diection of the esult is opposite that of the oiginal vecto A component is a pat It is useful to use ectangula components Components of a Vecto These ae the pojections of the vecto along the  and yaes 1
4 Vecto Component Teminology A and A y ae the component vectos of A They ae vectos and follow all the ules fo vectos A and A y ae scalas, and will be efeed to as the components of A Components of a Vecto The component of a vecto is the pojection along the ais A Acosθ The ycomponent of a vecto is the pojection along the yais A Asinθ y Components of a Vecto The pevious equations ae valid only if θ is measued with espect to the ais The components ae the legs of the ight tiangle whose hypotenuse is A 1 Ay A A + Ay and θ tan A May still have to find θ with espect to the positive ais Unit Vectos A unit vecto is a dimensionless vecto with a magnitude of eactly 1. Unit vectos ae used to specify a diection and have no othe physical significance
5 Unit Vectos, cont. Unit Vectos in Vecto Notation The symbols î, ĵ, andkˆ epesent unit vectos They fom a set of mutually pependicula vectos The complete vecto can be epessed as ĵ A A ˆi + A ˆj + A kˆ y z î Adding Vectos Using Unit Vectos Using R A + B Then ( A ˆ ˆ ) ( ˆ ˆ Ay B By ) R i + j + i + j ( A ) ˆ B ( Ay By ) R + i + + ˆj R R + Ry and so R A + B and R y A y + B y Tig Function Waning The component equations (A A cos θ and A y A sin θ) apply only when the angle is measued with espect to the ais (pefeably ccw fom the positive ais). The esultant angle (tan θ A y / A ) gives the angle with espect to the ais. R R + R θ tan 1 y R R y
6 Adding Vectos with Unit Vectos Adding Vectos Using Unit Vectos Thee Diections Using R A + B R A ˆi + A ˆj + A kˆ + B ˆi + B ˆj + B kˆ ( y z ) ( y z ) ( A B ) ( A B ) ( A B ) R + ˆi + + ˆj + + kˆ y y z z R R + Ry + Rz R A + B, R y A y + B y and R z A z + B z R R + R + R θ tan 1 y z R etc. R Which figue shows A + A + A? 1 3 Chapte 3. Questions
7 Which figue shows A + A + A? 1 3 Which figue shows A B? Which figue shows A B? What ae the  and ycomponents C and C y of vecto C? A. C 1, Cy 1 B. C 3, C y 1 C. C, Cy 1 D. C 4, C y E. C 3, C y 1
8 What ae the  and ycomponents C and C y of vecto C? Angle φ that specifies the diection of is given by C A. C 1, Cy 1 B. C 3, C y 1 C. C, Cy 1 D. C 4, C y E. C 3, C y 1 A. tan 1 (C y /C ) B. tan 1 (C / C y ) C. tan 1 (C y / C ) D. tan 1 (C /C y ) E. tan 1 ( C / C y ) Angle φ that specifies the diection of is given by C Back to the concepts of motion: Chapte 1 A. tan 1 (C y /C ) B. tan 1 (C / C y ) C. tan 1 (C y / C ) D. tan 1 (C /C y ) E. tan 1 ( C / C y )
9 Chapte 1. Concepts of Motion The univese we live in is one of change and motion. Although we all have intuition about motion, based on ou epeiences, some of the impotant aspects of motion tun out to be athe subtle. Chapte Goal: To intoduce the fundamental concepts of motion. Displacement  vecto Velocity  vecto Acceleation vecto Diffeent types of motion Diffeent types of motion Tanslational Motion Cicula Motion Pojectile Motion Rotational Motion
10 1 sec sec 3 sec 4 sec 1 sec sec 3 sec 4 sec How can we chaacteize the motion? 1 sec sec 3 sec 4 sec 1 sec sec 3 sec 4 sec What is the diffeence between these motions? How can we chaacteize these motions? The fist step: PARTICLE MODEL MOTION DIAGRAM We conside object as a single point without size o shape, disegad intenal motion of the object. How can we chaacteize the motion? 1 sec sec 3 sec 4 sec Diffeent oigins diffeent coodinates 1 Physical meaning displacement  1sec sec 3sec 4sec Oigin (0) The second step: POSITION OF THE OBJECT (POINT) COORDIANTE SYSTEM  DISPLACEMENT We intoduce coodinate system: fo motion along a line  only (which means that y0); fo a motion in a plane and y. Oigin (0) Oigin (0) ( 10) Copyight 008 Peason Education, Inc., publishing as 30 Peason AddisonWesley.
11 Displacement y a A b d B A  initial position of the object If O is an oigin then vecto chaacteizes initial position of the object B  final position of the O (oigin) object Vecto b chaacteizes the final position of the object Vecto d is a displacement (final position minus initial position does not depend on coodinate system) d b a Standad notation fo displacement is final initial a Displacement y initial O (oigin) final 1,initial 1, final Displacement does not depend on coodinate system final initial 1, final 1, initial Displacement Displacement is a vecto, it does not depend on coodinate system A B How can we chaacteize the motion? The fist step: PARTICLE MODEL MOTION DIAGRAM, The second step: POSITION OF THE OBJECT (POINT) DISPLACEMENT The thid step: (AVERAGE) VELOCITY 1sec sec 3sec 4sec Aveage velocity is a vecto: displacement v avg time t A B C Oigin 10 (0) vavg 1sec vavg 0 sec 30 Copyight 008 Peason Education, Inc., publishing sec as Peason AddisonWesley. Fo a motion along the line diection of velocity is along the line and the magnitude v avg t
12 AVERAGE VELOCITY v avg displacement time t The magnitude of velocity (vecto) is called speed Eample: We know initial position of the object (in some coodinate system) 1 We know the aveage velocity v of the object duing time Then: What is the final position of the object? displacement 1 v time t t + v t 1 How can we chaacteize the motion? The fist step: PARTICLE MODEL MOTION DIAGRAM The second step: POSITION OF THE OBJECT (POINT) DISPLACEMENT The thid step: (AVERAGE) VELOCITY The foth step: (AVERAGE) ACCELERATION Oigin (0) a v1, avg 0 sec avg v, avg 1sec 30 sec The change in position is chaacteized by aveage velocity, The change in velocity is chaacteized by aveage acceleation v a Copyight 008 Peason Education, Inc., publishing avg as Peason AddisonWesley. t v (30 0), avg v 1, avg sec 10 t 1sec sec y How can we chaacteize the motion? The fist step: PARTICLE MODEL MOTION DIAGRAM The second step: POSITION OF THE OBJECT (POINT) DISPLACEMENT The thid step: (AVERAGE) VELOCITY v v avg t aavg The foth step: (AVERAGE) ACCELERATION t 1 v1 t v 1 3 v t v 1 t v v v v 1 a1 a t t v3 t 3 v 3 Acceleation Because velocity is a vecto, it can change in two possible ways. 1.The magnitude can change, indicating a change in speed, o. The diection can change, indicating that the object has changed diection.
13 Acceleation is the change of velocity (speed can be the same) v 1 v v v a t 1 v 1 1sec sec 3sec 4sec v Velocity is the same zeo acceleation v aavg 0 t 1sec sec 3sec 4sec Velocity is inceasing acceleation has the same diection as velocity v v 1 sec sec v 3 sec v v 3 4 sec v 3 a 3 t a v v 3 Velocity is deceasing acceleation has the opposite diection v v a t 3 a v v 3 What is the diffeence between these motions? 1sec sec 3sec 4sec 1sec sec 3sec 4sec 1 sec a 0 sec 3 sec 4 sec v v a v a
14 SI units Units of velocity: Basic Units: m Time seconds (s) vavg t s Length metes (m) Units of acceleation: Mass kilogam v m / s m (kg) aavg t s s EXAMPLE 1.7 Intepeting a position gaph EXAMPLE 1.7 Intepeting a position gaph Geneal PoblemSolving Stategy
15 Chapte 1. Summay Slides Geneal Stategy Geneal Stategy
16 Impotant Concepts Impotant Concepts Impotant Concepts Applications
17 Applications Chapte 1. Questions Which ca is going faste, A o B? Assume thee ae equal intevals of time between the fames of both movies. Which ca is going faste, A o B? Assume thee ae equal intevals of time between the fames of both movies. B is going faste.
18 Thee motion diagams ae shown. Which is a dust paticle settling to the floo at constant speed, which is a ball dopped fom the oof of a building, and which is a descending ocket slowing to make a soft landing on Mas? A. (a) is ball, (b) is dust, (c) is ocket B. (a) is ball, (b) is ocket, (c) is dust C. (a) is ocket, (b) is dust, (c) is ball D. (a) is ocket, (b) is ball, (c) is dust E. (a) is dust, (b) is ball, (c) is ocket Thee motion diagams ae shown. Which is a dust paticle settling to the floo at constant speed, which is a ball dopped fom the oof of a building, and which is a descending ocket slowing to make a soft landing on Mas? A. (a) is ball, (b) is dust, (c) is ocket B. (a) is ball, (b) is ocket, (c) is dust C. (a) is ocket, (b) is dust, (c) is ball D. (a) is ocket, (b) is ball, (c) is dust E. (a) is dust, (b) is ball, (c) is ocket A paticle moves fom position 1 to position duing the inteval t. Which vecto shows the paticle s aveage velocity? A paticle moves fom position 1 to position duing the inteval t. Which vecto shows the paticle s aveage velocity?
19 A paticle undegoes acceleation a while moving fom point 1 to point. Which of the choices shows the velocity vecto v as the object moves away fom point? A paticle undegoes acceleation a while moving fom point 1 to point. Which of the choices shows the velocity vecto v as the object moves away fom point? Rank in ode, fom the most to the least, the numbe of significant figues in the following numbes. Fo eample, if b has moe than c, c has the same numbe as a, and a has moe than d, you could give you answe as b > c a > d. a. 800 b c d A. a b d > c B. b d > c > a C. d > c > b a D. d > c > a > b E. b > a c d Rank in ode, fom the most to the least, the numbe of significant figues in the following numbes. Fo eample, if b has moe than c, c has the same numbe as a, and a has moe than d, you could give you answe as b > c a > d. a. 800 b c d A. a b d > c B. b d > c > a C. d > c > b a D. d > c > a > b E. b > a c d
Displacement, Velocity And Acceleration
Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,
More informationUnit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0.
Unit Vectos What is pobabl the most common mistake involving unit vectos is simpl leaving thei hats off. While leaving the hat off a unit vecto is a nast communication eo in its own ight, it also leads
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.7. find the vecto defined
More informationAlgebra and Trig. I. A point is a location or position that has no size or dimension.
Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite
More informationRevision Guide for Chapter 11
Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams
More informationCoordinate Systems L. M. Kalnins, March 2009
Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean
More informationCHAT PreCalculus Section 10.7. Polar Coordinates
CHAT PeCalculus Pola Coodinates Familia: Repesenting gaphs of equations as collections of points (, ) on the ectangula coodinate sstem, whee and epesent the diected distances fom the coodinate aes to
More informationModel Question Paper Mathematics Class XII
Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationMultiple choice questions [70 points]
Multiple choice questions [70 points] Answe all of the following questions. Read each question caefull. Fill the coect bubble on ou scanton sheet. Each question has exactl one coect answe. All questions
More informationTrigonometry in the Cartesian Plane
Tigonomet in the Catesian Plane CHAT Algeba sec. 0. to 0.5 *Tigonomet comes fom the Geek wod meaning measuement of tiangles. It pimail dealt with angles and tiangles as it petained to navigation astonom
More information4.1  Trigonometric Functions of Acute Angles
4.1  Tigonometic Functions of cute ngles a is a halfline that begins at a point and etends indefinitel in some diection. Two as that shae a common endpoint (o vete) fom an angle. If we designate one
More information12. Rolling, Torque, and Angular Momentum
12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.
More informationUniform Rectilinear Motion
Engineeing Mechanics : Dynamics Unifom Rectilinea Motion Fo paticle in unifom ectilinea motion, the acceleation is zeo and the elocity is constant. d d t constant t t 111 Engineeing Mechanics : Dynamics
More informationPHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013
PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,
More informationMoment and couple. In 3D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r
Moment and couple In 3D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationLINES AND TANGENTS IN POLAR COORDINATES
LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More information2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES
. 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 informationChapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43
Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.
More informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
More informationMagnetic Field and Magnetic Forces. Young and Freedman Chapter 27
Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew  electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field
More information81 Newton s Law of Universal Gravitation
81 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,
More informationVoltage ( = Electric Potential )
V1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is
More informationNURBS Drawing Week 5, Lecture 10
CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu
More informationHour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and
Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon
More informationExam 3: Equation Summary
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P
More informationIn the lecture on double integrals over nonrectangular domains we used to demonstrate the basic idea
Double Integals in Pola Coodinates In the lectue on double integals ove nonectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example
More informationMechanics 1: Work, Power and Kinetic Energy
Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).
More informationTransformations in Homogeneous Coordinates
Tansfomations in Homogeneous Coodinates (Com S 4/ Notes) YanBin Jia Aug, 6 Homogeneous Tansfomations A pojective tansfomation of the pojective plane is a mapping L : P P defined as u a b c u au + bv +
More informationSolution Derivations for Capa #8
Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass
More informationThe force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges
The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee
More informationLab M4: The Torsional Pendulum and Moment of Inertia
M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disklike mass suspended fom a thin od o wie. When the mass is twisted about the
More informationChapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6
Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe
More informationCarterPenrose diagrams and black holes
CatePenose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example
More informationTECHNICAL DATA. JIS (Japanese Industrial Standard) Screw Thread. Specifications
JIS (Japanese Industial Standad) Scew Thead Specifications TECNICAL DATA Note: Although these specifications ae based on JIS they also apply to and DIN s. Some comments added by Mayland Metics Coutesy
More informationFigure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!
1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the
More informationrotation  Conservation of mechanical energy for rotation  Angular momentum  Conservation of angular momentum
Final Exam Duing class (13:55 pm) on 6/7, Mon Room: 41 FMH (classoom) Bing scientific calculatos No smat phone calculatos l ae allowed. Exam coves eveything leaned in this couse. Review session: Thusday
More informationGraphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.
Gaphs of Equations CHAT PeCalculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such
More informationVoltage ( = Electric Potential )
V1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationChapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.
Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More informationIntroduction to Electric Potential
Univesiti Teknologi MARA Fakulti Sains Gunaan Intoduction to Electic Potential : A Physical Science Activity Name: HP: Lab # 3: The goal of today s activity is fo you to exploe and descibe the electic
More informationGravitation. AP Physics C
Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What
More informationForces & Magnetic Dipoles. r r τ = μ B r
Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationExperiment 6: Centripetal Force
Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee
More informationMultiple choice questions [60 points]
1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions
More informationA couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance, d. F A F B (= F A
5 Moment of a Couple Ref: Hibbele 4.6, edfod & Fowle: Statics 4.4 couple is a pai of foces, equal in magnitude, oppositely diected, and displaced by pependicula distance, d. d (=  ) Since the foces ae
More informationSkills Needed for Success in Calculus 1
Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell
More informationProblems on Force Exerted by a Magnetic Fields from Ch 26 T&M
Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuentcaying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to
More informationSpirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project
Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationLab #7: Energy Conservation
Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 14 Intoduction: Pehaps one of the most unusual
More informationProblems of the 2 nd International Physics Olympiads (Budapest, Hungary, 1968)
Poblems of the nd ntenational Physics Olympiads (Budapest Hungay 968) Péte Vankó nstitute of Physics Budapest Univesity of Technical Engineeing Budapest Hungay Abstact Afte a shot intoduction the poblems
More informationCh. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth
Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate
More informationMechanics 1: Motion in a Central Force Field
Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.
More informationPhys 2101 Gabriela González. cos. sin. sin
1 Phys 101 Gabiela González a m t t ma ma m m T α φ ω φ sin cos α τ α φ τ sin m m α τ I We know all of that aleady!! 3 The figue shows the massive shield doo at a neuton test facility at Lawence Livemoe
More informationFXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More informationPY1052 Problem Set 8 Autumn 2004 Solutions
PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ighthand end. If H 6.0 m and h 2.0 m, what
More informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei iskewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationUNITS, PHYSICAL QUANTITIES AND VECTORS
UNITS, PHYSICAL QUANTITIES AND VECTORS 1 1.37. IDENTIFY: Vector addition problem. We are given the magnitude and direction of three vectors and are asked to find their sum. SET UP: A B C 3.25 km 2.90 km
More informationCRRC1 Method #1: Standard Practice for Measuring Solar Reflectance of a Flat, Opaque, and Heterogeneous Surface Using a Portable Solar Reflectometer
CRRC Method #: Standad Pactice fo Measuing Sola Reflectance of a Flat, Opaque, and Heteogeneous Suface Using a Potable Sola Reflectomete Scope This standad pactice coves a technique fo estimating the
More informationDeflection of Electrons by Electric and Magnetic Fields
Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An
More information9.5 Volume of Pyramids
Page of 7 9.5 Volume of Pyamids and Cones Goal Find the volumes of pyamids and cones. Key Wods pyamid p. 49 cone p. 49 volume p. 500 In the puzzle below, you can see that the squae pism can be made using
More information2. SCALARS, VECTORS, TENSORS, AND DYADS
2. SCALARS, VECTORS, TENSORS, AND DYADS This section is a eview of the popeties of scalas, vectos, and tensos. We also intoduce the concept of a dyad, which is useful in MHD. A scala is a quantity that
More informationChapter 13. VectorValued Functions and Motion in Space 13.6. Velocity and Acceleration in Polar Coordinates
13.6 Velocity and Acceleation in Pola Coodinates 1 Chapte 13. VectoValued Functions and Motion in Space 13.6. Velocity and Acceleation in Pola Coodinates Definition. When a paticle P(, θ) moves along
More informationAP Physics Electromagnetic Wrap Up
AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle
More informationVISCOSITY OF BIODIESEL FUELS
VISCOSITY OF BIODIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use
More informationQuantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w
1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a
More informationLesson 7 Gauss s Law and Electric Fields
Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual
More informationThe Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = W/q 0 1V [Volt] =1 Nm/C
Geneal Physics  PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit
More informationChapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom
Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in
More informationUNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Approximate time two 100minute sessions
Name St.No.  Date(YY/MM/DD) / / Section Goup# UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Appoximate time two 100minute sessions OBJECTIVES I began to think of gavity extending to the ob of the moon,
More informationNotes on Electric Fields of Continuous Charge Distributions
Notes on Electic Fields of Continuous Chage Distibutions Fo discete pointlike electic chages, the net electic field is a vecto sum of the fields due to individual chages. Fo a continuous chage distibution
More informationChapter 3 Vectors 3.1 Vector Analysis Introduction to Vectors Properties of Vectors Cartesian Coordinate System...
Chapter 3 Vectors 3.1 Vector Analsis... 1 3.1.1 Introduction to Vectors... 1 3.1.2 Properties of Vectors... 1 3.2 Cartesian Coordinate Sstem... 5 3.2.1 Cartesian Coordinates... 6 3.3 Application of Vectors...
More information92.131 Calculus 1 Optimization Problems
9 Calculus Optimization Poblems ) A Noman window has the outline of a semicicle on top of a ectangle as shown in the figue Suppose thee is 8 + π feet of wood tim available fo all 4 sides of the ectangle
More informationFluids Lecture 15 Notes
Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2D, this velocit
More informationEXPERIMENT 16 THE MAGNETIC MOMENT OF A BAR MAGNET AND THE HORIZONTAL COMPONENT OF THE EARTH S MAGNETIC FIELD
260 161. THEORY EXPERMENT 16 THE MAGNETC MOMENT OF A BAR MAGNET AND THE HORZONTAL COMPONENT OF THE EARTH S MAGNETC FELD The uose of this exeiment is to measue the magnetic moment μ of a ba magnet and
More informationCLASS XI CHAPTER 3. Theorem 1 (sine formula) In any triangle, sides are proportional to the sines of the opposite angles. That is, in a triangle ABC
CLASS XI Anneue I CHAPTER.6. Poofs and Simple Applications of sine and cosine fomulae Let ABC be a tiangle. By angle A we mean te angle between te sides AB and AC wic lies between 0 and 80. Te angles B
More information10. Collisions. Before During After
10. Collisions Use conseation of momentum and enegy and the cente of mass to undestand collisions between two objects. Duing a collision, two o moe objects exet a foce on one anothe fo a shot time: F(t)
More informationPhysics 505 Homework No. 5 Solutions S51. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z.
Physics 55 Homewok No. 5 s S5. Angula momentum uncetainty elations. A system is in the lm eigenstate of L 2, L z. a Show that the expectation values of L ± = L x ± il y, L x, and L y all vanish. ψ lm
More informationPhysics HSC Course Stage 6. Space. Part 1: Earth s gravitational field
Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe
More information1.4 Phase Line and Bifurcation Diag
Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationChapter 4: Fluid Kinematics
Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian
More informationESCAPE VELOCITY EXAMPLES
ESCAPE VELOCITY EXAMPLES 1. Escape velocity is the speed that an object needs to be taveling to beak fee of planet o moon's gavity and ente obit. Fo example, a spacecaft leaving the suface of Eath needs
More informationProblem Set 6: Solutions
UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 164 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente
More information7 Circular Motion. 71 Centripetal Acceleration and Force. Period, Frequency, and Speed. Vocabulary
7 Cicula Motion 71 Centipetal Acceleation and Foce Peiod, Fequency, and Speed Vocabulay Vocabulay Peiod: he time it takes fo one full otation o evolution of an object. Fequency: he numbe of otations o
More informationTheory and measurement
Gavity: Theoy and measuement Reading: Today: p11  Theoy of gavity Use two of Newton s laws: 1) Univesal law of gavitation: ) Second law of motion: Gm1m F = F = mg We can combine them to obtain the gavitational
More informationChapter 4. Electric Potential
Chapte 4 Electic Potential 4.1 Potential and Potential Enegy... 43 4.2 Electic Potential in a Unifom Field... 47 4.3 Electic Potential due to Point Chages... 48 4.3.1 Potential Enegy in a System of
More informationChapter 30: Magnetic Fields Due to Currents
d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.11 T) i to ue a lage cuent flowing though a wie.
More informationPHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013
PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0
More informationCHAPTER 10 Aggregate Demand I
CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income
More informationGauss Law. Physics 231 Lecture 21
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 informationTORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION
MISN034 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction..............................................
More informationImpulse and Linear Momentum 5
Implse and Linea Momentm 5 How does jet poplsion wok? How can yo mease the speed of a bllet? Wold a meteoite collision significantly change Eath s obit? In pevios chaptes we discoveed that the pshing inteaction
More information