Chapter 1: Introduction to Polar Coordinates
|
|
|
- Dulcie Flynn
- 6 years ago
- Views:
Transcription
1 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, we ve plotted the point = (, on the coodinate plane in Figue. ' Figue : The point = (, on the ectangula coodinate plane. Instead of using these ectangula coodinates, we can use a cicula coodinate sstem to descibe points on the plane: ola Coodinates. Odeed pais in pola coodinates have fom (, θ whee epesents the point s distance fom the oigin and θ epesents the angula displacement of the point with espect to the positive -ais. Let s find the pola coodinates that descibe in Figue. Fist let s find, the distance fom point to the oigin; in othe wods, we need to find the length of the segment labeled in Figue : Figue ' Figue We can use the thagoean Theoem to find :
2 Habeman MTH Section III: Chapte = + = = ( ( = + Now we need to find the angle between the positive -ais and the segment labeled ; this angle is labeled θ in Figue. = θ ' Figue We can use the ight tiangle induced b the angle θ and the side along with eithe sine o cosine to find the value of θ : sin( θ = θ = Thus, in pola coodinates, (, plane in Figue. =. We ve plotted the point on the pola coodinate = ' Figue : The point (, = on the pola coodinate l
3 Habeman MTH Section III: Chapte EXAMLE : lot the point A = on the pola coodinate plane and detemine the ectangula coodinates of point A. SOLUTION: To plot the point A (, θ, so A = we need to ecognize that pola odeed pais have fom = implies that We ve plotted the point A = 0 and θ =. = on the pola coodinate plane in Figue. θ = = 0 A Figue To find the ectangula coodinates of point A we can use the efeence angle fo θ, which is, and the induced ight tiangle; see Figue. A 0 Figue
4 Habeman MTH Section III: Chapte Using the tiangle in Figue, we can see that cos ( = 0 = 0cos = 0 = ( and sin ( = 0 = 0sin = 0 = ( Since point A is in Quadant III, we know that both and ae negative. Thus, the ectangula coodinates of point A ae (,. EXAMLE : Find the ectangula coodinates of a geneic point (, θ coodinate plane. = on the pola SOLUTION: In Figue 7, we've pointed plotted in the pola plane. θ = (, θ Figue 7 We can constuct a ight tiangle and use tigonomet to obtain epessions fo the hoizontal and vetical coodinates of point ; see Figue 8 below.
5 Habeman MTH Section III: Chapte θ Figue 8 Based on the tiangle in Figue 8, we can see that cos( θ = = cos( θ and sin( θ = = sin( θ = is epesents a point on the pola coodinate plane, then the =. (Notice that we obseved Thus, if (, θ ectangula coodinates of ae (, ( cos( θ, sin( θ essentiall the same fact in Section I: Chapte. We can use what we ve discoveed to tanslate pola coodinates into ectangula coodinates. The pola coodinates (, θ ae equivalent to the ectangula coodinates ( θ θ (, = cos(, sin(. Ke oint: ola and ectangula odeed pais cannot be set equal to each othe. When odeed pais ae descibed as being equal, it means that the have 0.7, 0. =, the same coodinates so we can wite something like ( ( 0.7 = and 0. = but we can t wite = (, since (fom Eample since 0 and. In ode to communicate that ectangula odeed pais and pola odeed pais descibe the same location, we need to compose sentences like, The ectangula 0,. odeed pai (, is equivalent to the pola odeed pai (
6 Habeman MTH Section III: Chapte EXAMLE : lot the point B (, SOLUTION: ectangula coodinates of the point. To plot the point B (, (, θ, so B (, = on the pola coodinate plane and find the = we need to ecognize that pola odeed pais have fom = implies that = and θ =. Hee, is negative. This means that when we get to the teminal side of θ =, instead of going fowad units into Quadant II, we need to go backwads units into Quadant IV; see Figue 9. θ = = B Figue 9 To find the ectangula coodinates of point B, we can use the convesion equations we deived in the pevious eample. = cos( θ = cos = = ( ( and = sin( θ = sin = = ( ( Thus, the ectangula coodinates of B ae (, = (,.
Section 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
radians). 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
Polar 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
MULTIPLE 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
Introduction 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
Chapter Eight Notes N P U1C8S4-6
Chapte Eight Notes N P UC8S-6 Name Peiod Section 8.: Tigonometic Identities An identit is, b definition, an equation that is alwas tue thoughout its domain. B tue thoughout its domain, that is to sa that
Chapter 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
3.6 Applied Optimization
.6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the
Graphs of Sine and Cosine Functions
Gaphs of Sine and Cosine Functions In pevious sections, we defined the tigonometic o cicula functions in tems of the movement of a point aound the cicumfeence of a unit cicle, o the angle fomed by the
1.6. 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
INTRODUCTION. 2. Vectors in Physics 1
INTRODUCTION Vectos ae used in physics to extend the study of motion fom one dimension to two dimensions Vectos ae indispensable when a physical quantity has a diection associated with it As an example,
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
Trigonometry 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
ENGR 1990 Engineering Mathematics Application of Trigonometric Functions in Mechanical Engineering: Part II
ENGR 990 Engineeing Mathematics pplication of Tigonometic Functions in Mechanical Engineeing: Pat II Poblem: Find the coodinates of the end-point of a two-link plana obot am Given: The lengths of the links
MATH 155/GRACEY CH. 10 PRACTICE. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
MATH /GRACEY CH. PRACTICE Name SHORT ANSWER. Wite the wod o phase that best completes each statement o answes the question. At the given point, find the line that is nomal to the cuve at the given point.
REVIEW 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
Name 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,
Trigonometric 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
The 1958 musical Merry Andrew starred Danny Kaye as
The 1958 musical Me Andew staed Dann Kae as Andew Laabee, a teache with a flai fo using unconventional methods in his classes. He uses a musical numbe to teach the Pthagoean theoem, singing and dancing
9.1 POLAR COORDINATES
9. Pola Coodinates Contempoay Calculus 9. POLAR COORDINATES The ectangula coodinate system is immensely useful, but it is not the only way to assign an addess to a point in the plane and sometimes it is
5.8 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
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Chapte 7-8 Review Math 1316 Name SHORT ANSWER. Wite the wod o phase that best completes each statement o answes the question. Solve the tiangle. 1) B = 34.4 C = 114.2 b = 29.0 1) Solve the poblem. 2) Two
Math Section 4.2 Radians, Arc Length, and Area of a Sector
Math 1330 - Section 4. Radians, Ac Length, and Aea of a Secto The wod tigonomety comes fom two Geek oots, tigonon, meaning having thee sides, and mete, meaning measue. We have aleady defined the six basic
COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT
COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT Link to: phsicspages home page. To leave a comment o epot an eo, please use the auilia blog. Refeence: d Inveno, Ra, Intoducing Einstein s Relativit
Vectors 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
Physics 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.
Tutorial Exercises: Central Forces
Tutoial Execises: Cental Foces. Tuning Points fo the Keple potential (a) Wite down the two fist integals fo cental motion in the Keple potential V () = µm/ using J fo the angula momentum and E fo the total
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,
A Tutorial on Multiple Integrals (for Natural Sciences / Computer Sciences Tripos Part IA Maths)
A Tutoial on Multiple Integals (fo Natual Sciences / Compute Sciences Tipos Pat IA Maths) Coections to D Ian Rud (http://people.ds.cam.ac.uk/ia/contact.html) please. This tutoial gives some bief eamples
Physics 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
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
THE 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
Homework # 3 Solution Key
PHYSICS 631: Geneal Relativity Homewok # 3 Solution Key 1. You e on you hono not to do this one by hand. I ealize you can use a compute o simply look it up. Please don t. In a flat space, the metic in
Lab #9: The Kinematics & Dynamics of. Circular Motion & Rotational Motion
Reading Assignment: Lab #9: The Kinematics & Dynamics of Cicula Motion & Rotational Motion Chapte 6 Section 4 Chapte 11 Section 1 though Section 5 Intoduction: When discussing motion, it is impotant to
PDF Created with deskpdf PDF Writer - Trial ::
A APPENDIX D TRIGONOMETRY Licensed to: jsamuels@bmcc.cun.edu PDF Ceated with deskpdf PDF Wite - Tial :: http://www.docudesk.com D T i g o n o m e t FIGURE a A n g l e s Angles can be measued in degees
Stress, 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
γ b =2 γ e In case there is no infiltration under the dam, the angle α is given by In case with infiltration under the dam, the angle a is given by
O γ e γ b α H γ b γ e Sol A B In case thee is no infiltation unde the dam, the angle α is given b So that Fom whee In case with infiltation unde the dam, the angle a is given b So that Fom whee A. Zeghloul
4.3 Right Triangle Trigonometry
Section. Right Tiangle Tigonomet 77. Right Tiangle Tigonomet The Si Tigonometic Functions Ou second look at the tigonometic functions is fom a ight tiangle pespective. Conside a ight tiangle, with one
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
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
ME 210 Applied Mathematics for Mechanical Engineers
Tangent and Ac Length of a Cuve The tangent to a cuve C at a point A on it is defined as the limiting position of the staight line L though A and B, as B appoaches A along the cuve as illustated in the
. 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
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),
Homework 7 Solutions
Homewok 7 olutions Phys 4 Octobe 3, 208. Let s talk about a space monkey. As the space monkey is oiginally obiting in a cicula obit and is massive, its tajectoy satisfies m mon 2 G m mon + L 2 2m mon 2
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
P.7 Trigonometry. What s round and can cause major headaches? The Unit Circle.
P.7 Tigonomet What s ound and can cause majo headaches? The Unit Cicle. The Unit Cicle will onl cause ou headaches if ou don t know it. Using the Unit Cicle in Calculus is equivalent to using ou multiplication
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
FREE 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
(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.
Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed
Chapter 5: Trigonometric Functions of Angles
Chapte 5: Tigonometic Functions of Angles In the pevious chaptes we have exploed a vaiety of functions which could be combined to fom a vaiety of shapes. In this discussion, one common shape has been missing:
In many engineering and other applications, the. variable) will often depend on several other quantities (independent variables).
II PARTIAL DIFFERENTIATION FUNCTIONS OF SEVERAL VARIABLES In man engineeing and othe applications, the behaviou o a cetain quantit dependent vaiable will oten depend on seveal othe quantities independent
Relative motion (Translating axes)
Relative motion (Tanslating axes) Paticle to be studied This topic Moving obseve (Refeence) Fome study Obseve (no motion) bsolute motion Relative motion If motion of the efeence is known, absolute motion
Transformations in Homogeneous Coordinates
Tansfomations in Homogeneous Coodinates (Com S 4/ Notes) Yan-Bin Jia Aug 4 Complete Section Homogeneous Tansfomations A pojective tansfomation of the pojective plane is a mapping L : P P defined as u a
B da = 0. Q E da = ε. E da = E dv
lectomagnetic Theo Pof Ruiz, UNC Asheville, doctophs on YouTube Chapte Notes The Maxwell quations in Diffeential Fom 1 The Maxwell quations in Diffeential Fom We will now tansfom the integal fom of the
Lecture 7: Angular Momentum, Hydrogen Atom
Lectue 7: Angula Momentum, Hydogen Atom Vecto Quantization of Angula Momentum and Nomalization of 3D Rigid Roto wavefunctions Conside l, so L 2 2 2. Thus, we have L 2. Thee ae thee possibilities fo L z
Rectilinea Motion. A foce P is applied to the initially stationay cat. Detemine the velocity and displacement at time t=5 s fo each of the foce histoi
Rectilinea Motion 1. Small objects ae deliveed to the m inclined chute by a conveyo belt A which moves at a speed v 1 =0.4 m/s. If the conveyo belt B has a speed v =0.9 m/s and the objects ae deliveed
1. Show that the volume of the solid shown can be represented by the polynomial 6x x.
7.3 Dividing Polynomials by Monomials Focus on Afte this lesson, you will be able to divide a polynomial by a monomial Mateials algeba tiles When you ae buying a fish tank, the size of the tank depends
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
Radian Measure CHAPTER 5 MODELLING PERIODIC FUNCTIONS
5.4 Radian Measue So fa, ou hae measued angles in degees, with 60 being one eolution aound a cicle. Thee is anothe wa to measue angles called adian measue. With adian measue, the ac length of a cicle is
Chapter 2: Basic Physics and Math Supplements
Chapte 2: Basic Physics and Math Supplements Decembe 1, 215 1 Supplement 2.1: Centipetal Acceleation This supplement expands on a topic addessed on page 19 of the textbook. Ou task hee is to calculate
Motions and Coordinates
Chapte Kinematics of Paticles Motions and Coodinates Motion Constained motion Unconstained motion Coodinates Used to descibe the motion of paticles 1 ectilinea motion (1-D) Motion Plane cuvilinea motion
Class #16 Monday, March 20, 2017
D. Pogo Class #16 Monday, Mach 0, 017 D Non-Catesian Coodinate Systems A point in space can be specified by thee numbes:, y, and z. O, it can be specified by 3 diffeent numbes:,, and z, whee = cos, y =
Physics 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 =
Radian and Degree Measure
CHAT Pe-Calculus Radian and Degee Measue *Tigonomety comes fom the Geek wod meaning measuement of tiangles. It pimaily dealt with angles and tiangles as it petained to navigation, astonomy, and suveying.
Cylindrical and Spherical Coordinate Systems
Clindical and Spheical Coodinate Sstems APPENDIX A In Section 1.2, we leaned that the Catesian coodinate sstem is deined b a set o thee mtall othogonal saces, all o which ae planes. The clindical and spheical
Motion in One Dimension
Motion in One Dimension Intoduction: In this lab, you will investigate the motion of a olling cat as it tavels in a staight line. Although this setup may seem ovesimplified, you will soon see that a detailed
Chapter 10 Sample Exam
Chapte Sample Exam Poblems maked with an asteisk (*) ae paticulaly challenging and should be given caeful consideation.. Conside the paametic cuve x (t) =e t, y (t) =e t, t (a) Compute the length of the
Motion in Two Dimensions
SOLUTIONS TO PROBLEMS Motion in Two Dimensions Section 3.1 The Position, Velocity, and Acceleation Vectos P3.1 x( m) 0!3 000!1 70!4 70 m y( m)!3 600 0 1 70! 330 m (a) Net displacement x + y 4.87 km at
Chapter 12: Kinematics of a Particle 12.8 CURVILINEAR MOTION: CYLINDRICAL COMPONENTS. u of the polar coordinate system are also shown in
ME 01 DYNAMICS Chapte 1: Kinematics of a Paticle Chapte 1 Kinematics of a Paticle A. Bazone 1.8 CURVILINEAR MOTION: CYLINDRICAL COMPONENTS Pola Coodinates Pola coodinates ae paticlaly sitable fo solving
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?
Permutations and Combinations
Pemutations and Combinations Mach 11, 2005 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication Pinciple
transformation Earth V-curve (meridian) λ Conical projection. u,v curves on the datum surface projected as U,V curves on the projection surface
. CONICAL PROJECTIONS In elementay texts on map pojections, the pojection sufaces ae often descibed as developable sufaces, such as the cylinde (cylindical pojections) and the cone (conical pojections),
A Bijective Approach to the Permutational Power of a Priority Queue
A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation
2. 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
OSCILLATIONS AND GRAVITATION
1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,
8 Separation of Variables in Other Coordinate Systems
8 Sepaation of Vaiables in Othe Coodinate Systems Fo the method of sepaation of vaiables to succeed you need to be able to expess the poblem at hand in a coodinate system in which the physical boundaies
Practice Problems Test 3
Pactice Poblems Test ********************************************************** ***NOTICE - Fo poblems involving ʺSolve the Tiangleʺ the angles in this eview ae given by Geek lettes: A = α B = β C = γ
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
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:,
Magnetic Field. Conference 6. Physics 102 General Physics II
Physics 102 Confeence 6 Magnetic Field Confeence 6 Physics 102 Geneal Physics II Monday, Mach 3d, 2014 6.1 Quiz Poblem 6.1 Think about the magnetic field associated with an infinite, cuent caying wie.
10.1 Angles and their Measure
0. Angles and thei Measue This section begins ou stud of Tigonomet and to get stated, we ecall some basic definitions fom Geomet. A a is usuall descibed as a half-line and can be thought of as a line segment
Related Rates - the Basics
Related Rates - the Basics In this section we exploe the way we can use deivatives to find the velocity at which things ae changing ove time. Up to now we have been finding the deivative to compae the
11.2. Area of a Circle. Lesson Objective. Derive the formula for the area of a circle.
11.2 Aea of a Cicle Lesson Objective Use fomulas to calculate the aeas of cicles, semicicles, and quadants. Lean Deive the fomula fo the aea of a cicle. A diamete divides a cicle of adius into 2 semicicles.
Static equilibrium requires a balance of forces and a balance of moments.
Static Equilibium Static equilibium equies a balance of foces and a balance of moments. ΣF 0 ΣF 0 ΣF 0 ΣM 0 ΣM 0 ΣM 0 Eample 1: painte stands on a ladde that leans against the wall of a house at an angle
11.2 Proving Figures are Similar Using Transformations
Name lass ate 11. Poving igues ae Simila Using Tansfomations ssential Question: How can similait tansfomations be used to show two figues ae simila? esouce ocke ploe onfiming Similait similait tansfomation
MATH 417 Homework 3 Instructor: D. Cabrera Due June 30. sin θ v x = v r cos θ v θ r. (b) Then use the Cauchy-Riemann equations in polar coordinates
MATH 417 Homewok 3 Instucto: D. Cabea Due June 30 1. Let a function f(z) = u + iv be diffeentiable at z 0. (a) Use the Chain Rule and the fomulas x = cosθ and y = to show that u x = u cosθ u θ, v x = v
Foundations of Trigonometry
Chapte 0 Foundations of Tigonomet 0. Angles and thei Measue This section begins ou stud of Tigonomet and to get stated, we ecall some basic definitions fom Geomet. A a is usuall descibed as a half-line
Describing Circular motion
Unifom Cicula Motion Descibing Cicula motion In ode to undestand cicula motion, we fist need to discuss how to subtact vectos. The easiest way to explain subtacting vectos is to descibe it as adding a
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.
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
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
Kinematics of rigid bodies
Kinematics of igid bodies elations between time and the positions, elocities, and acceleations of the paticles foming a igid body. (1) Rectilinea tanslation paallel staight paths Cuilinea tanslation (3)
Introduction to Systems of Differential Equations
Chapte 4 Intoduction to Systes of Diffeential Equations Poject 4.1 Keple's aws and Planetay Obits The Section 4.1 poject in the text stats with Newton's invese-squae law of gavitation and outlines a deivation
RECTIFYING THE CIRCUMFERENCE WITH GEOGEBRA
ECTIFYING THE CICUMFEENCE WITH GEOGEBA A. Matín Dinnbie, G. Matín González and Anthony C.M. O 1 Intoducction The elation between the cicumfeence and the adius of a cicle is one of the most impotant concepts
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
15 Solving the Laplace equation by Fourier method
5 Solving the Laplace equation by Fouie method I aleady intoduced two o thee dimensional heat equation, when I deived it, ecall that it taes the fom u t = α 2 u + F, (5.) whee u: [0, ) D R, D R is the
2 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,
The Archimedean Circles of Schoch and Woo
Foum Geometicoum Volume 4 (2004) 27 34. FRUM GEM ISSN 1534-1178 The Achimedean Cicles of Schoch and Woo Hioshi kumua and Masayuki Watanabe Abstact. We genealize the Achimedean cicles in an abelos (shoemake
Physics 11 Chapter 4: Forces and Newton s Laws of Motion. Problem Solving
Physics 11 Chapte 4: Foces and Newton s Laws of Motion Thee is nothing eithe good o bad, but thinking makes it so. William Shakespeae It s not what happens to you that detemines how fa you will go in life;
AY 7A - Fall 2010 Section Worksheet 2 - Solutions Energy and Kepler s Law
AY 7A - Fall 00 Section Woksheet - Solutions Enegy and Keple s Law. Escape Velocity (a) A planet is obiting aound a sta. What is the total obital enegy of the planet? (i.e. Total Enegy = Potential Enegy