Algebra and Trig. I. A point is a location or position that has no size or dimension.


 Debra Arnold
 1 years ago
 Views:
Transcription
1 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 amount of points. A plane is a flat, smooth suface that extends indefinitely in all diections, and contains an infinite numbe of points and lines. Plane A A B Line Segment AB AB Ray AB AB A B A line segment o segment is pat of a line and stats and stops at distinct points called endpoints. A ay consists of a point on a line and all points of the line on one side of the point. An angle is fomed by two ays (o lines) that have a common endpoint. One ay is called the initial side and the othe the teminal side. An acute angle: (0 <θ<90 ) An obtuse angle: (90 <θ<180 ) A ight angle: (θ=90, otation) A staight angle: (θ=180, otation) 1 P a g e
2 The clock to the left shows the hou hand (initial side) at twelve and the minute hand (teminal side) at 3. These two hands ae ays and they fom an angle. The common endpoint of the two ays is called the vetex of the angle. C Teminal Side θ B A Vetex Initial Side Angles ae often labeled with lowecase lette Geek lettes, such as alpha(α), beta(β), gamma(γ), and theta(θ). An angle is in standad position if  its vetex is at the oigin of a ectangula coodinate system  its initial side lies along the positive xaxis. α θ α is in standad position α is positive θ is in standad position θ is negative 2 P a g e
3 An angle is positive if geneated by counteclockwise otation. An angle is negative if geneated by clockwise otation. Measuing Angles A degee is a unit of measuing angles. It epesents of a complete otation about the vetex. (Thee ae 360 degees (360 ) in a complete otation o cicle). 90 A ight angle is an angle that measues 90 degees (90 ). Two lines that intesect to fom a ight angle ae pependicula lines. A small squae in the cone of the angle is used to indicate a ight angle. 180 A staight line has an angle measue of 180 degees (180 ). A vetical line uns up and down, and a hoizontal line uns left and ight. Angle Names: Acute Angle measues less than 90, but moe than 0 Staight Angle measues 180 Obtuse Angle measues moe than 90, but less than 180 Right Angle measues 90 Factional pats of angles ae measued in minutes and seconds.  One minute, witten 1 is degee: 1 =  One second, witten 1 is degee: 1 = Example 3 P a g e
4 Many calculatos ae able to change fom degeeminutesecond notation (D M S ) to a decimal and vice a vesa. Calculato: 2 nd Apps Measuing Angles in Radians Anothe way to measue an angle is in adians. Teminal Side One adian Initial Side One adian is the measue of the cental angle of a cicle that intecepts an ac equal to the lengths to the adius of the cicle. (the adius of the cicle is ) A cental angle is an angle whose vetex is at the cente of the cicle. β We find the length of an angle in adians by dividing the length of the intecepted ac by the adius. γ Radian Measue Conside an ac of length s on a cicle of adius. The measue of the cental angle, θ, that intecepts the ac is 4 P a g e
5 Example Find the adian measue of θ if the ac length is 15 inches and the adius is 6 inches? Example Find the adian measue of θ if the ac length is 42 inches and the adius is 12 inches? Relationship between Degees and Radians We know a full otation aound a cicle has 360 and we know that the cicumfeence of a cicle with adius,, is 2π. Thus the adian measue of a cental angle is the cicumfeence of the cicle divided by the cicles adius,. We use the fomula fo adian measue to find the adian measue of the 360 angle. Because one complete otation measues 360 and 2π adians, 360 = 2π adians Dividing both sides by 2, we have 180 = π adians 5 P a g e
6 Convesion between degees and adians: Using the basic elationship π adians = 180, 2 π adians = To convet degees to adians, multiply degees by  To convet adians to degees, multiply adians by Angles that ae factions of a complete otation ae usually expessed in adian measue as factional multiples of π, athe than decimal appoximations. Fo example θ = athe than using the decimal appoximation θ 1.57 Example Convet each angle in degees to adians Example Convet each angle in adians to degees P a g e
7 Dawing Angles in Standad Position To become comfotable with adian measue, conside angles in standad position. Each oigin is the vetex and each initial side is along the positive xaxis. Think of the teminal side as the side of the angle as evolving aound the oigin. Example Dawing angles in standad position; Note one way to do this is to convet to degees P a g e
8 Teminal Side Radian Measue of Angle Degee Measue of Angle 8 P a g e
9 The gaph below shows what is called the unit cicle. It contains the degee measuements and adian measuements Recall that the xaxis is initial side so when moving counteclockwise the angles ae positive, and when moving clockwise the angles ae negative, so instead of having 330 we would have 30, instead of 315 we would have 45, instead of 300 we would have 60, and so on. Also the adian measues would change also so instead of we would have, instead of we would have, and so on. 9 P a g e
10 Two angles with the same initial sides but possibly diffeent otations ae called coteminal angles. Evey angle has infinitely many coteminal angles. Coteminal Angles Inceasing o deceasing the degee of an angle in standad position by an intege multiply of 360 esults in a coteminal angle. Thus an angle of θ is coteminal with angles of θ ± 360 k, whee k is an intege. Inceasing o deceasing the adian measue of an angle in standad position by an intege multiply of 2π esults in a coteminal angle. Thus an angle of θ adians is coteminal with angles of θ ± 2πk, whee k is an intege. Two coteminal angles fo an angle of θ can be found by adding 360 to θ and by subtacting 360 fom θ. Example Assuming the following angles ae in standad position. Find a positive angle less than 360 that is coteminal with each of the following. 1. a 420 angle 2. a 120 angle 10 P a g e
11 Example Assuming the following angles ae in standad position. Find a positive angle less than 2π that is coteminal with each of the following. 1. a 2. a To find a positive coteminal angle less than 360 o 2π, it is sometimes necessay to add o subtact moe than one multiple of 360 o 2π. 11 P a g e
12 Example Assuming the following angles ae in standad position. Find a positive angle less than 360 o 2π that is coteminal with each of the following. 1. a 2. a 3. a 12 P a g e
13 θ x=ac length The Length of a Cicula Ac Let be the adius of a cicle and θ the nonnegative adian measue of a cental angle of the cicle. The length of the ac intecepted by the cental angle is Example A cicle has a adius a 10 inches. Find the length of the ac intecepted by a cental angle of 120. Example A cicle has a adius a 6 inches. Find the length of the ac intecepted by a cental angle of 45. Expess ac length in tems of π. Then ound you answe to the neaest hundeds. 13 P a g e
14 Linea and Angula Speed Think of a caousel it contains fou cicula ows of animals. As the caousel evolves, the animals in the oute ow tavel a geate distance pe unit time than those in the inne ows. By contast, all animals, egadless of ow, complete the same numbe of evolutions pe unit time. All animals in the fou ows tavel at the same angula speed. Linea and Angula Speed If a point is in motion on a cicle of adius though an angle of θ adians in time t, then its linea speed is, whee s is the ac length given by, and its angula speed is. Example If the had dive in a compute otates at 3600 otations pe minute. Expess the angula speed of a had dive in adians pe minute. (Note: 1 evolution = 2π adians) We can establish a elationship between linea speed and angula speed, by dividing both sides of the ac length fomula, by t Thus, linea speed is the poduct of the adius and the angula speed. 14 P a g e
15 Note we can wite linea speed in tems of angula speed. Recall and and. Example A windmill is used to geneate electicity has blades that ae 10 feet in length. The popelle is otating aound at 4 evolutions pe second. Find the linea speed, in feet pe second of the tips of the blades. 15 P a g e
TRIGONOMETRY REVIEW. The Cosines and Sines of the Standard Angles
TRIGONOMETRY REVIEW The Cosines and Sines of the Standad Angles P θ = ( cos θ, sin θ ) . ANGLES AND THEIR MEASURE In ode to define the tigonometic functions so that they can be used not only fo tiangula
More informationSection 53 Angles and Their Measure
5 5 TRIGONOMETRIC FUNCTIONS Section 5 Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,
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 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 informationTrigonometric Functions of Any Angle
Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008 Tigonometic Functions of An
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 informationIn order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.
Radians At school we usually lean to measue an angle in degees. Howeve, thee ae othe ways of measuing an angle. One that we ae going to have a look at hee is measuing angles in units called adians. In
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 information2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90
. Tigonometic Ratios of An Angle Focus on... detemining the distance fom the oigin to a point (, ) on the teminal am of an angle detemining the value of sin, cos, o tan given an point (, ) on the teminal
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 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 informationSo we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)
Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing
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 informationGeneral Physics (PHY 2130)
Geneal Physics (PHY 130) Lectue 11 Rotational kinematics and unifom cicula motion Angula displacement Angula speed and acceleation http://www.physics.wayne.edu/~apetov/phy130/ Lightning Review Last lectue:
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 8, Rotational Kinematics. Angular Displacement
Chapte 8, Rotational Kinematics Sections 1 3 only Rotational motion and angula displacement Angula velocity and angula acceleation Equations of otational kinematics 1 Angula Displacement! B l A The length
More informationChapter 3: Vectors and Coordinate Systems
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
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 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 informationSolutions to Homework Set #5 Phys2414 Fall 2005
Solution Set #5 1 Solutions to Homewok Set #5 Phys414 Fall 005 Note: The numbes in the boxes coespond to those that ae geneated by WebAssign. The numbes on you individual assignment will vay. Any calculated
More informationTh Po er of th Cir l. Lesson3. Unit UNIT 6 GEOMETRIC FORM AND ITS FUNCTION
Lesson3 Th Po e of th Ci l Quadilateals and tiangles ae used to make eveyday things wok. Right tiangles ae the basis fo tigonometic atios elating angle measues to atios of lengths of sides. Anothe family
More informationSECTION 53 Angles and Their Measure
53 Angle and Thei Meaue 357 APPLICATIONS Appoximating. Poblem 93 and 9 efe to a equence of numbe geneated a follow: If an nd egula polygon i incibed in a cicle of adiu, then it can be hown that the aea
More informationCircumference and Arc Length
11.4 icumfeence and c Length efoe Now You found the cicumfeence of a cicle. You will find ac lengths and othe measues. Why? So you can find a unning distance, as in Example 5. Key Vocabulay cicumfeence
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 informationDisplacement, 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 informationTALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS 14. NEWTON'S RINGS
4. NEWTON'S RINGS. Obective Detemining adius of cuvatue of a long focal length planoconvex lens (lage adius of cuvatue).. Equipment needed Measuing micoscope, planoconvex long focal length lens, monochomatic
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 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 informationWrite and Graph Equations of Circles
0.7 Wite and Gaph Equations of icles Befoe You wote equations of lines in the coodinate plane. Now You will wite equations of cicles in the coodinate plane. Wh? So ou can detemine zones of a commute sstem,
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 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 informationLesson 32: Measuring Circular Motion
Lesson 32: Measuing Cicula Motion Velocity hee should be a way to come up with a basic fomula that elates velocity in icle to some of the basic popeties of icle. Let s ty stating off with a fomula that
More informationSeventh Edition DYNAMICS 15Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University
Seenth 15Fedinand P. ee E. Russell Johnston, J. Kinematics of Lectue Notes: J. Walt Ole Texas Tech Uniesity Rigid odies CHPTER VECTOR MECHNICS FOR ENGINEERS: YNMICS 003 The McGawHill Companies, Inc. ll
More informationLab 5: Circular Motion
Lab 5: Cicula motion Physics 193 Fall 2006 Lab 5: Cicula Motion I. Intoduction The lab today involves the analysis of objects that ae moving in a cicle. Newton s second law as applied to cicula motion
More informationExam I. Spring 2004 Serway & Jewett, Chapters 15. Fill in the bubble for the correct answer on the answer sheet. next to the number.
Agin/Meye PART I: QUALITATIVE Exam I Sping 2004 Seway & Jewett, Chaptes 15 Assigned Seat Numbe Fill in the bubble fo the coect answe on the answe sheet. next to the numbe. NO PARTIAL CREDIT: SUBMIT ONE
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 informationSamples of conceptual and analytical/numerical questions from chap 21, C&J, 7E
CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known
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 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 information2008 QuarterFinal Exam Solutions
2008 Quatefinal Exam  Solutions 1 2008 QuateFinal Exam Solutions 1 A chaged paticle with chage q and mass m stats with an initial kinetic enegy K at the middle of a unifomly chaged spheical egion of
More informationmv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2 " 2 = GM . Combining the results we get !
Chapte. he net foce on the satellite is F = G Mm and this plays the ole of the centipetal foce on the satellite i.e. mv mv. Equating the two gives = G Mm i.e. v = G M. Fo cicula motion we have that v =!
More informationReview Module: Dot Product
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics 801 Fall 2009 Review Module: Dot Poduct We shall intoduce a vecto opeation, called the dot poduct o scala poduct that takes any two vectos and
More informationK.S.E.E.B., Malleshwaram, Bangalore SSLC MathematicsModel Question Paper1 (2015) Regular Private Candidates (New Syllabus)
K.S.E.E.B., Malleshwaam, Bangaloe SSLC MathematicsModel Question Pape1 (015) Regula Pivate Candidates (New Syllabus) Max Maks: 100 No. of Questions: 50 Time: 3 Hous Code No. : Fou altenatives ae given
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 informationReview of Coordinate Systems
Review o Coodinate Sstems good undestanding o coodinate sstems can be ve helpul in solving poblems elated to Mawell s Equations. The thee most common coodinate sstems ae ectangula (,, ), clindical (,,
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 informationThe Grating Spectrometer and Atomic Spectra
PHY 19 Gating Spectomete 1 The Gating Spectomete and Atomic Specta Intoduction In the pevious expeiment diffaction and intefeence wee discussed and at the end a diffaction gating was intoduced. In this
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 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 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 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 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 informationOriginally TRIGONOMETRY was that branch of mathematics concerned with solving triangles using trigonometric ratios which were seen as properties of
Oiginall TRIGONOMETRY was that banch of mathematics concened with solving tiangles using tigonometic atios which wee seen as popeties of tiangles athe than of angles. The wod Tigonomet comes fom the Geek
More informationUniversal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool
Univesal Cycles 2011 Yu She Wial Gamma School fo Gils Depatment of Mathematical Sciences Univesity of Livepool Supeviso: Pofesso P. J. Giblin Contents 1 Intoduction 2 2 De Buijn sequences and Euleian Gaphs
More information1.1 KINEMATIC RELATIONSHIPS
1.1 KINEMATIC RELATIONSHIPS Thoughout the Advanced Highe Physics couse calculus techniques will be used. These techniques ae vey poweful and knowledge of integation and diffeentiation will allow a deepe
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 informationRadian Measure and Dynamic Trigonometry
cob980_ch0_089.qd 0//09 7:0 Page 89 Debd MHDQNew:MHDQ:MHDQ.: CHAPTER CONNECTIONS Radian Measue and Dnamic Tigonomet CHAPTER OUTLINE. Angle Measue in Radians 90. Ac Length, Velocit, and the Aea of a
More information4.1 Cylindrical and Polar Coordinates
4.1 Cylindical and Pola Coodinates 4.1.1 Geometical Axisymmety A lage numbe of pactical engineeing poblems involve geometical featues which have a natual axis of symmety, such as the solid cylinde, shown
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 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 informationCHAPTER 4 POSITION, VELOCITY AND ACCELERATION ANALYSES FOR PLANAR MECHANISMS USING COMPLEX NUMBER METHOD
CHPTER POSITION, VELOCITY ND CCELERTION NLYSES FOR PLNR MECHNISMS USING COMPLEX NUMER METHOD Vecto nalysis: Fo the position vectos shown below, the positive angle is measued counteclock wise (ccw) fom
More informationChapter 23: Gauss s Law
Chapte 3: Gauss s Law Homewok: Read Chapte 3 Questions, 5, 1 Poblems 1, 5, 3 Gauss s Law Gauss s Law is the fist of the fou Maxwell Equations which summaize all of electomagnetic theoy. Gauss s Law gives
More informationMagnetism: a new force!
1 Magnetism: a new foce! o fa, we'e leaned about two foces: gaity and the electic field foce. F E = E, FE = E Definition of Efield kq Efields ae ceated by chages: E = 2 Efield exets a foce on othe
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 informationNew proofs for the perimeter and area of a circle
New poofs fo the peimete and aea of a cicle K. Raghul Kuma Reseach Schola, Depatment of Physics, Nallamuthu Gounde Mahalingam College, Pollachi, Tamil Nadu 64001, India 1 aghul_physics@yahoo.com aghulkumak5@gmail.com
More informationSurface Area of a Cone
8.4 Suface Aea of a Cone cone a theedimenional object with a cicula bae and a cuved lateal uface that extend fom the bae to a point called the vetex A cone i a familia hape to mot people. Many of u lean
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 information9. Mathematics Practice Paper for Class XII (CBSE) Available Online Tutoring for students of classes 4 to 12 in Physics, Chemistry, Mathematics
Available Online Tutoing fo students of classes 4 to 1 in Physics, 9. Mathematics Class 1 Pactice Pape 1 3 1. Wite the pincipal value of cos.. Wite the ange of the pincipal banch of sec 1 defined on the
More informationXIIth PHYSICS (C2, G2, C, G) Solution
XIIth PHYSICS (C, G, C, G) 6 Solution. A 5 W, 0 V bulb and a 00 W, 0 V bulb ae connected in paallel acoss a 0 V line nly 00 watt bulb will fuse nly 5 watt bulb will fuse Both bulbs will fuse None of
More informationPhysics 111 Fall 2007 Electrostatic Forces and the Electric Field  Solutions
Physics 111 Fall 007 Electostatic Foces an the Electic Fiel  Solutions 1. Two point chages, 5 µc an 8 µc ae 1. m apat. Whee shoul a thi chage, equal to 5 µc, be place to make the electic fiel at the
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 informationChapter 22 The Electric Field II: Continuous Charge Distributions
Chapte The lectic Field II: Continuous Chage Distibutions 1 [M] A unifom line chage that has a linea chage density l equal to.5 nc/m is on the x axis between x and x 5. m. (a) What is its total chage?
More informationWeek 34: Permutations and Combinations
Week 34: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication
More informationSet 3: 120 N/m units 0.25 Kg. 1A The unit of the damping constant is a) Kg/s b) N/m c) N d) Joule/s e) NA
05/05/04 PHYSICS 3 Exa #1 NAME Please wite down you nae also on the back side of this exa 1. Hee ae thee sets of values fo the sping constant k, daping constant b, and ass fo thee daped oscillatos: Set
More informationThank you for participating in Teach It First!
Thank you fo paticipating in Teach It Fist! This Teach It Fist Kit contains a Common Coe Suppot Coach, Foundational Mathematics teache lesson followed by the coesponding student lesson. We ae confident
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 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 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 informationPY1052 Problem Set 3 Autumn 2004 Solutions
PY1052 Poblem Set 3 Autumn 2004 Solutions C F = 8 N F = 25 N 1 2 A A (1) A foce F 1 = 8 N is exeted hoizontally on block A, which has a mass of 4.5 kg. The coefficient of static fiction between A and the
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 informationExperiment MF Magnetic Force
Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuentcaying conducto is basic to evey electic moto  tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating
More informationIn order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.
Radians mctyadians20091 Atschoolweusuallyleantomeasueanangleindegees. Howeve,theeaeothewaysof measuinganangle. Onethatweaegoingtohavealookatheeismeasuinganglesinunits called adians. In many scientific
More informationPhysics 107 HOMEWORK ASSIGNMENT #14
Physics 107 HOMEWORK ASSIGNMENT #14 Cutnell & Johnson, 7 th edition Chapte 17: Poblem 44, 60 Chapte 18: Poblems 14, 18, 8 **44 A tube, open at only one end, is cut into two shote (nonequal) lengths. The
More informationF = kq 1q 2 r 2. F 13 = k( q)(2q) 2a 2 cosθˆx + sinθŷ F 14 = k( 2q)(2q) F 12 = k(q)(2q) a 2. tanθ = a a
.1 What ae the hoizontal and vetical components of the esultant electostatic foce on the chage in the lowe left cone of the squae if q =1. 1 7 and a =5.cm? +q q a +q a q F = kq 1q F 1 = k(q)(q) a F 13
More informationThe Critical Angle and Percent Efficiency of Parabolic Solar Cookers
The Citical Angle and Pecent Eiciency o Paabolic Sola Cookes Aiel Chen Abstact: The paabola is commonly used as the cuve o sola cookes because o its ability to elect incoming light with an incoming angle
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 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 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 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 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 information11.5 Graphs of Polar Equations
9 Applications of Tigonomet.5 Gaphs of Pola Equations In this section, we discuss how to gaph equations in pola coodinates on the ectangula coodinate plane. Since an given point in the plane has infinitel
More informationIntroduction to Fluid Mechanics
Chapte 1 1 1.6. Solved Examples Example 1.1 Dimensions and Units A body weighs 1 Ibf when exposed to a standad eath gavity g = 3.174 ft/s. (a) What is its mass in kg? (b) What will the weight of this body
More informationChapter 26  Electric Field. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University
Chapte 6 lectic Field A PowePoint Pesentation by Paul. Tippens, Pofesso of Physics Southen Polytechnic State Univesity 7 Objectives: Afte finishing this unit you should be able to: Define the electic field
More informationThe Prime Coe cient Formulas for Rotation of Conics
The Pime Coe cient Fomulas fo Rotation of Conics It is known that the gah of a conic section in the x; y coodinate lane has a second degee equation in the vaiables x and y: I.e., Ax + xy +Cy +Dx+Ey +F
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 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 informationL19 Geomagnetic Field Part I
Intoduction to Geophysics L191 L19 Geomagnetic Field Pat I 1. Intoduction We now stat the last majo topic o this class which is magnetic ields and measuing the magnetic popeties o mateials. As a way o
More information1. CIRCULAR MOTION. ω =
1. CIRCULAR MOION 1. Calculate the angula elocity and linea elocity of a tip of minute hand of length 1 cm. 6 min. 6 6 s 36 s l 1 cm.1 m ω?? Fomula : ω π ω ω π 3.14 36 ω 1.744 1 3 ad/s ω.1 1.74 1 3 1.74
More information4.1: Angles and Radian Measure
4.1: Angles and Radian Measure An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other is called the terminal side. The endpoint that they share is
More information