Section 53 Angles and Their Measure


 Kristian Summers
 1 years ago
 Views:
Transcription
1 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, degee and adian. FIGURE Angle o angle PVQ o V. n Q Teminal V Initial P m Angles The study of tigonomety depends on the concept of angle. An angle is fomed by otating a half line, called a ay, aound its endpoint. One ay m, called the initial of the angle, emains fixed; a second ay n, called the teminal of the angle, stats in the initial position and otates aound the common endpoint V in a plane until it eaches its teminal position. The common endpoint V is called the vetex (see Fig. ). A counteclockwise otation poduces a positive angle, and a clockwise otation poduces a negative angle, as shown in Figues (a) and (b). The amount of otation in eithe diection is not esticted. Two diffeent angles may have the same initial and teminal s, as shown in Figue (c). Such angles ae said to be coteminal. FIGURE Angles and otation. Teminal Initial Teminal Teminal Initial negative positive Initial and coteminal (a) (b) (c) An angle in a ectangula coodinate system is said to be in standad position if its vetex is at the oigin and the initial is along the positive x axis. If the teminal of an angle in standad position lies along a coodinate axis, the angle is said to be a quadantal angle. If the teminal does not lie along a coodinate axis, then the angle is often efeed to in tems of the quadant in which the teminal lies (Fig. ). FIGURE Angles in standad positions. II III Teminal y Initial I IV x II III Teminal y Initial is a quadantal angle is a thidquadant angle is a secondquadant angle (a) (b) (c) I IV x II III Teminal y Initial I IV x
2 Degees and Radian Measue 5 Angles and Thei Measue 55 Just as line segments ae measued in centimetes, metes, inches, o miles, angles ae measued in diffeent units. The two most commonly used units fo angle measue ae degee and adian. DEFINITION DEGREE MEASURE An angle fomed by one complete otation is said to have a measue of 60 degees (60 ). An angle fomed by 60 of a complete otation is said to have a measue of degee ( ). The symbol denotes degees. Cetain angles have special names. Figue shows a staight angle, a ight angle, an acute angle, and an obtuse angle. FIGURE Types of angles Staight angle Right angle Acute angle Obtuse angle otation otation (0 90) (90 80) (a) (b) (c) (d) Two positive angles ae complementay if thei sum is 90 ; they ae supplementay if thei sum is 80. A degee can be divided futhe using decimal notation. Fo example,.75 epesents an angle of degee measue plus theequates of degee. A degee can also be divided futhe using minutes and seconds just as an hou is divided into minutes and seconds. Each degee is divided into 60 equal pats called minutes, and each minute is divided into 60 equal pats called seconds. Symbolically, minutes ae epesented by and seconds by. Thus, is a concise way of witing degees, minutes, and seconds. Decimal degees (DD) ae useful in some instances and degees minutes seconds (DMS) ae useful in othes. You should be able to go fom one fom to the othe as demonstated in Example. CONVERSION ACCURACY If an angle is measued to the neaest second, the conveted decimal fom should not go beyond thee decimal places, and vice vesa.
3 56 5 TRIGONOMETRIC FUNCTIONS EXAMPLE Fom DMS to DD and Back (A) Convet 7 to decimal degees. (B) Convet 05.8 to degee minute second fom. Solutions (A) , (B) (0.8.60) ( ) MATCHED PROBLEM (A) Convet 9 7 to DD fom. (B) Convet 7.65 to DMS fom. Some scientific and some gaphic calculatos can convet the DD and DMS foms automatically, but the pocess diffes significantly among the vaious types of calculatos. Check you owne s manual fo you paticula calculato. The convesion methods outlined in Example show you the easoning behind the pocess, and ae sometimes easie to use than the automatic methods fo some calculatos. Degee measue of angles is used extensively in engineeing, suveying, and navigation. Anothe unit of angle measue, called the adian, is bette suited fo cetain mathematical developments, scientific wok, and engineeing applications. DEFINITION RADIAN MEASURE If the vetex of an angle is placed at the cente of a cicle with adius 0, and the length of the ac opposite on the cicumfeence is s, then the adian measue of is given by Also, s adians s s O
4 5 Angles and Thei Measue 57 DEFINITION continued If s, then adian Thus, one adian is the size of the cental angle of a cicle that intecepts an ac the same length as the adius of the cicle. [Note: s and must be measued in the O adian same units. Also note that is being used in two ways: is the name of an angle and is the measue of the angle. The context detemines the choice. Thus, when we wite s/, we mean the adian measue of angle is s/.] s EXAMPLE Solution Computing Radian Measue What is the adian measue of a cental angle opposite an ac of metes in a cicle of adius 6 metes? s metes 6 metes adians MATCHED PROBLEM Remak What is the adian measue of a cental angle opposite an ac of 60 feet in a cicle of adius feet? Radian measue is a unitless numbe. The units in which the ac length and adius ae measued cancel; hence, we ae left with a unitless, o pue, numbe. Fo this eason, the wod adian is often omitted when we ae dealing with the adian measue of angles unless a special emphasis is desied. Exploe/Discuss Discuss why the adian measue of an angle is independent of the size of the cicle having the angle as a cental angle. Fom Degees to Radians and Vice Vesa What is the adian measue of an angle of 80? A cental angle of 80 is subtended by an ac of the cicumfeence of a cicle. Thus, if C is the cicumfeence of a cicle, then of the cicumfeence is given by s C and s ad
5 58 5 TRIGONOMETRIC FUNCTIONS Hence, 80 coesponds to * ad. This is impotant to emembe, since the adian measues of many special angles can be obtained fom this coespondence. Fo example, 90 is 80 /; theefoe, 90 coesponds to / ad. Exploe/Discuss Wite the adian measue of each of the following angles in the fom a a, whee a and b ae positive integes and faction is educed to b b lowest tems: 5, 0, 5, 60, 75, 90, 05, 0, 5, 50, 65, 80. Some key esults fom Exploe/Discuss ae summaized in Figue 5 fo easy efeence. These coespondences and multiples of them will be used extensively in wok that follows. FIGURE 5 Radian degee coespondences / 60 / 5 / 0 / / In geneal, the following popotion can be used to convet degee measue to adian measue and vice vesa. RADIAN DEGREE CONVERSION FORMULAS deg 80 ad ad Radians to degees deg 80 ad ad o ad ad 80 deg Degees to adians [Note: The popotion on the left is usually easie to emembe. Also we will omit units in calculations until the final answe. If you calculato does not have a key labeled, use.59.] *The constant has a long and inteesting histoy; a few impotant dates ae listed below: 650 B.C. Rhind Papyus 56 0 B.C. Achimedes ( ) A.D. 6 Liu Hui.59 A.D. 70 Tsu Ch ungchih A.D. 67 Leibniz ( ) (This and othe seies can be used to compute to any decimal accuacy desied.) A.D. 76 Johann Lambet Showed to be iational ( as a decimal is nonepeating and nonteminating)
6 5 Angles and Thei Measue 59 Some scientific and gaphic calculatos can automatically convet adian measue to degee measue, and vice vesa. Check the owne s manual fo you paticula calculato. EXAMPLE Radian Degee Convesions (A) Find the adian measue, exact and to thee significant digits, of an angle of 75. (B) Find the degee measue, exact and to fou significant digits, of an angle of 5 adians. (C) Find the adian measue to two decimal places of an angle of. Solutions Exact Thee significant digits ad 5 (A) ad. 80 deg (75) 80 FIGURE 6 Automatic convesion. Exact Fou significant digits (B) 900 deg 80 ad ad 80 (5) 86.5 (C) 60. ad ad 80 deg (.) Change to DD fist. To two decimal places Figue 6 shows the thee peceding convesions done automatically on a gaphing calculato. MATCHED PROBLEM (A) Find the adian measue, exact and to thee significant digits, of an angle of 0. (B) Find the degee measue, exact and to thee significant digits, of an angle of adian. (C) Find the adian measue to thee decimal places of an angle of 5. Remak We will wite in place of deg and ad when it is clea fom the context whethe we ae dealing with degee o adian measue. EXAMPLE Engineeing A belt connects a pulley of inch adius with a pulley of 5inch adius. If the lage pulley tuns though 0 adians, though how many adians will the smalle pulley tun?
7 60 5 TRIGONOMETRIC FUNCTIONS Solution Fist we daw a sketch (Fig. 7). FIGURE 7 P 5 in. Q in. When the lage pulley tuns though 0 adians, the point P on its cicumfeence will tavel the same distance (ac length) that point Q on the smalle cicle tavels. Fo the lage pulley, s s (5)(0) 50 inches Fo the smalle pulley, s 50 5 adians MATCHED PROBLEM In Example, though how many adians will the lage pulley tun if the smalle pulley tuns though adians? Answes to Matched Poblems 80. (A) 9.9 (B) ad. (A).9 (B) 57. (C) ad EXERCISE 5 In all poblems, if angle measue is expessed by a numbe that is not in degees, it is assumed to be in adians. A Find the degee measue of each of the angles in Poblems 6, keeping in mind that an angle of one complete otation coesponds to otation. 5 otation. otation. 8 otation 5. 8 otations 6. Find the adian measue of a cental angle opposite an ac s in a cicle of adius, whee and s ae as given in Poblems centimetes, s centimetes 8. 8 inches, s 6 inches 9. feet, s 0 feet 0. 8 metes, s 7 metes otations
8 5 Angles and Thei Measue 6 Find the adian measue of each angle in Poblems 6, keeping in mind that an angle of one complete otation coesponds to adians.. 8 otation. 6 otation. 5. otation 5. otations 6. B Find the exact adian measue, in tems of, of each angle in Poblems , 60, 90, 0, 50, , 0, 80, 0, 00, , 90, 5, , 80, 70, 60 Find the exact degee measue of each angle in Poblems...,,,, 5,..,,,,,, Convet each angle in Poblems 5 8 to decimal degees to thee decimal places Convet each angle in Poblems 9 to degee minute second fom ,,,, 5 6, Find the adian measue to thee decimal places fo each angle in Poblems Find the degee measue to two decimal places fo each angle in Poblems otation 8 otations Indicate whethe each angle in Poblems 5 6 is a I, II, III, o IV quadant angle o a quadantal angle. All angles ae in standad position in a ectangula coodinate system. (A sketch may be of help in some poblems.) Vebally descibe the meaning of a cental angle in a cicle with adian measue. 66. Vebally descibe the meaning of an angle with degee measue. C Which angles in Poblems 67 7 ae coteminal with 0 if all angles ae placed in standad position in a ectangula coodinate system? Which angles in poblems 7 78 ae coteminal with / if all angles ae placed in standad position in a ectangula coodinate system? APPLICATIONS 79. Cicumfeence of the Eath. The ealy Geeks used the popotion s/c /60, whee s is an ac length on a cicle, is degee measue of the coesponding cental angle, and C is the cicumfeence of the cicle (C ). Eatosthenes (0 B.C.), in his famous calculation of the cicumfeence of the eath, easoned as follows: He knew at Syene (now Aswan) duing the summe solstice the noon sun was diectly ovehead and shined on the wate 6
9 6 5 TRIGONOMETRIC FUNCTIONS staight down a deep well. On the same day at the same time, 5,000 stadia (appox. 500 mi) due noth in Alexandia, sun ays cossed a vetical pole at an angle of 7.5 as indicated in the figue. Cay out Eatosthenes calculation fo the cicumfeence of the eath to the neaest thousand miles. (The cuent calculation fo the equatoial cicumfeence is,90 mi.) Eath Alexandia 7.5 Syene Well Sun ays 80. Cicumfeence of the Eath. Repeat Poblem 79 with the sun cossing the vetical pole in Alexandia at Cicumfeence of the Eath. In Poblem 79, vebally explain how in the figue was detemined. 8. Cicumfeence of the Eath. Vebally explain how the adius, suface aea, and volume of the eath can be detemined fom the esult of Poblem Radian Measue. What is the adian measue of the lage angle made by the hands of a clock at :0? Expess the answe exactly in tems of. 8. Radian Measue. What is the adian measue of the smalle angle made by the hands of a clock at :0? Expess the answe exactly in tems of. 85. Engineeing. Though how many adians does a pulley of 0centimete diamete tun when 0 metes of ope ae pulled though it without slippage? 86. Engineeing. Though how many adians does a pulley of 6inch diamete tun when feet of ope ae pulled though it without slippage? 87. Astonomy. A line fom the sun to the Eath sweeps out an angle of how many adians in week? Assume the Eath s obit is cicula and thee ae 5 weeks in a yea. Expess the answe in tems of and as a decimal to two decimal places. 88. Astonomy. A line fom the cente of the Eath to the equato sweeps out an angle of how many adians in 9 hous? Expess the answe in tems of and as a decimal to two decimal places. 89. Engineeing. A tail bike has a font wheel with a diamete of 0 centimetes and a back wheel of diamete 60 centimetes. Though what angle in adians does the font wheel tun if the back wheel tuns though 8 adians? 90. Engineeing. In Poblem 89, though what angle in adians will the back wheel tun if the font wheel tuns though 5 adians? The ac length on a cicle is easy to compute if the coesponding cental angle is given in adians and the adius of the cicle is known (s ). If the adius of a cicle is lage and a cental angle is small, then an ac length is often used to appoximate the length of the coesponding chod as shown in the figue. If an angle is given in degee measue, conveting to adian measue fist may be helpful in cetain poblems. This infomation will be useful in Poblems 9 9. c s 9. Astonomy. The sun is about mi fom the eath. If the angle subtended by the diamete of the sun on the suface of the eath is 9. 0 ad, appoximately what is the diamete of the sun to the neaest thousand miles in standad decimal notation? 9. Astonomy. The moon is about 8,000 km fom the eath. If the angle subtended by the diamete of the moon on the suface of the eath is ad, appoximately what is the diamete of the moon to the neaest hunded kilometes? 9. Photogaphy. The angle of view of a,000mm telephoto lens is.5. At 750 ft, what is the width of the field of view to the neaest foot? 9. Photogaphy. The angle of view of a 00mm lens is 8. At 500 ft, what is the width of the field of view to the neaest foot? c s
SECTION 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 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 informationTRIGONOMETRY 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationCHAPTER 17 MAGNETIC DIPOLE MOMENT
1 CHAPTER 17 MAGNETIC DIPOLE MOMENT 17.1 Intoduction A numbe of diffeent units fo expessing magnetic dipole moment (heeafte simply magnetic moment ) ae commonly seen in the liteatue, including, fo example,
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 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 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 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 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 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 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 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 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 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 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 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 informationThe Binomial Distribution
The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between
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 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 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 informationRoad tunnel. Road tunnel information sheet. Think about. Using the information
Road tunnel This activity is about using a gaphical o algebaic method to solve poblems in eal contets that can be modelled using quadatic epessions. The fist poblem is about a oad tunnel. The infomation
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 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 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 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 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 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 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 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 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 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 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 informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation
More informationGeostrophic balance. John Marshall, Alan Plumb and Lodovica Illari. March 4, 2003
Geostophic balance John Mashall, Alan Plumb and Lodovica Illai Mach 4, 2003 Abstact We descibe the theoy of Geostophic Balance, deive key equations and discuss associated physical balances. 1 1 Geostophic
More informationA) 2 B) 2 C) 2 2 D) 4 E) 8
Page 1 of 8 CTGavity1. m M Two spheical masses m and M ae a distance apat. The distance between thei centes is halved (deceased by a facto of 2). What happens to the magnitude of the foce of gavity between
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 information6.2 Orbits and Kepler s Laws
Eath satellite in unstable obit 6. Obits and Keple s Laws satellite in stable obit Figue 1 Compaing stable and unstable obits of an atificial satellite. If a satellite is fa enough fom Eath s suface that
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 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 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 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 informationReview of Vectors. Appendix A A.1 DESCRIBING THE 3D WORLD: VECTORS. 3D Coordinates. Basic Properties of Vectors: Magnitude and Direction.
Appendi A Review of Vectos This appendi is a summa of the mathematical aspects of vectos used in electicit and magnetism. Fo a moe detailed intoduction to vectos, see Chapte 1. A.1 DESCRIBING THE 3D WORLD:
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 F. Magnetism. Blinn College  Physics Terry Honan
Chapte F Magnetism Blinn College  Physics 46  Tey Honan F.  Magnetic Dipoles and Magnetic Fields Electomagnetic Duality Thee ae two types of "magnetic chage" o poles, Noth poles N and South poles S.
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 informationChapter 13 Gravitation
Chapte 13 Gavitation Newton, who extended the concept of inetia to all bodies, ealized that the moon is acceleating and is theefoe subject to a centipetal foce. He guessed that the foce that keeps the
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 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 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 informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
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 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 informationPhysics: Electromagnetism Spring PROBLEM SET 6 Solutions
Physics: Electomagnetism Sping 7 Physics: Electomagnetism Sping 7 PROBEM SET 6 Solutions Electostatic Enegy Basics: Wolfson and Pasachoff h 6 Poblem 7 p 679 Thee ae si diffeent pais of equal chages and
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 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 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 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 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 informationChapter 6. GraduallyVaried Flow in Open Channels
Chapte 6 GaduallyVaied Flow in Open Channels 6.. Intoduction A stea nonunifom flow in a pismatic channel with gadual changes in its watesuface elevation is named as gaduallyvaied flow (GVF). The backwate
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 information(a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of
Homewok VI Ch. 7  Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the
More informationA discus thrower spins around in a circle one and a half times, then releases the discus. The discus forms a path tangent to the circle.
Page 1 of 6 11.2 Popeties of Tangents Goal Use popeties of a tangent to a cicle. Key Wods point of tangency p. 589 pependicula p. 108 tangent segment discus thowe spins aound in a cicle one and a half
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 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 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 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 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 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 information