UNIT-05 VECTORS. 3. Utilize the characteristics of two or more vectors that are concurrent, or collinear, or coplanar.
|
|
- Steven Kelly
- 6 years ago
- Views:
Transcription
1 UNIT-05 VECTORS Introduction: physical quantity that can be specified by just a number the magnitude is known as a scalar. In everyday life you deal mostly with scalars such as time, temperature, length and mass of objects, etc. vector is a physical quantity that requires both a magnitude and a direction for its specification. simple example of a vector is the displacement a change of position in a given direction - of an object. nother example of a vector quantity is the velocity of an object, which is defined as the rate of change of displacement with time. Since many engineering situations require us to represent quantities in terms of magnitudes and directions, engineering students must acquire the ability to represent the appropriate quantities as vectors and be able to manipulate these vector quantities. Learning Objectives of this UNIT. 1. Understand the characteristics of a vector quantity. 2. The polar description: Represent a vector quantity in terms of its magnitude and direction, given a description of the physical situation. 3. Utilize the characteristics of two or more vectors that are concurrent, or collinear, or coplanar. 4. Learn to add similar vector quantities. e able to combine (add or subtract) two or more vectors into a single, resultant vector using the graphical tailto-tip and the parallelogram methods. 5. Resolve (break-up) a vector into components in specified directions. 6. The unit vector notation: Represent a vector in terms of its magnitude and a unit vectors in the principal directions of a given reference frame. Combine vector components into a resultant vector, giving magnitude and direction in terms of orthogonal unit vectors in the reference frame. Representation of a vector: vector quantity is written as a letter with an arrow on top,!. In printed material it becomes cumbersome to type an arrow and therefore, purely as a matter of convenience, vectors are represented by bold-type letters i.e. where is the magnitude of. Graphically a vector is represented by a straight line drawn to a scale with the length representing the magnitude and the arrow giving the direction of. In Figure 1 below, the vector represents a displacement of 10.0 m 1
2 along the x-axis where a scale of 1.0cm represents a displacement of 5.0m. The scaling factor in representing vectors is arbitrary and is dictated largely by convenience and spatial constraints. Y 2.0 cm o X FIGURE 1. Graphical representation of a vector. ddition of Vectors Graphical Methods Only similar vectors (vectors representing the same physical quantity can be added Velocity to velocity, force to force and so on) can be added. The Tail-to-Tip Method: To add two vectors and, place the tail of the second vector () at the head of the first vector (). third vector let us say C drawn from the tail of the first () to the head of the second () gives the sum of the two vectors and. Graphically, the sum of two vectors is C = +. C =+ h α β d FIGURE 2. dding two vectors using the tail-to-tip method. Notes: 1. You can convince yourself that C = + = + 2. The sum of two vectors is also a vector. The magnitude of C can be calculated as follows: From the Pythagorean theorem: C 2 = h 2 + (+d) 2 = h d 2 + 2d substitute, d = cosβ and h 2 + d 2 = 2 in the above equation to get C 2 = cosβ = cos (180-α) = cosα C = ( cos α) 1/2 [1] 2
3 We identify C as the sum of vectors and or alternately we can also identify and as component vectors of C. y a similar argument, we see that d is the component of along the direction of whereas h is the component of along a direction perpendicular to. When the two vectors and are at right angles, α = 90 0, we get C = ( cos 90 0 ) 1/2 = ( ) 1/2...[2] (Note from eqs.[1] and [2] that the magnitude of the resultant vector C! +. When and are collinear, C = +, and C = when and are antiparallel) To add more than two vectors we simply extend the tail over tip method. In Figure 3 below we have added three vectors, and D to get E = + + D = C + D. D E = C+D C =+ FIGURE 3. dding three vectors,, and D using the tip-to-tail method. The Parallelogram Method: This method is different in appearance but is fully equivalent to the tail-to-tip method. In this method you put the tails of the two vectors together, complete a parallelogram as shown below. The diagonal of the parallelogram is then the resultant of the two vectors. In the figure below we have added and to obtain C = +. C =+ FIGURE 4. dding two vectors and using the parallelogram method. Subtraction of Vectors and Multiplication by a Scalar: 3
4 The subtraction of a vector from can be viewed as adding to a direction-reversed. D = = + (- ) [3] Graphically, eq.[3] is illustrated below for the tip-to-tail method: C =+ D - FIGURE 5. Subtracting vectors from vector using the parallelogram method. Exercise 1: In the diagram below, which of the following vectors does X represent? [a] [b] [c] [ + ] X Multiplication by a Scalar: Using the tail-to-tip method, it follows that + = 2. Thus, multiplying a vector by a number simply increases the magnitude of the vector by a factor equal to the number but leaves the direction of the vector unchanged. Similarly, - 2 = 2(- ). Therefore, multiplying a vector by a negative number reverses the direction of the vector and increases the magnitude of the vector by a factor equal to the number modulus. In the example above, 2 is parallel to. In general, if = n, where n is a number, then and are parallel to each other with = n. When n = 1, then =, and =. This means that two parallel 2-2 4
5 vectors of the same magnitude are identical, i.e., they are the same vector. Rectangular Components of a Vector If is along x-axis and is along y-axis one can add them to get C = +. We call and as rectangular component vectors of C. See figure below. y C! x FIGURE 6. The rectangular components of a vector in the diagram and are, respectively, the x- and y-components of vector C. From the definition of the sine and cosine functions, /C = cosθ and /C = sinθ Therefore, = C cosθ and =C sinθ [4] = C cosθ along x-axis is also called the x-component of C (= C x ). Similarly, = C sinθ is called the y-component of C, and written as C y. Magnitude of C can be obtained from its rectangular components as follows: C x 2 + C y 2 = C 2 [cos 2 θ + sin 2 θ]= C 2, since cos 2 θ + sin 2 =1 Therefore, C = (C x 2 + C y 2 ) 1/2 The direction of C is given by θ, the angle between C and the x-axis. We see that C y / C x = C sinθ / C cosθ = tanθ θ = tan -1 [C y / C x ] 5
6 Thus, if we know the rectangular components C x and C y of a vector C, we can determine the magnitude C = (C x 2 + C y 2 ) 1/2 and direction from θ = tan -1 (C y / C x ). Y C =+ y x y C y x C x X FIGURE 7. In the diagram C = +. Notice that C x = x + x and C y = y + y. The method of adding two vectors depicted in Fig. 7 can be extended to adding more than two vectors. When adding many similar vectors, say,, C we can resolve each vector into its rectangular components (x- and y- components) and add all the x- components as scalars to find the resultant x-component (R x = x + x + C x +.) and add all the y-components to find the resultant y-component (R y = y + y + C y ). Once we know these resultant x- and y-components, we can get the magnitude of the resultant vector R = (R x 2 + R y 2 ) 1/2 and its direction from tanθ = (R y / R x ). Unit Vector Representation: So far we have been representing vectors graphically we have been drawing them. Vectors can also be expressed algebraically or analytically we can express them in the written form. convenient way to do this is by using the concept of the unit vectors. unit vector is defined as a vector of magnitude one (1). We define unit vectors that point along the three axes of a rectangular co-ordinate system. Z i k j Y X FIGURE 8.The unit vectors. 6
7 The unit vectors along the x-, y-, and z-axis are, respectively, called the i, j, and k. (Note: we will mostly work with two-dimensional vectors in the x-y plane and hence deal with only the i and j unit vectors. In the print form the three unit vectors are written as î, ĵ, and ˆk and are read as i-cap, j-cap, and k-cap. In the typewritten format the unit vectors are written in lower case bold type.) Thus, C = C x i + C y j. In this representation, we can identify the components C x and C y and the direction directly from the equation itself. For example, a displacement vector d = (10.0 i j) m implies it has an x-component of 10.0 m and a y-component of magnitude 12.0 m. (Note: when you specify a vector in terms of its magnitude and orientation, it is called the polar description of a vector. When you describe a vector in terms of its components, it s called the rectangular-component description. The unit vector notation or the (i,j,k)-notation is a rectangular-components description of a vector). Exercise 2. displacement vector in the xy plane is 25.0m long and directed at angle θ =30 o as shown. Determine the x- and y-components of and express in the unit vector notation. Solution: x = cos30 o = 25.0x0.87 = 21.6m y = sin30 o = 25.0x0.5 = 12.5m o y θ x y x = x i + y j = 21.6i j Exercise 3. Find the magnitude and direction of = 45.0i j Solution: = = ! = tan "1 # & $ % 45.0' ( = Therefore, the magnitude of is 75.0 and it points above the x-axis. dding Vectors Problem Solving Strategy Here is an outline of how to proceed when adding vectors. We will illustrate this with an example. Example: bus travels m due east from the bus depot at O due east to station. From station, the bus proceeds to station travelling southeast (45 0 ) for m and then to station C for a distance of 400.0m in a direction 53 0 south of west. What is the net displacement of the bus? 7
8 Step 1. Choose x- and y-axes. Choose them in a way that will make your work easier. This is often done by choosing one of the axes along one of the given vectors. In our case we have aligned the first leg of the bus s journey along the x-axis. Draw a welllabeled diagram(see the diagram on the left below). y y 0 D 1 45 o x 0 D 1 θ D 2x 45 o x D 2 D 2 D 2y D 53 o D D 3x 53 o D 3y D 3 D 3 C C Step 2. Find the components. Resolve each component into its x- and y-components (see the diagram on the right above). D 1x = m D 1y = 0.0 m D 2x = cos45 0 m = m D 2y = sin45 0 m = m D 3x = cos53 0 m = m D 2y = sin53 0 m = m Step 3. dd the components. D x = D 1x + D 2x + D 3x = = m D y = D 1y + D 2y + D 3y = = m Step 4. Find the magnitude and direction. 8
9 D = = 764.7m # D! = tan "1 y & $ % ' ( = # "673.1& tan"1 $ % ' ( = " D x Thus the total displacement of the bus is m and it points below the x-axis. Note : In the unit vector notation, the solution to this problem would be written as follows: D 1 = i m D 2 = cos45 0 i sin45 0 j = i j m D 3 = cos53 0 i sin53 0 j = i j m The net displacement: D = D 1 + D 2 + D 3 = ( )i + ( )j = 362.9i j m The magnitude and direction can now be determined as outlined above. 9
10 12.0 km Solved Examples: Example 1. vector in the xy plane is 25.0m in magnitude and directed at angle θ =30 o as shown. nother vector is 30.0m in magnitude and perpendicular to. [a] What are the x- and y-components of the resultant R = +? [b] Determine the magnitude and direction of the resultant vector R. Solution: [a] x = cos30 o = 25.0x0.866 = 21.6m y y = sin30 o = 25.0x0.5 = 12.5m x = - sin30 o = x0.5 = m θ y = cos30 o = 30.0x0.866 = 25.98m R x = x + x = ( )m = 6.6m o θ x R y = y + y = ( )m = 38.5m [b] Magnitude, R = = 39.1m! = tan "1 (38.5 / 6.6) = 80.3 o Thus the resultant is 39.1m and points above the x-axis. Example 2. fishing boat sets out to sail to a point 12.0km due north. Without catching many fish, the boat sails further to a point 9.0km due west for better fishing. From the second spot, how far and in which direction must the boat sail to reach its original starting point? Solution: To get back to O, the boat must travel along C. From the diagram: + + C = 0 or C = - ( + ) = - ( 12.0 i j) km Magnitude of C = [(12.0) 2 + (9.0) 2 ] 1/2 = 15.0 km. θ = - tan -1 [9.0/12.0] = - 37 o 9.0 km θ C y(north) o x(east) 10
11 Thus the boat must sail 15.0 km, 37 o south of east. Example 3. Loosening a nut on a bolt is a common experience and we see how a force applied may be split into various components. In order to loosen a nut, a person holding a horizontal wrench exerts a downward force F = 50.0 lb at an angle of 30 to the vertical. [a] What are the horizontal and vertical components of the force F? The vertical component, F V = 50.0 cos 30 = 43.3 lb The horizontal component, F H = 50.0 sin 30 = 25.0 lb Negative signs indicate the components are along the negative x- and y-axes. [b] Express F in the unit vector i and j notation. F = F H i + F V j = [ 43.3 i 25.0 j ] lb Example 4: In the first leg of its flight an airplane flies from city to city in a direction due east for mi (mi = miles). Next, it flies from city to city C, in a direction 53 north of east for mi. D C N W S E 500 miles 53 o 600 miles [a] Determine the components, along the easterly and northerly directions, of the resultant displacement of the plane from city to city C. Let d x and d y represent, respectively, the components of the plane s displacement along the east and the north. 11
12 d x = (cos53 ) mi = mi. d y = (sin 53 ) mi = mi [b] What are the magnitude and direction of the resultant displacement of the plane from city to city C? Magnitude: d = ( ) 1/2 = miles Direction: tan θ = 400.0/900.0 = 4/9, so θ = tan -1 (4/9) = This direction is 24.0 North of East. The plane then flies directly from city C to city D directly north of city, a distance of miles in the last segment of its flight. [c] What is the magnitude and the direction of the displacement of the plane from city C to city D? Magnitude = miles. This direction is westerly; see the vector representation below. D d 3 = 900 miles C 29 o o d f = 400 miles d = 985 miles d 2 = 500 miles 24 o d 1 = 600 miles 53 o COMPONENT DUE EST [d] What is the net displacement of the plane as it flies from city to city D? The net displacement, R = i mi or mi pointing north. [e] What is the total distance the plane has traveled as it flew from to D. The total distance = = mi 12
13 Example 5. disabled automobile is pulled to the right by means of two cables and C as shown. The tension in the cable C is T C = 6.0 kn. If it has to be pulled along the direction X, the axis of the automobile, determine the magnitude of the resultant force, R, in that direction and the tension, T, in cable. Solution: R = T + T C In the x-direction: R x = T C (cos 30 ) + T (cos 37 ) [1] In the y-direction: 0 = T sin 37 - T sin 30 [2] From eq.[2]: T = 6000(sin 30 ) /0.6 = 5000 N. From eq.[1]: R x = T C (cos 30 ) + T (cos 37 )= (6000 N )(0.866)+(5000 N)(0.8)= 9200 N. 13
Chapter 2 A Mathematical Toolbox
Chapter 2 Mathematical Toolbox Vectors and Scalars 1) Scalars have only a magnitude (numerical value) Denoted by a symbol, a 2) Vectors have a magnitude and direction Denoted by a bold symbol (), or symbol
More informationNew concepts: scalars, vectors, unit vectors, vector components, vector equations, scalar product. reading assignment read chap 3
New concepts: scalars, vectors, unit vectors, vector components, vector equations, scalar product reading assignment read chap 3 Most physical quantities are described by a single number or variable examples:
More informationPhysics 12. Chapter 1: Vector Analysis in Two Dimensions
Physics 12 Chapter 1: Vector Analysis in Two Dimensions 1. Definitions When studying mechanics in Physics 11, we have realized that there are two major types of quantities that we can measure for the systems
More informationMECHANICS. Prepared by Engr. John Paul Timola
MECHANICS Prepared by Engr. John Paul Timola MECHANICS a branch of the physical sciences that is concerned with the state of rest or motion of bodies that are subjected to the action of forces. subdivided
More informationKinematics in Two Dimensions; Vectors
Kinematics in Two Dimensions; Vectors Vectors & Scalars!! Scalars They are specified only by a number and units and have no direction associated with them, such as time, mass, and temperature.!! Vectors
More informationSECTION 6.3: VECTORS IN THE PLANE
(Section 6.3: Vectors in the Plane) 6.18 SECTION 6.3: VECTORS IN THE PLANE Assume a, b, c, and d are real numbers. PART A: INTRO A scalar has magnitude but not direction. We think of real numbers as scalars,
More information4/13/2015. I. Vectors and Scalars. II. Addition of Vectors Graphical Methods. a. Addition of Vectors Graphical Methods
I. Vectors and Scalars A vector has magnitude as well as direction. Some vector quantities: displacement, velocity, force, momentum A scalar has only a magnitude. Some scalar quantities: mass, time, temperature
More informationCHAPTER 3 KINEMATICS IN TWO DIMENSIONS; VECTORS
CHAPTER 3 KINEMATICS IN TWO DIMENSIONS; VECTORS OBJECTIVES After studying the material of this chapter, the student should be able to: represent the magnitude and direction of a vector using a protractor
More informationTenth Edition STATICS 1 Ferdinand P. Beer E. Russell Johnston, Jr. David F. Mazurek Lecture Notes: John Chen California Polytechnic State University
T E CHAPTER 1 VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinand P. Beer E. Russell Johnston, Jr. David F. Mazurek Lecture Notes: Introduction John Chen California Polytechnic State University! Contents
More informationVector components and motion
Vector components and motion Objectives Distinguish between vectors and scalars and give examples of each. Use vector diagrams to interpret the relationships among vector quantities such as force and acceleration.
More informationPhysics 40 Chapter 3: Vectors
Physics 40 Chapter 3: Vectors Cartesian Coordinate System Also called rectangular coordinate system x-and y- axes intersect at the origin Points are labeled (x,y) Polar Coordinate System Origin and reference
More informationNotes: Vectors and Scalars
A particle moving along a straight line can move in only two directions and we can specify which directions with a plus or negative sign. For a particle moving in three dimensions; however, a plus sign
More informationChapter 8 Vectors and Scalars
Chapter 8 193 Vectors and Scalars Chapter 8 Vectors and Scalars 8.1 Introduction: In this chapter we shall use the ideas of the plane to develop a new mathematical concept, vector. If you have studied
More informationVectors. Introduction. Prof Dr Ahmet ATAÇ
Chapter 3 Vectors Vectors Vector quantities Physical quantities that have both n u m e r i c a l a n d d i r e c t i o n a l properties Mathematical operations of vectors in this chapter A d d i t i o
More informationVectors and 2D Kinematics. AIT AP Physics C
Vectors and 2D Kinematics Coordinate Systems Used to describe the position of a point in space Coordinate system consists of a fixed reference point called the origin specific axes with scales and labels
More information2-9. The plate is subjected to the forces acting on members A and B as shown. If θ = 60 o, determine the magnitude of the resultant of these forces
2-9. The plate is subjected to the forces acting on members A and B as shown. If θ 60 o, determine the magnitude of the resultant of these forces and its direction measured clockwise from the positie x
More informationIshik University / Sulaimani Civil Engineering Department. Chapter -2-
Ishik University / Sulaimani Civil Engineering Department Chapter -- 1 orce Vectors Contents : 1. Scalars and Vectors. Vector Operations 3. Vector Addition of orces 4. Addition of a System of Coplanar
More information(arrows denote positive direction)
12 Chapter 12 12.1 3-dimensional Coordinate System The 3-dimensional coordinate system we use are coordinates on R 3. The coordinate is presented as a triple of numbers: (a,b,c). In the Cartesian coordinate
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Two Dimensions; Vectors Vectors and Scalars Units of Chapter 3 Addition of Vectors Graphical Methods Subtraction of Vectors, and Multiplication of a Vector by a Scalar Adding Vectors
More informationVectors. Introduction
Chapter 3 Vectors Vectors Vector quantities Physical quantities that have both numerical and directional properties Mathematical operations of vectors in this chapter Addition Subtraction Introduction
More informationVector Algebra August 2013
Vector Algebra 12.1 12.2 28 August 2013 What is a Vector? A vector (denoted or v) is a mathematical object possessing both: direction and magnitude also called length (denoted ). Vectors are often represented
More information9.1. Basic Concepts of Vectors. Introduction. Prerequisites. Learning Outcomes. Learning Style
Basic Concepts of Vectors 9.1 Introduction In engineering, frequent reference is made to physical quantities, such as force, speed and time. For example, we talk of the speed of a car, and the force in
More informationQuiz No. 1: Tuesday Jan. 31. Assignment No. 2, due Thursday Feb 2: Problems 8.4, 8.13, 3.10, 3.28 Conceptual questions: 8.1, 3.6, 3.12, 3.
Quiz No. 1: Tuesday Jan. 31 Assignment No. 2, due Thursday Feb 2: Problems 8.4, 8.13, 3.10, 3.28 Conceptual questions: 8.1, 3.6, 3.12, 3.20 Chapter 3 Vectors and Two-Dimensional Kinematics Properties of
More informationChapter 3. Vectors. 3.1 Coordinate Systems 3.2 Vector and Scalar Quantities 3.3 Some Properties of Vectors 3.4 Components of a Vector and Unit Vectors
Chapter 3 Vectors 3.1 Coordinate Systems 3.2 Vector and Scalar Quantities 3.3 Some Properties of Vectors 3.4 Components of a Vector and Unit Vectors 1 Vectors Vector quantities Physical quantities that
More informationUnit 1 Representing and Operations with Vectors. Over the years you have come to accept various mathematical concepts or properties:
Lesson1.notebook November 27, 2012 Algebra Unit 1 Representing and Operations with Vectors Over the years you have come to accept various mathematical concepts or properties: Communative Property Associative
More informationCoordinate Systems. Chapter 3. Cartesian Coordinate System. Polar Coordinate System
Chapter 3 Vectors Coordinate Systems Used to describe the position of a point in space Coordinate system consists of a fixed reference point called the origin specific axes with scales and labels instructions
More informationEngineering Mechanics: Statics in SI Units, 12e Force Vectors
Engineering Mechanics: Statics in SI Units, 1e orce Vectors 1 Chapter Objectives Parallelogram Law Cartesian vector form Dot product and angle between vectors Chapter Outline 1. Scalars and Vectors. Vector
More informationFORCE TABLE INTRODUCTION
FORCE TABLE INTRODUCTION All measurable quantities can be classified as either a scalar 1 or a vector 2. A scalar has only magnitude while a vector has both magnitude and direction. Examples of scalar
More informationOmm Al-Qura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics
LECTURE OUTLINE CHAPTER 3 Vectors in Physics 3-1 Scalars Versus Vectors Scalar a numerical value (number with units). May be positive or negative. Examples: temperature, speed, height, and mass. Vector
More informationVector Addition and Subtraction: Graphical Methods
Vector Addition and Subtraction: Graphical Methods Bởi: OpenStaxCollege Displacement can be determined graphically using a scale map, such as this one of the Hawaiian Islands. A journey from Hawai i to
More informationMathematical review trigonometry vectors Motion in one dimension
Mathematical review trigonometry vectors Motion in one dimension Used to describe the position of a point in space Coordinate system (frame) consists of a fixed reference point called the origin specific
More informationChapter 3. Vectors and Two-Dimensional Motion
Chapter 3 Vectors and Two-Dimensional Motion 1 Vector vs. Scalar Review All physical quantities encountered in this text will be either a scalar or a vector A vector quantity has both magnitude (size)
More informationChapter 7.4: Vectors
Chapter 7.4: Vectors In many mathematical applications, quantities are determined entirely by their magnitude. When calculating the perimeter of a rectangular field, determining the weight of a box, or
More informationChapter 2 Mechanical Equilibrium
Chapter 2 Mechanical Equilibrium I. Force (2.1) A. force is a push or pull 1. A force is needed to change an object s state of motion 2. State of motion may be one of two things a. At rest b. Moving uniformly
More informationMechanics: Scalars and Vectors
Mechanics: Scalars and Vectors Scalar Onl magnitude is associated with it Vector e.g., time, volume, densit, speed, energ, mass etc. Possess direction as well as magnitude Parallelogram law of addition
More informationIntroduction to Vectors
Introduction to Vectors Why Vectors? Say you wanted to tell your friend that you re running late and will be there in five minutes. That s precisely enough information for your friend to know when you
More informationGround Rules. PC1221 Fundamentals of Physics I. Coordinate Systems. Cartesian Coordinate System. Lectures 5 and 6 Vectors.
PC1221 Fundamentals of Phsics I Lectures 5 and 6 Vectors Dr Ta Seng Chuan 1 Ground ules Switch off our handphone and pager Switch off our laptop computer and keep it No talking while lecture is going on
More informationPhys 221. Chapter 3. Vectors A. Dzyubenko Brooks/Cole
Phs 221 Chapter 3 Vectors adzubenko@csub.edu http://www.csub.edu/~adzubenko 2014. Dzubenko 2014 rooks/cole 1 Coordinate Sstems Used to describe the position of a point in space Coordinate sstem consists
More informationKinematics in Two Dimensions; 2D- Vectors
Kinematics in Two Dimensions; 2D- Vectors Addition of Vectors Graphical Methods Below are two example vector additions of 1-D displacement vectors. For vectors in one dimension, simple addition and subtraction
More informationChapter 3. Vectors and. Two-Dimensional Motion Vector vs. Scalar Review
Chapter 3 Vectors and Two-Dimensional Motion Vector vs. Scalar Review All physical quantities encountered in this text will be either a scalar or a vector A vector quantity has both magnitude (size) and
More informationChapter 2 Statics of Particles. Resultant of Two Forces 8/28/2014. The effects of forces on particles:
Chapter 2 Statics of Particles The effects of forces on particles: - replacing multiple forces acting on a particle with a single equivalent or resultant force, - relations between forces acting on a particle
More informationENT 151 STATICS. Statics of Particles. Contents. Resultant of Two Forces. Introduction
CHAPTER ENT 151 STATICS Lecture Notes: Azizul bin Mohamad KUKUM Statics of Particles Contents Introduction Resultant of Two Forces Vectors Addition of Vectors Resultant of Several Concurrent Forces Sample
More informationWelcome back to Physics 215
Welcome back to Physics 215 Lecture 2-2 02-2 1 Last time: Displacement, velocity, graphs Today: Constant acceleration, free fall 02-2 2 2-2.1: An object moves with constant acceleration, starting from
More informationScalar Quantities - express only magnitude ie. time, distance, speed
Chapter 6 - Vectors Scalar Quantities - express only magnitude ie. time, distance, speed Vector Quantities - express magnitude and direction. ie. velocity 80 km/h, 58 displacement 10 km (E) acceleration
More informationCHAPTER 1 MEASUREMENTS AND VECTORS
CHPTER 1 MESUREMENTS ND VECTORS 1 CHPTER 1 MESUREMENTS ND VECTORS 1.1 UNITS ND STNDRDS n phsical quantit must have, besides its numerical value, a standard unit. It will be meaningless to sa that the distance
More informationUNCORRECTED PAGE PROOFS
TOPIC 3 Motion in two dimensions 3.1 Overview 3.1.1 Module 1: Kinematics Motion on a Plane Inquiry question: How is the motion of an object that changes its direction of movement on a plane described?
More informationVECTORS. 3-1 What is Physics? 3-2 Vectors and Scalars CHAPTER
CHAPTER 3 VECTORS 3-1 What is Physics? Physics deals with a great many quantities that have both size and direction, and it needs a special mathematical language the language of vectors to describe those
More informationVectors (Trigonometry Explanation)
Vectors (Trigonometry Explanation) CK12 Editor Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive
More information11.8 Vectors Applications of Trigonometry
00 Applications of Trigonometry.8 Vectors As we have seen numerous times in this book, Mathematics can be used to model and solve real-world problems. For many applications, real numbers suffice; that
More informationProjectile Motion and 2-D Dynamics
Projectile Motion and 2-D Dynamics Vector Notation Vectors vs. Scalars In Physics 11, you learned the difference between vectors and scalars. A vector is a quantity that includes both direction and magnitude
More informationEngineering Mechanics: Statics in SI Units, 12e
Engineering Mechanics: Statics in SI Units, 12e 2 Force Vectors 1 Chapter Objectives Parallelogram Law Cartesian vector form Dot product and an angle between two vectors 2 Chapter Outline 1. Scalars and
More informationDepartment of Physics, Korea University
Name: Department: Notice +2 ( 1) points per correct (incorrect) answer. No penalty for an unanswered question. Fill the blank ( ) with (8) if the statement is correct (incorrect).!!!: corrections to an
More informationLecture Notes (Vectors)
Lecture Notes (Vectors) Intro: - up to this point we have learned that physical quantities can be categorized as either scalars or vectors - a vector is a physical quantity that requires the specification
More informationChapter 2: Statics of Particles
CE297-A09-Ch2 Page 1 Wednesday, August 26, 2009 4:18 AM Chapter 2: Statics of Particles 2.1-2.3 orces as Vectors & Resultants orces are drawn as directed arrows. The length of the arrow represents the
More informationVectors A Guideline For Motion
AP Physics-1 Vectors A Guideline For Motion Introduction: You deal with scalar quantities in many aspects of your everyday activities. For example, you know that 2 liters plus 2 liters is 4 liters. The
More informationChapter 3 Vectors. 3.1 Vector Analysis
Chapter 3 Vectors 3.1 Vector nalysis... 1 3.1.1 Introduction to Vectors... 1 3.1.2 Properties of Vectors... 1 3.2 Coordinate Systems... 6 3.2.1 Cartesian Coordinate System... 6 3.2.2 Cylindrical Coordinate
More informationPhysics 170 Lecture 2. Phys 170 Lecture 2 1
Physics 170 Lecture 2 Phys 170 Lecture 2 1 Phys 170 Lecture 2 2 dministrivia Registration issues? Web page issues? On Connect? http://www.physics.ubc.ca/~mattison/courses/phys170 Mastering Engineering
More informationGraphical Vector Addition
Vectors Chapter 4 Vectors and Scalars Measured quantities can be of two types Scalar quantities: only require magnitude (and proper unit) for description. Examples: distance, speed, mass, temperature,
More informationChapter 3 Vectors Prof. Raymond Lee, revised
Chapter 3 Vectors Prof. Raymond Lee, revised 9-2-2010 1 Coordinate systems Used to describe a point s position in space Coordinate system consists of fixed reference point called origin specific axes with
More informationPlease Visit us at:
IMPORTANT QUESTIONS WITH ANSWERS Q # 1. Differentiate among scalars and vectors. Scalars Vectors (i) The physical quantities that are completely (i) The physical quantities that are completely described
More information2.1 Scalars and Vectors
2.1 Scalars and Vectors Scalar A quantity characterized by a positive or negative number Indicated by letters in italic such as A e.g. Mass, volume and length 2.1 Scalars and Vectors Vector A quantity
More informationscalar and - vector - - presentation SCALAR AND VECTOR
http://www.slideshare.net/fikrifadzal/chapter-14scalar-and-vector- and presentation SCLR ND VECTOR Scalars Scalars are quantities which have magnitude without directioni Examples of scalars temperaturere
More informationAdding Vectors in Two Dimensions
Slide 37 / 125 Adding Vectors in Two Dimensions Return to Table of Contents Last year, we learned how to add vectors along a single axis. The example we used was for adding two displacements. Slide 38
More informationSection 1.4: Adding and Subtracting Linear and Perpendicular Vectors
Section 1.4: Adding and Subtracting Linear and Perpendicular Vectors Motion in two dimensions must use vectors and vector diagrams. Vector Representation: tail head magnitude (size): given by the length
More informationVector Addition INTRODUCTION THEORY
Vector Addition INTRODUCTION All measurable quantities may be classified either as vector quantities or as scalar quantities. Scalar quantities are described completely by a single number (with appropriate
More informationChapter 3. Kinematics in Two Dimensions
Chapter 3 Kinematics in Two Dimensions 3.1 Trigonometry 3.1 Trigonometry sin! = h o h cos! = h a h tan! = h o h a 3.1 Trigonometry tan! = h o h a tan50! = h o 67.2m h o = tan50! ( 67.2m) = 80.0m 3.1 Trigonometry!
More information6. Vectors. Given two points, P 0 = (x 0, y 0 ) and P 1 = (x 1, y 1 ), a vector can be drawn with its foot at P 0 and
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationUNIT V: Multi-Dimensional Kinematics and Dynamics Page 1
UNIT V: Multi-Dimensional Kinematics and Dynamics Page 1 UNIT V: Multi-Dimensional Kinematics and Dynamics As we have already discussed, the study of the rules of nature (a.k.a. Physics) involves both
More informationCHAPTER 2: VECTORS IN 3D
CHAPTER 2: VECTORS IN 3D 2.1 DEFINITION AND REPRESENTATION OF VECTORS A vector in three dimensions is a quantity that is determined by its magnitude and direction. Vectors are added and multiplied by numbers
More informationGeneral Physics I, Spring Vectors
General Physics I, Spring 2011 Vectors 1 Vectors: Introduction A vector quantity in physics is one that has a magnitude (absolute value) and a direction. We have seen three already: displacement, velocity,
More informationVectors. In kinematics, the simplest concept is position, so let s begin with a position vector shown below:
Vectors Extending the concepts of kinematics into two and three dimensions, the idea of a vector becomes very useful. By definition, a vector is a quantity with both a magnitude and a spatial direction.
More informationOpenStax-CNX module: m Vectors. OpenStax College. Abstract
OpenStax-CNX module: m49412 1 Vectors OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section you will: Abstract View vectors
More informationa Particle Forces the force. of action its sense is of application. Experimen demonstra forces ( P Resultant of Two Note: a) b) momentum)
Chapter 2 : Statics of a Particle 2.2 Force on a Particle: Resultant of Two Forces Recall, force is a vector quantity whichh has magnitude and direction. The direction of the the force. force is defined
More information9/29/2014. Chapter 3 Kinematics in Two Dimensions; Vectors. 3-1 Vectors and Scalars. Contents of Chapter Addition of Vectors Graphical Methods
Lecture PowerPoints Chapter 3 Physics: Principles with Applications, 7 th edition Giancoli Chapter 3 Kinematics in Two Dimensions; Vectors This work is protected by United States copyright laws and is
More informationBELLWORK feet
BELLWORK 1 A hot air balloon is being held in place by two people holding ropes and standing 35 feet apart. The angle formed between the ground and the rope held by each person is 40. Determine the length
More informationLesson 7. Chapter 3: Two-Dimensional Kinematics COLLEGE PHYSICS VECTORS. Video Narrated by Jason Harlow, Physics Department, University of Toronto
COLLEGE PHYSICS Chapter 3: Two-Dimensional Kinematics Lesson 7 Video Narrated by Jason Harlow, Physics Department, University of Toronto VECTORS A quantity having both a magnitude and a direction is called
More informationVectors in Physics. Topics to review:
Vectors in Physics Topics to review: Scalars Versus Vectors The Components of a Vector Adding and Subtracting Vectors Unit Vectors Position, Displacement, Velocity, and Acceleration Vectors Relative Motion
More informationThe geometry of least squares
The geometry of least squares We can think of a vector as a point in space, where the elements of the vector are the coordinates of the point. Consider for example, the following vector s: t = ( 4, 0),
More informationVectors a vector is a quantity that has both a magnitude (size) and a direction
Vectors In physics, a vector is a quantity that has both a magnitude (size) and a direction. Familiar examples of vectors include velocity, force, and electric field. For any applications beyond one dimension,
More informationReview. Projectile motion is a vector. - Has magnitude and direction. When solving projectile motion problems, draw it out
Projectile Motion Review Projectile motion is a vector - Has magnitude and direction When solving projectile motion problems, draw it out Two methods to drawing out vectors: 1. Tail-to-tip method 2. Parallelogram
More informationGraphical Analysis; and Vectors
Graphical Analysis; and Vectors Graphs Drawing good pictures can be the secret to solving physics problems. It's amazing how much information you can get from a diagram. We also usually need equations
More informationIntroduction to Engineering Mechanics
Introduction to Engineering Mechanics Statics October 2009 () Introduction 10/09 1 / 19 Engineering mechanics Engineering mechanics is the physical science that deals with the behavior of bodies under
More informationVectors Primer. M.C. Simani. July 7, 2007
Vectors Primer M.. Simani Jul 7, 2007 This note gives a short introduction to the concept of vector and summarizes the basic properties of vectors. Reference textbook: Universit Phsics, Young and Freedman,
More informationWhat you will learn today
What you will learn today The Dot Product Equations of Vectors and the Geometry of Space 1/29 Direction angles and Direction cosines Projections Definitions: 1. a : a 1, a 2, a 3, b : b 1, b 2, b 3, a
More informationA SCALAR is ANY quantity in physics that has MAGNITUDE, but NOT a direction associated with it. Magnitude A numerical value with units.
Vectors and Scalars A SCALAR is ANY quantity in physics that has MAGNITUDE, but NOT a direction associated with it. Magnitude A numerical value with units. Scalar Example Speed Distance Age Heat Number
More informationEngineering Mechanics Statics
Mechanical Systems Engineering- 2016 Engineering Mechanics Statics 2. Force Vectors; Operations on Vectors Dr. Rami Zakaria MECHANICS, UNITS, NUMERICAL CALCULATIONS & GENERAL PROCEDURE FOR ANALYSIS Today
More informationDefinitions In physics we have two types of measurable quantities: vectors and scalars.
1 Definitions In physics we have two types of measurable quantities: vectors and scalars. Scalars: have magnitude (magnitude means size) only Examples of scalar quantities include time, mass, volume, area,
More informationChapter 3. Table of Contents. Section 1 Introduction to Vectors. Section 2 Vector Operations. Section 3 Projectile Motion. Section 4 Relative Motion
Two-Dimensional Motion and Vectors Table of Contents Section 1 Introduction to Vectors Section 2 Vector Operations Section 3 Projectile Motion Section 4 Relative Motion Section 1 Introduction to Vectors
More informationPrecalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers. A: Initial Point (start); B: Terminal Point (end) : ( ) ( )
Syllabus Objectives: 5.1 The student will explore methods of vector addition and subtraction. 5. The student will develop strategies for computing a vector s direction angle and magnitude given its coordinates.
More informationAP Physics C Mechanics Vectors
1 AP Physics C Mechanics Vectors 2015 12 03 www.njctl.org 2 Scalar Versus Vector A scalar has only a physical quantity such as mass, speed, and time. A vector has both a magnitude and a direction associated
More informationChapter 2: Force Vectors
Chapter 2: Force Vectors Chapter Objectives To show how to add forces and resolve them into components using the Parallelogram Law. To express force and position in Cartesian vector form and explain how
More informationChapter 2 One-Dimensional Kinematics
Review: Chapter 2 One-Dimensional Kinematics Description of motion in one dimension Copyright 2010 Pearson Education, Inc. Review: Motion with Constant Acceleration Free fall: constant acceleration g =
More information**Answers may or may not be the same due to differences in values of original question. Answers in bold and figures are not provided.
PCS106 Assignment # 2: Vectors **Answers may or may not be the same due to differences in values of original question. Answers in bold and figures are not provided.** Component of Vectors 1. Shown is a
More informationChapter 3. Vectors. θ that the vector forms with i ˆ is 15. I. Vectors and Scalars
Chapter 3. Vectors I. Vectors and Scalars 1. What type of quantity does the odometer of a car measure? a) vector; b) scalar; c) neither scalar nor vector; d) both scalar and vector. 2. What type of quantity
More informationChapter Objectives. Copyright 2011 Pearson Education South Asia Pte Ltd
Chapter Objectives To show how to add forces and resolve them into components using the Parallelogram Law. To express force and position in Cartesian vector form and explain how to determine the vector
More informationtwo forces and moments Structural Math Physics for Structures Structural Math
RHITETURL STRUTURES: ORM, EHVIOR, ND DESIGN DR. NNE NIHOLS SUMMER 05 lecture two forces and moments orces & Moments rchitectural Structures 009abn Structural Math quantify environmental loads how big is
More information5.) Unit Vectors https://www.youtube.com/watch?v=iaekl5h2sjm (Mario s Math Tutoring)
This review covers the definition of a vector, graphical and algebraic representations, adding vectors, scalar multiples, dot product, and cross product for two and three dimensional vectors, along with
More informationReview of Coordinate Systems
Vector in 2 R and 3 R Review of Coordinate Systems Used to describe the position of a point in space Common coordinate systems are: Cartesian Polar Cartesian Coordinate System Also called rectangular coordinate
More information9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b
vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component
More informationh p://edugen.wileyplus.com/edugen/courses/crs1404/pc/c05/c2hlch... CHAPTER 5 MOMENTS 1 of 3 10-Sep-12 16:35
Peter Christopher/Masterfile... 1 of 3 10-Sep-12 16:35 CHAPTER 5 MOMENTS Peter Christopher/Masterfile... 2 of 3 10-Sep-12 16:35 Peter Christopher/Masterfile In Chapter 4 we considered the forces that push
More information