1. FOR SCIENTISTS AND ENGINEERS physics a strategic approach THIRD EDITION randall d. knight © 2013 Pearson Education, Inc. Chapter 3 Lecture

2. © 2013 Pearson Education, Inc. Chapter Goal: To learn how vectors are represented and used. Chapter 3 Vectors and Coordinate Systems Slide 3-2

3. © 2013 Pearson Education, Inc. Chapter 3 Preview Slide 3-3

4. © 2013 Pearson Education, Inc. Chapter 3 Preview Slide 3-4

5. © 2013 Pearson Education, Inc. Chapter 3 Preview Slide 3-5

6. © 2013 Pearson Education, Inc. Chapter 3 Reading Quiz Slide 3-6

7. © 2013 Pearson Education, Inc. What is a vector? A.A quantity having both size and direction. B.The rate of change of velocity. C.A number defined by an angle and a magnitude. D.The difference between initial and final displacement. E.None of the above. Reading Question 3.1 Slide 3-7

8. © 2013 Pearson Education, Inc. What is a vector? A.A quantity having both size and direction. B.The rate of change of velocity. C.A number defined by an angle and a magnitude. D.The difference between initial and final displacement. E.None of the above. Reading Question 3.1 Slide 3-8

9. © 2013 Pearson Education, Inc. What is the name of the quantity represented as ? A.Eye-hat. B.Invariant magnitude. C.Integral of motion. D.Unit vector in x-direction. E.Length of the horizontal axis. Reading Question 3.2 Slide 3-9

10. © 2013 Pearson Education, Inc. What is the name of the quantity represented as ? A.Eye-hat. B.Invariant magnitude. C.Integral of motion. D.Unit vector in x-direction. E.Length of the horizontal axis. Reading Question 3.2 Slide 3-10

11. © 2013 Pearson Education, Inc. This chapter shows how vectors can be added using A.Graphical addition. B.Algebraic addition. C.Numerical addition. D.Both A and B. E.Both A and C. Reading Question 3.3 Slide 3-11

12. © 2013 Pearson Education, Inc. This chapter shows how vectors can be added using A.Graphical addition. B.Algebraic addition. C.Numerical addition. D.Both A and B. E.Both A and C. Reading Question 3.3 Slide 3-12

13. © 2013 Pearson Education, Inc. To decompose a vector means A.To break it into several smaller vectors. B.To break it apart into scalars. C.To break it into pieces parallel to the axes. D.To place it at the origin. E.This topic was not discussed in Chapter 3. Reading Question 3.4 Slide 3-13

14. © 2013 Pearson Education, Inc. To decompose a vector means A.To break it into several smaller vectors. B.To break it apart into scalars. C.To break it into pieces parallel to the axes. D.To place it at the origin. E.This topic was not discussed in Chapter 3. Reading Question 3.4 Slide 3-14

15. © 2013 Pearson Education, Inc. Chapter 3 Content, Examples, and QuickCheck Questions Slide 3-15

16. © 2013 Pearson Education, Inc.  A quantity that is fully described by a single number is called a scalar quantity (i.e., mass, temperature, volume).  A quantity having both a magnitude and a direction is called a vector quantity.  The geometric representation of a vector is an arrow with the tail of the arrow placed at the point where the measurement is made.  We label vectors by drawing a small arrow over the letter that represents the vector, i.e.,: for position, for velocity, for acceleration. Vectors Slide 3-16 The velocity vector has both a magnitude and a direction.

17. © 2013 Pearson Education, Inc.  Suppose Sam starts from his front door, takes a walk, and ends up 200 ft to the northeast of where he started.  We can write Sam’s displacement as:  The magnitude of Sam’s displacement is S  |S|  200 ft, the distance between his initial and final points. Properties of Vectors Slide 3-17

18. © 2013 Pearson Education, Inc.  Sam and Bill are neighbors.  They both walk 200 ft to the northeast of their own front doors.  Bill’s displacement B  (200 ft, northeast) has the same magnitude and direction as Sam’s displacement S.  Two vectors are equal if they have the same magnitude and direction.  This is true regardless of the starting points of the vectors.  B  S. Properties of Vectors Slide 3-18

19. © 2013 Pearson Education, Inc.  A hiker’s displacement is 4 miles to the east, then 3 miles to the north, as shown.  Vector is the net displacement:  Because and are at right angles, the magnitude of is given by the Pythagorean theorem:  To describe the direction of, we find the angle:  Altogether, the hiker’s net displacement is: Vector Addition Slide 3-19

20. © 2013 Pearson Education, Inc. Which of the vectors in the second row shows ? QuickCheck 3.1 Slide 3-20

21. © 2013 Pearson Education, Inc. Which of the vectors in the second row shows ? QuickCheck 3.1 Slide 3-21

22. © 2013 Pearson Education, Inc. Example 3.1 Using Graphical Addition to Find a Displacement Slide 3-22

23. © 2013 Pearson Education, Inc. Example 3.1 Using Graphical Addition to Find a Displacement Slide 3-23

24. © 2013 Pearson Education, Inc.  It is often convenient to draw two vectors with their tails together, as shown in (a) below.  To evaluate F  D  E, you could move E over and use the tip-to-tail rule, as shown in (b) below.  Alternatively, F  D  E can be found as the diagonal of the parallelogram defined by D and E, as shown in (c) below. Parallelogram Rule for Vector Addition Slide 3-24

25. © 2013 Pearson Education, Inc.  Vector addition is easily extended to more than two vectors.  The figure shows the path of a hiker moving from initial position 0 to position 1, then 2, 3, and finally arriving at position 4.  The four segments are described by displacement vectors D 1, D 2, D 3 and D 4.  The hiker’s net displacement, an arrow from position 0 to 4, is:  The vector sum is found by using the tip-to-tail method three times in succession. Addition of More than Two Vectors Slide 3-25

26. © 2013 Pearson Education, Inc. More Vector Mathematics Slide 3-26

27. © 2013 Pearson Education, Inc. Which of the vectors in the second row shows 2  ? QuickCheck 3.2 Slide 3-27

28. © 2013 Pearson Education, Inc. Which of the vectors in the second row shows 2  ? QuickCheck 3.2 Slide 3-28

29. © 2013 Pearson Education, Inc.  A coordinate system is an artificially imposed grid that you place on a problem.  You are free to choose: Where to place the origin, and How to orient the axes.  Below is a conventional xy -coordinate system and the four quadrants I through IV. Coordinate Systems and Vector Components Slide 3-29 The navigator had better know which way to go, and how far, if she and the crew are to make landfall at the expected location.

30. © 2013 Pearson Education, Inc.  The figure shows a vector A and an xy -coordinate system that we’ve chosen.  We can define two new vectors parallel to the axes that we call the component vectors of A, such that:  We have broken A into two perpendicular vectors that are parallel to the coordinate axes.  This is called the decomposition of A into its component vectors. Component Vectors Slide 3-30

31. © 2013 Pearson Education, Inc.  Suppose a vector A has been decomposed into component vectors A x and A y parallel to the coordinate axes.  We can describe each component vector with a single number called the component.  The component tells us how big the component vector is, and, with its sign, which ends of the axis the component vector points toward.  Shown to the right are two examples of determining the components of a vector. Components Slide 3-31

32. © 2013 Pearson Education, Inc. Tactics: Determining the Components of a Vector Slide 3-32

33. © 2013 Pearson Education, Inc. What are the x- and y- components of this vector? A.3, 2 B.2, 3 C.–3, 2 D.2, –3 E.–3, -2 QuickCheck 3.3 Slide 3-33

34. © 2013 Pearson Education, Inc. What are the x- and y- components of this vector? A.3, 2 B.2, 3 C.–3, 2 D.2, –3 E.–3, -2 QuickCheck 3.3 Slide 3-34

35. © 2013 Pearson Education, Inc. What are the x- and y- components of this vector? A.3, 4 B.4, 3 C.–3, 4 D.4, –3 E.–3, –4 QuickCheck 3.4 Slide 3-35

36. © 2013 Pearson Education, Inc. What are the x- and y- components of this vector? A.3, 4 B.4, 3 C.–3, 4 D.4, –3 E.–3, –4 QuickCheck 3.4 Slide 3-36

37. © 2013 Pearson Education, Inc. What are the x- and y- components of vector C ? A.1, –3 B.–3, 1 C.1, –1 D.–4, 2 E.2, –4 QuickCheck 3.5 Slide 3-37

38. © 2013 Pearson Education, Inc. What are the x- and y- components of vector C ? A.1, –3 B.–3, 1 C.1, –1 D.–4, 2 E.2, –4 QuickCheck 3.5 Slide 3-38

39. © 2013 Pearson Education, Inc.  We will frequently need to decompose a vector into its components.  We will also need to “reassemble” a vector from its components.  The figure to the right shows how to move back and forth between the geometric and component representations of a vector. Moving Between the Geometric Representation and the Component Representation Slide 3-39

40. © 2013 Pearson Education, Inc.  If a component vector points left (or down), you must manually insert a minus sign in front of the component, as done for B y in the figure to the right.  The role of sines and cosines can be reversed, depending upon which angle is used to define the direction.  The angle used to define the direction is almost always between 0  and 90 . Moving Between the Geometric Representation and the Component Representation Slide 3-40

41. © 2013 Pearson Education, Inc. Find the x- and y- components of the acceleration vector a shown below. Example 3.3 Finding the Components of an Acceleration Vector Slide 3-41

42. © 2013 Pearson Education, Inc. Example 3.3 Finding the Components of an Acceleration Vector Slide 3-42

43. © 2013 Pearson Education, Inc. Example 3.4 Finding the Direction of Motion Slide 3-43

44. © 2013 Pearson Education, Inc. Example 3.4 Finding the Direction of Motion Slide 3-44

45. © 2013 Pearson Education, Inc.  Each vector in the figure to the right has a magnitude of 1, no units, and is parallel to a coordinate axis.  A vector with these properties is called a unit vector.  These unit vectors have the special symbols:  Unit vectors establish the directions of the positive axes of the coordinate system. Unit Vectors Slide 3-45

46. © 2013 Pearson Education, Inc.  When decomposing a vector, unit vectors provide a useful way to write component vectors:  The full decomposition of the vector A can then be written: Vector Algebra Slide 3-46

47. © 2013 Pearson Education, Inc. Vector C can be written QuickCheck 3.6 Slide 3-47 A. –3î + ĵ. B. –4î + 2ĵ. C. î – 3ĵ. D. 2î – 4ĵ. E. î – ĵ.

48. © 2013 Pearson Education, Inc. Vector C can be written QuickCheck 3.6 Slide 3-47 A. –3î + ĵ. B. –4î + 2ĵ. C. î – 3ĵ. D. 2î – 4ĵ. E. î – ĵ.

49. © 2013 Pearson Education, Inc. Example 3.5 Run Rabbit Run! Slide 3-49

50. © 2013 Pearson Education, Inc. Example 3.5 Run Rabbit Run! Slide 3-50

51. © 2013 Pearson Education, Inc.  We can perform vector addition by adding the x- and y- components separately.  This method is called algebraic addition.  For example, if D  A  B  C, then:  Similarly, to find R  P  Q we would compute:  To find T  cS, where c is a scalar, we would compute: Working With Vectors Slide 3-51

52. © 2013 Pearson Education, Inc. Example 3.6 Using Algebraic Addition to Find a Displacement Slide 3-52

53. © 2013 Pearson Education, Inc. Example 3.6 Using Algebraic Addition to Find a Displacement Slide 3-53

54. © 2013 Pearson Education, Inc. Example 3.6 Using Algebraic Addition to Find a Displacement Slide 3-54

55. © 2013 Pearson Education, Inc.  For some problems it is convenient to tilt the axes of the coordinate system.  The axes are still perpendicular to each other, but there is no requirement that the x- axis has to be horizontal.  Tilted axes are useful if you need to determine component vectors “parallel to” and “perpendicular to” an arbitrary line or surface. Tilted Axes and Arbitrary Directions Slide 3-55

56. © 2013 Pearson Education, Inc. Example 3.7 Muscle and Bone Slide 3-56

57. © 2013 Pearson Education, Inc. Example 3.7 Muscle and Bone Slide 3-57

58. © 2013 Pearson Education, Inc. The angle Φ that specifies the direction of vector is A.tan –1 (C x /C y ). B.tan –1 (C y /C x ). C.tan –1 (|C x |/C y ). D.tan –1 (|C x |/|C y |). E.tan –1 (|C y |/|C x |). QuickCheck 3.7 Slide 3-58

59. © 2013 Pearson Education, Inc. The angle Φ that specifies the direction of vector is A.tan –1 (C x /C y ). B.tan –1 (C y /C x ). C.tan –1 (|C x |/C y ). D.tan –1 (|C x |/|C y |). E.tan –1 (|C y |/|C x |). QuickCheck 3.7 Slide 3-59

60. © 2013 Pearson Education, Inc. Chapter 3 Summary Slides Slide 3-60

61. © 2013 Pearson Education, Inc. Important Concepts Slide 3-61

62. © 2013 Pearson Education, Inc. Important Concepts Slide 3-62