1. Principles of Technology/Physics in Context (PT/PIC) Chapter 4 Vector Quantities 1 Text p. 57-62

2. Key Objectives At the conclusion of this chapter you’ll be able to: Define the terms scalar and vector and list scalar and vector quantities. Represent a vector quantity by an arrow drawn to scale. Relate the direction of a vector to compass directions. Define the term resultant vector. Add vector quantities (1) graphically and (2) algebraically.

3. 4.1 INTRODUCTION TO VECTORS If we want to measure the mass of an object, it makes no difference whether we face in any particular direction while taking the measurement. Quantities such as mass, time, and temperature are called scalar quantities because they can be described solely in terms of their magnitudes (sizes).

4. 4.1 INTRODUCTION TO VECTORS If we are traveling in a car at 60 miles per hour, however, it makes a great deal of difference which way the car is facing: A car traveling east from Los Angeles might end up in New York, but if the car faced west it could end up in the Pacific! Quantities such as velocity, acceleration, and force are called vector quantities because they must be described in terms of their magnitudes and directions.

5. Assessment Question 1 All of the following are true EXCEPT: A.Scalar quantities can be described solely in terms of their magnitudes (sizes). B.Vector quantities must be described in terms of their magnitudes and directions. C.Director quantities must be described in terms of their direction only? D.Quantities such as mass, time, and temperature are called scalar quantities E.Quantities such as velocity, acceleration, and force are called vector quantities.

6. 4.2 DISPLACEMENT & REPRESENTATION OF VECTOR QUANTITIES In our study of motion in Chapter 2, we defined the simplest vector, called displacement, as a directed change in the position of an object. In other words, displacement is a distance (magnitude) in a given direction. example, the quantity 30 meters [east] represents a displacement.

7. 4.2 DISPLACEMENT & REPRESENTATION OF VECTOR QUANTITIES In the book, we will indicate the direction in brackets following the magnitude of the vector. We can easily represent a vector quantity by using an arrow. The length of the arrow represents the magnitude of the vector, and the direction of the vector points from the tail of the arrow toward its tip.

8. 4.2 DISPLACEMENT & REPRESENTATION OF VECTOR QUANTITIES The diagrams illustrate two displacement vectors.

9. 4.2 DISPLACEMENT & REPRESENTATION OF VECTOR QUANTITIES It should be obvious that the arrows shown above are not really 30 meters and 50 meters long. They are drawn to scale, as a map maker does when creating a map. In this instance, 1 inch of arrow length represents 20 meters of distance, and the arrows are 1.5 and 2.5 inches long, respectively.

10. Assessment Question 2 All of the following are true EXCEPT: A.The quantity 30 meters [east] represents a displacement. B.In the quantity 30 meters [east], 30 meters represents magnitude. C.In the quantity 30 meters [east], east represents a direction. D.The quantity 30 meters [east] represented as a vector must be drawn the same size as the magnitude indicates. E.The quantity 30 meters [east] represented as a vector can be drawn to scale, such as 1 cm representing 10 m.

11. 4.2 DISPLACEMENT & REPRESENTATION OF VECTOR QUANTITIES PROBLEM If 0.20 meter represents a displacement of 30. kilometers, what displacement is represented by a vector 0.80 meter long?

12. Assessment Question 3 If 0.02 meter represents a displacement of 10. kilometers, what displacement is represented by a vector 0.32 meter long? 0.32 m (10. km / 0.02 m) = A.0.64 km B.1.5 km C.10. km D.160 km E.320 km

13. 4.2 DISPLACEMENT & REPRESENTATION OF VECTOR QUANTITIES There are many ways to represent the direction of a vector. We will use the method that we believe to be simplest in the long run. The direction is indicated by the angle that the vector makes with the x-axis (horizontal). We use the compass points to describe how many degrees the vector is north (N) or south (S) of the east-west (E-W) baseline.

14. 4.2 DISPLACEMENT & REPRESENTATION OF VECTOR QUANTITIES The diagram illustrates how we do this.

15. 4.2 DISPLACEMENT & REPRESENTATION OF VECTOR QUANTITIES On occasion, it may be necessary to measure the angle that the vector makes with the y-axis (vertical). The conversion is an easy matter since the angles are complementary to one another. In the diagram above, the angles’ would be 60° east of north, 20° west of north, and 45° west of south.

16. Assessment Question 4 All of the following are correctly labeled vector quantities EXCEPT: A.A B.B C.C D.D

17. 4.3 VECTOR ADDITION An airplane traveling at 300 meters per second [east] enters the jet stream, whose velocity is 100 meters per second [north] What is the velocity of the airplane when measured by a person on the ground?

18. 4.3 VECTOR ADDITION This problem, as well as a host of related problems, can be solved by a process called vector addition. When applied to vector quantities, the term addition means calculating the net effect of two or more vectors acting on the same object.

19. 4.3 VECTOR ADDITION It does not matter whether the vector quantities are velocities, displacements, forces, or others. The techniques of vector addition are the same in each case.

20. Assessment Question 5 All of the following are true EXCEPT: A.Many physics problems can be solved by a process called vector addition. B.The type of vector quantity determines the vector addition techniques applied. C.When applied to vector quantities, the term addition means calculating the net effect of two or more vectors acting on the same object. D.The techniques of vector addition are the same for all vector quantities. E.The techniques of vector addition are the same whether the vector quantities are velocities, displacements, or forces.

21. 4.3 VECTOR ADDITION Suppose a person walks 3.0 meters [east] and then walks 4.0 meters [east]. What is the net displacement of the person?

22. 4.3 VECTOR ADDITION This is a simple problem in vector addition.

23. 4.3 VECTOR ADDITION The addition is performed by placing the two vectors in a line, head to tail, as shown in the diagram. The sum of the vectors (indicated by the bold arrow) is called the resultant vector (R) and is determined by drawing an arrow beginning at the tail of the first vector and extending to the head of the second vector.

24. 4.3 VECTOR ADDITION The vector equation for this sum is:

25. Assessment Question 6 All of the following are representations of the same vector quantities EXCEPT: A.Suppose a person walks 30 meters [east] and then walks 40 meters [east]. B.40 east – 30 = 10 west C.T E. D.

26. 4.3 VECTOR ADDITION Note that symbols representing vector quantities are set in boldface type (A). Another of representing a vector quantity is to place an arrow above the symbol (A). In this book we will use boldface type for vectors. When vectors are oriented in opposite directions, their magnitudes are subtracted, as shown in the following problem.

27. 4.3 VECTOR ADDITION PROBLEM A bird flies north 3.0 kilometers and then south 4.0 kilometers. What is the resultant displacement of the bird? SOLUTION In this case, as the diagram shows, the addition yields a resultant of 1.0 kilo meter south:

28. 4.3 VECTOR ADDITION SOLUTION In this case, as the diagram shows, the addition yields a resultant of 1.0 kilometer south:

29. Assessment Question 7 A bird flies south 40 kilometers and then north 30 kilometers. What is the resultant displacement of the bird? 40 km [South] + 30 km [North] = A.10 km [South] B.70 km [North] C.20 km [North] D.100 km [East] E.1000 km [West]

30. 4.3 VECTOR ADDITION Now we will consider how vectors at right angles are added. PROBLEM A plane flies with a velocity of 300. meters per second [east] and enters the jet stream, whose velocity is 100. meters per second [north]. What is the resultant velocity of the plane?

31. 4.3 VECTOR ADDITION SOLUTION The solution to this problem also involves a diagram drawn to scale:

32. 4.3 VECTOR ADDITION SOLUTION If we used a ruler to measure the magnitude of the resultant velocity R, we would find that the length of the resultant arrow translates to 320 m/s. Using a protractor, we see that the angle made by the resultant with the x-axis is approximately 18° N of E.

33. 4.3 VECTOR ADDITION SOLUTION We could have solved this problem algebraically by recognizing that the resultant is the hypotenuse of a right triangle.

34. 4.3 VECTOR ADDITION SOLUTION Using the Pythagorean theorem, we have

35. 4.3 VECTOR ADDITION SOLUTION To calculate angle θ, we note that

36. Assessment Question 8 A person runs with a velocity of 4 meters per second [east] and then runs 3 meters per second [north]. What is the magnitude of the resultant velocity of the person? √[(4 m/s) 2 + (3 m/s) 2 ] = A.1 m/s B.5 m/s C.7 m/s D.25 m/s E.100 m/s

37. Assessment Question 9 A person runs with a velocity of 4 meters per second [east] and then runs 3 meters per second [north]. What is the angle (θ) of the resultant velocity of the person? θ = tan -1 [(3 m/s) / (4 m/s)] = A.25 o B.37 o C.45 o D.60 o E.90 o

38. 4.3 VECTOR ADDITION Finally, let’s consider the effect of two forces that act concurrently (i.e., at the same point) at an angle other than 90°. PROBLEM As shown in the diagram, an object is subjected to two concurrent forces: A = 50. newtons [east] B = 30. newtons [30 o north of east]. What is the resultant force on the object?

39. 4.3 VECTOR ADDITION PROBLEM

40. 4.3 VECTOR ADDITION SOLUTION We place the vectors head to tail by displacing vector B to the right, as shown in the diagram:

41. 4.3 VECTOR ADDITION SOLUTION Then we draw R from the tail of A to the head of B. Using a ruler, a pro tractor, and a suitable scale factor, we find that the magnitude of R is 77 N and the angle θ is 11°.

42. 4.3 VECTOR ADDITION SOLUTION This problem may also be solved algebraically by using the relationships known as the law of cosines and the law of sines. First we re-label the triangle as shown:

43. 4.3 VECTOR ADDITION SOLUTION Then we have:

44. 4.3 VECTOR ADDITION SOLUTION Using both diagrams and applying these two laws, we get:

45. 4.3 VECTOR ADDITION This is a complicated procedure indeed! We will soon see that there is an easier— and more powerful—way to add vectors algebraically.

46. Conclusion Vectors are quantities that have both magnitude and direction. Forces and displacements are examples of vector quantities. A vector is best represented by an arrow: it points in the vector’s direction, and its length is proportional to the vector’s magnitude.

47. Conclusion Vector quantities can be combined with one another. The process is known as vector addition, and the sum is equal to the combined result of the interaction of the individual vectors. The vector sum is also known as the resultant vector. Vectors can be added geometrically by using a measured scale or mathematically by using the laws of algebra and trigonometry.

48. Assessment Question 10 An object is subjected to two forces: A = 10 newtons [east] B = 15 newtons [30 o north of east]. What is the resultant force on the object? c = √[a 2 + b 2 -2ab cosC] c = √[10 2 + 15 2 – 2(10∙15) cos 150 o ] A.12 N B.17 N C.24 N D.35 N