Lecture 3- Vectors Chapter 3

Transcription

1 1 / 36 Lecture 3- Vectors Chapter 3 Instructor: Prof. Noronha-Hostler Course Administrator: Prof. Roy Montalvo PHY-123 ANALYTICAL PHYSICS IA Phys- 123 Sep. 21 th, 2018

2 2 / 36 Course Reminders The course entry survey link is up and will be up until next Friday at 11:59PM. All students are required to take the survey. Students having any issues with doing the homework that are not physics related (e.g. how to enter answers, exponents, graphing, etc.) should contact Wiley Student Support at the link on the course website. Office hours:

3 3 / 36 Objectives Difference between scalars and vectors Vector addition and subtraction Vector multiplication (dot product and cross product)

4 4 / 36 What are vectors? Scalars Speed Temperature Mass Power Energy Vectors Displacement, Velocity, and Acceleration Force Weight Friction Momentum

5 5 / 36 Vector terminology

6 6 / 36 Only magnitude and direction matters

7 7 / 36 Displacement Vector

8 8 / 36 Components of a vector Resolving a vector a x = a cos θ a y = a sin θ Note, the components may be negative! Are a x and a y scalars or vectors?

9 9 / 36 Components of a vector Resolving a vector a x = a cos θ a y = a sin θ Note, the components may be negative! Are a x and a y scalars or vectors? Scalars

10 10 / 36 Components of a vector

11 Magnitude angle notation If we know the vector components a x and a y, we can calculate it s magnitude and angle a = ax 2 + ay 2 tan θ = a y a x 3-Dimensions Either a magnitude and two angles (a, θ, φ) or (a x, a y, a z ). 11 / 36

12 12 / 36 Unit Vector Right-handed coordinate system Unit Vector î = 1 the magnitude of a unit vector is always equal to one. Points in a specific direction.

13 13 / 36 Unit vector example 1.0 y j i a x x a y a

14 14 / 36 Unit vector example a = a x î + a y ĵ = 2î + 2ĵ 1.0 y j i a x =-2 x a y =2 a

15 15 / 36 Adding Vectors

16 16 / 36 Adding Vectors s = a + b

17 17 / 36 Rules of Vector Addition/Subtaction ( a + b = b + a a + ) ( ) b + c = a + b + c d = a ( b = a + ) b commutative law associative law vector subtraction

18 18 / 36 commutative law a + b = b + a

19 19 / 36 associative law ( a + ) ( ) b + c = a + b + c

20 20 / 36 subtraction b = b Use same head-to-tail arrangement (just like for addition)

21 21 / 36 Addition/subtraction by components c = a + b Can add the components separately: c x = a x + b x (1) c y = a y + b y (2) or in other words c = (a x + b x ) î + (a y + b y ) ĵ

22 22 / 36 Addition example Path of an ant on the ground. Total displacement? y b c 3 2 a x -1

23 Addition example y 6 5 b c 4 3 a = 5î + 1ĵ b = 3 î + 4ĵ a 2 1 c = ( 5 + 3)î + (1 + 4)ĵ c = 2î + 5ĵ x / 36

24 Which expression is false concerning the vectors shown in the sketch? r r r a) C A B r r r b) C A B r r r c) A B C 0 d) C < A + B e) A 2 + B 2 = C 2

25 Which expression is false concerning the vectors shown in the sketch? r r r a) C A B r r r b) C A B r r r c) A B C 0 d) C < A + B e) A 2 + B 2 = C 2

26 What is the minimum number of vectors with unequal magnitudes whose vector sum can be zero? a) 2 b) 3 c) 4 d) 5 e) 6

27 What is the minimum number of vectors with unequal magnitudes whose vector sum can be zero? a) 2 b) 3 c) 4 d) 5 e) 6

28 During the execution of a play, a football player carries the ball for a distance of 33 m in the direction 76 north of east. To determine the number of meters gained on the play, find the northward component of the ball s displacement. a) 8.0 m b) 16 m c) 24 m d) 28 m e) 32 m

29 During the execution of a play, a football player carries the ball for a distance of 33 m in the direction 76 north of east. To determine the number of meters gained on the play, find the northward component of the ball s displacement. a) 8.0 m b) 16 m c) 24 m d) 28 m e) 32 m

30 24 / 36 Course Reminders The course entry survey link is up and will be up until next Friday at 11:59PM. All students are required to take the survey. Students having any issues with doing the homework that are not physics related (e.g. how to enter answers, exponents, graphing, etc.) should contact Wiley Student Support at the link on the course website. Office hours:

31 25 / 36 Multiplying vectors Multiplying 2 vectors can either produce Scalar a b = a b cos φ Vector Magnitude: a b = a b sin φ Direction: Right-hand rule a b (3) a b (4)

32 26 / 36 Dot Product (scalar) Scalar a b = a b cos φ By components a b = ( ) ( ) a x î + a y ĵ + a z ˆk b x î + b y ĵ + b z ˆk = a x b x + a y b y + a z b z

33 27 / 36 Dot Product Laws Commutative law a b = b a Associative law ( a b ) ( ) c = a b c Distributive law ( ) a b + c = a b + a c

34 28 / 36 Cross Product (vector) Vector Magnitude: a b = a b sin φ Direction: Right-hand rule a b = ( ) ( ) a x î + a y ĵ + a z ˆk b x î + b y ĵ + b z ˆk = (a y b z b y a z )î + (a zb x b z a x )ĵ + (a xb y b x a y )ˆk

35 29 / 36 Cross Product Laws NOT Commutative a b b a Rather, a b = b a NOT necessarily Associative ( a ) ( ) b c a b c Distributive law ( ) a b + c = a b + a c

36 30 / 36 Unit Vectors Dot Product i i = j j = k k = 1 i j = i k = 0 j i = j k = 0 k i = k j = 0 Cross Product i i = j j = k k = 0 i j = k j k = i k i = j

37 31 / 36 Unit Vectors Dot Product i i = j j = k k = 1 i j = i k = 0 j i = j k = 0 k i = k j = 0 product of perpendicular vectors=0! Cross Product i i = j j = k k = 0 j i = k k j = i i k = j product of parallel vectors=0!

38 32 / 36 Right Hand Rule

39 33 / 36 Dot Product question What is the dot product of A B? Where A = 1.0î + 2.0ĵ and B = 1.5î + 2.0ĵ? A 3.0 B -0.5 C 0.5 D 2.5 E -2.5

40 34 / 36 Dot Product question What is the dot product of A B? Where A = 1.0î + 2.0ĵ and B = 1.5î + 2.0ĵ? A 3.0 B -0.5 C 0.5 D 2.5 E -2.5

41 Consider the various vectors given in the choices below. The cross product of which pair of vectors is equal to zero?

42 Consider the various vectors given in the choices below. The cross product of which pair of vectors is equal to zero?

43 35 / 36 Examples

44 Two vectors a r and b r are added together to form a vector c r. The relationship between the magnitudes of the vectors is given by a + b = c. Which one of the following statements concerning these vectors is true? a) a r and b r must point in the same direction. b) a r and b r must be displacements. c) a r and b r must be at right angles to each other. d) a r and b r must point in opposite directions. e) a r and b r must have equal lengths.

45 Two vectors a r and b r are added together to form a vector c r. The relationship between the magnitudes of the vectors is given by a + b = c. Which one of the following statements concerning these vectors is true? a) a r and b r must point in the same direction. b) a r and b r must be displacements. c) a r and b r must be at right angles to each other. d) a r and b r must point in opposite directions. e) a r and b r must have equal lengths.

46 A physics student adds two displacement vectors with magnitudes of 8.0 km and 6.0 km. Which one of the following statements is true concerning the magnitude of the resultant displacement? a) The magnitude must be 14.0 km. b) The magnitude must be 10.0 km. c) The magnitude could be equal to zero kilometers, depending on how the vectors are oriented. d) The magnitude could have any value between 2.0 km and 14.0 km, depending on how the vectors are oriented. e) No conclusion can be reached without knowing the directions of the vectors.

47 A physics student adds two displacement vectors with magnitudes of 8.0 km and 6.0 km. Which one of the following statements is true concerning the magnitude of the resultant displacement? a) The magnitude must be 14.0 km. b) The magnitude must be 10.0 km. c) The magnitude could be equal to zero kilometers, depending on how the vectors are oriented. d) The magnitude could have any value between 2.0 km and 14.0 km, depending on how the vectors are oriented. e) No conclusion can be reached without knowing the directions of the vectors.

48 36 / 36 Next Week Projectile Motion