Prep for Engineering Studies PHYS-100

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1 Prep for Engineering Studies PHYS-100 Winter Dr. Joseph Trout

2 Pythagoras was a Greek philosopher who made important developments in mathematics, astronomy, and the theory of music. The theorem now known as Pythagoras's theorem was known to the Babylonians 1000 years earlier but he may have been the first to prove it. 2

3 r 2 =x 2 y 2 y x 3

4 r= x 2 y 2 y x 4

5 r= 4 m 2 3m 2 = 16 m 2 9m 2 = 25m 2 =5m r y=3 m x=4 m 5

6 r= 6 m 2 7m 2 = 36 m 2 49m 2 = 8 2 =9.22m r y=7m x=6 m 6

7 r y sin = y r cos = x r x tan = y x r= x 2 y 2 7

8 r=5 y=4 sin = y r =4 5 cos = x r =3 5 x=3 tan = y x =4 3 r= x 2 y 2 r= =5 8

9 hyp opp sin = opp hyp cos = adj hyp adj tan = opp adj 9

10 hyp opp sin = opp hyp cos = adj hyp adj tan = opp adj 10

11 y x 11

12 y 90 o 180 o 0 o 360 o x 270 o 12

13 r s=r = s r [ ]=radians 13

14 y 90 o 180 o 0 o 0 rad 360 o 2 rad x Whole Circle 270 o = s r =2 r r =2 radians 14

15 y 90 o 2 rad 180 o rad 0 o 0 rad x 360 o 2 rad Whole Circle 270 o 3 2 rad = s r =2 r r =2 radians 15

16 Scalar Magnitude only. Example: mass, distance, speed Example: m, x, v Vector Magnitude and Direction. Example: displacement,velocity, acceleration, force Example: x, v, a, F

17 Distance scalar magnitude of the total distance traveled. Displacement vector distance between final position and initial position AND the direction.

18 Displacement in One Dimension: Direction will either be positive or negative.

19 Distance vs. Displacement: distance= x=4.0 m X= -4.0 m X=-2.0 m X= 0.0 m X= 2.0 m X=4.0 m

20 Distance vs. Displacement: distance= x=4.0 m displacement= x=x f x i =4.0 m 0.0 m= 4.0 m X= -4.0 m X=-2.0 m X= 0.0 m X= 2.0 m X=4.0 m

21 Distance vs. Displacement: distance=4.0 m 8.0 m=12m X= -4.0 m X=-2.0 m X= 0.0 m X= 2.0 m X=4.0 m displacement x=x f x i = 4m 0m= 4m

22 Marathon distance = 26 miles Marathon displacement = -0.2iles Start Finish +x

23 Displacement = x= x f x i

24 Scalar Magnitude ONLY distance=iles mass=6 kg Vector Magnitude and Direction Example :Velocity v=40m/s North v=40m/s in positive x direction. 30 o v=3m/ s i 7 m/ s j 6 m/s k 24

25 y o =60 o x 25

26 j Vector Notation y axis z axis x axis i k 26

27 j Vector Notation B=3 m j i A=4 m i 27

28 j Vector Notation R= A B B=3 m j i A=4 m i 28

29 j Vector Notation R= A B R=4m i 3m j B=3 m j i A=4 m i 29

30 Pythagoras was a Greek philosopher who made important developments in mathematics, astronomy, and the theory of music. The theorem now known as Pythagoras's theorem was known to the Babylonians 1000 years earlier but he may have been the first to prove it. 30

31 r 2 =x 2 y 2 y x 31

32 r= x 2 y 2 y x 32

33 r= 4 m 2 3m 2 = 16 m 2 9m 2 = 25m 2 =5m r y=3 m x=4 m 33

34 r= 6 m 2 7m 2 = 36 m 2 49m 2 = 8 2 =9.22m r y=7m x=6 m 34

35 r y sin = y r cos = x r x tan = y x r= x 2 y 2 35

36 hyp opp sin = opp hyp cos = adj hyp adj tan = opp adj 36

37 j Vector Notation R= A B R=4m i 3 m j 3 m i 4 m R = 4m 2 3 m 2 = 2 2 =5m tan = opp =tan adj 1 3m 4m =36.87o 37

38 j R= A B R=4m i 3 m j o Vector Notation 3 m i 4 m R = 4m 2 3 m 2 = 2 2 =5m tan = opp =tan adj 1 3m 4m =36.87o 38

39 j R=10 40 o Vector Notation R =10 m R y R =40 o i R x 39

40 j R=10 40 o Vector Notation R =10 m R y R =40 o i R x cos = adj hyp cos R = R x R R x = R cos R =10 m cos 40 o =7.66m 40

41 j R=10 40 o Vector Notation R =10 m R y R =40 o R x =7.66 m sin = opp hyp i sin R = R y R R y = R sin R =10m sin 40 o =6.43m 41

42 j Vector Notation R =10 m R y =6.43m R =40 o R x =7.66 m i R=10 40 o R=R x i R y j=7.66 m i 6.43m j 42

43 j Vector Notation Check: R = R x 2 R y 2 R = 7.66 m m 2 =10 m R =10 m R =40 o R x =7.66 m R=10 40 o R y =6.43m i tan = R y R x 6.43 m =tan m =40o R=R x i R y j=7.66 m i 6.43m j 43

44 Vector Notation II A y =4 m j I A A= A x i A y j=3m i 4 m j A x =3 m i III IV 44

45 Vector Notation II A y =4 m j I A A= A x i A y j=3m i 4 m j A x =3 m i III IV A = 3m 2 4m 2 = A =tan 1 4 m 3 m =53.13o 45

46 Vector Notation II A y =4 m j I A A= A x i A y j=3m i 4 m j A= A =5m@53.13 o A A x =3 m i III IV A = 3m 2 4m 2 = A =tan 1 4 m 3 m =53.13o 46

47 Vector Notation II j I B B y =2m B=B x i B y j= 4m i 2m j B x =4 m B i III IV 47

48 Vector Notation II j I B B y =2m B=B x i B y j= 4m i 2m j B x = 4 m B i B = 4m 2 2m 2 =4.47m III IV 48

49 Vector Notation II j I B B y =2m B=B x i B y j= 4m i 2m j B x = 4 m B B ' i III IV B = 4m 2 2m 2 =4.47m B =tan 1 2m 4 m = 26.57o????? 49

50 Vector Notation II j I B B x = 4 m III B y =2m B IV B ' B=B x i B y j= 4m i 2m j i =180 o ' B B =180 o o = o B B = 4m 2 2m 2 =4.47m B =tan 1 2m 4 m = 26.57o????? 50

51 Vector Notation j II I B=B x i B y j= 4m i 2 m j B= B =4.47m@ o B B y =2m B x = 4 m B B ' i =180 o ' B B B =180 o o = o III IV B = 4m 2 2m 2 =4.47m B =tan 1 2m 4 m = 26.57o????? 51

52 Vector Notation j II I C=C x i C y j= 3m i 6 m j C x = 3m C i C C y = 6 m III IV 52

53 Vector Notation j II C C=C x i C y j= 3m i 6 m j C x = 3m C i C C y = 6 m III IV C = 3m 2 6m 2 =6.71m C =tan 1 6m 3 m =71.57o????? 53

54 Vector Notation j II I C=C x i C y j= 3m i 6 m j C x = 3m C C ' i C =180 o C ' C C y = 6 m C =180 o o = o III IV C = 3m 2 6m 2 =6.71m C =tan 1 6m 3 m =71.57o????? 54

55 Vector Notation j II I C=C x i C y j= 3m i 6 m j C= C =6.71m@ o C x = 3m C C ' i C =180 o C ' C C y = 6 m C =180 o o = o III IV C = 3m 2 6m 2 =6.71m C =tan 1 6m 3 m =71.57o????? 55

56 Vector Notation j II I D=D x i D y j=3m i 3 m j D= D = o D D x =3 m i D y = 3 m D III IV D = 3m 2 3m 2 =4.24 m D =tan 1 3m 3m = 45o????? 56

57 Vector Notation j II I D=D x i D y j=3m i 3 m j D= D = o D D y = 3 m D ' D x =3 m D i III IV D = 3m 2 3m 2 =4.24 m D =tan 1 3m 3m = 45o????? 57

58 Vector Notation II j I D=D x i D y j=3m i 3 m j D= D = o = o D D y = 3 m D ' D x =3 m D i D =360 o C ' D =360 o 45 o =315 o III IV D = 3m 2 3m 2 =4.24 m D =tan 1 3m 3m = 45o????? 58

59 Adding Vectors 59

60 Adding Vectors A=5m i 0 m j B=0m i 6 m j B A 60

61 Adding Vectors C B A=5m i 0 m j B=0m i 6 m j C= A B=5m i 6 m j A 61

62 Adding Vectors C B A=5m i 0 m j B=0m i 6 m j C= A B=5m i 6 m j A C = 5m 2 6m 2 =7.81 m C =tan 1 6m =50.19o 62

63 Adding Vectors B A 63

64 Adding Vectors A=5m i 0m j B=3m i 5m j B B y A A x B x 64

65 Adding Vectors A=5m i 0m j B=3m i 5m j C= A B=8m i 5m j C B A C = 8m 2 5m 2 =9.43m C =tan 1 5m 8 m =32.00o 65

66 Adding Vectors Vectors may be moved so that the tail of each is at the origin. B A 66

67 Adding Vectors B A 67

68 Adding Vectors C=C x i C y j B C A 68

69 Adding Vectors C=C x i C y j B C C y = A C x =8 m 69

70 Adding Vectors C=C x i C y j C=8m i 5m j B C C y = A C x =8 m 70

71 Adding Vectors C=C x i C y j C=8m i 5m j B C C y = A C x =8 m C = 8m 2 5m 2 =9.43m C =tan 1 5m 8 m =32.00o 71

72 Adding Vectors B A 72

73 Adding Vectors A=5m i 0m j B= 5m i j B A 73

74 Adding Vectors A=5m i 0m j B= 5m i j B A 74

75 Adding Vectors A=5m i 0m j B= 5m i j B C A 75

76 Adding Vectors A=5m i 0m j B= 5m i j C= A B=0m i j B C A C = 0m 2 5m 2 =5.00m C =90.00 o 76

77 Adding Vectors B A 77

78 Adding Vectors B A 78

79 Adding Vectors B A 79

80 Adding Vectors B C= A B C=C x i C y j C=2m i 8m j A C = 2 m 2 8m 2 =8.2 C =tan 1 8m 2 m =75.96o 80

81 Adding Vectors A=7m i 3m j B= 5m i j B A 81

82 Adding Vectors A=7m i 3m j B= 5m i j C B A 82

83 Adding Vectors A=7m i 3m j B= 5m i j C C= A B=2 m i 8 m j B C A C = 2 m 2 8 m 2 =8.2 C =tan 1 8m 2 m =75.96o 83

84 Adding Vectors B C A=7m i 3m j B= 5m i j C=4 m i 2m j A 84

85 Adding Vectors A=7m i 3m j C B= 5m i j C=4 m i 2m j B D D= A B C=6m i 6 m j A 85

86 Adding Vectors A=7m i 3m j B= 5m i j C=4 m i 2m j D= A B C=6m i 6 m j B A C 86

87 Adding Vectors A=7m i 3m j B= 5m i j C=4 m i 2m j A B D= A B C=6m i 6 m j B A C 87

88 Adding Vectors A B A=7m i 3m j B= 5m i j C=4 m i 2m j D= A B C=6m i 6 m j A B C C 88

89 Adding Vectors A=7m i 3m j B= 5m i j C=4 m i 2m j D= A B C=6m i 6 m j A B C= D C 89

90 Adding Vectors B C F A D E 90

91 Adding Vectors A=5m i 2 m j B C F A D E 91

92 Adding Vectors B C A=5m i 2 m j B=3m i 6m j F A D E 92

93 Adding Vectors B C A=5m i 2 m j B=3m i 6m j C=0m i 3 m j F A D E 93

94 Adding Vectors B C A=5m i 2 m j B=3m i 6m j C=0m i 3 m j D= 8m i 10 m j F A D E 94

95 Adding Vectors B C A=5m i 2 m j B=3m i 6m j C=0m i 3 m j D= 8m i 10 m j F A D E= i 0 m j E 95

96 Adding Vectors B C A=5m i 2 m j B=3m i 6m j C=0m i 3 m j D= 8m i 10 m j F A D E= i 0 m j F= 3 m i 10 m j E 96

97 Adding Vectors B C A=5m i 2 m j B=3m i 6m j C=0m i 3 m j D= 8m i 10 m j F R A D E= i 0 m j F= 3 m i 10 m j R= 2 m i 5m j E 97

98 Adding Vectors B C A=5m i 2 m j B=3m i 6m j C=0m i 3 m j D= 8m i 10 m j F R A D E= i 0 m j F= 3 m i 10 m j R= 2 m i 5m j E R = 2 m 2 2 =5.39 m R =tan 1 5m 2m = 68.20o 180 o = o 98

99 Adding Vectors F B E A C D 99

100 Adding Vectors F B E A C D 100

101 Adding Vectors F D 101

102 Adding Vectors F D 102

103 Adding Vectors F R R= 2 m i 5m j D 103

104 Adding Vectors F R R= 2 m i 5m j D 104

105 More Vectors: s Displacement [ s]=m Distance between the starting point and final point and direction. v Velocity [ v]=m/ s Speed and direction. The time derivative of position a Acceleration [ a]=m/s 2 The time derivative of velocity 105

106 x=100 m Tugboat with broken rudder. Can only go straight. It takes ten seconds to cross calm lake. v x = x t =100 m 10 s =10 m/s 106

107 x=100 m Tugboat with broken rudder. Can only go straight. It takes ten seconds to cross calm lake. v x =10m/ s v x = x t =100 m 10 s =10 m/s 107

108 x=100 m Tugboat with broken rudder. Can only go straight. It takes ten seconds to cross lake. Now consider strong current. v x =10m/ s x=100 m v y =/s y=v y t= /s 10 s =50 m 108

109 x=100 m Tugboat with broken rudder. Can only go straight. It takes ten seconds to cross lake. Now consider strong current. v y=50 m x=100 m v y =/s 109

110 x=100 m Tugboat with broken rudder. Can only go straight. It takes ten seconds to cross lake. Now consider strong current. v x=100 m v y =/s y=50 m v=v x i v y j v=10 m/s i 5m/s j 110

111 x=100 m Tugboat with broken rudder. Can only go straight. It takes ten seconds to cross lake. Now consider strong current. v x=100 m v y =/s y=50 m v=v x i v y j v=10m/s i / s j v = 10 m/s 2 /s 2 =11.18 m/s /s =tan 1 10 m/s =26.57o 111

112 x=100 m Tugboat with broken rudder. Can only go straight. It takes ten seconds to cross lake. Now consider strong current. v x=100 m v y =/s y=50 m o r= x 2 y 2 r= 100 m 2 50 m 2 = m 112

113 x=100 m Tugboat with broken rudder. Can only go straight. It takes ten seconds to cross lake. Now consider strong current. v x=100 m v y =/s y=50 m v=11.18 m/ s@26.57 o r= m v= r t = m =11.18 m/ s 10 s 113

114 x=100 m Tugboat with broken rudder. Can only go straight. It takes ten seconds to cross lake. Now consider strong current. v x=100 m y=50 m v y =/s v =11.18 m/ s =26.57 o v x =10 m/s v y =/s v=11.18 m/ s@26.57 o r= m v= r t = m =11.18 m/ s 10 s 114

115 Adding Vectors v=17.20 o 10m/ s 10m/ s 10m/ s 10m/ s 115

116 Adding Vectors v y = v sin v y =17.20 m/s sin o v y =14 m/s 10m/ s v=17.20m/s@54.46 o 10m/ s 10m/ s 10m/ s v x = v cos v x =17.20 m/s cos54.46 o v x =10 m/s 116

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