ME 201 Engineering Mechanics: Statics

Transcription

1 ME 01 Engineering Mechanics: Statics Unit.1 Cartesian Vectors Addition of Cartesian Vectors

2 Mastery Quiz Mastery Quiz available directly on I-Learn, i.e., NOT in Testing Center Mastery Quizzes are not to be taken in class. Quizzes are Closed Book/Notes with the eception of: One side of one 3" 5" note card Calculators Ecel or Mathematica may be used as a calculator (no eternal files) Quizzes pull from a pool of questions, i.e., no two students will get the same quiz Quiz problems are analytical, multiple choice (no distractors) Typically 5 quiz problems, some problems may require multiple answers

3 Mastery Quiz Typically 5 quiz problems, some problems may require multiple answers Required to achieve at least 50% before being allowed to move on to net unit (no minimum requirement on Advanced Mastery Quizzes) Quizzes may be retaken once (without remediation) After nd attempt will be required to meet with instructor to review quiz prior to subsequent attempt. Only the highest quiz is used in grade calculation You are limited to hours on the quiz and it must be completed in a single sitting, i.e., no save option.

4 Mastery Quiz

5 Vector undamentals 3D Vector Visualization Right Hand Coordinate System orce Vector Unit Vector Direction Angles

6 Right-Hand Rule Used to determine positive direction of X,Y, & Z aes

7 Unit Vectors Direction of a Vector Magnitude of 1 A A A u k j i A A k j i u A Au A A

8 z Direction Cosines The orientation of vector is defined by direction angles α, β, and γ, measured from the tail of to positive, y, and z ais, respectively. y cos z cos cos y cos z y cos z z cos z α y β γ y y z y

9 Direction Cosines Another way to obtain the directions cosines is to form a unit vector. Given: Dividing by magnitude of or k j i U z y k j i k y k j i U cos cos cos

10 Direction Angle Identity Also, since Dividing each term by and substituting z y z y cos cos cos 1 This identity is useful to determining direction angles

11 Direction Angle Identity Also, since Dividing each term by and substituting z y z y cos cos cos 1 This identity is useful to determining direction angles

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17 Video a Right Hand Coordinate System undamental Concepts Right Hand Rule

18 Right Hand Coordinate System While you are seated in Statics class, the instructor draws a coordinate system on the board, as shown below. If the coordinate system follows right hand rule, which direction is the positive Z-ais? A. Out of the board or towards the classroom B. Into the board or away from the classroom C. Neither a or b D. Cannot tell from given information

19 Video b Unit Vector undamental Concepts Vector with a Magnitude of 1 Essentially Direction of a Vector

20 Right Hand Coordinate System What is the unit vector of the following vector: A 1i j k A. 1 B. 3 C. < > D. <..4.4 > E. None of the above

21 Video c Direction Angles undamental Concepts Angles α, β, and γ, measured from the vector to the positive, y, and z ais, respectively Used to define the direction of a Vector See 3D Vector Visualization Animation z z z α y β γ y y z y

22 Video c - Other Angles Used to ind Determine Vector Direction

23 Direction Cosines Given the vector below, what is the cos γ? i 3j 6k A..655 B..857 C..96 D E. None of the above

24 Key Concepts:.5-.6 Cartesian Vectors Right-Handed Coordinate System Unit Vector Direction Cosines of a Vector Summation of orces

25 Right-Handed Coordinate System Eplain Right-Hand Rule as it applies to Statics Why is it useful?

26 Right Hand Coordinate System Right Hand Rule A rectangular or Cartesian coordinate system is said to be right-handed provided the thumb of the right hand points in the direction of the positive z ais when the right hand fingers are curled about the ais and directed from positive to positive y ais

27 Right-Hand Rule Used to determine positive direction of X,Y, & Z aes

28 Right-Hand Rule Also used to determine positive rotational direction

29 Right Hand Coordinate System While you are seated in Statics class, the instructor draws a coordinate system on the board, as shown below. If the coordinate system follows right hand rule, which direction is the positive Z-ais? A. Out of the board or towards the classroom B. Into the board or away from the classroom C. Neither a or b D. Cannot tell from given information

30 Unit Vectors What is a Unit Vector? How are they useful in Statics?

31 Right Hand Coordinate System What is the unit vector of the following vector: A 1i j k A. 1 B. 3 C. < > D. <..4.4 > E. None of the above

32 Unit Vectors Direction of a Vector Magnitude of 1 A A A u k j i A A k j i u A Au A A

33 Direction Cosines Angles What are the Direction Cosine Angles of a Vector? How are they useful in Statics? Show 3D Vector Visualization Animation

34 z Direction Cosines The orientation of vector is defined by direction angles α, β, and γ, measured from the tail of to positive, y, and z ais, respectively. y cos z cos cos y cos z y cos z z cos z α y β γ y y z y

35 Direction Cosines Another way to obtain the directions cosines is to form a unit vector. Given: Dividing by magnitude of or k j i U z y k j i k y k j i U cos cos cos

36 Direction Cosines Given the vector below, what is the cos γ? i 3j 6k A..655 B..857 C..96 D E. None of the above

37 Direction Angle Identity Also, since Dividing each term by and substituting z y z y cos cos cos 1 This identity is useful to determining direction angles

38 Other Angles Used to ind Determine Vector Direction

39 Other Angles Used to ind Determine Vector Direction Use right triangle relationships to calculate, y, and z components of force. cos cos y cos sin z sin Note that these formulas are valid only for this angle configuration.

40 Summation of orces What do we mean by the term resultant? How do we sum forces? R i y j z k

41 Eample Problems #1 ind Resultant/angles z 1 #3 ind direction angles of 1 y # ind as a vector z γ β y #4-ind Resultant vector

42 Eample Problem Given: 1 = 60j + 80k lb = 50i -100j + 100k lb ind: R, R, α, β, γ z 1 y

43 Eample Problem Solution Given: 1 = 60j + 80k lb = 50i -100j + 100k lb ind: R, R, α, β, γ z 1 y

44 Eample Problem Solution Given: 1 = 60j + 80k lb = 50i -100j + 100k lb ind: R, R, α, β, γ Solution: 1 = 60j + 80k lb = 50i -100j + 100k lb R = 50i - 40j + 180k lb R 50 ( 40) R 191lb 180 z 1 50 cos cos cos y

45 In Class Eercise Given: =00 N β = 60º γ = 45º ind: (vector notation) z γ β y 100i 100j141. 4k N

46 In Class Eercise Solution Solution: cos cos cos cos cos cos 60 cos z γ β or 10 00(cos60i cos60j cos45k) 100i 100j141. 4k N (α=60º, by inspection) y

47 In Class Eercise

48 Solution: -66

49 In Class Eercise

50 Solution: -68