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ENGI 1313 Mechanics I

ENGI 1313 Mechanics I . Lecture 05: Cartesian Vectors. Chapter 2 Objectives. to review concepts from linear algebra to sum forces, determine force resultants and resolve force components for 2D vectors using Parallelogram Law to express force and position in Cartesian vector form

gwendolyn-leblanc
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ENGI 1313 Mechanics I

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  1. ENGI 1313 Mechanics I Lecture 05: Cartesian Vectors

  2. Chapter 2 Objectives • to review concepts from linear algebra • to sum forces, determine force resultants and resolve force components for 2D vectors using Parallelogram Law • to express force and position in Cartesian vector form • to introduce the concept of dot product

  3. Lecture 05 Objectives • to further examine Cartesian vector notation and extend to representation of a 3D vector • to sum 3D concurrent force systems

  4. ^ ^ Unit Vector; i = FX Unit Vector; j = Fy FX Fy Recall 2D Cartesian Vector • Coplanar Force Vector Summation • Unit vector  dimensionless Resultant Force Vector ComponentVectors Force Vectors

  5. Extend to 3D • Why Use Vectors • Simplifies mathematical operations • Rectangular Coordinate System • Right-hand rule

  6. Cartesian Vector in 3D • Three Unit Vectors • Component magnitude • Ax, Ay, Az scalar • Component sense • +, - • Cartesian quadrant • Component direction • i, j, k unit vectors

  7. Summation Cartesian Vectors in 3D • Use 2D Principles • Vector A • Vector B

  8. Cartesian Vector Magnitude • Successive Application of 2D Principle • Pythagorean theorem • Find components Ax and Ay • Combine with z component • Magnitude A of vector A

  9. Cartesian Vector Direction • Coordinate Direction Angle • Vector tail with coordinate axis • , , and  • Range from 0 to 180 • Visualization aid using right rectangular prism

  10. Cartesian Vector Direction (cont.) • Coordinate Direction Angle,  • Measured +x-axis to tail of vector A

  11. Cartesian Vector Direction (cont.) • Coordinate Direction Angle,  • Measured +y-axis to tail of vector A

  12. Cartesian Vector Direction (cont.) • Coordinate Direction Angle,  • Measured +z-axis to tail of vector A

  13. Cartesian Vector Direction (cont.) • Direction Cosines • Coordinate direction angles (, , & ) determined by cos-1

  14. ^ Unit Vector; uA = A A Cartesian Vector Direction (cont.) • Express as Cartesian Vector, A • Recall Unit Vector • Form Unit Vector, uA ^ uA

  15. Cartesian Vector Direction (cont.) ^ • Unit Vector, uA • Relate to Direction Cosines • Therefore uA

  16. Cartesian Vector Direction (cont.) ^ • Find Unit Vector Amplitude, |uA | • Recall general case for vector and magnitude • Where the unit vector and magnitude is • Therefore

  17. Cartesian Vector Orientation (cont.) • Typical Cartesian Vector Problems • Magnitude and coordinate angles • Example 2.8 • Magnitude and projection angles • Example 2.10 • Only need to know 2 angles

  18. Comprehension Quiz 5-01 • If you only know uA (unit vector) you can determine the ________ of A uniquely. • A) magnitude (A) • B) angles (, , and ) • C) components (Ax, Ay, & Az) • D) All of the above • Answer • B Unit vector (uA) defines direction |A| defines magnitude

  19. G= 80lb • = 111  = 69.3 Example Problem 5-01 • Forces F and G are applied to a hook. Force F makes 60° angle with the X-Y plane. Force G has a magnitude of 80 lb with  = 111° and  = 69.3°. • Find the resultant force in Cartesian vector form

  20. G= 80lb • = 111  = 69.3 Example Problem 5-01 (cont.) • Resolve Force F components • Cartesian Vector Notation Fz F Fx Fy

  21. G= 80lb • = 111  = 69.3 Example Problem 5-01 (cont.) • Determine  for Vector G 

  22. Example Problem 5-01 (cont.) • Coordinate Direction Angles =111 -x G = 80lb z  = 111 y Unit circle  x

  23. Example Problem 5-01 (cont.) • Coordinate Direction Angles  and  -x G = 80lb z  = 30.2  = 69.3 y x

  24. Example Problem 5-01 (cont.) • Cartesian Vector G -x G = 80lb z  = 111  = 30.2  = 69.3 y x

  25. G= 80lb • = 111  = 69.3 Example Problem 5-01 (cont.) • Combine Force Vectors • Resultant Vector

  26. Group Problem 5-01 • Problem 2-57 (Hibbeler, 2007) • Determine the magnitude and coordinate direction angles of F1 and F2. Sketch each force on an x, y, z reference.

  27. Group Problem 5-01 (cont.) • Force F1

  28. Group Problem 5-01 (cont.) • Force F2

  29. Group Problem 5-02 • Problem 2-59 (Hibbeler, 2007) • Determine themagnitude and coordinate anglesof F acting on the stake.

  30. Group Problem 5-02 • Determine F and components F Fz Fx Fy

  31. Group Problem 5-02 • Determine Coordinate Direction Angles

  32. Classification of Textbook Problems • Hibbeler (2007)

  33. References • Hibbeler (2007) • http://wps.prenhall.com/esm_hibbeler_engmech_1 • en.wikipedia.org

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