1. Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@engr.mun.ca ENGI 1313 Mechanics I Lecture 05:Cartesian Vectors

2. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 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. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 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. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 4 Recall 2D Cartesian Vector Coplanar Force Vector Summation  Unit vector  dimensionless Force Vectors Component Vectors Resultant Force Vector Unit Vector; i = F X FXFX ^ Unit Vector; j = F y FyFy ^

5. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 5 Extend to 3D Why Use Vectors  Simplifies mathematical operations Rectangular Coordinate System  Right-hand rule

6. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 6 Cartesian Vector in 3D Three Unit Vectors  Component magnitude A x, A y, A z scalar  Component sense +, - Cartesian quadrant  Component direction i, j, k unit vectors

7. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 7 Summation Cartesian Vectors in 3D Use 2D Principles  Vector A  Vector B

8. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 8 Cartesian Vector Magnitude Successive Application of 2D Principle  Pythagorean theorem  Find components A x and A y  Combine with z component Magnitude A of vector A

9. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 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. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 10 Cartesian Vector Direction (cont.) Coordinate Direction Angle,   Measured +x-axis to tail of vector A

11. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 11 Cartesian Vector Direction (cont.) Coordinate Direction Angle,   Measured +y-axis to tail of vector A

12. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 12 Cartesian Vector Direction (cont.) Coordinate Direction Angle,   Measured +z-axis to tail of vector A

13. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 13 Cartesian Vector Direction (cont.) Direction Cosines  Coordinate direction angles ( , , &  ) determined by cos -1

14. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 14 Cartesian Vector Direction (cont.) Express as Cartesian Vector, A Recall Unit Vector Form Unit Vector, u A uAuA A ^ Unit Vector; u A = A ^

15. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 15 Cartesian Vector Direction (cont.) Unit Vector, u A Relate to Direction Cosines Therefore uAuA ^

16. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 16 Cartesian Vector Direction (cont.) Find Unit Vector Amplitude, | u A |  Recall general case for vector and magnitude  Where the unit vector and magnitude is Therefore ^

17. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 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. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 18 Comprehension Quiz 5-01 If you only know u A (unit vector) you can determine the ________ of A uniquely.  A) magnitude (A)  B) angles ( , , and  )  C) components (A x, A y, & A z )  D) All of the above Answer  BUnit vector (u A ) defines direction |A| defines magnitude

19. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 19 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 G= 80lb  = 111   = 69.3 

20. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 20 G= 80lb  = 111   = 69.3  Example Problem 5-01 (cont.) Resolve Force F components Cartesian Vector Notation FzFz F FyFy FxFx

21. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 21 G= 80lb  = 111   = 69.3  Example Problem 5-01 (cont.) Determine  for Vector G 

22. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 22 Example Problem 5-01 (cont.) Coordinate Direction Angles  =111   = 111  y z x -x G = 80lb Unit circle 

23. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 23 Example Problem 5-01 (cont.) Coordinate Direction Angles  and  y z x -x G = 80lb  = 69.3   = 30.2 

24. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 24 Example Problem 5-01 (cont.) Cartesian Vector G  = 111  y z x -x  = 69.3   = 30.2  G = 80lb

25. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 25 G= 80lb  = 111   = 69.3  Example Problem 5-01 (cont.) Combine Force Vectors Resultant Vector

26. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 26 Group Problem 5-01 Problem 2-57 (Hibbeler, 2007) 12  Determine the magnitude and coordinate direction angles of F 1 and F 2. Sketch each force on an x, y, z reference.

27. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 27 Group Problem 5-01 (cont.) 1 Force F 1

28. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 28 Group Problem 5-01 (cont.) 2 Force F 2

29. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 29 Group Problem 5-02 Problem 2-59 (Hibbeler, 2007)  Determine the magnitude and coordinate angles of F acting on the stake.

30. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 30 Group Problem 5-02 Determine F and components FzFz FyFy FxFx F

31. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 31 Group Problem 5-02 Determine Coordinate Direction Angles

32. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 32 Classification of Textbook Problems Hibbeler (2007) Problem Set Concept Degree of Difficulty Estimated Time 2-57Cartesian: Magnitude & directionMedium10-15min 2-58 to 2-60Cartesian: ForcesEasy5-10min 2-61 to 2-69Cartesian; Force components & resultantEasy5-10min 2-70 to 2-71Cartesian; Force components & resultantHard20-25min 2-72 to 2-78Cartesian; Force components & resultantMedium15-20min

33. ENGI 1313 Statics I – Lecture 05© 2007 S. Kenny, Ph.D., P.Eng. 33 References Hibbeler (2007) http://wps.prenhall.com/esm_hibbeler_eng mech_1 en.wikipedia.org