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 06:Cartesian and Position Vectors

2. ENGI 1313 Statics I – Lecture 06© 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 06© 2007 S. Kenny, Ph.D., P.Eng. 3 Lecture 06 Objectives to further examine 3D Cartesian vectors to define a position vector in Cartesian coordinate system to determine force vector directed along a line

4. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 4 Example Problem 6-01 Problem 2-77 (Hibbeler, 2007). The bolt is subjected to the force F, which has components acting along the x, y, z axes as shown. If the magnitude of F is 80 N, and  = 60° and  = 45°, determine the magnitudes of its components.

5. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 5 Example Problem 6-01 Known Find

6. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 6 Example Problem 6-01 (cont.) Find Angle  Find component magnitudes FzFz FyFy FxFx  = 60   = 45 

7. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 7 Position Vectors – General 3D Coordinates  Unique position in space  Right-hand coordinate system A(4,2,-6) B(0,2,0) C(6,-1,4)

8. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 8 Position Vectors – Origin to a Point Fixed vector locating a point P(x,y,z) in space relative to another point (origin) within a defined coordinate system.  Right-hand Cartesian coordinate system  Tip-to-tail vector component technique zk ^ yj ^ xi ^

9. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 9 Position Vector – General Case Two Points in Space  Rectangular Cartesian coordinate system Origin O  Point A and Point B A(x A, y A, z A ) B(x B, y B, z B ) x y z O(0, 0, 0)

10. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 10 Position Vector – General Case Establish Position Vectors  From Point O to Point A (r OA = r A )  From Point O to Point B (r OB = r B )  From Point A to Point B (r AB = r ) A(x A, y A, z A ) B(x B, y B, z B ) x y z O(0, 0, 0) r OA r OB r AB Recall “tip-to-tail” vector addition laws 

11. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 11 Position Vector – General Case Define Position Vector (r AB = r )  “tip – tail” or B(x B, y B, z B ) – A(x A, y A, z A ) A(x A, y A, z A ) x y z O(0, 0, 0) r OA r OB r AB ^ (x B – x A ) i ^ (z B – z A ) k ^ (y B – y A ) j r = r AB B(x B, y B, z B )

12. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 12 Comprehension Quiz 6-01 Two points in 3D space have coordinates of P(1, 2, 3) and Q (4, 5, 6) meters. The position vector r QP is given by  A) { 3 i + 3 j + 3 k} m  B) {-3 i - 3 j - 3 k} m  C) { 5 i + 7 j + 9 k} m  D) {-3 i + 3 j + 3 k} m  E) { 4 i + 5 j + 6 k} m Answer: B  {-3 i - 3 j - 3 k} m

13. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 13 Comprehension Quiz 6-02 P and Q are two points in a 3-D space. How are the position vectors r PQ and r QP related?  A) r PQ = r QP  B) r PQ = -r QP  C) r PQ = 1/r QP  D) r PQ = 2r QP Answer: B Q(x B, y Q, z Q ) x y z P(x P, y P, z P ) r PQ = -r QP

14. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 14 Comprehension Quiz 6-03 If F is a force vector (N) and r is a position vector (m), what are the units of the expression  A) N  B) Dimensionless  C) m  D) N  m  E) The expression is algebraically illegal Answer: A

15. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 15 Example 6-01 Express the force vector F DA in Cartesian form Known:  A(0,0,14) ft  D(2,6,0) ft  F DA = 400 lb

16. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 16 Example 6-01 (cont.) Find Position Vector r DA  Through point coordinates r DA

17. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 17 Example 6-01 (cont.) Find Position Vector |r DA | Magnitude r DA

18. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 18 Example 6-01 (cont.) Find unit vector u DA u DA

19. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 19 Example 6-01 (cont.) Find Unit Vector u DA Magnitude  Confirm unity u DA

20. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 20 Example 6-01 (cont.) Find Force Vector F DA  or

21. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 21 Group Problem 6-01 Find the resultant force magnitude and coordinate direction Plan  Cartesian vector form of F CA and F CB  Sum concurrent forces  Obtain solution

22. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 22 Group Problem 6-01 (cont.) Position Vectors and Magnitude  r CA  r CB

23. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 23 Group Problem 6-01 (cont.) Force Vectors and Magnitude  F CA  F CB

24. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 24 Group Problem 6-01 (cont.) Force Resultant Vector Magnitude & Orientation

25. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 25 Group Problem 6-01 (cont.) Force Resultant Vector Magnitude & Orientation F1F1 F2F2 FRFR

26. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 26 Classification of Textbook Problems Hibbeler (2007) Problem SetConcept Degree of Difficulty Estimated Time 2-79 to 2-84Position vectorsEasy5-10min 2-85 to 2-90Resultant force vectorsMedium15-20min 2-91 to 2-96Resultant force & position vectorsMedium15-20min 2-97 to 2-99Position vectorsEasy10-15min 2-100Resultant force & position vectorsHard30min 2-101 to 2-106Resultant force & position vectorsMedium15-20min

27. ENGI 1313 Statics I – Lecture 06© 2007 S. Kenny, Ph.D., P.Eng. 27 References Hibbeler (2007) http://wps.prenhall.com/esm_hibbeler_eng mech_1