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 04:Force Vectors and System of Coplanar Forces

2. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 2 Tutorial Questions SI Units and Use  Section 1.4 Page 9  Use a single prefix  Magnitude between 0.1 and 1000

3. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 3 Tutorial Questions SI Units and Use  Section 1.4 Page 9  Use a single prefix  Magnitude between 0.1 and 1000

4. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 4 Tutorial Questions SI Units and Use  Section 1.4 Page 9  Do not use compound prefixes

5. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 5 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

6. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 6 Lecture 04 Objectives to sum force vectors, determine force resultants, and resolve force components for 2D vectors using Scalar or Cartesian Vector Notation to demonstrate by example

7. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 7 Why an Alternate Approach? Application of Parallelogram Law  Cumbersome with a large number of coplanar forces due to successive application Recall Lecture 02 (Slide 13)

8. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 8 Parallelogram Law (Lecture 02) Multiple Force Vectors

9. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 9 Recall Lecture 02 (Slide 9) What is the Alternate Approach? Resolve Force Components Algebraic Summation Force Vectors Component Vectors

10. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 10 What is the Alternate Approach? Resolve Force Components Algebraic Summation Form Resultant Force Force Vectors Component Vectors Resultant Force Vector

11. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 11 Coplanar Force Vector Summation How to resolve a system of forces into rectangular components and determine the resultant force? Two Notations Used  (1) Scalar Notation More familiar approach  (2) Cartesian Vector Notation Useful in applications of linear algebra Advantageous over scalar notation for 3D

12. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 12 Cartesian Coordinate System Characteristics  Rectangular coordinate system  Unique spatial position  Vector algebra  Analytical geometry Abscissa Ordinate

13. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 13 F = F x + F y Rectangular Force Components Axes Must be Orthogonal Axes Orientation Does not Matter F = F x + F y

14. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 14 Resolve Force Components Known:Force Vector and Orientation Angle 

15. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 15 Resolve Force Components Known: Force Vector and Slope LxLx LyLy LhLh

16. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 16 Determine Resultant Force Known:Force Components Resultant Force Magnitude  Pythagorean theorem Resultant Force Direction  Trigonometry  y x FxFx FyFy F

17. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 17 Notation – Summation Coplanar Forces Scalar Notation Cartesian Vector Notation +X +Y FRFR -X -Y F R  F X + F Y F  = F X i + F Y j ^ ^ FXFX FYFY Common 1.Magnitude: F X & F Y 2.Sense: + & - 3. Direction: Orthogonal X & Y axes 3. Direction: Unit vectors Unit Vector; i = F X FXFX ^ Unit Vector; j = F Y FYFY ^ +i-i +j -j

18. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 18 Unit Vector Lecture 3  Scalar Magnitude and sense (+,-)  Vector Magnitude, sense (+,-) and direction Unit Vector  Vector Magnitude  4 units Sense  Positive Direction  X-axis x +

19. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 19 Coplanar Force Vector Summation Step 1: Define System of Forces  Rectangular coordinate system  Force vectors F 1, F 2 and F 3 Force Vectors

20. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 20 Coplanar Force Vector Summation Step 2: Resolve Component Forces Force Vectors Component Vectors

21. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 21 Coplanar Force Vector Summation Step 3: Sum System Force Components  Obtain resultant force vector components Force Vectors Component Vectors Resultant Force Vector

22. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 22 Coplanar Force Vector Summation Step 3: Sum System Force Components  Scalar notation Force Vectors Component Vectors Resultant Force Vector

23. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 23 Coplanar Force Vector Summation Step 4: Determine Resultant Force Vector  Magnitude, sense and direction Force Vectors Component Vectors Resultant Force Vector

24. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 24 Coplanar Force Vector Summation Step 3: Sum System Force Components  Cartesian vector notation Force Vectors Component Vectors Resultant Force Vector Unit Vector; i = F X FXFX ^

25. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 25 Comprehension Quiz 4-01 Resolve F along x and y axes in Cartesian vector notation. F = { ___________ } N  A) 80 cos 30° i - 80 sin 30° j  B) 80 sin 30° i + 80 cos 30° j  C) 80 sin 30° i - 80 cos 30° j  D) 80 cos 30° i + 80 sin 30° j 30° x y F = 80 N F x = 80 sin30  F y ĵ = -80 cos 30 

26. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 26 Comprehension Quiz 4-02 Determine the magnitude of the resultant force when F1 = { 10 î + 20 ĵ } N F2 = { 20 î + 20 ĵ } N A) 30 N B) 40 N C) 50 N D) 60 N E) 70 N i j F1F1 F2F2 C) 50 N FRFR F1F1

27. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 27 Example Problem 4-01 Find the magnitude and angle of the resultant force acting on the bracket. Solution Plan  Step 1: Define system of forces  Step 2: Resolve component forces  Step 3: Sum system force components  Step 4: Determine resultant force vector, magnitude and direction

28. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 28 Example Problem 4-01 (cont.) Step 2: Resolve Components  Cartesian vector form, F 1 F 1x = 15kN sin 40  F 1y = 15kN cos 40  F 1x F 1y

29. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 29 Example Problem 4-01 (cont.) Step 2: Resolve Components  Cartesian vector form, F 2 F 2x = -26kN (12/13) F 2y = 26kN (5/13) F 2x F 2y

30. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 30 Example Problem 4-01 (cont.) Step 2: Resolve Components  Cartesian vector form, F 3 F 3x = 36kN cos 30  F 3y = 36kN sin 30  F 3x F 3y

31. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 31 Example Problem 4-01 (cont.) Step 2: Resolve Components  Cartesian vector form  Therefore F 3x F 3y F 2x F 2y F 1x F 1y

32. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 32 Example Problem 4-01 (cont.) Step 3: Sum Collinear Forces  Collinear Cartesian vector form F 1x F 3y F 2x F 2y F 3x F 1y

33. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 33 Example Problem 4-01 (cont.) Step 3: Sum Collinear Forces  Resultant components Cartesian vector form F Rx F Ry

34. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 34 Example Problem 4-01 (cont.) Step 4: Determine Resultant Force Vector FRFR  = 11.7  F Rx F Ry

35. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 35 Group Problem 4-01 Find the magnitude and angle of the resultant force acting on the bracket. Solution Plan  Step 1: Define system of forces  Step 2: Resolve component forces  Step 3: Sum system force components  Step 4: Determine resultant force vector, magnitude and direction

36. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 36 Group Problem 4-01 (cont.) Step 2: Resolve Force Components F 1x F 1y

37. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 37 Group Problem 4-01 (cont.) Step 2: Resolve Force Components F 2x F 2y

38. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 38 Group Problem 4-01 (cont.) Step 2: Resolve Force Components F 3x F 3y

39. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 39 Group Problem 4-01 (cont.) Step 3: Sum Collinear Forces FRFR F Ri F Rj

40. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 40 Group Problem 4-01 (cont.) Step 4: Determine Resultant Force Vector FRFR

41. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 41 Classification of Textbook Problems Hibbeler (2007) Problem SetConcept Degree of Difficulty Estimated Time 2-31 to 2-32Vector Addition Parallelogram LawMedium10-15min 2-33 to 2-38Vector Addition Parallelogram LawEasy5-10min 2-39 to 2-41Resultant ForceEasy5-10min 2-42 to 2-55Resultant & ComponentsMedium10-15min 2-56Resultant & ComponentsHard20min

42. ENGI 1313 Statics I – Lecture 04© 2007 S. Kenny, Ph.D., P.Eng. 42 References Hibbeler (2007) http://wps.prenhall.com/esm_hibbeler_eng mech_1 en.wikipedia.org