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ENGI 1313 Mechanics I

ENGI 1313 Mechanics I . Lecture 04: Force Vectors and System of Coplanar Forces. Tutorial Questions. SI Units and Use Section 1.4 Page 9 Use a single prefix Magnitude between 0.1 and 1000. Tutorial Questions. SI Units and Use Section 1.4 Page 9 Use a single prefix

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ENGI 1313 Mechanics I

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  1. ENGI 1313 Mechanics I Lecture 04: Force Vectors and System of Coplanar Forces

  2. Tutorial Questions • SI Units and Use • Section 1.4 Page 9 • Use a single prefix • Magnitude between 0.1 and 1000

  3. Tutorial Questions • SI Units and Use • Section 1.4 Page 9 • Use a single prefix • Magnitude between 0.1 and 1000

  4. Tutorial Questions • SI Units and Use • Section 1.4 Page 9 • Do not use compound prefixes

  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. 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. 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. Parallelogram Law (Lecture 02) • Multiple Force Vectors

  9. Recall Lecture 02 (Slide 9) What is the Alternate Approach? • Resolve Force Components • Algebraic Summation ComponentVectors Force Vectors

  10. What is the Alternate Approach? • Resolve Force Components • Algebraic Summation • Form Resultant Force Resultant Force Vector ComponentVectors Force Vectors

  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. Cartesian Coordinate System • Characteristics • Rectangular coordinate system • Unique spatial position • Vector algebra • Analytical geometry Ordinate Abscissa

  13. F = Fx + Fy F = Fx + Fy Rectangular Force Components • Axes Must be Orthogonal • Axes Orientation Does not Matter

  14. Resolve Force Components • Known: Force Vector and Orientation Angle 

  15. Resolve Force Components • Known: Force Vector and Slope Lh Ly Lx

  16. Determine Resultant Force • Known: Force Components • Resultant Force Magnitude • Pythagorean theorem • Resultant Force Direction • Trigonometry y F Fy  x Fx

  17. ^ ^ F= FX i + FY j ^ ^ Unit Vector; j = FY Unit Vector; i = FX FX FY Notation – Summation Coplanar Forces • Common • Magnitude: FX & FY • Sense: + & - • Scalar Notation • Cartesian Vector Notation FR FX + FY 3. Direction: Orthogonal X & Y axes +j +Y FR FY -X +X 3. Direction: Unit vectors -i +i FX -j -Y

  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. Coplanar Force Vector Summation • Step 1: Define System of Forces • Rectangular coordinate system • Force vectors F1, F2 and F3 Force Vectors

  20. Coplanar Force Vector Summation • Step 2: Resolve Component Forces ComponentVectors Force Vectors

  21. Coplanar Force Vector Summation • Step 3: Sum System Force Components • Obtain resultant force vector components Resultant Force Vector ComponentVectors Force Vectors

  22. Coplanar Force Vector Summation • Step 3: Sum System Force Components • Scalar notation Resultant Force Vector ComponentVectors Force Vectors

  23. Coplanar Force Vector Summation • Step 4: Determine Resultant Force Vector • Magnitude, sense and direction Resultant Force Vector ComponentVectors Force Vectors

  24. ^ Unit Vector; i = FX FX Coplanar Force Vector Summation • Step 3: Sum System Force Components • Cartesian vector notation Resultant Force Vector ComponentVectors Force Vectors

  25. Comprehension Quiz 4-01 y • Resolve F along x and y axes in Cartesianvector 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 x F = 80 N 30° Fyĵ = -80 cos 30 Fx = 80 sin30

  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 FR F1 j F1 F2 i C) 50 N

  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. Example Problem 4-01 (cont.) • Step 2: Resolve Components • Cartesian vector form, F1 F1x = 15kN sin 40 F1y = 15kN cos 40 F1x F1y

  29. Example Problem 4-01 (cont.) • Step 2: Resolve Components • Cartesian vector form, F2 F2x = -26kN (12/13) F2y = 26kN (5/13) F2y F2x

  30. Example Problem 4-01 (cont.) • Step 2: Resolve Components • Cartesian vector form, F3 F3x = 36kN cos 30 F3y = 36kN sin 30 F3x F3y

  31. Example Problem 4-01 (cont.) • Step 2: Resolve Components • Cartesian vector form • Therefore F1x F1y F2y F3x F2x F3y

  32. Example Problem 4-01 (cont.) • Step 3: Sum Collinear Forces • Collinear Cartesian vector form F1y F2y F1x F3x F2x F3y

  33. Example Problem 4-01 (cont.) • Step 3: Sum Collinear Forces • Resultant components Cartesian vector form FRy FRx

  34. Example Problem 4-01 (cont.) • Step 4: Determine Resultant Force Vector FR  = 11.7 FRy FRx

  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. Group Problem 4-01 (cont.) • Step 2: Resolve Force Components F1x F1y

  37. Group Problem 4-01 (cont.) • Step 2: Resolve Force Components F2y F2x

  38. Group Problem 4-01 (cont.) • Step 2: Resolve Force Components F3x F3y

  39. Group Problem 4-01 (cont.) • Step 3: Sum Collinear Forces FRi FRj FR

  40. Group Problem 4-01 (cont.) • Step 4: Determine Resultant Force Vector FR

  41. Classification of Textbook Problems • Hibbeler (2007)

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

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