Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland ENGI 1313 Mechanics I Lecture 03:Force Vectors and Parallelogram Law

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 2 Revised – Course Method of Evaluation 6 Tutorial Quizzes15%  During week 38, 39, 40, 43, 44, & 45  Best 5 out of 6 toward final Mid-Term Exam30%  Oct. 18 Final Exam55%  Dec. 6

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 3 Tutorial Sessions Teaching Assistants  Kenton Pike  Nasser Daiyan  YanZhen Ou Section Day MonThu Fri Time 3–3:502–2:504–4:5010–10:503–3:504–4:50 Room EN1040 EN2007EN1040

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 4 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

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 5 Lecture 03 Objectives to review concepts from linear algebra to sum force vectors, determine force resultants, and resolve force components for 2D vectors using Parallelogram Law

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 6 Introductory Concepts Scalar  Magnitude (value) and sense (positive, negative)  No direction Examples  Mass  Volume  Length  Temperature  Speed

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 7 Direction Magnitude  ►Sense Vector  Magnitude  Sense (+, -)  Direction or orientation  Convention Textbook is boldface, A PowerPoint notation typically  A  Examples Force Velocity Introductory Concepts

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 8 Scalar Multiplication and Division Change in Magnitude Change in Sense

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 9 Vector Operations Engineering Need  Determine resultant force due to applied forces  Resolve force into components Method  Parallelogram law Triangle construction

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 10 Vector Addition Parallelogram Law  Graphical construction Vector Tip Vector Tail Resultant Vector (F R ) Component Vectors (F 1, F 2 ) Resultant Vector forms the Parallelogram Diagonal

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 11 Vector Addition Parallelogram Law  Special case Collinear vectors Algebraic addition

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 12 Vector Addition Parallelogram Law  Triangle construction “Tip-to-Tail” technique Parallelogram

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 13 Vector Addition Parallelogram Law  Triangle construction “Tip-to-Tail” technique Parallelogram

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 14 Vector Subtraction Parallelogram Law  Triangle Construction “Tip-to-Tail” technique

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 15 Parallelogram Law Multiple Force Vectors

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 16 Vector Summation Resultant Force Magnitude  Cosine law

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 17 Vector Summation Resultant Force Direction or Magnitude of Component Forces  Sine law

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 18 Applications Lifting Devices

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 19 Applications Guyed Towers

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 20 Applications Cable Stayed Bridge

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 21 Applications Offshore Platform Foundation Connections

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 22 Applications Towing

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 23 Comprehension Quiz 2-01 Scalar or Vector?  Force  Time  Mass  Position Scalar or Vector?  Force  Vector  Time  Scalar  Mass  Scalar  Position  Vector

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 24 Comprehension Quiz 2-02 Q: Is this the correct application of the parallelogram law to determine the resultant force vector (F R )? F 1 = 4 kN F 2 = 10 kN 30  4 kN 90  FRFR X Y

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 25 Comprehension Quiz 2-02 (cont.) A: No  “Tip-to-Tail” triangle construction technique F 1 = 4 kN  R = 180  – (180  – 30  – 90  ) = 120  FRFR F 1 = 4 kN F 2 = 10 kN X Y 30  RR 11 22

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 26 Comprehension Quiz 2-02 (cont.) Determine Resultant Force Magnitude  Cosine Law Therefore F R = 12.5 kN F 1 = 4 kN FRFR F 2 = 10 kN X Y 30  RR 11 22  R = 120 

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 27 Comprehension Quiz 2-02 (cont.) Determine Resultant Force Direction  Sine Law F 1 = 4 kN FRFR F 2 = 10 kN X Y 30  RR 11 22  R = 120  Therefore 43.9  from horizontal (clockwise) 43.9 

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 28 Example Problem 3-01 Determine the component magnitudes (F X and F Y ) of the 700-lb force resultant (F R ) FxFx FYFY XX YY RR F R = 700 lb X Y 60  30  F R = 700 lb X Y Vector Triangle 60  30 

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 29 Example Problem 3-01 (cont.) Determine Interior Angles of Vector Triangle XX YY RR X Y 60  30   Y = 60  - 30  = 30  30   = 90  - 30  = 60   R = 180  - 60  - 30  = 90    X =  = 60 

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 30 Example Problem 3-01 (cont.) Determine the component magnitudes (F x and F y ) of the resultant 700-lb force FxFx FYFY 60  30  90  F R = 700 lb X Y 60  30 

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 31 Example Problem 3-02 Problem 2-12 from Hibbeler (2007)  The component of force F acting along line aa is required to be 30 lb. Determine the magnitude of F and its component along line bb.  Given:

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 32 Example Problem 3-02 (cont.) Problem 2-12 from Hibbeler (2007)  Draw force vectors a a b b F 80  60  F a = 30lb FbFb bb FF  F = 180  -  1 -  b = 180  - 80  - 60  = 40   2 =  b = 60 

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 33 Example Problem 3-02 (cont.) Problem 2-12 from Hibbeler (2007)  Magnitude of F & F b from sine law F 80  FbFb F a = 30lb 60  40   F = 40   2 =  b = 60   1 =  a = 80 

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 34 Vector Summation Methods Studied  Parallelogram Law Vector triangle construction Sine law Cosine law Limitations  Resultant of multiple vectors determined through successive summation of two vectors Cumbersome for large systems

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 35 Representative Problems Hibbeler (2007) Textbook Problem SetConcept Degree of Difficulty Estimated Time 2-1 to 2-10Vector Addition Parallelogram LawEasy5-10min 2-11 to 2-19Vector Addition Parallelogram LawMedium10-15min 2-20 to 2-24Vector Addition Parallelogram LawEasy5-10min 2-25 to 2-30Vector Addition Parallelogram LawMedium10-15min

ENGI 1313 Statics I – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng. 36 References Hibbeler (2007) en.wikipedia.org