ENGI 1313 Mechanics I Lecture 03 Force Vectors

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

Revised – Course Method of Evaluation n 6 Tutorial Quizzes Ø During week 38, 39, 40, 43, 44, & 45 Ø n n 30% Oct. 18 Final Exam Ø 2 Best 5 out of 6 toward final Mid-Term Exam Ø 15% 55% Dec. 6 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 03

Tutorial Sessions n Teaching Assistants Ø Ø Ø Kenton Pike (kenton@engr. mun. ca) Nasser Daiyan (daiyann@engr. mun. ca) Yan. Zhen Ou (yanzhen@engr. mun. ca) 1 2 3 4 5 6 Day Mon Thu Thu Fri Time 3– 3: 50 2– 2: 50 4– 4: 50 10– 10: 50 3– 3: 50 4– 4: 50 Room EN 1040 EN 2007 EN 1040 Section 3 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 03

Chapter 2 Objectives to review concepts from linear algebra n to sum forces, determine force resultants and resolve force components for 2 D vectors using Parallelogram Law n to express force and position in Cartesian vector form n to introduce the concept of dot product n 4 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 03

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Comprehension Quiz 2 -02 (cont. ) n Determine Resultant Force Magnitude Ø Cosine Law F 1 = 4 k. N 30 X 1 F 2 = 10 k. N Y FR R 2 F 1 = 4 k. N R = 120 Therefore FR = 12. 5 k. N 26 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 03

Comprehension Quiz 2 -02 (cont. ) n F 1 = 4 k. N Determine Resultant Force Direction Ø X 43. 9 30 Sine Law 1 F 2 = 10 k. N Y FR R 2 F 1 = 4 k. N R = 120 Therefore 43. 9 from horizontal (clockwise) 27 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 03

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

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

Example Problem 3 -01 (cont. ) Determine the component magnitudes (Fx and Fy) of the resultant 700 -lb force FY 60 FR = 700 lb Y n 90 Fx 30 60 30 X 30 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 03

Example Problem 3 -02 n Problem 2 -12 from Hibbeler (2007) Ø 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. Ø 31 Given: © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 03

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

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

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

Representative Problems n 35 Hibbeler (2007) Textbook Problem Set Concept Degree of Difficulty Estimated Time 2 -1 to 2 -10 Vector Addition Parallelogram Law Easy 5 -10 min 2 -11 to 2 -19 Vector Addition Parallelogram Law Medium 10 -15 min 2 -20 to 2 -24 Vector Addition Parallelogram Law Easy 5 -10 min 2 -25 to 2 -30 Vector Addition Parallelogram Law Medium 10 -15 min © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 03

References n n n n n 36 Hibbeler (2007) http: //wps. prenhall. com/esm_hibbeler_engmech_1 www. hanessupply. com www. sabrecom. com en. wikipedia. org www. caldwellinc. com www. atlantia. com www. c-core. ca www. straylight. ca/greglocke/hibernia. htm www. hibernia. ca © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 03