ENGI 1313 Mechanics I Lecture 06 Cartesian and

Chapter 2 Objectives to review concepts from linear algebra n to sum forces, determine

Lecture 06 Objectives to further examine 3 D Cartesian vectors n to define a

Example Problem 6 -01 n 4 Problem 2 -77 (Hibbeler, 2007). The bolt is

Example Problem 6 -01 n Known n Find 5 © 2007 S. Kenny, Ph.

Example Problem 6 -01 (cont. ) n Find Angle = 60 = 45 Fz

Position Vectors – General n 3 D Coordinates Unique position in space Ø Right-hand

Position Vectors – Origin to a Point n Fixed vector locating a point P(x,

Position Vector – General Case n Two Points in Space Ø Rectangular Cartesian coordinate

Position Vector – General Case n Establish Position Vectors From Point O to Point

Position Vector – General Case n Define Position Vector (r. AB = r )

Comprehension Quiz 6 -01 n Two points in 3 D space have coordinates of

Comprehension Quiz 6 -02 n P and Q are two points in a 3

Comprehension Quiz 6 -03 n If F is a force vector (N) and r

Example 6 -01 n Express the force vector FDA in Cartesian form n Known:

Example 6 -01 (cont. ) n Find Position Vector r. DA Ø 16 Through

Example 6 -01 (cont. ) n Find Position Vector |r. DA| Magnitude r. DA

Example 6 -01 (cont. ) n Find unit vector u. DA 18 © 2007

Example 6 -01 (cont. ) n Find Unit Vector u. DA Magnitude Ø 19

Example 6 -01 (cont. ) n Find Force Vector FDA Ø 20 or ©

Group Problem 6 -01 n Find the resultant force magnitude and coordinate direction n

Group Problem 6 -01 (cont. ) n Position Vectors and Magnitude Ø r. CA

Group Problem 6 -01 (cont. ) n Force Vectors and Magnitude FCA Ø FCB

Group Problem 6 -01 (cont. ) n 24 Force Resultant Vector Magnitude & Orientation

Group Problem 6 -01 (cont. ) n Force Resultant Vector Magnitude & Orientation F

Classification of Textbook Problems n Hibbeler (2007) Problem Set Degree of Difficulty Concept Estimated

References Hibbeler (2007) n http: //wps. prenhall. com/esm_hibbeler_eng mech_1 n 27 © 2007 S.

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ENGI 1313 Mechanics I Lecture 06: Cartesian and Position Vectors Shawn Kenny, Ph. D. , P. Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@engr. mun. ca

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 2 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 06

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

Example Problem 6 -01 n 4 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. © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 06

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

Position Vectors – Origin to a Point n 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 Ø Tip-to-tail vector component technique Ø ^ zk ^ xi yj^ 8 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 06

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

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

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

Comprehension Quiz 6 -01 n Two points in 3 D space have coordinates of P(1, 2, 3) and Q (4, 5, 6) meters. The position vector r. QP is given by Ø Ø Ø n 12 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 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 06

Comprehension Quiz 6 -02 n 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 Ø B) r. PQ = -r. QP Ø C) r. PQ = 1/r. QP Ø D) r. PQ = 2 r. QP Ø z Q(x. B, y. Q, z. Q) P(x. P, y. P, z. P) r. PQ = -r. QP y n 13 Answer: B © 2007 S. Kenny, Ph. D. , P. Eng. x ENGI 1313 Statics I – Lecture 06

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

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

Classification of Textbook Problems n Hibbeler (2007) Problem Set Degree of Difficulty Concept Estimated Time 2 -79 to 2 -84 Position vectors Easy 5 -10 min 2 -85 to 2 -90 Resultant force vectors Medium 15 -20 min 2 -91 to 2 -96 Resultant force & position vectors Medium 15 -20 min 2 -97 to 2 -99 Position vectors Easy 10 -15 min 2 -100 Resultant force & position vectors Hard 30 min Medium 15 -20 min 2 -101 to 2 -106 Resultant force & position vectors 26 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 06