ENGI 1313 Mechanics I Lecture 15 Force System

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ENGI 1313 Mechanics I Lecture 15: Force System Resultant and Cross Product Shawn Kenny,

ENGI 1313 Mechanics I Lecture 15: Force System Resultant and Cross Product Shawn Kenny, Ph. D. , P. Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@engr. mun. ca

Chapter 4 Objectives n n n 2 to discuss the concept of the moment

Chapter 4 Objectives n n n 2 to discuss the concept of the moment of a force and show to calculate it in two and three dimensions. to provide a method for finding the moment of a force about a specified axis. to define the moment of a couple. to present methods for determining the resultants of non-concurrent force systems. to indicate how to reduce a simple distributed loading to a resultant force having a specified location. © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Lecture 15 Objectives n to examine concepts of the moment of a force in

Lecture 15 Objectives n to examine concepts of the moment of a force in scalar notation Ø n 3 Slides in Lecture 13 revised for consistent notation with the text and provide summary classification of textbook problems to examine concepts on the cross product with respect to the moment of a force © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Moment Force Line of Action (cont. ) n Find Line of Action x O

Moment Force Line of Action (cont. ) n Find Line of Action x O y d F 4 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Moment Force Line of Action (cont. ) n Find Components x O y Fx

Moment Force Line of Action (cont. ) n Find Components x O y Fx F 5 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15 Fy

Comprehension Quiz 15 -01 n If a force of magnitude F can be applied

Comprehension Quiz 15 -01 n If a force of magnitude F can be applied in four different 2 -D configurations (P, Q, R, & S), select the cases resulting in the maximum and minimum (Max, Min) moment (torque) values on the wrench about Point A. Ø Ø n 6 A) (Q, P) B) (R, S) C) (P, R) D) (Q, S) Answer: D P S A R © 2007 S. Kenny, Ph. D. , P. Eng. Q ENGI 1313 Statics I – Lecture 15

Example 15 -01 n For each case illustrated, determine the moment of the force

Example 15 -01 n For each case illustrated, determine the moment of the force about point O. - 7 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Example 15 -01 (cont. ) n For each case illustrated, determine the moment of

Example 15 -01 (cont. ) n For each case illustrated, determine the moment of the force about point O. - 8 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Example 15 -01 (cont. ) n For each case illustrated, determine the moment of

Example 15 -01 (cont. ) n For each case illustrated, determine the moment of the force about point O. + 9 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Example 15 -01 (cont. ) n For each case illustrated, determine the moment of

Example 15 -01 (cont. ) n For each case illustrated, determine the moment of the force about point O. + 10 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Moment of Coplanar Force System n 11 System of Forces Acting on a Single

Moment of Coplanar Force System n 11 System of Forces Acting on a Single Plane © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Moment of Coplanar Force System n 12 System of Forces Acting on a Single

Moment of Coplanar Force System n 12 System of Forces Acting on a Single Plane © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Cross Product n 13 Vector Resultant Ø Magnitude Ø Direction © 2007 S. Kenny,

Cross Product n 13 Vector Resultant Ø Magnitude Ø Direction © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Cross Product (cont. ) n 14 Commutative Law © 2007 S. Kenny, Ph. D.

Cross Product (cont. ) n 14 Commutative Law © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Cross Product (cont. ) n Vector Direction + - 15 © 2007 S. Kenny,

Cross Product (cont. ) n Vector Direction + - 15 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Cross Product (cont. ) n Cartesian Vectors Terms go to zero More convenient determinant

Cross Product (cont. ) n Cartesian Vectors Terms go to zero More convenient determinant form 16 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Cross Product (cont. ) n Cartesian Components from Determinant Element i 17 © 2007

Cross Product (cont. ) n Cartesian Components from Determinant Element i 17 © 2007 S. Kenny, Ph. D. , P. Eng. Element j Element k ENGI 1313 Statics I – Lecture 15

Applications of Cross Product n 18 Vector Notation – Moment of a Force ©

Applications of Cross Product n 18 Vector Notation – Moment of a Force © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Applications of Cross Product n 19 Vector Notation – Moment of a Force ©

Applications of Cross Product n 19 Vector Notation – Moment of a Force © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Principle of Transmissibility n 20 The force vector F can act at any point

Principle of Transmissibility n 20 The force vector F can act at any point along the force vector line of action (“sliding vector”) and still create the same moment about point O. © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Principle of Moments n Varignon’s Theorem Ø 21 The moment of a force about

Principle of Moments n Varignon’s Theorem Ø 21 The moment of a force about a point is equal to the sum of the moments of the force’s components about the point. © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Comprehension Quiz 15 -01 n Using the CCW direction as positive, the net moment

Comprehension Quiz 15 -01 n Using the CCW direction as positive, the net moment of the two forces about point P is: Ø Ø Ø n 22 A) 10 N·m B) 20 N·m C) -20 N·m D) 40 N·m E) -40 N·m 10 N 3 m P 2 m Answer: B © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15 5 N

Comprehension Quiz 15 -02 n If r = {5 j } m and F

Comprehension Quiz 15 -02 n If r = {5 j } m and F = {10 k} N, the moment r x F equals: Ø Ø Ø n 23 A) 50 i N·m B) 50 j N·m C) -50 i N·m D) -50 j N·m E) 0 N·m Answer: A © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

Example 15 -01 n 24 Given a = 3 in, b = 6 in

Example 15 -01 n 24 Given a = 3 in, b = 6 in and c = 2 in. Find the moment of F on the block about point O. © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15

References Hibbeler (2007) n http: //wps. prenhall. com/esm_hibbeler_eng mech_1 n 25 © 2007 S.

References Hibbeler (2007) n http: //wps. prenhall. com/esm_hibbeler_eng mech_1 n 25 © 2007 S. Kenny, Ph. D. , P. Eng. ENGI 1313 Statics I – Lecture 15