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

2. ENGI 1313 Statics I – Lecture 15© 2007 S. Kenny, Ph.D., P.Eng. 2 Chapter 4 Objectives to discuss the concept of the moment of a force and show how 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.

3. ENGI 1313 Statics I – Lecture 15© 2007 S. Kenny, Ph.D., P.Eng. 3 Lecture 15 Objectives to examine concepts of the moment of a force in scalar notation  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

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

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

6. ENGI 1313 Statics I – Lecture 15© 2007 S. Kenny, Ph.D., P.Eng. 6 Comprehension Quiz 15-01 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.  A) (Q, P)  B) (R, S)  C) (P, R)  D) (Q, S) Answer: D Q S P R A

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

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

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

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

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

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

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

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

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

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

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

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

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

20. ENGI 1313 Statics I – Lecture 15© 2007 S. Kenny, Ph.D., P.Eng. 20 Principle of Transmissibility 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.

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

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

23. ENGI 1313 Statics I – Lecture 15© 2007 S. Kenny, Ph.D., P.Eng. 23 Comprehension Quiz 15-02 If r = {5 j } m and F = {10 k} N, the moment r x F equals:  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

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

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