ENGI 1313 Mechanics I

1. ENGI 1313 Mechanics I Lecture 15: Force System Resultant and Cross Product

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. 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. - Moment Force Line of Action (cont.) • Find Line of Action x O y d F

5. - Moment Force Line of Action (cont.) • Find Components x O y Fx Fy F

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 P S A R Q

7. - Example 15-01 • For each case illustrated, determine the moment of the force about point O.

8. - Example 15-01 (cont.) • For each case illustrated, determine the moment of the force about point O.

9. + Example 15-01 (cont.) • For each case illustrated, determine the moment of the force about point O.

10. + Example 15-01 (cont.) • For each case illustrated, determine the moment of the force about point O.

11. Moment of Coplanar Force System • System of Forces Acting on a Single Plane

12. Moment of Coplanar Force System • System of Forces Acting on a Single Plane

13. Cross Product • Vector Resultant • Magnitude • Direction

14. Cross Product (cont.) • Commutative Law

15. + - Cross Product (cont.) • Vector Direction

16. Cross Product (cont.) • Cartesian Vectors Terms go to zero More convenient determinant form

17. Cross Product (cont.) • Cartesian Components from Determinant Element i Element j Element k

18. Applications of Cross Product • Vector Notation – Moment of a Force

19. Applications of Cross Product • Vector Notation – Moment of a Force

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. 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. 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 5 N 3 m P 2 m

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. 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. References • Hibbeler (2007) • http://wps.prenhall.com/esm_hibbeler_engmech_1