1. ENGR 3340: Fundamentals of Statics and Dynamics Fundamentals of Statics and Dynamics - ENGR 3340 Professor: Dr. Omar E. Meza Castillo omeza@bayamon.inter.edu http://facultad.bayamon.inter.edu/omeza Department of Mechanical Engineering

2. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 2 Tentative Lectures Schedule TopicLecture Moment of a Force – Vector Formulation 9

3. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Vector Formulation Vector Formulation Topic 9: Moment of a Force 3 One thing you learn in science is that there is no perfect answer, no perfect measure. A. O. Beckman

4. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 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 specific axis  To define the moment of a couple  To present methods for determining the resultants of nonconcurrent force systems  To indicate how to reduce a simple distributed loading to a resultant force having a specified location 4

5. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Moment of a Force. Moment in 3-D – Vector Formulation Moments in 3-D can be calculated using scalar (2-D) approach but, it can be difficult and time consuming. The resultant moment is the same when acts over the line of action, also called principle of transmissibility. Thus, it is often easier to use a mathematical approach called the vector cross product. Here r is the position vector from point O to any point on the line of action of F. 5

6. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Cross Product In general, the cross product of two vectors A and B results in another vector C, i.e., C = A  B. The magnitude and direction of the resulting vector can be written as Here u C is the unit vector perpendicular to both A and B vectors as shown (or to the plane containing the A and B vectors). 6

7. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Cross Product. Cartesian Vector Formulation The right hand rule is a useful tool for determining the direction of the vector resulting from a cross product. For example: i  j = k In general for expressed in Cartesian Coordinates: 7

8. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Cross Product. Cartesian Vector Formulation Of even more utility, the cross product can be written as. Each component can be determined using 2  2 determinants x -direction or i -element y -direction or j -element z -direction or k -element (-)(+) (-)(+) (-) (+) (-)(+) (-)(+) 8

9. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Moment of a Force. Moment in 3-D – Vector Formulation So, using the cross product, a moment can be expressed as. By expanding the above equation using 2  2 determinants Sample units are N·m or lb·ft. The physical meaning of the above equation becomes evident by considering the force components separately and using a 2-D formulation 9

10. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 10

11. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 11

12. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 12

13. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 13

14. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Principle of Moments States that the moment of a force about a point is equal to the sum of the moment of the force’s component about the point. By expanding the above equation using 2  2 determinants An application of the principle of moments and the principle of transmissibility shows that the moment about A is, Also called to as Varignon’s theorem. 14

15. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 15

16. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 16

17. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 17

18. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 18

19. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Moment of a Couple A couple is defined as two parallel forces with the same magnitude but opposite in direction separated by a perpendicular distance d. The moment of a couple is defined as: M O = F d(using a scalar analysis) or as M O = r  F (using a vector analysis). Here r is any position vector from the line of action of – F to the line of action of F. 19

20. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Moment of a Couple. (Continued) The net external effect of a couple is that the net force equals zero and the magnitude of the net moment equals: F d Since the moment of a couple depends only on the distance between the forces, the moment of a couple is a free vector. It can be moved anywhere on the body and have the same external effect on the body. Moments due to couples can be added using the same rules as adding any vectors. 20

21. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 21

22. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 22

23. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 23

24. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 24

25. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 25

26. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 26

27. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Homework5  http://facultad. bayamon.inter.edu/omeza/http://facultad. bayamon.inter.edu/omeza/ Omar E. Meza Castillo Ph.D. 27

28. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 28 ¿Preguntas? Comentarios

29. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 29 GRACIAS