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 – Scalar Formulation 8

3. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Scalar Formulation Scalar Formulation Topic 8: 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. Applications What is the net effect of the two forces on the wheel? What is the effect of the 30 N force on the lug nut? 5

6. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Moment of a Force. Moment in 2-D The moment of a force about a point provides a measure of the tendency for rotation (sometimes called a torque). Moment about z-axis Moment about x-axis No moment 6

7. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Moment of a Force. Moment in 2-D – Scalar Formulation As shown, d is the perpendicular distance from point O to the line of action of the force. The moment is a scalar or vector? In the 2-D case, the magnitude of the moment is In 2-D, the direction of M O depends on the tendency for rotation, clockwise,or counter-clockwise + + The resultant moment of a system of coplanar forces is defined as: + 7

8. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Moment of a Force. Moment in 2-D – Scalar Formulation + For example, magnitude,M O = F d direction, counter-clockwise + a F b d O Often it is easier to determine M O by using the components of F a FyFy b O FxFx Note the different signs on the terms! The typical sign convention for a moment in 2-D is that counter-clockwise is considered positive. We can determine the direction of rotation by imagining the body pinned at O and deciding which way the body would rotate because of the force. 8

9. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 9

10. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Quiz 10 1. What is the moment of the 10 N force about point A (M A )? A) 3 N·m B) 36 N·m C) 12 N·m D) (12/3) N·m E) 7 N·m A d = 3 m F = 12 N

11. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Example 1 11

12. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Example 1 12

13. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Example 2 13

14. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Example 3 14  A 100-lb vertical force is applied to the end of a lever which is attached to a shaft at O.  Determine: a)Moment about O, b)Horizontal force at A which creates the same moment, c)Smallest force at A which produces the same moment, d)Location for a 240-lb vertical force to produce the same moment, e)Whether any of the forces from b, c, and d is equivalent to the original force.

15. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Solution 3 15 a)Moment about O is equal to the product of the force and the perpendicular distance between the line of action of the force and O. Since the force tends to rotate the lever clockwise, the moment vector is into the plane of the paper.

16. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Solution 3 16 b)Horizontal force at A that produces the same moment,

17. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Solution 3 17 c)The smallest force at A to produce the same moment occurs when the perpendicular distance is a maximum or when F is perpendicular to OA.

18. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Solution 3 18 d)To determine the point of application of a 240 lb force to produce the same moment,

19. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Solution 3 19 e)Although each of the forces in parts b), c), and d) produces the same moment as the 100 lb force, none are of the same magnitude and sense, or on the same line of action. None of the forces is equivalent to the 100 lb force.

20. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Example 4 20 Plan: Since this is a 2-D problem: 1) Resolve the 20 lb force along the handle’s x and y axes. 2) Determine M A using a scalar analysis. Given: A 20 lb force is applied to the hammer. Find: The moment of the force at A. x y

21. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Solution 4 21 Solution : +  F y = 20 sin 30° lb +  F x = 20 cos 30° lb x y + M A = { – (20 cos 30°)lb (18 in) – (20 sin 20°)lb (5 in)} = – 351.77 lb·in = 352 lb·in (clockwise)

22. 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. 22

23. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 23 ¿Preguntas? Comentarios

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