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 2-D Vector Addition 2

3. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Vectors Topic 2: 2-D Vector Addition 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 Chapter Objectives  Up on completion of this chapter, the student will be able to To show how add forces and resolve them into components using the Parallelogram Law To express force and position in Cartesian vector form and explain how to determine the vector’s magnitude and direction To introduce the dot product in order to determine the angle between two vectors or the projection of one vector onto another 4

5. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 2. Application of Vector Addition. There are four concurrent cable forces acting on the bracket. How do you determine the resultant force acting on the bracket ? 5

6. ENGR 3340: Fundamentals of Statics and Dynamics 2.1 Scalars and Vectors. ScalarsVectors Examples: mass, volumeforce, velocity Characteristics: It has a magnitude (positive or negative) It has a magnitude and direction Addition rule:Simple arithmeticParallelogram law Special Notation: None (italic sometimes) Bold font, a line, an arrow or a “carrot” 6

7. ENGR 3340: Fundamentals of Statics and Dynamics 2.2 Vector Operations. Scalar Multiplication and Division 7

8. ENGR 3340: Fundamentals of Statics and Dynamics 2.2 Vector Operations. Vector Addition Parallelogram Law (graphical method). Triangle Construction. How do you subtract a vector? How can you add more than two concurrent vectors graphically ? 8

9. ENGR 3340: Fundamentals of Statics and Dynamics 2.2 Vector Operations. Resolution of a Vector Resolution of a vector is breaking up a vector into components. It is kind of like using the parallelogram law in reverse. 9

10. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 2.3 Vector Addition of Forces. Procedure for Analysis Method 1. Parallelogram Law, graphical method Method 2. Trigonometry, use the Cosine or Sine Law ( 3 forces ). 10

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14. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 2.4 Addition of a System of Coplanar Forces. We resolve vectors into components using the x and y axes system. Each component of the vector is shown as a magnitude and a direction. The directions are based on the x and y axes. We use the “ unit vectors ” i and j to designate the x and y axes. F = F x i + F y jF’ = F’ x i - F’ y j The x and y axes are always perpendicular to each other. Together, they can be directed at any inclination. 14

15. ENGR 3340: Fundamentals of Statics and Dynamics 2.4 Addition of a System of Coplanar Forces. Coplanar Force Resultants Step 1 is to resolve each force into its x-y components. Step 3 is to find the magnitude and angle of the resultant vector. Step 2 is to add all the x components together and add all the y components together. These two totals become the resultant vector. Method 3. Cartesian Vector Method, three steps. 15

16. ENGR 3340: Fundamentals of Statics and Dynamics 2.4 Addition of a System of Coplanar Forces. Coplanar Force Resultants Step 1 Step 2 Scalar Notation : 16

17. ENGR 3340: Fundamentals of Statics and Dynamics 2.4 Addition of a System of Coplanar Forces. Coplanar Force Resultants 17 QuadrantValue of Tan -1 IUse the calculator value IIAdd 180 o to the calculator value IIIAdd 180 o to the calculator value IVAdd 180 o to the calculator value Step 3

18. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics If the rectangular coordinates of a point are (-3, -5.2), what are the polar coordinates (R, ) of the point? 18

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26. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Homework2  http://facultad. bayamon.inter.edu/omeza/http://facultad. bayamon.inter.edu/omeza/ Omar E. Meza Castillo Ph.D. 26 Extra  http://www.ae.metu.edu.tr/~ae261/ch02.pdfhttp://www.ae.metu.edu.tr/~ae261/ch02.pdf

27. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 27 ¿Preguntas? Comentarios

28. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 28 GRACIAS