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 Equilibrium of a Particle in 2-D 5

3. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Equilibrium Topic 5: Equilibrium of a Particle in 2-D 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 introduce the concept of the free-body diagram for a particle  To show how to solve particle equilibrium problems using the equations of equilibrium 4

5. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Equilibrium of a Particle Applications For a spool of given weight, what are the forces in cables AB and AC ? 5

6. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Equilibrium of a Particle Applications (continued) For a given cable strength, what is the maximum weight that can be lifted ? 6

7. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Equilibrium of a Particle Applications (continued) 7 For a given weight of the motor, what are the forces in the cables? What size of cable must you use ?

8. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Equilibrium of a Particle Applications (continued) 8 Static Equilibrium

9. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Balancing Forks 9  Materials:

10. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Balancing Forks 10  Step 1:

11. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Balancing Forks 11  Step 2:

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

13. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Coke Can Balancing 13  Materials:

14. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Coke Can Balancing 14  Step 1:

15. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Coke Can Balancing 15  Step 2:

16. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Coke Can Balancing 16  Step 3:

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

18. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 3.1 Condition for the Equilibrium of a Particle. 18 EQUILIBRIUM  Particle is said to be in EQUILIBRIUM if: (Static Equilibrium) It remains at rest (v=0) if originally at rest. (Static Equilibrium) It has a constant velocity if originally in motion. EQUILIBRIUM Resultant force zero  To maintain EQUILIBRIUM, it is necessary to satisfy Newton’s first law of motion. Resultant force acting on a particle require to be zero. where ∑F is the vector sum of all the forces acting on the particle. Equation of Equilibrium

19. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 3.1 Condition for the Equilibrium of a Particle. 19  Consequently, the particle indeed moves with constant velocity or remains at rest.  For Equilibrium:

20. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 3.2 The Free-Body Diagram (FBD) 20 EQUATION OF EQUILIBRIUM ALL THE KNOWN UNKNOWN FORCESON  To apply the EQUATION OF EQUILIBRIUM, we must account for ALL THE KNOWN and UNKNOWN FORCES which act ON the particle FREE-BODY DIAGRAM (FBD).  In order to account for all the forces that act on the particle, it is necessary to draw its FREE-BODY DIAGRAM (FBD). SKETCH “FREE” ALL  The free-body diagram is simple a SKETCH which shows the particle “FREE” from its surrounding with ALL the forces that act on it. FBD

21. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Types of Connections 21 SPRINGSCABLES and PULLEYS.  Two types of connections often encountered in particle equilibrium SPRINGS and CABLES and PULLEYS.  SPRINGS:  SPRINGS: The magnitude of force exerted on a linearly elastic spring is: which has a stiffness k and is deformed (elongate or compressed) a distance s, measured from its unloaded position. Where l o the original length and l the final length

22. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Types of Connections 22  CABLES and PULLEYS: negligible weight cannot stretch.  CABLES and PULLEYS: All cables are assumed to have negligible weight and they cannot stretch.  A cable can support only a tension or “pulling” force, and this force always act in the direction of the cable.  The tension in a cable is constant throughout its length.

23. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Procedure for Drawing a Free-Body Diagram: 23  Draw Outlined Shape:  Draw Outlined Shape: Imagine the particle to be isolated or cut “free” from its surrounding with all the forces outlined shape.  Show All Forces:  Show All Forces: Indicate on this sketch all the forces that act on the particle.  Identify Each Force:  Identify Each Force: The forces which are known should be labeled with their proper magnitudes and directions. Letter are used to represent the magnitudes and directions of forces that are unknown.

24. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Example 1 6 kg free-body diagram spherecord CEknot at C  The sphere has a mass of 6 kg and is supported as shown. Draw a free-body diagram of the sphere, the cord CE, and the knot at C. 24

25. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Solution 1  Sphere:  Sphere: There are two forces acting on the sphere: W=6kg(9.81m/s 2 )=58.9N The weight of the sphere W=6kg(9.81m/s 2 )=58.9N The force F CE of the cord CE acting on the sphere  The Free-Body Diagram 25

26. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Solution 1  Cord CE:  Cord CE: When the cord CE is isolated from its surroundings, its free-body diagram shows only two forces acting on it: The force of the sphere and F CE. The force of the knot F EC  The Free-Body Diagram 26 F CE =F EC

27. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Solution 1  Knot:  Knot: The knot at C is subjected to three forces. They are caused by the cords CBA and CE and the spring CD:  The Free-Body Diagram 27

28. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 3.3 Coplanar Force Systems 28 SYSTEM OF COPLANAR FORCES resolvedij components  If a particle is subjected to a SYSTEM OF COPLANAR FORCES that lie in the x-y plane. Then each force can be resolved into i and j components EQUILIBRIUMSUM ZERO RESULTANT  For EQUILIBRIUM, these forces must SUM to produce a ZERO RESULTANT. Hence:

29. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Example 2 tension BABC 60-kg  Determine the tension in cables BA and BC necessary to support the 60-kg cylinder. 29

30. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Solution 2 EQUILIBRIUMweight tensionBD  Due to EQUILIBRIUM, the weight of the cylinder causes the tension in cable BD to be: BABC equilibrium of the ring B  The forces in cable BA and BC can be determined by investigating the equilibrium of the ring B. Its free body diagram is then: 30

31. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Solution 2  The magnitudes of T A and T C are unknown, but their directions are known. 31

32. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Solution 2  Equations of equilibrium along the x and y axes: 32

33. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 33

34. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Homework3  http://facultad. bayamon.inter.edu/omeza/http://facultad. bayamon.inter.edu/omeza/ Omar E. Meza Castillo Ph.D. 34

35. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 35 ¿Preguntas? Comentarios

36. MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 36 GRACIAS