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ENGR 3340: Fundamentals of Statics and Dynamics Fundamentals of Statics and Dynamics - ENGR 3340 Professor: Dr. Omar E. Meza Castillo [email protected] http://facultad.bayamon.inter.edu/omeza Department of Mechanical Engineering of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Tentative Lectures Schedule Topic Lecture Equilibrium of a Particle in 2-D 5 2 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón One thing you learn in science is that there is no perfect answer, no perfect measure. A. O. Beckman Topic 5: Equilibrium of a Particle in 2-D Equilibrium 3 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón 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 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Equilibrium of a Particle Applications For a spool of given weight, what are the forces in cables AB and AC ? 5 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Equilibrium of a Particle Applications (continued) For a given cable strength, what is the maximum weight that can be lifted ? 6 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Equilibrium of a Particle Applications (continued) For a given weight of the motor, what are the forces in the cables? What size of cable must you use ? 7 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Equilibrium of a Particle Applications (continued) Static Equilibrium 8 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Balancing Forks Materials: 9 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Balancing Forks Step 1: 10 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Balancing Forks Step 2: 11 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Balancing Forks 12 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Coke Can Balancing Materials: 13 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Coke Can Balancing Step 1: 14 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Coke Can Balancing Step 2: 15 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Coke Can Balancing Step 3: 16 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Coke Can Balancing 17 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón 3.1 Condition for the Equilibrium of a Particle. Particle is said to be in EQUILIBRIUM if: It remains at rest (v=0) if originally at rest. (Static Equilibrium) It has a constant velocity if originally in motion. To maintain EQUILIBRIUM, it is necessary to satisfy Newton’s first law of motion. Resultant force acting on a particle require to be zero. F 0 Equation of Equilibrium where ∑F is the vector sum of all the forces acting on the particle. 18 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón 3.1 Condition for the Equilibrium of a Particle. a =0 ∑F = ma = 0 Consequently, the particle indeed moves with constant velocity or remains at rest. For Equilibrium: ∑Fx = 0 ∑F y = 0 19 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón 3.2 The Free-Body Diagram (FBD) To apply the EQUATION OF EQUILIBRIUM, we must account for ALL THE KNOWN and UNKNOWN FORCES which act ON the particle In order to account for all the forces that act on the particle, it is necessary to draw its FREE-BODY DIAGRAM (FBD). 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 20 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Types of Connections Two types of connections often encountered in particle equilibrium SPRINGS and CABLES and PULLEYS. SPRINGS: The magnitude of force exerted on a linearly elastic spring is: F = ks which has a stiffness k and is deformed (elongate or compressed) a distance s, measured from its unloaded position. s = l _ lo Where lo the original length and l the final length 21 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Types of Connections 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. 22 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Procedure for Drawing a Free-Body Diagram: Draw Outlined Shape: Imagine the particle to be isolated or cut “free” from its surrounding with all the forces outlined shape. Show All Forces: Indicate on this sketch all the forces that act on the particle. 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. 23 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Example 1 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 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Solution 1 Sphere: There are two forces acting on the sphere: The weight of the sphere W=6kg(9.81m/s2)=58.9N The force FCE of the cord CE acting on the sphere The Free-Body Diagram 25 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Solution 1 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 FCE. The force of the knot FEC The Free-Body Diagram FCE=FEC 26 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Solution 1 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 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón 3.3 Coplanar Force Systems 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 For EQUILIBRIUM, these forces must SUM to produce a ZERO RESULTANT. Hence: ∑F = 0 ∑F x i +∑F y j + = 0 ∑F x = 0 ∑F y = 0 28 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Example 2 Determine the tension in cables BA and BC necessary to support the 60-kg cylinder. 29 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Solution 2 Due to EQUILIBRIUM, the weight of the cylinder causes the tension in cable BD to be: TBD = 60(9.81) = 588.6N 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 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Solution 2 The magnitudes of TA and TC are unknown, but their directions are known. 31 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Solution 2 Equations of equilibrium along the x and y axes: x y 32 ∑F =0 ∑F =0 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón 33 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Homework3 http://facultad. bayamon.inter.edu/omeza/ Omar E. Meza Castillo Ph.D. 34 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón ¿Preguntas? Comentarios 35 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón GRACIAS 36