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Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@engr.mun.ca ENGI 1313 Mechanics I Lecture 11:2D and 3D Particle Equilibrium

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 2 Chapter 3 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

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 3 Lecture 11 Objectives to further examine and apply Chapter 3 objectives in 2D and 3D space

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 4 Note on Free Body Diagram Force Sense and Solution Negative sign indicates the force sense is opposite that shown on the FBD F 1 = mg F2F2 F2F2

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 5 Omit Ch.3 Spring Problems

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 6 Example 11-01 Each cord can sustain a maximum tension of 200 N Determine the largest weight of the sack that can be supported. Also, determine θ of cord DC for equilibrium.

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 7 Example 11-01 (cont.) Where to Start? F BE F BC F BA B 30 45 F CD F CB F CA C 60 F AC F AB F AH A 60 45 Point B 2 Equations 3 Unknowns Point A 2 Equations 3 Unknowns Point C 2 Equations 3 Unknowns W = mg H F AH Point H 1 Equation 2 Unknowns but…. Newton’s 3 rd Law

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 8 Example 11-01 (cont.) FBD at Point H What Cord Will Have the Maximum Tension? Educated guess Experience Theoretical approach Assume W = 1N Maximum cord tension 200 N W = mg H F AH

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 9 Example 11-01 (cont.) FBD at Point A F AC F AB F AH = W = 1N A 60 45

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 10 Example 11-01 (cont.) F BE F BC B 30 45 FBD at Point B F BA = F AB = 0.5176N

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 11 Example 11-01 (cont.) FBD at Point C F CD F CB = F BC = 0.268N F CA = F AC = 0.7321N C 60

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 12 Example 11-01 (cont.) Cord Forces Analysis summary unit force Maximum force 200 N

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 13 Example 11-01 (cont.) Use of Vector Algebra in Mathematical Software to Solve Mechanics Problems Mathcad www.mathcad.com Engineering calculations This discussion on the use of Mathcad is just for knowledge It is not part of any course requirement

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 14 Example 11-01 (cont.) Mathcad Solution Set-up equilibrium equations

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 15 Example 11-01 (cont.) Mathcad Solution Uses a command Find to solves a system of linear equations This system of linear equations is based on the FBD analysis that defines the equilibrium equations ( F x and F y ) The Find command function requires an initial guess or estimate of the forces and angle ( ) to start the mathematical search of the solution

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 16 Example 11-01 (cont.) Mathcad Solution Solve system of equations

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 17 Particle Equilibrium in 3D 3 Equations Solve for at most 3 unknowns Scalar components = 0 Cartesian Vector Newton’s 1 st Law

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 18 Comprehension Quiz 11-01 In 3-D, the direction of a force is known but not the force magnitude, how many unknowns corresponding to that force remain? A) One B) Two C) Three D) Four Answer: A

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 19 Comprehension Quiz 11-02 In 3-D, when you don’t know either the direction or magnitude of a force, how many unknowns do you have corresponding to that force? A) One B) Two C) Three D) Four Answer: C

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 20 Comprehension Quiz 11-03 Four forces act at point A and the system is in equilibrium. Select the correct force vector F 4 to balance the system. Answer: D z x A y F 3 = 10 N F 1 = 20 N F 2 = 10 N

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 21 Classification of Textbook Problems Hibbeler (2007)

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ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 22 References Hibbeler (2007) http://wps.prenhall.com/esm_hibbeler_eng mech_1

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