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

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

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

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

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

ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 6 Example 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.

ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 7 Example (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

ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 8 Example (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

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

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

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

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

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

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

ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 15 Example (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

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

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

ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 18 Comprehension Quiz 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

ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 19 Comprehension Quiz 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

ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 20 Comprehension Quiz 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

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

ENGI 1313 Statics I – Lecture 11© 2007 S. Kenny, Ph.D., P.Eng. 22 References Hibbeler (2007) mech_1