ENGI 1313 Mechanics I

ENGI 1313 Mechanics I Lecture 11: 2D and 3D Particle Equilibrium

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

Lecture 11 Objectives • to further examine and apply Chapter 3 objectives in 2D and 3D space

F2 F1= mg F2 F1= mg Note on Free Body Diagram • Force Sense and Solution • Negative sign indicates the force sense is opposite that shown on the FBD

Omit Ch.3 Spring Problems

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.

FBE FCD B C 30  FBC FCB 60 45 FCA FBA FAC FAH FAB A 60 45 H FAH W = mg Example 11-01 (cont.) • Where to Start? Point C 2 Equations 3 Unknowns Point B 2 Equations 3 Unknowns Point H 1 Equation 2 Unknowns but…. Newton’s 3rd Law Point A 2 Equations 3 Unknowns

FAH H W = mg 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

FAC FAB A 60 45 FAH = W = 1N Example 11-01 (cont.) • FBD at Point A

Example 11-01 (cont.) • FBD at Point B FBE B 30 FBC 45 FBA = FAB = 0.5176N

Example 11-01 (cont.) • FBD at Point C FCB = FBC = 0.268N FCD C  60 FCA = FAC = 0.7321N

Example 11-01 (cont.) • Cord Forces • Analysis summary unit force • Maximum force • 200 N

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

Example 11-01 (cont.) • Mathcad Solution • Set-up equilibriumequations

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 (Fx and Fy) • The Find command function requires an initial guess or estimate of the forces and angle ()to start the mathematical search of the solution

Example 11-01 (cont.) • Mathcad Solution • Solve system of equations

Particle Equilibrium in 3D Newton’s 1st Law Cartesian Vector  Scalar components = 0 3 Equations  Solve for at most 3 unknowns

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

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

z F3 = 10 N F2 = 10 N A y F1 = 20 N x Comprehension Quiz 11-03 • Four forces act at point A and the system is in equilibrium. Select the correct force vector F4 to balance the system. • Answer: D

Classification of Textbook Problems Hibbeler (2007)

References • Hibbeler (2007) • http://wps.prenhall.com/esm_hibbeler_engmech_1

ENGI 1313 Mechanics I