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Engineering 36 Chp 3: Particle Equilibrium Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] Engineering-36: Engineering Mechanics - Statics 1 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Learning Goals Determine WHEN “Particle” Analysis can be Applied (even to Large Systems) • Determine if a Point of Concurrency Exists – Body has NO Tendency to “Twist” Draw Free Body Diagrams for Particles • Isolate the “Particle” and show Forces acting there-on Use Particle-Equilibrium Criteria to solve for Unknown Forces Engineering-36: Engineering Mechanics - Statics 2 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Rigid Bodies Most Bodies In Elementary Mechanics Are Assumed To Be RIGID • i.e., Actual Deformations Are Small And Do Not Affect The Force and/or Moment analysis of the System DEFORMABLE Body Mechanics are the Subject of Later Courses • Intro to this in ENGR45 • More Full Treatment in a 3rd Year Mechanics of Materials Course Engineering-36: Engineering Mechanics - Statics 3 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Full Mechanical Equilibrium A Rigid Body in Static Mechanical Equilibrium is Characterized by • Balanced External Forces & Torques A Body/Force/Moment System will have no Tendency to Toward TRANSLATIONAL (forces) or ROTATIONAL (torques) Motion of the Body Engineering-36: Engineering Mechanics - Statics 4 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Special Case Particles In Mechanics even very Large Bodies can be regarded as “Particles” if the Body meets Certain Criteria A 3D (or 2D) Rigid Body may be regarded as a Particle If: • There are No APPLIED Torques • ALL Forces acting on the Body are CONCURRENT Concurrent Forces – That is, all the Force LoA’s Pass Thru a COMMON Point Engineering-36: Engineering Mechanics - Statics 5 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Special Case: Particles The Common Point can be Called the Point of Concurrency (PoC) Use The PoC as the Point that represents the Entire Body. • That is, the action of all forces act on a PARTICLE located at the PoC Note that Concurrent forces Generate NO Tendency to Twist the Body • Thus the Body is NOT Subjected to any Torques Engineering-36: Engineering Mechanics - Statics 6 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Particle Analysis → Need PoC Particle Analysis is MUCH easier than non-Particle Analysis However Improper Application of the Particle methods produce Incorrect results The Particle Idealization Applies ONLY when the LoA’s of ALL Forces applied to the Body Pass thru ONE Point • This Pt is called the Point of Concurrency Engineering-36: Engineering Mechanics - Statics 7 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Particle Equilibrium Recall Newton’s F m a First Law A Similar Law applies T I α to Twisting Actions Bodies with a Point of Concurrency are NOT subject to Torques so Only the Force Equation Applies For NonMoving (static) or ConstantVelocity systems a = dv/dt = 0 Engineering-36: Engineering Mechanics - Statics 8 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Particle Equilibrium For Static or Constant-Velocity “Particles” the Condition of Equilibrium F m 0 F 0 By Component DeComposition: 2D : 3D : F 0 0 F x F x Engineering-36: Engineering Mechanics - Statics 9 F 0 0 F y y z Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx 0 Particle Equilibrium Summary The 2D Case • Note the PoC Engineering-36: Engineering Mechanics - Statics 10 The 3D Case • Note the PoC Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Particle Example • The Loads at B & D are known at 500 lb & 1200 lb • Assume weights of the members and Gusset plate are negligible The Gusset Plate above is used to connect 4 members of a planar truss that is in equlibrium Engineering-36: Engineering Mechanics - Statics 11 Find the loads FC and FA acting on the Gusset Plate Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Particle Example Thus PARTICLE ANALYSIS applies in this situation Start with ΣFx = 0 F x 0 FA FC cos 60 1200lbs 0 Note that All the Force LoA’s have a Point of Concurrency (PoC) Engineering-36: Engineering Mechanics - Statics 12 or FA FC cos 60 1200lbs Also ReCall cos 60° = 1 2 → 𝐹𝐴 = − 𝐹𝐶 2 + 1200 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Particle Example Thus FC 577.4lbs Sub FC into previous eqn for FA FA FC cos 60 1200lb Now by ΣFy = 0 F y 0 500lbs FC sin 60 0 or FC 500lbs sin 60 Engineering-36: Engineering Mechanics - Statics 13 500lb FA cos 60 1200lb sin 60 500lb FA 1200lb sin 60 cos 60 500lb FA 1200lb tan 60 FA 288.7lb 1200lb or FA 911.3Bruce lb Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Graphical Solution (1) Use Known Mag & Dir to Draw scaled versions of FB & FD • Scaling Factor = 150 lb/inch Draw “X-lines” for the known LoA’s for FA & FC • FC LoA is 60° off the Horizontal Engineering-36: Engineering Mechanics - Statics 14 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Graphical Solution (2) Connect the intersecting LoA’s to Define the Scaled-Magnitudes for FA & FC Then Measure with inch-Ruler Scale-Up using 150 lbs/inch → 6.1 inches Engineering-36: Engineering Mechanics - Statics 15 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Scaling Compared to Calculation for Gusset Plate Scale 585 FC Calc Gusset Force Load ID FA 577.4 915 911.3 0 100 200 300 400 500 600 700 800 Load (lbs) Engineering-36: Engineering Mechanics - Statics 16 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx 900 1,000 Special Case: Frictionless Pulley A FrictionLess Pulley is Typically used to Change the Direction of a Cable or Rope in Tension Pulley with PERFECT Axel (FrictionLess) Engineering-36: Engineering Mechanics - Statics 17 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Special Case: Frictionless Pulley FrictionLess Pulleys (Atwoods Machines) will Change the DIRECTION of a Tension-Force, but NOT its MAGNITUDE The Direction is determined by the TANGENT-Point of the Cord as it passes over the Pulley Circumference Engineering-36: Engineering Mechanics - Statics 18 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Special Case: Frictionless Pulley For a frictionless pulley in static equilibrium, the tension in the cable is the same on both sides of the pulley T2 T1 Engineering-36: Engineering Mechanics - Statics 19 T1 T2 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Special Case: Frictionless Pulley In this Case The forces have NO tendency for rotation about the axel • They can thus be effectively Moved to the Axel T2 T1 Engineering-36: Engineering Mechanics - Statics 20 T1 T2 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Special Case: Frictionless Pulley Graphically for FrictionLESS Pulley: T1 = T2 T1 = T2 = T3 T1 = T2 T1 = T2 T3 • Equivalent to a POINT OF CONCURENCE Engineering-36: Engineering Mechanics - Statics 21 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx FrictionLess Pulley – Special Case Since the Cables/Ropes passing over a FrictionLess Pulley generate NO Moment About the Pulley Axel, then for this case the ΣMaxel = 0 by Definition. Thus in this case, as in the Particle Case: F 0 Engineering-36: Engineering Mechanics - Statics 22 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Example: FrictionLess Pulley Consider the Multiple Pulley System at Right • Assume the Pulleys are Frictionless & Massless For this System Determine the Weight of the Block, W Engineering-36: Engineering Mechanics - Statics 23 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Example: FrictionLess Pulley Using T1 = T2 Draw the FBD for Pulley-C 50 lb 50 lb By the ΣFy = 0 find TC = 100 lbs Pulley-B FBD 100 lb 100 lb TC Engineering-36: Engineering Mechanics - Statics 24 TB Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Example: FrictionLess Pulley By the ΣFy = 0 find TB = 200 lbs Pulley-A FBD 200 lb 200 lb W Engineering-36: Engineering Mechanics - Statics 25 By the ΣFy = 0 find W= 400 lbs 400 lbs Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Special Cases Summarized Particle: 2D 3D F x F x 0 0 F FrictionLess Pulley: Engineering-36: Engineering Mechanics - Statics 26 y F y 0 0 F z T1 T2 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx 0 WhiteBoard Work Lets Work a Pulley Problem 25 Both Pulleys may be Regarded as Free-Wheeling (FrictionLess) Find for EQUILIBRIUM • ||P|| • Angle α Engineering-36: Engineering Mechanics - Statics 27 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx 25 Engineering-36: Engineering Mechanics - Statics 28 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Engineering-36: Engineering Mechanics - Statics 29 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Engineering-36: Engineering Mechanics - Statics 30 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Engineering-36: Engineering Mechanics - Statics 31 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Unit Circle – individual objects Im Re Unit_Circle_Template_1609.pptx Engineering-36: Engineering Mechanics - Statics 32 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Unit Circle – Picture Objects 𝑦 Im Re 𝑥 Unit_Circle_Template_1609.pptx Engineering-36: Engineering Mechanics - Statics 33 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx Engineering 36 Appendix Bruce Mayer, PE Registered Electrical & Mechanical Engineer [email protected] Engineering-36: Engineering Mechanics - Statics 34 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx