Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

1. Engineering 36 Chp3: ParticleEquilibrium Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. Learning Goals • Determine WHEN “Particle” Analysis can be Applied (even to Large Systems) • Determine if a Point of Concurrency Exists • Body and NO Tendency to “Twist” • Draw Free Body Diagrams for Particles • Isolate particle and show Forces acting on the particle • Use Particle-Equilibrium Criteria to solve for Unknown Forces

3. 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

4. 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

5. 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 • That is, all the Force LoA’s Pass Thru a COMMON Point ConcurrentForces

6. 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

7. Particle Analysis → Need PoC • Particle Analysis is MUCHeasier than non-ParticleAnalysis • However ImproperApplication of the Particlemethods 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

8. Particle Equilibrium • Recall Newton’s First Law • A Similar Law appliesto Twisting Actions • Bodies with a Point of Concurrency are NOT subject to Torques so Only the Force Equation Applies • For NonMoving (static) or Constant-Velocity systems a = dv/dt = 0

9. Particle Equilibrium • For Static or Constant-Velocity “Particles” the Condition of Equilibrium • By Component DeComposition:

10. Particle Equilibrium Summary • The 2D Case • Note the PoC • The 3D Case • Note the PoC

11. Particle Example • The Gusset Plate above is used to connect 4 members of a planar truss that is in equlibrium • The Loads at B & D are known at 500 lb & 1200 lb • Assume weights of the members and Gusset plates are negligible • Find the loads FC and FA acting on the Gusset Plate

12. Particle Example • Note that All the Force LoA’s have a Point of Concurrency (PoC) • Thus PARTICLE ANALYSIS applies in this situation • Start with ΣFx = 0

13. Particle Example • Now by ΣFy= 0 • Thus • Sub FC into previous eqn for FA

14. Graphical Solution (1) • Use Known Mag & Dir to Drawscaled versions of FB & FD • Scaling Factor = 150 lb/inch • Draw “X-lines” for the know LoA’sfor FA& FC • FCLoA is 60° off the Horizontal

15. Graphical Solution (2) • Connect the intersecting LoA’s to Definethe Scaled-Magnitudes for FA & FC • Then Measure with inch-Ruler • Scale-Up using 150 lbs/inch

17. 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)

18. 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

19. 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

20. FrictionLess Pulley – Special Case • Since the Cables/Ropes passing over a FrictionLess Pulley generate NO MomentAbout the Pulley Axel, then for this case theΣMaxel = 0 by Definition. • Thus in this case, as inthe Particle Case:

21. 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

22. Example: FrictionLess Pulley • Using T1 = T2 Draw the FBD for Pulley-C • By the ΣFy = 0 find TC = 100 lbs • Pulley-B FBD 50 lb 50 lb 100 lb 100 lb TC TB

23. Example: FrictionLess Pulley • By the ΣFy = 0 find TB = 200 lbs • Pulley-A FBD • By the ΣFy = 0 find W = 400 lbs 200 lb 200 lb 400 lbs W

24. Special Cases Summarized • Particle: • FrictionLess Pulley:

25. WhiteBoard Work Lets Work a PulleyProblem Both Pulleys may be Regarded as Free-Wheeling (FrictionLess) • Find for EQUILIBRIUM • ||P|| • Angle α

30. Engineering 36 Appendix Bruce Mayer, PE Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu