ENGR-36_Lec-06_Particle-Equilibrium.pptx 1 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engineering 36 Chp 3: Particle Equilibrium

ENGR-36_Lec-06_Particle-Equilibrium.pptx 2 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-06_Particle-Equilibrium.pptx 3 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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 3 rd Year Mechanics of Materials Course

ENGR-36_Lec-06_Particle-Equilibrium.pptx 4 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-06_Particle-Equilibrium.pptx 5 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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 Concurrent Forces

ENGR-36_Lec-06_Particle-Equilibrium.pptx 6 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-06_Particle-Equilibrium.pptx 7 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-06_Particle-Equilibrium.pptx 8 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Particle Equilibrium  Recall Newton’s First Law  A Similar Law applies to 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

ENGR-36_Lec-06_Particle-Equilibrium.pptx 9 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Particle Equilibrium  For Static or Constant-Velocity “Particles” the Condition of Equilibrium  By Component DeComposition:

ENGR-36_Lec-06_Particle-Equilibrium.pptx 10 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Particle Equilibrium Summary  The 2D Case Note the PoC  The 3D Case Note the PoC

ENGR-36_Lec-06_Particle-Equilibrium.pptx 11 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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 F C and F A acting on the Gusset Plate

ENGR-36_Lec-06_Particle-Equilibrium.pptx 12 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Particle Example  Note that All the Force LoA’s have a Point of Concurrency (PoC)  Thus PARTICLE ANALYSIS applies in this situation  Start with ΣF x = 0

ENGR-36_Lec-06_Particle-Equilibrium.pptx 13 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Particle Example  Now by ΣF y = 0  Thus  Sub F C into previous eqn for F A

ENGR-36_Lec-06_Particle-Equilibrium.pptx 14 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Graphical Solution (1)  Use Known Mag & Dir to Draw scaled versions of F B & F D Scaling Factor = 150 lb/inch  Draw “X-lines” for the know LoA’s for F A & F C F C LoA is 60° off the Horizontal

ENGR-36_Lec-06_Particle-Equilibrium.pptx 15 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Graphical Solution (2)  Connect the intersecting LoA’s to Define the Scaled-Magnitudes for F A & F C  Then Measure with inch-Ruler  Scale-Up using 150 lbs/inch

ENGR-36_Lec-06_Particle-Equilibrium.pptx 16 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-06_Particle-Equilibrium.pptx 17 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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)

ENGR-36_Lec-06_Particle-Equilibrium.pptx 18 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-06_Particle-Equilibrium.pptx 19 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-06_Particle-Equilibrium.pptx 20 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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 ΣM axel = 0 by Definition.  Thus in this case, as in the Particle Case:

ENGR-36_Lec-06_Particle-Equilibrium.pptx 21 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-06_Particle-Equilibrium.pptx 22 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: FrictionLess Pulley  Using T 1 = T 2 Draw the FBD for Pulley-C  By the ΣF y = 0 find T C = 100 lbs  Pulley-B FBD 50 lb TCTC 100 lb TBTB

ENGR-36_Lec-06_Particle-Equilibrium.pptx 23 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example: FrictionLess Pulley  By the ΣF y = 0 find T B = 200 lbs  Pulley-A FBD  By the ΣF y = 0 find W = 400 lbs 200 lb W 400 lbs

ENGR-36_Lec-06_Particle-Equilibrium.pptx 24 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Special Cases Summarized  Particle:  FrictionLess Pulley:

ENGR-36_Lec-06_Particle-Equilibrium.pptx 25 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics WhiteBoard Work Lets Work a Pulley Problem Both Pulleys may be Regarded as Free-Wheeling (FrictionLess)  Find for EQUILIBRIUM ||P|| Angle α

ENGR-36_Lec-06_Particle-Equilibrium.pptx 26 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-06_Particle-Equilibrium.pptx 27 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-06_Particle-Equilibrium.pptx 28 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-06_Particle-Equilibrium.pptx 29 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-06_Particle-Equilibrium.pptx 30 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bruce Mayer, PE Registered Electrical & Mechanical Engineer Engineering 36 Appendix