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 and NO Tendency to “Twist”  Draw

F

ree

B

ody

D

iagrams for Particles • Isolate particle and show Forces acting on the particle  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 3 • More Full Treatment in a 3 rd Year Mechanics of Materials Course Bruce Mayer, PE

Engineering-36: Engineering Mechanics - Statics

[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 – That is, all the Force LoA’s Pass Thru a COMMON Point

Concurrent Forces 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 6 • Thus the Body is NOT Subjected to any Torques

Engineering-36: Engineering Mechanics - Statics

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 First Law A Similar Law applies to Twisting Actions 

F

 

T

m

a



I

α

 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 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx

Engineering-36: Engineering Mechanics - Statics

8

Particle Equilibrium

 For Static or Constant-Velocity “Particles” the Condition of Equilibrium 

F



m

 0  

F

 0  By Component DeComposition: 2D : 

F x

 0 

F y

 0 3D : 

F x

 0 

F y

 0 

F z

 0 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx

Engineering-36: Engineering Mechanics - Statics

9

Particle Equilibrium Summary

 The 2D Case  The 3D Case • Note the PoC

Engineering-36: Engineering Mechanics - Statics

10 • 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 plates are negligible 11  The Gusset Plate above is used to connect 4 members of a planar truss that is in equlibrium

Engineering-36: Engineering Mechanics - Statics

 Find the loads

F

C

and

F

A

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 ΣF

x

= 0  Note that All the Force LoA’s have a Point of Concurrency (PoC)  

F A F x

 

F C

0 or

F A

 cos 60   1200

lbs

 

F C

cos 60   0  1200

lbs

Engineering-36: Engineering Mechanics - Statics

12 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx

Particle Example

  Now by ΣF

y F y

 0  500

lbs



F C

sin 60  = 0  0 or

F C

 500

lbs

sin 60 

Engineering-36: Engineering Mechanics - Statics

13  Thus

F C

 577 .

4

lbs

 Sub

F C

into previous eqn for

F A F A

 

F C

cos 60   1200

lb F A

  500 sin

lb

60  cos 60   1200

lb F A

  sin 500

lb

60  cos 60   1200

lb F A F A

or    500 tan

lb

60   288 .

7

lb

 1200

lb

 1200

lb F A

 911 .

3

lb

Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx

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

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

F

A

&

F

C

Then Measure with inch-Ruler Scale-Up using 150 lbs/inch

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 585 Scale Calc 577.4

915 911.3

0 100 200 300

Engineering-36: Engineering Mechanics - Statics

16 400 500

Load (lbs)

600 700 800 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

T

2

T

1 

T

2

T

1

Engineering-36: Engineering Mechanics - Statics

19 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 ΣM axel = 0 by Definition.

 Thus in this case, as in 20 the Particle Case: 

F Engineering-36: Engineering Mechanics - Statics

 0 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

21 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx

Example: FrictionLess Pulley

 Using T 1 = T 2 Draw the FBD for Pulley-C

50 lb 50 lb

 By the ΣF y T C = 0 find = 100 lbs  Pulley-B FBD

100 lb 100 lb T C Engineering-36: Engineering Mechanics - Statics

22

T B

Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx

Example: FrictionLess Pulley

  By the ΣF y T B = 0 find = 200 lbs Pulley-A FBD

200 lb 200 lb

 By the ΣF y = 0 find W = 400 lbs

W Engineering-36: Engineering Mechanics - Statics

23

400 lbs

Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx

Special Cases Summarized

 2D 3D Particle: 

F x

  0

F x

  0

F y

  0

F y

  0

F z

 0  FrictionLess Pulley:

T

1 

T

2

Engineering-36: Engineering Mechanics - Statics

24 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx

WhiteBoard Work

Lets Work a Pulley Problem

25 

Both Pulleys may be Regarded as Free-Wheeling (FrictionLess) Engineering-36: Engineering Mechanics - Statics

25  Find for EQUILIBRIUM • ||

P

|| • Angle α Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx

25 

Engineering-36: Engineering Mechanics - Statics

26 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx

Engineering-36: Engineering Mechanics - Statics

27 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx

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

Appendix

Bruce Mayer, PE Registered Electrical & Mechanical Engineer [email protected]

Engineering-36: Engineering Mechanics - Statics

30 Bruce Mayer, PE [email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx