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Engineering 36
Chp 3: Particle
Equilibrium
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
Engineering-36: Engineering Mechanics - Statics
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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
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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
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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
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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
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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
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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
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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
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[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
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F
0
0
F
y
y
z
Bruce Mayer, PE
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0
Particle Equilibrium Summary
 The 2D Case
• Note the PoC
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 The 3D Case
• Note the PoC
Bruce Mayer, PE
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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
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 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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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 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:
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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
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25
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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
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[email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx
Unit Circle – Picture Objects
𝑦
Im
Re
𝑥
Unit_Circle_Template_1609.pptx
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Bruce Mayer, PE
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Engineering 36
Appendix
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer
[email protected]
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Bruce Mayer, PE
[email protected] • ENGR-36_Lec-06_Particle-Equilibrium.pptx