Rigid body examples .power point presentation

rigid body equilibrium

1. Lecture 6Equilibrium of a Particle & a Rigid Body
1
Aparticleatrest:Aparticleisatrestiforiginallyatrestorhasaconstantvelocityif
originallyinmotion.
Aparticleisinequilibrium:iftheresultantofALLforcesactingontheparticleisequalto
zero.(Newton’sfirstlawisthatabodyatrestisnotsubjectedtoanyunbalancedforces).
Equilibrium of a particle:
Equilibrium equations for a particle:
Sum of all forces acting on a particle = F= 0
Inarectangularcoordinatesystemtheequilibriumequationscanberepresentedbythree
scalarequations:0
0
0
x
y
z
F
F
F






Forces on a
particle
Equilibrium

2. 2
Lecture 6Equilibrium of a Particle & a Rigid Body
Equilibrium of a Rigid Body:
Incontrasttotheforcesonaparticle,theforcesonarigid-bodyarenotusually
concurrentandmaycauserotationofthebody(duetothemomentscreatedbytheforces).
Forarigidbodytobeinequilibrium,thenetforceas
wellasthenetmomentaboutanyarbitrarypoint(O)
mustbeequaltozero.x y z
0 =0 0F F F  
And:
F=0(notranslation)M
O=0(norotation)
Equilibrium in 2D:0,0,0   oyx
MFF
3 D
Equilibrium in 3D:
F=0(notranslation)M
O=0(norotation)
M
X= 0,M
Y= 0,M
Z= 0
x
y
2 D

3. THE PROCESS OF SOLVING RIGID BODY EQUILIBRIUM
PROBLEMS
Lecture 6
1.Foranalyzinganactualphysicalsystem,firstwe
needtocreateanidealizedmodel(aboveright).
Foranalyzinganyactualsystemforequilibrium,
thebestwayistofollowtheprocedureforthe
examplementionedbelow:
2.Draw a free-body diagram (FBD) showing all the
external (active and reactive) forces.
5.Finally,applytheequationsofequilibriumto
solveforanyunknowns.
3.Showalltheexternalforcesandcouplemoments.
Thesetypicallyinclude:
a)appliedloads,b)supportreactions,and,
c)theweightofthebody.
4.LabelloadsanddimensionsontheFBD:
Allknownforcesandcouplemomentsshouldbe
labeledwiththeirmagnitudesanddirections.
3
Given:Anoperatorapplies(20lb)tothefootpedal.A
springwith(k=20lb/in)isstretched(1.5in).
Idealized model
FBD

4. 4
Lecture 6
EXAMPLES of (2 D) Equilibrium:
Example1:Abeam(AB)issupportedbypins(A&
C),averticalforce(4kN)isappliedatpoint(B).Find
thesupportreactionsat(AandC).
Solution procedures:
1. Put the x and y axes respectively.
2. Determine if there are any two-force members.
3. Draw a complete FBD of the boom.
FBD of the beam:
A
X
A
Y
A
1.5 m
C B
4 kN
F
C
45°
1.5 m
x
y
Note:UponrecognizingCDasatwo-forcemember,
thenumberofunknownsatCarereducedfromtwo
toone.Now,usingE-of-E,weget,
+ M
A= F
Csin 451.5 –4 3 = 0
F
c= 11.31 kN or 11.3 kN
A
X
A
Y
A
1.5 m
C B
4 kN
1.5 m
x
y
F
Csin 45
o
F
Ccos 45
o
+ F
X= A
X+ 11.31 cos 45= 0;
A
X= –8.00 kN
+ F
Y= A
Y+ 11.3 sin 45–4 = 0;
+ A
Y= –4.00 kN
Note:thatthenegativesignsmeansthatthereactions
havetheoppositedirectiontothatshownonFBD.

5. Lecture 6
Example2:Twosmoothpipes,eachhavingamass
of300kg,aresupportedbythetinesofthetractor
forkattachment.Determineallthereactiveforces?
EXAMPLES of (2 D) Equilibrium:
1.Isolatetheobjectfromitssurroundings,
2.Drawtheoutlineoftheobject;considerall
dimensionsandangles,
3.Includeallforcesandcouplemoments,
4.Labelknownforcesandmomentswiththeirproper
magnitudesanddirections,
5.Unknownforcesandmomentsshouldberepresented
withletters.
Solution procedures:
Idealized
model
5
Themomentequationscanbedeterminedabout
anypoint.Choosingthepointwherethemaximum
numberofunknownforcesarepresentsimplifies
thesolution.
For Pipe A:
O
C
D
x
y
2943 cos30
o
2943 sin30
o

6. 6
Lecture 6
For Pipe B:
WhenpipesAandBareconsideredasoneobject,
youneglectthereactionforcesbetweenthem.
For both Pipes ( A & B ):
Continue Example 2:
Or:
2943 cos30
o
2943 sin30
o
D
x
y
O
0.7 m
C
x
y

7. Lecture 6
7
Example3:TheleverABCinpinsupportedatAand
connectedtoashortlinkBDasshowninfigure.Ifthe
weightofthemembersisnegligible,determinetheforce
ofthepinontheleveratA?
Asshowninfigure,theshortlinkBDisatwoforce
member,sotheresultantforcesatpinsDandBmustbe
equal,oppositeandcollinear.Althoughthemagnitudeof
theforceisunknown,thelineofactionisknownsinceit
passesthroughBandD.
Solution
LeverABCisathreeforcemember,andtherefore,in
ordertosatisfymomentequilibrium,thethreenonparallel
forcesactingonitmustbeconcurrentatPoint(O)as
showninbelowfigure.
Note,thattheforceFontheleverat(B)isequalbut
oppositetotheforce(F)actingat(B)onthelink(BD).
thedistance(CO)mustbeequalto(0.5M),sincethe
linesofactionof(F)andthe(400N)forceareknown.

8. 8
Lecture 6Continue Example 3:

9. 9
Lecture 6
Example4:Thehomogenousplateshowninfigurehasa
massof(100kg)andissubjectedtoaforceandcouple
alongitsedges,ifitissupportedinthehorizontalplaneby
arollerat(A),aballandsocketjointat(B),andacordat
(C),determinethecomponentsofreactionatthese
supports?
Solution
FBD.Therearefiveunknownreactionsactingonthe
plate,asshowninfigure.Eachofthesereactionsisassumed
toactinapositivecoordinatedirection.
F
z= 0; A
z+ B
z+ T
z–300 N –981 N = 0
F
z= 0; A
z+ B
z+ T
z–1281 N = 0 …. ( 1 )

10. 10
Lecture 6
Forcesthatareparalleltoanaxisorpassthroughit,createnomomentaboutthataxes.
Hence,summingmomentsaboutthepositive(xandyaxes),wehave:
Continue Example 4: