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8/9/2019 6-Moments Couples and Force Couple Systems_PartA
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Lecture #6 part a (refChapter 4)
Force System Resultants
Moments, Couples, and Force
Couple Systems
Note, we will not being using cross-
products as shown in section 4.2 and
4. ! "ust scalars.
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Equivalent Forces
• #e de$ined e%ui&alent $orces as being $orces
with the same magnitude acting in the same
direction and acting along the same line o$
action 'this is through the (rinciple o$
)ransmissibility*, but why do the $orces need
to act along the same line+
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4.1 Introduction to Moents
)he tendency o$ a $orce to rotate a rigid body
about any de$ined ais is called the Moment
o$ the $orce about the ais
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M!ME" !F $ F!%CE & 'C$L$% F!%ML$I!" ('ection 4.1)
)he moment, M, o$ a $orce about a point pro&ides a measure o$ the tendency $or
rotation 'sometimes called a tor%ue*.
M
M F d
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Moent caused * a Force
)he Moment o$ Force 'F* about an ais
through (oint '/* or $or short, the Moment o$
F about /, is the product o$ the magnitude o$
the $orce and the perpendicular distance
between (oint '/* and the line o$ action o$
Force 'F*
M / Fd
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nits of a Moent
)he units o$ a Moment are0 N·m in the S1 system
ft·lbs or in·lbs in the S Customary system
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$++LIC$I!"'
3eams are o$ten used to bridge gaps in walls. #e ha&e tonow what the e$$ect o$ the $orce on the beam will ha&e on
the beam supports.
#hat do you thin those impacts are at points / and 3+
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$++LIC$I!"'
Carpenters o$ten use a hammer in this way to pull a stubborn nail.)hrough what sort o$ action does the $orce F5 at the handle pull the nail+
5ow can you mathematically model the e$$ect o$ $orce F5 at point 6+
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+roperties of a Moent
Moments not only ha&e
a magnitude, they also
ha&e a sense to them.
)he sense o$ a moment
is clocwise or counter-
clocwise depending
on which way it willtend to mae the ob"ect
rotate
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+roperties of a Moent
)he sense o$ a Moment is de$ined by the
direction it is acting on the /is and can be
$ound using Right 5and Rule.
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,ari-nons heore
The moment of a Force about any axis is
equal to the sum of the moments of its
components about that axis
)his means that resol&ing or replacing $orces
with their resultant $orce will not a$$ect the
moment on the ob"ect being analy7ed
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M!ME" !F $ F!%CE & 'C$L$% F!%ML$I!" 'continued*
/s shown, d is the perpendicular distance $rom point 6 to the line o$ action o$
the $orce.
1n 2-8, the direction o$ M6 is either clocwise or
counter-clocwise, depending on the tendency $or rotation.
1n the 2-8 case, the magnitude o$ the moment is Mo F d
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%E$/I"0 I2
9. #hat is the moment o$ the 9: N $orce about point /
'M /*+
/* N;m 3* < N;m C* 92 N;m
8* '92=* N;m >* ? N;m@ /
d m
F 92 N
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E3aple #1 / 9::-lb &ertical $orce is applied to
the end o$ a le&er which is attachedto a sha$t at O.
8etermine0a* Moment about O,
b* 5ori7ontal $orce at A whichcreates the same moment,
c* Smallest $orce at / whichproduces the same moment,
d* Aocation $or a 24:-lb &ertical
$orce to produce the samemoment,
e* #hether any o$ the $orces $romb, c, and d is e%ui&alent to theoriginal $orce.
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E3aple #1
( )
( ) ( )in.12lb100
in.1260cosin.24
=
=°=
=
O
O
M
d
Fd M
a) Moment about O is equal to the product of the
force and the perpendicular distance between the
line of action of the force and O. Since the force
tends to rotate the lever clockwise the moment
vector is into the plane of the paper.
inlb1200 ⋅=O M
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E3aple #1
( )
( )
in.!.20
in.lb1200
in.!.20in.lb1200
in.!.2060sinin.24
⋅=
=⋅=
=°=
F
F
Fd M
d
O
b) "ori#ontal force at A that produces the same
moment
lb$.%$= F
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E3aple #1
( )
in.42
in.lb1200
in.42in.lb1200
⋅=
=⋅
=
F
F
Fd M O
c) &he smallest force at A to produce the same
moment occurs when the perpendicular distance is
a ma'imum or when F is perpendicular to OA.
lb%0= F
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E3aple #1
( )
in.%cos60
in.%lb402
in.lb1200lb240in.lb1200
=°
=⋅
=
=⋅
=
OB
d
d
Fd M O
d) &o determine the point of application of a 240 lb
force to produce the same moment
in.10=OB
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E3aple #1
e) (lthouh each of the forces in parts b) c) and d)
produces the same moment as the 100 lb force none
are of the same manitude and sense or on the same
line of action. *one of the forces is equivalent to the
100 lb force.
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4.4 +rinciple of Moents
Barignons )heorem0 )he moment o$ a $orce
about a point is e%ual to the sum o$ moments
o$ the components o$ the $orce about the
point0
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M!ME" !F $ F!%CE & 'C$L$% F!%ML$I!" 'continued*
6$ten it is easier to determine M6 by using the components o$ F as shown
'Barignons )heorem*.
)hen M6 'FD a* ! 'FE b*. Note the di$$erent signs on the terms )he typical
sign con&ention $or a moment in 2-8 is that counter-clocwise is considered
positi&e. #e can determine the direction o$ rotation by imagining the body
pinned at 6 and deciding which way the body would rotate because o$ the
$orce.
For eample, M6
F d and the
direction is counter-clocwise.
Fa
b
d
6
ab
6
F
F
F y
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0%!+ +%!LEM '!L,I"0
Since this is a 2-8 problem0
9* Resol&e the 2: lb $orce along the
handles and y aes.
2* 8etermine M / using a scalar
analysis.
0iven5 / 2: lb $orce is applied to the
hammer. Find5 )he moment o$ the $orce at /.
+lan5
y
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0%!+ +%!LEM '!L,I"0 (cont.)
'olution0
G ↑ Fy 2: sin :H lb
G → F 2: cos :H lb
y
G M / I –'2: cos :H*lb '9J in* – '2: sin 2:H*lb 'K in*L
– K9.?? lb;in K2 lb;in 'clocwise*
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Moents in /
4.7 Moent of a Force aout a
'pecific $3is
1n 28 bodies the moment is due to a $orce
contained in the plane o$ action perpendicular
to the ais it is acting around. )his maes the
analysis &ery easy.
1n 8 situations, this is &ery seldom $ound to
be the case.
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Moents in /
)he moment about an ais is still calculatedthe same way 'by a $orce in the planeperpendicular to the ais* but most $orces are
acting in abstract angles. 3y resol&ing the abstract $orce into its
rectangular components 'or into itscomponents perpendicular to the ais o$
concern* the moment about the ais can thenbe $ound the same way it was $ound in 28 !M Fd 'where d is the distance between the$orce and the ais o$ concern*
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"otation for Moents
1n simpler terms the Moment o$ a Force about
the y -ais 'My * can be $ound by using the
pro"ection o$ the Force on the x-z (lane
)he Notation used to denote Moments about
the Cartesian /es are 'M x , My , and Mz *
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8 Moment0
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@i&en the tension in cable 3C is ?:: N, $ind M, My, and M7 about point /.
8 Moments >ample0