6-moments couples and force couple systems_parta

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

6-moments couples and force couple systems_parta

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