engr-36 lec-23 fa12 center of gravity h13e
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7/23/2019 ENGR-36 Lec-23 Fa12 Center of Gravity H13e
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[email protected] • ENGR-36_Lec-232_Center_o_Gra!"ty.##t$%Bruce Mayer& 'EEngineering-36: Engineering Mechanics - Statics
Bruce Mayer& 'E
L"cen(ed Electr"cal ) Mechan"cal Eng"neer [email protected]
Engineering 36
Chp09: Center
of Gravity
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[email protected] • ENGR-36_Lec-232_Center_o_Gra!"ty.##t$2Bruce Mayer& 'EEngineering-36: Engineering Mechanics - Statics
Introduction: Center of Gravity
*he earth e$ert( a gra!"tat"onal orce oneach o the #art"cle( or+"ng a body.
• *he(e orce( can be re#laced by a
,NGLE eu"!alent orce eual to the
/e"ght o the body and a##l"ed at the
CEN*ER 01 GR*4 5CG or the body
*he CEN*R07 o an RE "(
analogou( to the CG o a body.• *he conce#t o the 1R,* M0MEN* o
an RE "( u(ed to locate the centro"d
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Total Mass – General Case
G"!en a Ma(("!eBody "n 37 ,#ace
7"!"de the Body "n to
ery ,+all olu+e(8 d
Each dn "( located at
#o("t"on 5$n&yn&9n
*he 7EN,*4& :& can
be a unct"on o
'0,*0N 8 ( )nnnn z y x ,, ρ ρ =
k z j yi x nnnˆˆˆr ++=
( )nnnn z y xdV ,,at
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Total Mass – General Case
No/ ("nce + < :•& then the"ncre+ental +a((& d+
ntegrate d+ o!er the ent"rebody to obta"n the total
Ma((& M
( ) nnnnn dV z y xdm ⋅= ,, ρ k z j yi x nnnˆˆˆr ++=
( )nnnn z y xdV ,,at
( )∫ ∫ ⋅==volume
nnnn
body
n dV z y xdm M ,, ρ
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Total WEIGT – General Case
Recall that > < Mg ?("ng the the #re!"ou(
e$#re(("on or M
No/ 7e"ne ,'EC1C
>EG*& γ& a( :•g 8
k z j yi x nnnˆˆˆr ++=
( )nnnn z y xdV ,,at
( )
( )
( )[ ]∫
∫ ∫ ∫
⋅=
⋅=
⋅===
volume
nnnn
volume
nnnn
volume
nnnn
body
n
dV g z y x
dV g z y x
dV z y x g gdm Mg W
,,
,,
,,
ρ
ρ
ρ
( )[ ]
( )∫
∫ ⋅=
⋅=
volume
nnnn
volume
nnnn
dV z y x
dV g z y xW
,,
,,
γ
ρ
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!nifor" #ensity Case
Con("der a body /"th?N10RM 7EN,*4A ".e.
*hen M ) >8
k z j yi x nnnˆˆˆr ++=
( )nnnn z y xdV ,,at
( ) ( )
( ) ( ) γ γ γ
ρ ρ ρ
==
==
nmmnnn
nmmnnn
z y x z y x
z y x z y x
,,,,
,,,,
V dV dV W
V dV dV M
volume
n
volume
n
volumen
volumen
γ γ γ
ρ ρ ρ
==⋅=
==⋅=
∫ ∫ ∫ ∫
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Center of Mass $ocation
?(e M0MEN*, to LocateCenter o Ma((Gra!"ty
Recall 7e"nt"on o a
M0MEN*
n the General Center-o-Ma(( Ca(e
• Le!err+ D 'o("t"on ector& r n& or "t( co+#onent(• nten("ty D ncre+ental Ma((& d+n
k z j yi x nnnˆˆˆr ++=
( )nnnn z y xdV ,,at
( ) ( )IntensityleverArmMoment ⋅=
( ) ( )[ ] ∑∑ ⋅= sIntensitiesIntensitieLeverArms M R
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Center of Mass $ocation
RM "n Co+#onent or+
No/ 7e"ne the ncre+ental
Mo+ent& dFn
LeverArm Intensity
k z j yi x nnnˆˆˆr ++=
( )nnnn z y xdV ,,at
k Z jY i X R M M M M ˆˆˆ ++=
( )( )nnnnn gdV k z j yi xd ρ ˆˆˆΩ ++=
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Center of Mass $ocation
ntegrat"ng d% to 1"nd % orthe ent"re body
k z j yi x nnnˆˆˆr ++=
( )nnnn z y xdV ,,at
( )( )∫
∫
++
==
volume
nnnn
body
n
gdV k z j yi x
d
ρ ˆˆˆ
ΩΩ
( ) ( ) ( )∫ ∫ ∫ ++=Ω+Ω+Ω=
yall zallxall
nnnnnn
x y x
dV z k g dV y j g dV xi g
k ji
ρ ρ ρ ˆˆˆ
ˆˆˆΩ
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Center of Mass $ocation
No/ Euate % to &M•Mg
Cancel"ng g& and euat"ng
Co+#onent( y"eld(& or e$a+#le& "n the I-7"r
LeverArm Intensity
k z j yi x nnnˆˆˆr ++=
( )nnnn z y xdV ,,at
( ) ( )( ) ( ) ( )∫ ∫ ∫ ++
=++=⋅
yall zallxall
ˆˆˆ
ˆˆˆ
nnnnnn
M M M M
dV z k g dV y j g dV xi g
k Z jY i X Mg Mg
ρ ρ ρ
R
( ) ( )∫ =xall
nn M dV x M X ρ
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Center of Mass $ocation
7"!"de out the *otal nten("ty&M& to (olate the 0!erall
I-d"rected Le!er r+& IM
nd the ,"+"lar e$#re(("on(
or the other Co0rd 7"rect"on(
k z j yi x nnnˆˆˆr ++=
( )nnnn z y xdV ,,at
( ) ( )
M
dV x
X nn
M ∫ = xall
ρ
( ) ( ) ( ) ( )
M
dV z
Z M
dV y
Y
nn
M
nn
M
∫ ∫ == xallyall
ρ ρ
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Center of Gravity of a '# (ody
Centro"d o an rea
*aJ"ng ncre+ental 'late
rea(& 1or+"ng the ΣF$ )
ΣFy& long >"th the Σ19<>
4"eld( the E$#re(("on or the
Eu"!alent '0N* o >
a##l"cat"on• Note F ?n"t( < n-lb or N-+
Centro"d o a L"ne
∫ ∑∑ ∫
∑∑
=
∆=⇒Ω
=
∆=⇒Ω
dW y
W yW y
dW x
W xW x
y
x
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Centroids of )reas * $ines
Centro"d o an rea
1or 'late o ?n"or+
*h"cJne((
• γ ≡ ,#ec""c >e"ght
• t ≡ 'late *h"cJne((
•
d> < γ td
>"re o
?n"or+
*h"cJne((
• γ ≡ ,#ec""c
>e"ght
• a ≡ I-,ec rea
• d> < γ a5dL
Centro"d o a L"ne
( ) ( )
axisxto.t.moment w.r 1st
axisy.t.moment w.r 1st
−=
Ω==
−=Ω==
=
=
∫
∫
∫ ∫
y
x
dA y A y
dA x A x
dAt x At x
dW xW x
γ γ ( ) ( )
∫ ∫ ∫ ∫
=
=
=
=
dL y L y
dL x L x
dLa x La x
dW xW x
γ γ
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+irst Mo"ents of )reas * $ines• n area "( (y++etr"c /"th re(#ect to an a$"(
BB’ " or e!ery #o"nt P there e$"(t( a #o"nt
P’ (uch that PP’ "( #er#end"cular to BB’ andthe rea "( d"!"ded "nto eual #art( by BB’.
• *he "r(t +o+ent o an area /"th re(#ect to
a l"ne o (y++etry "( KER0.
• an area #o((e((e( a l"ne o ,4MME*R4&
"t( centro"d LE, on ** I,
• an area #o((e((e( t/o l"ne( o (y++etry&
"t( centro"d l"e( at the"r N*ER,EC*0N.
• n area "( (y++etr"c /"th re(#ect to a
center O " or e!ery ele+ent dA at 5 x,y
there e$"(t( an area dA’ o eual area
at 5−x, −y .• *he centro"d o the area co"nc"de(
/"th the center o (y++etry& O.
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Centroids of Co""on )rea Sha,es
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Centroids of Co""on $ine Sha,es
Recall that or a ,MLL ngle&
α α ≅sin r x ≅⇒
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Co",osite lates and )reas
• Co+#o("te #late(
• Co+#o("te area
k k
k k
W yW Y
W xW X
∑∑∑∑
=
=
∑∑
∑∑=
=
k k
k k
A y AY
A x A X
332211 W xW xW xW X ++=
∑
332211 A y A y A y AY ++=∑
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E.a",le: Co",osite late
1or the #lane area
(ho/n& deter+"ne the
"r(t +o+ent( /"thre(#ect to the $ and y
a$e(& and the locat"on
o the centro"d.
,olut"on 'lan• 7"!"de the area "nto a tr"angle&
rectangle& (e+"c"rcle& and a
c"rcular cutout
• Calculate the "r(t +o+ent( o
each area / re(#ect to the a$e(• 1"nd the total area and "r(t
+o+ent( o the tr"angle&
rectangle& and (e+"c"rcle.
,ubtract the area and "r(t
+o+ent o the c"rcular cutout• Calc the coord"nate( o the area
centro"d by d"!"d"ng the
NE* "r(t +o+ent by
the total area
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E.a",le: Co",osite late
1"nd the total area and "r(t +o+ent( o the
tr"angle& rectangle& and (e+"c"rcle. ,ubtract
the area and "r(t +o+ent o the c"rcular cutout33
33
mm102506
mm107757
×+=Ω
×+=Ω
.
.
y
x
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E.a",le: Co",osite late
1"nd the coord"nate(
o the area centro"dby d"!"d"ng the"r(t +o+ent total(by the total area
,olut"on
23
33
mm1013.2
mm107.757
×
×+==
∑∑
A
A x X
mm.5!= X
23
33
mm1013.2
mm102.506
×
×+==
∑∑
A
A yY
mm6.36=Y
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Centroids /y Stri, Integration• 7ouble "ntegrat"on to "nd the "r(t
+o+ent +ay be avoided by de"n"ng
dA a( a th"n rectangle or (tr"#.
( )
( ) ydx y
dA y A y
ydx x
dA x A x
el
el
∫
∫ ∫
∫
=
=
=
=
2
( )[ ]
( )[ ]dy xa y
dA y A y
dy xa xa
dA x A x
el
el
−=
=
−+=
=
∫ ∫ ∫
∫
2
=
=
=
=
∫
∫ ∫
∫
θ θ
θ θ
d r r
dA y A y
d r
r
dA x A x
el
el
2
2
2
1sin
3
2
2
1
"os3
2
∫ ∫∫ ∫
∫ ∫∫ ∫
===
===
dA ydydx ydA y A y
dA xdydx xdA x A x
el
el
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E.a",le: Centroid /y Integration
7eter+"ne by d"rect
"ntegrat"on the
locat"on o thecentro"d o a
#arabol"c (#andrel.
,olut"on 'lan• 7eter+"ne Con(tant J
• Calculate the *otal rea
• ?("ng e"ther vertical or
hori0ontal ,*R',&#eror+ a ("ngle
"ntegrat"on to "nd the
"r(t +o+ent(
•
E!aluate the centro"dcoord"nate( by d"!"d"ng
the *otal %(t Mo+ent
by *otal rea.
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E.a",le: Centroid /y Integration ,olut"on
• 7eter+"ne Con(tant J
• Calculate the *otal rea
3
3
#tri$s%erti"al &se
0
3
2
0
2
2
ab
x
a
bdx x
a
bdx y
dA A
aa
=
===
⇒=
∫ ∫
∫
21
21
2
2
2
2
2 aw'en (an)*
yba xor x
ab y
a
bk ak b
x y xk y
==
=⇒=
===
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Bruce Mayer& 'EEngineering-36: Engineering Mechanics - Statics
E.a",le: Centroid /y Integration
,olut"on
• Calc the %(t Mo+ent(& F"
?("ng !ert"cal (tr"#(&
#eror+ a ("ngle
"ntegrat"on to "ndthe "r(t +o+ent(.
( )
!!
2
0
!
2
0
3
2
0
2
2
ba x
a
bdx x
a
b
dx xa
b xdx y xdA x
aa
a
el x
=
==
===Ω
∫
∫ ∫ ∫
( )
1052
2
1
2
2
0
5
!
2
0
2
2
2
ab x
a
b
dx x
a
bdx y
ydA y
a
a
el y
=
=
===Ω
∫ ∫ ∫
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E.a",le: Centroid /y Integration
E!aluate the
centro"d coord"nate(
• 7"!"de F$ and Fy by the total rea
1"nally the n(/er(
!3
2baab
x
A x x
=
Ω=
a x!
3=
103
2abab y
A y y
=
Ω=
b y10
3=
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Theore"s of a,,us-Guldinus ,urace o
re!olut"on "(generated by
rotat"ng a #lane
cur!e about a "$ed
a$"(.
rea o a (urace o re!olut"on
"( eual to the length o the
generat"ng cur!e& L& t"+e( the
d"(tance tra!eled by the
centro"d through the rotat"on.
∫ ∫ ∫
=Ω==
==
ydL L y L y A
ydL ydL A
yas2
22
π
π π
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Theore"s of a,,us-Guldinus Body o re!olut"on "(
generated byrotat"ng a #lane
area about a "$ed
a$"(.
olu+e o a body o re!olut"on
"( eual to the generat"ng area&
& t"+e( the d"(tance tra!eled
by the centro"d through the
rotat"on.
A y ydAV
ydAdV V
π π
π
22
2
==
==
∫ ∫ ∫
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E.a",le: a,,us-Guldinus
*he out("de d"a+eter o a
#ulley "( H. +& and the
cro(( (ect"on o "t( r"+ "( a(
(ho/n. no/"ng that the
#ulley "( +ade o (teel and
that the den("ty o (teel& ρ <
=H Jg+3& deter+"ne the
+a(( and /e"ght o the r"+.
,olut"on 'lan• ##ly the theore+ o 'a##u(-
Guld"nu( to e!aluate the
!olu+e( o re!olut"on or the
rectangular r"+ (ect"on and
the "nner cutout (ect"on.
• Mult"#ly by den("ty and
accelerat"on o gra!"ty to get
the +a(( and /e"ght.
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E.a",le: -G ##ly 'a##u(-Guld"nu(
to ,ect"on( ) ,ubtract -
( )( )
××== − 33+3633 mmm10mm1065.7m,-105.7V m ρ ,-0.60=m
( ) 2sm1.+,-0.60== mg W 5+=W
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White(oard Wor1
Find theAreal &
LinealCentroids
AA /
Areaorin)
&
LL /
Line4t#i)eorin)
&
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Bruce Mayer& 'E
Reg"(tered Electr"cal ) Mechan"cal Eng"neer [email protected]
Engineering 36
Appendix 00
sin'T
µs
T
µx
dx
dy==
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