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



[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 ++=


( )∫ ∫ ⋅==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



( )

( )

( )[ ]∫

∫ ∫ ∫

⋅=

⋅=

⋅===

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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Bruce Mayer& 'EEngineering-36: Engineering Mechanics - Statics

!nifor" #ensity Case

Con("der a body /"th?N10RM 7EN,*4A ".e.

*hen M ) >8



( ) ( )

( ) ( ) 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



( ) ( )IntensityleverArmMoment ⋅=

( ) ( )[ ] ∑∑ ⋅= sIntensitiesIntensitieLeverArms M R






RM "n Co+#onent or+

No/ 7e"ne the ncre+ental

Mo+ent& dFn

LeverArm Intensity



k Z jY i X R M M M M ˆˆˆ ++=

( )( )nnnnn gdV k z j yi xd ρ ˆˆˆΩ ++=






ntegrat"ng d% to 1"nd % orthe ent"re body



( )( )∫

∫

++

==

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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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



( ) ( )( ) ( ) ( )∫ ∫ ∫ ++

=++=⋅

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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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(



( ) ( )

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





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





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 ++=∑





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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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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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





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





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

==

=⇒=

===






,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

∫ ∫ ∫






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=





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





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





White(oard Wor1

Find theAreal &

LinealCentroids

AA /

Areaorin)

&

LL /

Line4t#i)eorin)

&





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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ENGR-36 Lec-23 Fa12 Center of Gravity H13e