ENGI 1313 Mechanics I - Memorial University of …spkenny/Courses/Undergraduate/ENGI1313/...Center...
Transcript of ENGI 1313 Mechanics I - Memorial University of …spkenny/Courses/Undergraduate/ENGI1313/...Center...
Shawn Kenny, Ph.D., P.Eng.Assistant ProfessorFaculty of Engineering and Applied ScienceMemorial University of [email protected]
ENGI 1313 Mechanics I
Lecture 40: Center of Gravity, Center of Mass and Geometric Centroid
2 ENGI 1313 Statics I – Lecture 40© 2007 S. Kenny, Ph.D., P.Eng.
Material Coverage for Final ExamIntroduction (Ch.1: Sections 1.1–1.6)Force Vectors (Ch.2: Sections 2.1–2.9) Particle Equilibrium (Ch.3: Sections 3.1–3.4) Force System Resultants (Ch.4: Sections 4.1–4.10)
Omit Wrench (p.174)Rigid Body Equilibrium (Ch.5: Sections 5.1–5.7)Structural Analysis (Ch.6: Sections 6.1–6.4 & 6.6)Friction (Ch.8: Sections 8.1–8.3)Center of Gravity and Centroid (Ch.9: Sections 9.1–9.3)
Ignore problems involving closed-form integrationSimple shapes such as square, rectangle, triangle and circle
3 ENGI 1313 Statics I – Lecture 40© 2007 S. Kenny, Ph.D., P.Eng.
Lecture 40 Objective
to understand the concepts of center of gravity, center of mass, and geometric centroidto be able to determine the location of these points for a system of particles or a body
4 ENGI 1313 Statics I – Lecture 40© 2007 S. Kenny, Ph.D., P.Eng.
Center of Gravity
Point locating the equivalent resultant weight of a system of particles or bodyExample: Solid Blocks
Are both final configurations stable?
WR WR
w1
w2
w3
w5
w1
w2
w3
w5
5 ENGI 1313 Statics I – Lecture 40© 2007 S. Kenny, Ph.D., P.Eng.
Center of Gravity (cont.)
Resultant Weight
Coordinates
Key Property
41iwW iR K== ∑
w1
w2
w3
w4
WR
x
x1~
41iwx~Wx iiR K== ∑41iwz~Wz iiR K== ∑
zz1~
( ) 41i0x~xwM iGiG K==−= ∑∑
xG
L/2
6 ENGI 1313 Statics I – Lecture 40© 2007 S. Kenny, Ph.D., P.Eng.
Center of Gravity (cont.)
Generalized Formulae∑
=
=n
1iiR wW
R
n
1iii
W
wx~
x∑
==
R
n
1iii
W
wy~
y∑
==
R
n
1iii
W
wz~
z∑
==
Moment about y-axis
Moment about x-axis
“Moment”about x-axisor y-axis
z2~
zn~ z1
~
7 ENGI 1313 Statics I – Lecture 40© 2007 S. Kenny, Ph.D., P.Eng.
Center of Mass
Point locating the equivalent resultant mass of a system of particles or body
Generally coincides with center of gravity (G)Center of mass coordinates
∑∑=
i
ii
mmx~
x∑
∑=i
ii
mmy~
y∑
∑=i
ii
mmz~
z
8 ENGI 1313 Statics I – Lecture 40© 2007 S. Kenny, Ph.D., P.Eng.
Center of Mass (cont.)
Can the Center of Mass be Outside the Body?
Fulcrum / Balance
Center of Mass
9 ENGI 1313 Statics I – Lecture 40© 2007 S. Kenny, Ph.D., P.Eng.
Center of Gravity & Mass – Applications
DynamicsInertial termsVehicle roll-over and stability
10 ENGI 1313 Statics I – Lecture 40© 2007 S. Kenny, Ph.D., P.Eng.
Geometric Centroid
Point locating the geometric center of an object or bodyHomogeneous body
Body with uniform distribution of density or specific weight• Center of mass and center of gravity coincident• Centroid only dependent on body dimensions and
not weight terms
∑∑=
i
ii
AAx~
x∑
∑=i
ii
AAy~
y∑
∑=i
ii
AAz~
z
11 ENGI 1313 Statics I – Lecture 40© 2007 S. Kenny, Ph.D., P.Eng.
Geometric Centroid (cont.)
Common Geometric ShapesSolid structure or frame elements
GC & CM
GC & CM
Median Lines
GC & CM
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Composite Body
Find center of gravity or geometric centroid of complex shape based on knowledge of simpler geometric forms
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Example 40-01
Determine the location (x, y) of the 7-kg particle so that the three particles, which lie in the x−yplane, have a center of mass located at the origin O.
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Example 40-01 (cont.)
Center of Mass
∑∑=
i
ii
mmx~
x
∑∑=
i
ii
mmy~
y
( ) ( )( ) ( )( )kg7kg5kg3
m4kg5m3kg3xkg70x++
−+== m57.1x =⇒
( ) ( )( ) ( )( )kg7kg5kg3
m2kg5m2kg3ykg70y++
++−== m29.2x =⇒
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Example 40-02 A rack is made from roll-formed sheet steel and has the cross section shown. Determine the location (x,y) of the centroid of the cross section. The dimensions are indicated at the center thickness of each segment.
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Example 40-02 (cont.)
Assume Unit ThicknessIgnore bend radiiCenter-to-center distance
Centroid Equations
∑∑=
i
ii
AAx~
x∑
∑=i
ii
AAy~
y∑
∑=i
ii
AAz~
z
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Example 40-02 (cont.)
Centroid Equations
∑∑=
i
ii
AAx~
x∑
∑=i
ii
AAy~
y
−(15)(7.5) = 112.5−15/2 = 7.5(15mm)(1mm) = 15mm21
Area (mm2)# ( )mmx~ ( )mmy~ ( )3mmAx~ ( )3mmAy~
1x1 = 7.5mm~
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Example 40-02 (cont.)
Centroid Equations
∑∑=
i
ii
AAx~
x∑
∑=i
ii
AAy~
y
(50)(25) = 1250−50/2 = 25−(50mm)(1mm) = 50mm25
Area (mm2)# ( )mmx~ ( )mmy~ ( )3mmAx~ ( )3mmAy~
5y5 = 25mm~
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Example 40-02 (cont.)
Centroid Equations
∑∑=
i
ii
AAx~
x∑
∑=i
ii
AAy~
y
195045050+30/2 = 6515(30mm)(1mm) = 30mm26
Area (mm2)# ( )mmx~ ( )mmy~ ( )3mmAx~ ( )3mmAy~
6
y6 = 65mm~
x6 = 15mm~
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Example 40-02 (cont.)
Centroid Equations
∑∑=
i
ii
AAx~
x∑
∑=i
ii
AAy~
y
195045050+30/2 = 6515(30mm)(1mm) = 30mm26
9550mm35737.5mm3−−235mm2Sum3200360080/2 = 4045(80mm)(1mm) = 80mm27
1250−50/2 = 25−(50mm)(1mm) = 50mm2524009008015+30/2 = 30(30mm)(1mm) = 30mm24750112.55015/2 = 7.5(15mm)(1mm) = 15mm23
−562.5−30+15/2 = 37.5(15mm)(1mm) = 15mm22−112.5−15/2 = 7.5(15mm)(1mm) = 15mm21
Area (mm2)# ( )mmx~ ( )mmy~ ( )3mmAx~ ( )3mmAy~
1 2
3
4
5
6
7
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Example 40-02 (cont.)
Centroid Equations
2
3
i
ii
mm235mm5.5737
AAx~
x ==∑
∑
2
3
i
ii
mm235mm9550
AAy~
y ==∑
∑
24.4 mm
40.6 mmmm4.24x =
mm6.40y =
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Example 40-03
Two blocks of different materials are assembled as shown. The densities of the materials are: ρA = 150 lb/ft3 and ρA = 400 lb/ft3. The center of gravity of this assembly.
23 ENGI 1313 Statics I – Lecture 40© 2007 S. Kenny, Ph.D., P.Eng.
Example 40-03 (cont.)
Center of Gravity
∑∑=
i
ii
wwx~
x
∑∑=
i
ii
wwy~
y
∑∑=
i
ii
wwz~
z
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Example 40-03 (cont.)
Center of Gravity
∑∑=
i
ii
wwx~
x∑
∑=i
ii
wwy~
y∑
∑=i
ii
wwz~
z
505016.6733116.67B6.253.12512.52143.125A
56.2553.1329.1719.79 Σ
Weight (lb)#
( )( )( )( )( )
lb125.3ft/in12
in2in6in62/1ft/lb150w 3
3
A ==
( )( )( )( )
lb67.16ft/in12
in2in6in6ft/lb400w 3
3
B ==
( )inx~ ( )iny~ ( )inz~ ( )inlbwx~ ⋅ ( )inlbwy~ ⋅ ( )inlbwz~ ⋅
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Example 40-03 (cont.)
Center of Gravity
in47.179.1917.29
wwx~
xi
ii ===∑
∑
in68.279.1913.53
wwy~
yi
ii ===∑
∑
in84.279.1925.56
wwz~
zi
ii ===∑
∑
26 ENGI 1313 Statics I – Lecture 40© 2007 S. Kenny, Ph.D., P.Eng.
Chapter 9 ProblemsUnderstand principles for simple geometric shapes
Rectangle, square, triangle and circleNo closed form integration knowledge required
ReviewExample 9.9 and 9.10Problems 9-44 to 9-61
OmitExample 9.1 through 9.8Problems 9-1 through 9-43, 9-62, 9-67 to 9-83
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References
Hibbeler (2007)http://wps.prenhall.com/esm_hibbeler_engmech_1