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9.1) Center of Gravity & Center of
Mass for a system of Particles
Center of Gravity:point locates the resultant weight
nnR Wz~Wz~Wz~Wz
2211
WWz~
z
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9.1) Center of Gravity & Center of
Mass for a system of Particles
Center of Gravity:point locates the resultant weight
Center of Mass:
W
Wz~
z
W
Wx~
x
W
Wy~
y
mmz~z mmx~x mmy~y
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9.2) Center of Gravity & Center of
Mass, and Centroid for a Body
Center of Gravity:
V
V
dV
dVx~x
dW
dWx~x
V
V
dV
dVy~
ydW
dWy~
y
V
V
dV
dVz~
z
dW
dWz~
z
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9.2) Center of Gravity & Center of
Mass, and Centroid for a Body
Center of Mass:
V
V
dV
dVx~x
V
V
dV
dVy~
y
V
V
dV
dVz~
z
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9.2) Center of Gravity & Center of
Mass, and Centroid for a Body
Centroid:point defines the geometric center
If the material composing a body is uniform or
homogeneous, the density or specific weight will be
constant throughout the body, then the centroid is the
same as the center of gravity or center of mass
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9.2) Center of Gravity & Center of
Mass, and Centroid for a Body
Volume Centroid:
V
V
dV
dVx~x
V
V
dV
dVy~
y
V
V
dV
dVz~
z
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9.2) Center of Gravity & Center of
Mass, and Centroid for a Body
Area Centroid:
A
A
dA
dAx~x
A
A
dA
dAy~
y
A
A
dA
dAz~
z
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9.2) Center of Gravity & Center of
Mass, and Centroid for a Body
Line Centroid:
L
L
dL
dLx~x
L
L
dL
dLy~
y
L
L
dL
dLz~
z
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9.2) Center of Gravity & Center of
Mass, and Centroid for a Body
Symmetry:
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Mass, and Centroid for a Body
Symmetry:
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9.2) Center of Gravity & Center of
Mass, and Centroid for a Body
Symmetry:
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9.2) Center of Gravity & Center of
Mass, and Centroid for a Body
Example 9-3:
Determine the distance
from the x-axis to the
centroid of the area of
the triangle shown inFig. 9-11
y
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Mass, and Centroid for a Body
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Mass, and Centroid for a Body
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9.3) Composite Areas
Procedure for Analysis:
1) Composite Parts:
Divide the area into finite number of simpler shapes
For a hole or a composite part that has no material,
then this part is considered to have negative area2) Moment
Arms:
Determine the centroid of each composite part, .
3) Summations: Determine the centroid of the whole area, ,
by
applying the centroid equations
If the area is symmetrical about an axis, then the
centroid lies on this axis
y~,x~ y,x
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9.3) Composite Areas
Example 9.10:
Locate the centroid of the plate area shown in Fig9-18a
D A l Al O i
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9.3) Composite Areas
= -
D A l Al O i
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9.3) Composite Areas
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9.3) Composite Areas
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9.3) Composite Areas
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9.3) Composite Areas
