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ME 201Engineering Mechanics: Statics
Unit 9.1
Center of Gravity and Center of Mass
Composite Bodies
Centroid
Point which defines geometrical center of an
object
Determined by geometry for simple shapes
and by computing moments of an element
about an axis for more complex shapes
Also referred to as First Moment Area
Centroid of Simple Shapes
2
~ wx
2
~ hy
hwA
Rectangle
h
wx
y
c
Triangle
h
bx
y
c
3
~ bx
3
~ hy
hbA 2
1
Circle
x
y
c
r
0~ x 0~ y
4
22 d
rA
Centroid of Simple Shapes
0~ x3
4~ ry
2
2
1rA
Semi-Circle
x
y
cr
Quarter-Circle
3
4~ rx
3
4~ ry
2
4
1rA
x
y
cr
Centroids of Composite Areas
Centroids of more complex shapes can
be found by the following procedure
Divide into numbered, simple areas
Find centroid of each area
Sum moment areas relative to origin
Shape Area x y xA yA
1
2
3
1
2
3
A
AxX
~
A
AyY
~
x
y
Example Problem Solution
252/50~502/100~
5000100*50
1
1
1
y
x
A
50~2.121)*3/(50*4100~
39272/50*
3
3
2
3
y
pix
piA
7.663/5050~7.663/2*100~
25002/100*50
2
2
2
y
x
A
Given:
Dhole = 50 mm
Rarc = 50 mm
Find:
Centroid of composite
Solution:
Divide into simple areas
Find centroid of each area
50
100
+(70,40)1
23
4
40~70~
19634/50*
4
4
2
4
y
x
piA
Example Problem Solution
shape
1 5000 50.0 25.0 250000 125000
2 2500 66.7 66.7 166667 166667
3 3927 121.2 50.0 476032 196350
4 -1963 70.0 40.0 -137445 -78540
∑ 9463 755254 409476
mmA
AxX 8.79
9463
755254~
mmA
AyY 3.43
9463
409476~
x~ y~ Ax~ Ay~A
Solution:
Divide into simple areas
Find centroid of each area
Sum area & moment areas
Divide summations to find
composite centroid
Centroid
Point which defines geometrical center of an
object
Determined by geometry for simple shapes
and by computing moments of an element
about an axis for more complex shapes
Also referred to as First Moment Area
Centroid of Simple Shapes
2
~ wx
2
~ hy
hwA
Rectangle
The centroid of a rectangle or parallelogram lies
at the intersection of its diagonals.
h
wx
y
c
Centroid of Simple Shapes
3
~ bx
3
~ hy
hbA 2
1
Triangle
The centroid of a triangle lies at the intersection
of its median.
h
bx
y
c
Centroid of Simple Shapes
0~ x 0~ y
4
22 d
rA
Circle
The centroid of a circle lies at its center point.
x
y
c
r
Centroid of Simple Shapes
0~ x3
4~ ry
2
2
1rA
Semi-Circle
x
y
cr
Quarter-Circle
3
4~ rx
3
4~ ry
2
4
1rA
x
y
cr
Centroids of Composite Areas
Centroids of more complex shapes can
be found by the following procedure
Divide into numbered, simple areas
Find centroid of each area
Sum moment areas relative to origin
Shape Area x y xA yA
1
2
3
1
2
3
A
AxX
~
A
AyY
~
x
y
Example ProblemCentroid of Composite Area
50
100
Given:
Dhole = 50 mm
Rarc = 50 mm
Find:
Centroid of composite
+(70,40)
Example ProblemGiven:
Dhole = 50 mm
Rarc = 50 mm
Find:
Centroid of composite50
100
+(70,40)
Example Problem Solution
252/50~502/100~
5000100*50
1
1
1
y
x
A
50~2.121)*3/(50*4100~
39272/50*
3
3
2
3
y
pix
piA
7.663/5050~7.663/2*100~
25002/100*50
2
2
2
y
x
A
Given:
Dhole = 50 mm
Rarc = 50 mm
Find:
Centroid of composite
Solution:
Divide into simple areas
Find centroid of each area
50
100
+(70,40)1
23
4
40~70~
19634/50*
4
4
2
4
y
x
piA
Example Problem Solution
shape
1 5000 50.0 25.0 250000 125000
2 2500 66.7 66.7 166667 166667
3 3927 121.2 50.0 476032 196350
4 -1963 70.0 40.0 -137445 -78540
∑ 9463 755254 409476
mmA
AxX 8.79
9463
755254~
mmA
AyY 3.43
9463
409476~
x~ y~ Ax~ Ay~A
Solution:
Divide into simple areas
Find centroid of each area
Sum area & moment areas
Divide summations to find
composite centroid
Center of Gravity & Mass
Centroid (Center of Area)
Center of Gravity
Center of Mass
A
AxX
~
A
AyY
~
W
WxX
~
W
WyY
~
W
WzZ
~
m
mxX
~
m
myY
~
m
mzZ
~
In Class Exercise
Compute the y centroid of object if the radius
of the circle is 3”
3” 3”
6”
9”
y = 4.42 “
Solution
Shape Area ybar yA
rect 108 4.5 486
2-tri -27 3 -81
cir -14.14 7.73 -109.23
66.86 4.42 295.77
In Class Exercise
Solution
I-Clicker QuestionIn-Class Exercise
Indicate the number of In-Class problems
which you were able to complete.
A. 0
B. 1
C. 2
