determine the moment of inertia about the centroidal axis of the area shown below.
Answers
Answer:
From the figure,
The length
A
B
and
E
F
is;
A
B
=
E
F
=
2
i
n
The length
D
C
and
H
G
is;
D
C
=
H
G
=
2
i
n
The length
A
D
and
E
H
is;
A
D
=
E
H
=
4
i
n
The length
D
G
and
I
J
is;
D
G
=
I
J
=
10
i
n
The length
C
H
is;
C
H
=
6
i
n
The length
D
I
and
G
J
is;
D
I
=
G
J
=
2
i
n
Find the area of rectangle 1.
A
1
=
A
B
×
A
D
=
2
i
n
×
4
i
n
=
8
i
n
2
Both rectangles 1 and 2 are geometrically similar. So, the area of rectangle 2 will be equal to the area of rectangle 1.
A
1
=
A
2
=
8
i
n
2
Find the area of rectangle 3.
A
3
=
D
G
×
D
I
=
10
i
n
×
2
i
n
=
20
i
n
2
Find the centroid of the composite area from the base along
y
-axis.
¯¯¯
y
=
A
1
y
1
+
A
2
y
2
+
A
3
y
3
A
1
+
A
2
+
A
3
¯¯¯
y
=
A
1
(
D
I
+
A
D
2
)
+
A
2
(
G
J
+
E
H
2
)
+
A
3
(
D
I
2
)
A
1
+
A
2
+
A
3
.
.
.
(
1
)
Substitute the given value in equation (1).
¯¯¯
y
=
(
8
i
n
2
)
(
2
i
n
+
4
i
n
2
)
+
(
8
i
n
2
)
(
2
i
n
+
4
i
n
2
)
+
(
20
i
n
2
)
(
2
i
n
2
)
8
i
n
2
+
8
i
n
2
+
20
i
n
2
=
(
8
i
n
2
)
(
4
i
n
)
+
(
8
i
n
2
)
(
4
i
n
)
+
(
20
i
n
2
)
(
1
i
n
)
36
i
n
2
≈
2.3
i
n
Find the moment of inertia of rectangle 1 along the centroidal axis by using the parallel axis theorem.
I
1
x
=
D
C
×
A
D
3
12
+
A
1
(
(
D
I
+
A
D
2
)
−
¯¯¯
y
)
2
=
(
2
i
n
)
×
(
4
i
n
)
3
12
+
(
8
i
n
2
)
(
(
2
i
n
+
4
i
n
2
)
−
2
.3
i
n
)
2
≈
33.8
i
n
4
Find the moment of inertia of rectangle 2 along the centroidal axis by using the parallel axis theorem.
I
2
x
=
H
G
×
E
H
3
12
+
A
2
(
(
G
J
+
E
H
2
)
−
¯¯¯
y
)
2
=
(
2
i
n
)
×
(
4
i
n
)
3
12
+
(
8
i
n
2
)
(
(
2
i
n
+
4
i
n
2
)
−
2
.3
i
n
)
2
≈
33.8
i
n
4
Find the moment of inertia of rectangle 3 along the centroidal axis by using the parallel axis theorem.
I
3
x
=
I
J
×
D
I
3
12
+
A
3
(
¯¯¯
y
−
D
I
2
)
2
=
(
10
i
n
)
×
(
2
i
n
)
3
12
+
(
20
i
n
2
)
(
2
.3
i
n
−
2
i
n
2
)
2
≈
40.5
i
n
4
Find the moment of inertia of the composite area about the centroidal axis.
I
X
X
=
I
1
x
+
I
2
x
+
I
3
x
=
33.8
i
n
4
+
33.8
i
n
4
+
40.5
i
n
4
=
108.1
i
n
4
So, the moment of inertia of the composite area is
108.1
i
n
4
.