1010
SOLUTION
y 1
n. The area of the differential element parallel
Moment of Inertia. Perform the integration,
Ix=
L
A
y2dA =
Lb
0
y2
a
a–a
b
1
n
y1
n
b
dy
Lb
b
10–1.
Determine the moment of inertia about the x axis. y
x
a
b
y xn
an
b
1011
10–2.
Determine the moment of inertia about the y axis.
SOLUTION
Differential Element. The area of the differential element parallel to the y axis
Moment of Inertia. Perform the integration,
y
x
a
b
y xn
an
b
1012
10–3.
Determine the moment of inertia for the shaded area about
the x axis.
SOLUTION
Moment of Inertia. Perform the integration,
Ix=
L
A
y2dA =
L200 mm
0
y2
c
2
2
50y1
2dy
d
100 mm
200 mm
y
y x2
1
50
1013
Ans:
SOLUTION
Differential Element. Here x
=2
50
y
1
2. The moment of inertia of the differential
element parallel to x axis shown in Fig. a about y axis is
*10–4.
Determine the moment of inertia for the shaded area about
the y axis.
100 mm
200 mm
y
x
y x2
1
50
1014
Ans:
SOLUTION
Differential Element. The area of the differential element parallel to the y axis
shown shaded in Fig. a is
dA =ydx
. The moment of inertia of this element about
the x axis is
Moment of Inertia. Perform the integration.
Ix=
L
dIx=
1
3x3
2dx
10–5.
Determine the moment of inertia for the shaded area about
the x axis.
y
x
y x1/2
1 m
1 m
Ans:
10–6.
Determine the moment of inertia for the shaded area about
the y axis.
SOLUTION
Differential Element. The area of the differential element parallel to the y axis
y
x
y x1/2
1 m
1 m
1016
I
x=
0.267 m
10–7.
Determine the moment of inertia for the shaded area about
the x axis.
SOLUTION
Differential Element. Here
x=2(1 –y2)
. The area of the differential element
parallel to the x axis shown shaded in Fig. a is
dA =xdy
=2(1 –y2)dy
.
y
x
2 m
1 m
y2 1 0.5x
1017
Ans:
*10–8.
Determine the moment of inertia for the shaded area about
the y axis.
SOLUTION
Differential Element. Here
x=2(1 –y2)
. The moment of inertia of the differential
element parallel to the x axis shown shaded in Fig. a about the y axis is
dI
y=
dI
y‘
+dAx
∼2
Moment of Inertia. Perform the integration,
y
x
2 m
1 m
y2 1 0.5x
1018
10–9.
SOLUTION
(a) Differential Element:The area of the differential element parallel to yaxis is
The moment of inertia of this element about xaxis is
Moment of Inertia:Performing the integration, we have
(b) Differential Element:Here,The area of the differential
element parallel to xaxis is
dA =2xdy =2225 –10ydy.
x=225 –10y.
dIx=dIx¿+dAy
‘2
dA =ydx.
Determine the moment of inertia of the area about the x
axis. Solve the problem in two ways,using rectangular
differential elements: (a) having a thickness dx and
(b) having a thickness of dy.
y
x
y=2.5 – 0.1x2
5ft
2.5 ft
Ans:
10–10.
Determine the moment of inertia for the shaded area about
the xaxis.
SOLUTION
Also,
dA =(b–x)dy =(b–
b
h2y2)dy
dI
x=
1
3y3dx
b
x
y
y
2
—x
h
h
2
b
1020
10–11.
Determine the moment of inertia for the shaded area about
the x axis.
y x3
y
8 m
4 m
x
1
8
1021
*10–12.
Determine the moment of inertia for the shaded area about
the y axis.
SOLUTION
Differential Element. The area of the differential element parallel to the y axis,
y x3
y
8 m
4 m
x
1
8
1022
10–13.
Determine the moment of inertia about the x axis.
SOLUTION
Moment of Inertia. Perform the integration.
y
x
2 m
1 m
x2 4y2 4
1023
10–14.
Determine the moment of inertia about the y axis.
SOLUTION
Differential Element. Here, y=
1
22
4–x2. The area of the differential element
parallel to the y axis shown shaded in Fig. a is dA =ydx =
1
22
4–x2dx
y
x
2 m
1 m
x2 4y2 4
10–15.
Determine the moment of inertia for the shaded area about
the x axis.
SOLUTION
Differential Element. The area of the differential element parallel with the x axis
shown shaded in Fig. a is
dA =x dy =y2 dy
.
y
x
16 in.
4 in.
y2 x
1025
*10–16.
Determine the moment of inertia for the shaded area about
the y axis.
SOLUTION
Differential Element. The moment of inertia of the differential element parallel to
the x axis shown shaded in Fig. a about the y axis is
dI
y=
dI
y
+dAx
∼2
Moment of Inertia. Perform the integration,
y
x
16 in.
4 in.
y2 x
Ans:
1026
10–17.
Determine the moment of inertia for the shaded area about
the x axis.
SOLUTION
Differential Element. The moment of inertia of the differential element parallel to
the y axis shown shaded in Fig. a about the x axis is
dIx=dIx′+dAy
∼2
y
x
h
b
y x3
h
b3
Ans:
10–18.
Determine the moment of inertia for the shaded area about
the y axis.
SOLUTION
Differential Element. The area of the differential element parallel to the y axis
y
x
h
b
y x3
h
b3
1028
10–19.
Determine the moment of inertia for the shaded area about
the x axis.
SOLUTION
Differential Element. Here y
=
(1
–
x)
1
2. The moment of inertia of the
y
y2 1 x
x
1 m
1 m
1 m
1029
*10–20.
Determine the moment of inertia for the shaded area about
the y axis.
SOLUTION
y
y2 1 x
x
1 m
1 m
1 m
Ans: