Problem 8.136 The thick plate consists of steel of
density D15 slug/ft3. Determine its moment of inertia
about the zaxis.
4 in
2 in 2 in
x
yy
z
Solution: Divide the object into three parts: Part (1) the rectangle
Problem 8.137 Determine the moment of inertia of the
plate in Problem 8.136 about the xaxis.
Solution: Use the same divisions of the object as in
Problem 8.136.
Part (1):
I1xaxis D1
Problem 8.138 Determine Iyand ky.
y
(1, 1)
Solution:
Problem 8.139 Determine Ixand kx.
Solution: (See figure in Problem 8.138.) dA Ddx dy,
IxDA
y2dA D1
dx x2
y2dy D1
x6dx
AD1
Problem 8.140 Determine JOand kO.
Solution: (See figure in Problem 8.138.)
Problem 8.141 Determine Ixy.
Solution: (See figure in Problem 8.138.) dA Ddx dy
674
Problem 8.142 Determine Iyand ky.
y
y = x – x2
1
–
Solution: By definition,
Problem 8.143 Determine Ixand kx.
Solution: By definition,
IxDA
y2dA,
From Problem 8.142,
AD0.4781
Problem 8.144 Determine Ixy.
Solution:
Ixy DA
xy dA,
Problem 8.145 Determine Iy0and ky0.
y
y’
1
Solution: The limits on the variable xare 0 x4. By definition,
From Problem 8.142 the area is AD2.667, from which yD0.3999 D
0.4. Similarly,
AxD4
0
xdxXX2/4
0
dy
Problem 8.146 Determine Ix0and kx0.
Solution: Using the results of Problems 8.143 and 8.145, IxD
0.6095 and yD0.4. The area moment of inertia about the centroid is
Problem 8.147 Determine Ix0y0.
676
Problem 8.148 Determine Iyand ky.
y
Solution: Divide the section into two parts: Part (1) is the upper
y2D0.08 m,
x2D0,
and kyDIy
Problem 8.149 Determine Ixand kxfor the area in
Ix1D1
y2D0.08 m,
Ix2D1
Problem 8.150 Determine Ixand kx.
x
y
mm
80
80
40
Solution: Use the results of the solutions to Problems 8.148–
xcDx1A1Cx2A2
AD0.1356 m.
The moment of inertia about the xaxis is
AD59.1mm
Problem 8.151 Determine JOand kOfor the area in
Problem 8.150.
Solution: Use the results of the solutions to Problems 8.148–
Ixc D5.028 ð105m4
and Iyc DIyD2.752 ð105m4,
AD0.0735 m
Problem 8.152 Determine Iyand ky.
y
2 ft
Solution: For a semicircle about a diameter:
678
Problem 8.153 Determine JOand kO. for the area in
Problem 8.152.
Solution: For a semicircle:
Iyy DIxx D1
8R4.
Problem 8.154 Determine Ixand kx.
y
6 ft
3 ft3 ft
Solution: Break the area into three parts: Part (1) The rectangle
with base 2aand altitude h; Part (2) The triangle on the right with
base ⊲b a⊳ and altitude h, and Part (3) The triangle on the left with
base ⊲b a⊳ and altitude h. Part (1) The area is
3D2.3333 ft,
IxD⊲y1⊳2A1CIxc1C⊲y2⊳2A2CIxc2C⊲y3⊳2A3CIxc3,
Problem 8.155 Determine Iyand kyfor the area in
Problem 8.154.
Solution: Divide the area as in the solution to Problem 8.154.
Part (1) The area is A1D2ah D24 ft2. The centroid is x1D0 and
3D2.3333 ft,
y2D2
3hD4ft,
Iyc2D1
AD1.472 ft
Problem 8.156 The moments of inertia of the area are
IxD36 m4,IyD145 m4, and Ixy D44.25 m4. Deter-
mine a set of principal axes and the principal moment
y
680
Problem 8.157 The moment of inertia of the 31-oz bat
about a perpendicular axis through point Bis 0.093 slug-
ft2. What is the bat’s moment of inertia about a perpen-
dicular axis through point A? (Point Ais the bat’s “instan-
A
14 in
12 in
B
C
Solution: The mass of the bat is mD31
16⊲32.17⊳D0.06023 slugs.
Use the parallel axis theorem to obtain the moment of inertia about the
center of mass C, and then use the parallel axis theorem to translate
to the point A.
12 2
Problem 8.158 The mass of the thin homogenous plate
is 4 kg. Determine its moment of inertia about the yaxis.
y
100 mm
140 mm
Solution: Divide the object into two parts: Part (1) is the semi-
AD9.9275 ð105kg/mm2.
For Part (1) x1Dy1D0,
Iy1D1
Problem 8.159 Determine the moment of inertia of the
plate in Problem 8.158 about the zaxis.
Solution: Use the same division of the parts and the results of the
D0.10265 kg-m2
Problem 8.160 The homogenous pyramid is of mass m.
Determine its moment of inertia about the zaxis.
x
y
Solution: The mass density is
Dm
Izaxis Dmw2
2h5h
0
z4dz D1
10 mw2
Problem 8.161 Determine the moment of inertia of the
homogenous pyramid in Problem 8.160 about the xand
yaxes.
Solution: Use the results of the solution of Problem 8.160 for the
Noting that ωDw
682
Problem 8.162 The homogenous object weighs
400 lb. Determine its moment of inertia about the xaxis.
y
9 in
z6 in
x
y
Solution: The volumes are
Problem 8.163 Determine the moments of inertia of
the object in Problem 8.162 about the yand zaxes.
Solution: See the solution of Problem 8.162. The position of the
y
y′
Problem 8.164 Determine the moment of inertia of the
14-kg flywheel about the axis L.
50 mm
100 mm
L
120 mm
70 mm
D144.3ð105mm3,
VD9.704 ð107kg/mm3.
The moment of inertia is
684