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PROBLEM B.56
Determine the mass moment of inertia of the steel fixture of
Problems 9.145 and 9.149 with respect to the axis through
the origin that forms equal angles with the x, y, and z axes.
SOLUTION
From the solutions to Problems 9.145 and 9.149, we have
OL
PROBLEM B.57
The thin bent plate shown is of uniform density and weight W. Determine its
mass moment of inertia with respect to the line joining the origin O and
Point A.
PROBLEM B.57 (Continued)
Now observe that the centroidal products of inertia, ,,
II
and ,
I
36 4
g
2
OA
g
PROBLEM B.58
A piece of sheet steel of thickness t and specific weight
is cut and
bent into the machine component shown. Determine the mass
moment of inertia of the component with respect to the joining the
origin O and Point A.
SOLUTION
1
22
2
22
gg
t
mta a
gg
Using Figure 9.28 for component 1 and the equations derived above (following the solution to Problem
9.134) for a semicircular plate for component 2, we have
12
22 2 222
() ()
14[(2)(2)]4( )
18.91335
xx x
II I
tt
g
PROBLEM B.58 (Continued)
2
222 2
22
4
24 2 3
9
1 1 16 16
41 4
324
99
12.00922
gg
tt
aaa a
gg
ta
g
Now observe that the centroidal products of inertia, ,,
x
yyz
II
and ,
zx
I
of both components are zero
because of symmetry. Also 10.x
PROBLEM B.58 (Continued)
Substituting into Eq. (9.46)
222
22
222
OA x x y y z z xy x y yz y z zx z x
II I I I I I
OA
g
PROBLEM B.59
Determine the mass moment of inertia of the machine
component of Problems 9.136 and 9.155 with respect to
the axis through the origin characterized by the unit vector
(4 8 )/9.
λijk
SOLUTION
From the solutions to Problems 9.136 and 9.155. We have
32
32
12.8950 10 kg m
94.0266 10 kg m
xy
yz
zx
I
I
Substituting into Eq. (9.46)
OL
PROBLEM B.60
For the wire figure of Problem 9.148, determine the mass
moment of inertia of the figure with respect to the axis
through the origin characterized by the unit vector
(3 6 2)/7.
ijk
SOLUTION
0.0672 kg
Now observe that the centroidal products of inertia,
,,and,
x
yyz zx
II I
for each
component are zero because of symmetry.
Also
PROBLEM B.60 (Continued)
OL
PROBLEM B.61
For the wire figure of Problem 9.147, determine the mass moment of inertia of
the figure with respect to the axis through the origin characterized by the unit
vector
(3 6 2)/7.
ijk
SOLUTION
First compute the mass of each component. We have
ST
ST
mV AL
g
2
1.9453 lb s /ft
Now observe that the centroidal products of inertia,
,,and,
x
yyz zx
II I
for each component are zero
because of symmetry.
PROBLEM B.61 (Continued)
From the solution to Problem 9.147, we have
32
32
39.1721 10 lb ft s
x
I
OL
PROBLEM B.62
For the wire figure of Problem 9.146, determine the mass moment
of inertia of the figure with respect to the axis through the origin
characterized by the unit vector
(3 6 2)/7.
ijk
SOLUTION
First compute the mass of each component. We have
AL
1(/)
W
mWLL
32
0.6832 10 lb ft/s
Now observe that the centroidal products of inertia,
,,and,
x
yyz zx
II I
of each component are zero
because of symmetry. Also
zx z x
PROBLEM B.62 (Continued)
From the solution to Problem 9.146, we have
32
32
10.3642 10 lb ft s
19.1097 10 lb ft s
xz
y
II
I
Substituting into Eq. (9.46)
222
222
222
362
10.3642 19.1097 10.3642
777
OL x x y y z z xy x y yz y z zx z x
II I I I I I
OL
0
PROBLEM B.63
For the homogeneous circular cylinder shown, of radius
a
and length
L
,
determine the value of the ratio
a/L
for which the ellipsoid of inertia of
the cylinder is a sphere when computed (
a
) at the centroid of the cylinder,
(
b
) at Point
A
.
SOLUTION
212
3
L
(
b
) Using Figure 9.28 and the parallel-axis theorem, we have
2
1
2448
12
L
PROBLEM B.64
For the rectangular prism shown, determine the values of the
ratios
b
/
a
and
c
/
a
so that the ellipsoid of inertia of the prism is a
sphere when computed (
a
) at Point
A
, (
b
) at Point
B
.
SOLUTION
(
a
) Using Figure 9.28 and the parallel-axis theorem, we have
2
22
22
2
22 22
12
1()
12 2
1(4 )
12
11
() (4)
12 2 12
x
y
z
a
Imacm
ma c
a
I mab m mab
Now observe that symmetry implies
a
PROBLEM B.64 (Continued)
(b) Using Figure 9.28 and the parallel-axis theorem, we have at Point B
2
22 2 2
2
22 2 2
22
11
() (4)
12 12
2
11
12 2 12
1()
12
x
z
c
I
mb c m mb c
c
Imab
Now observe that symmetry implies
2
a
PROBLEM B.65
For the right circular cone of Sample Problem 9.11, determine the
value of the ratio
a/h
for which the ellipsoid of inertia of the cone is a
sphere when computed (
a
) at the apex of the cone, (
b
) at the center of
the base of the cone.
10 20
3
h
PROBLEM B.66
Given an arbitrary body and three rectangular axes x, y, and z, prove that the mass moment of inertia of
the body with respect to any one of the three axes cannot be larger than the sum of the mass moments of
inertia of the body with respect to the other two axes. That is, prove that the inequality
x
yz
I
II
and
the two similar inequalities are satisfied. Further, prove that 1
2
yx
IIif the body is a homogeneous solid
of revolution, where x is the axis of revolution and y is a transverse axis.
2
yx
PROBLEM B.67
Consider a cube of mass
m
and side
a
. (
a
) Show that the ellipsoid of inertia at the center of the cube is a
sphere, and use this property to determine the moment of inertia of the cube with respect to one of its
diagonals. (
b
) Show that the ellipsoid of inertia at one of the corners of the cube is an ellipsoid of
revolution, and determine the principal moments of inertia of the cube at that point.
SOLUTION
PROBLEM B.67 (Continued)
First note that at corner A
2
2
3
xyz
I
II ma
2
23
11
12
kk ma
