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PROBLEM B.68
Given a homogeneous body of mass
m
and of arbitrary shape and three rectangular axes
x
,
y
, and
z
with
origin at
O
, prove that the sum
Ix
+
Iy
+
Iz
of the mass moments of inertia of the body cannot be smaller
than the similar sum computed for a sphere of the same mass and the same material centered at
O
.
Further, using the result of Problem 9.176, prove that if the body is a solid of revolution, where
x
is the
axis of revolution, its mass moment of inertia
Iy
about a transverse axis
y
cannot be smaller than 3
ma2
/10,
where
a
is the radius of the sphere of the same mass and the same material.
body sphere
xyz xyz
PROBLEM B.68 (Continued)
(ii) First note from Figure 9.28 that for a sphere
2
2
5
xyz
I
II ma
6
body
10
y
PROBLEM B.69*
The homogeneous circular cylinder shown has a mass
m
, and
the diameter
OB
of its top surface forms 45 angles with the
x
and
z
axes. (
a
) Determine the principal mass moments of
inertia of the cylinder at the origin
O
. (
b
) Compute the angles
that the principal axes of inertia at
O
form with the coordinate
axes. (
c
) Sketch the cylinder, and show the orientation of the
principal axes of inertia relative to the
x
,
y
, and
z
axes.
2
23
13 1 1 1 1
2()0
12 2
22 22 22
ma
PROBLEM B.69* (Continued)
(b) To determine the direction cosines , ,
x
yz
of each principal axis, we use two of the equations
of Equations (9.54) and (9.57).
Thus,
x
yyz
PROBLEM B.69* (Continued)
Thus, a third independent equation will be needed when the direction cosines associated with 2
K
12 x
1:K Substituting the value of 1
into Eq. (iii):
2
2
7
xz y
3:K Substituting the value of 3
into Eq. (iii):
2
2
7
xz y
PROBLEM B.69* (Continued)
K
2
: For this case, the set of equations to be solved consists of Equations (9.54a), (9.54b),
and (9.57).
PROBLEM B.70
For the component described in Problem 9.165, determine
(a) the principal mass moments of inertia at the origin, (b) the
principal axes of inertia at the origin. Sketch the body and show
the orientation of the principal axes of inertia relative to the x, y,
and z axes.
SOLUTION
(a) From the solutions to Problems 9.141 and 9.165 we have
33 32 32
(13.98800 10 )(20.55783 10 ) (13.98800 20.55783)(10 ) (0.39460 10 ) 0KK
320.58145 10 kg mK
320.6 10 kg mK
PROBLEM B.70 (Continued)
(b) To determine the direction cosines ,,
x
yz
of each principal axis, use two of the equations of
Eq. (9.54) and Eq. (9.57). Then
PROBLEM B.70 (Continued)
PROBLEM B.71*
For the component described in Problems 9.145 and 9.149,
determine (a) the principal mass moments of inertia at the
origin, (b) the principal axes of inertia at the origin. Sketch
the body and show the orientation of the principal axes of
inertia relative to the x, y, and z axes.
SOLUTION
(a) From the solutions to Problems 9.145 and 9.149, we have
332.2541 10 kg mK
3
PROBLEM B.71* (Continued)
1
0.45357( )
x
Now substitute into Eq. (9.57):
222
xyz
PROBLEM B.71* (Continued)
Simplifying
1.3405( ) ( ) 3.5222( ) 0
xy z
3
2.5795( )
x
