PROBLEM 9.175
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.
PROBLEM 9.176
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.
I
PROBLEM 9.177
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.
PROBLEM 9.177 (Continued)
PROBLEM 9.178
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 I
x
+ I
y
+ I
z
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 I
y
about a transverse axis y cannot be smaller than 3ma
2
/10,
where a is the radius of the sphere of the same mass and the same material.
PROBLEM 9.178 (Continued)
PROBLEM 9.179*
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
1
2
22
zx z x
aa
I I mz x m ma
Substituting into Equation (9.56)
322
13 3 13
12 2 12
KmaK
22
222
13 3 3 13 13 13 1 1 1 ()
12 2 2 12 12 12 2
22 22 ma K
22
223
13 3 13 13 1 3 1
12 2 12 12 2 2
22
13 1 1 1 1
2()0
12 2
22 22 22 ma
12 2 12
PROBLEM 9.179* (Continued)
xz y
PROBLEM 9.179* (Continued)
22 2
22 2
() () ( ) 1
xy x
or
22
11
() and()
22
xz
222
() 45.0()90.0()135.0
xyz
(c) Principal axes 1 and 3 lie in the vertical plane of symmetry passing through Points O and B.
Principal axis 2 lies in the xz plane.