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PROBLEM 9.169
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
PROBLEM 9.170
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
Then
222 333
()
xy x y
IImxymxymxy
2
(0.0672 kg)(0.6 m)(1.2 m) (0.0672 kg)(1.2 m)(0.6 m)
0.096768 kg m
()0
yz y z
IImyz
444 555
2
()
(0.0672 kg)(0.6 m)(1.2 m) (0.0672 kg)(1.2 m)(0.6 m)
0.096768 kg m
zx z x
IImzxmzxmzx
From the solution to Problem 9.148, we have
2
2
0.32258 kg m
0.41933 kg m
x
yz
I
II
0
0
PROBLEM 9.170 (Continued)
222
222
2
222
362
0.32258 0.41933 0.41933
777
36 23
2(0.096768) 2(0.096768) kg m
77 77
OL x x y y z z xy x y yz y z zx z x
II I I I I I
2
(0.059249 0.30808 0.034231 0.071095 0.023698) kg m
OL
I
or
2
0.354 kg m
OL
I
PROBLEM 9.171
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
333 444
()
yz y z
IImyzmyzmyz
Now
34 34 4 3
,, 0
yz
mm yy z z I
()or0
zx z x zx
IImzx I
0
PROBLEM 9.171 (Continued)
222
222
32
32
222
362
39.1721 36.2542 30.4184
777
36
2( 8.75480) 10 lb ft s
77
(7.19488 26.6357 2.48313 6.43210) 10 lb ft s
OL x x y y z z xy x y yz y z zx z x
II I I I I I
or
2
0.0427 lb ft s
OL
I
PROBLEM 9.172
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
()0
zx z x
IImzx
PROBLEM 9.172 (Continued)
222
222
32
222
362
10.3642 19.1097 10.3642
777
36 62
2(0.75911) 2(0.75911) 10 lb ft s
77 77
(1.90663 14.03978 0.8
OL x x y y z z xy x y yz y z zx z x
II I I I I I
32
4606 0.55771 0.37181) 10 lb ft s
or
32
16.61 10 lb ft s
OL
I
PROBLEM 9.173
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.
PROBLEM 9.174
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.
PROBLEM 9.174 (Continued)
