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PROBLEM B.20
Knowing that the thin hemispherical shell shown has a mass m and thickness t,
determine the mass moment of inertia and the radius of gyration of the shell with
respect to the x axis. (Hint: Consider the shell as formed by removing a hemisphere
of radius r from a hemisphere of radius r + t; then neglect the terms containing t2
and t3 and keep those terms containing t.)
PROBLEM B.21
A square hole is centered in and extends through the aluminum
machine component shown. Determine (a) the value of a for
which the mass moment of inertia of the component with respect
to the axis AA, which bisects the top surface of the hole, is
maximum, (b) the corresponding values of the mass moment of
inertia and the radius of gyration with respect to the axis AA.
(The specific weight of aluminum is 0.100 lb/in
3
.)
PROBLEM B.21 (Continued)
max
()
AA
I
occurs when 3
10
ab
Then 3
(4.2 in.) 10
a
2
(15 in.)(4.2 in.)
Lb
where 22
12 ()mm m Lb a
2
2
3
10
Then
7
10
(4.2 in.)
24 24
AA
kb
Lb
PROBLEM B.22
The cups and the arms of an anemometer are fabricated from
a material of density
. Knowing that the mass moment of
inertia of a thin, hemispherical shell of mass m and thickness
t with respect to its centroidal axis GG is
2
5/12,ma
determine (a) the mass moment of inertia of the anemometer
with respect to the axis AA, (b) the ratio of a to l for which
the centroidal moment of inertia of the cups is equal to 1
percent of the moment of inertia of the cups with respect to
the axis AA.
PROBLEM B.22 (Continued)
(b) We have cup
cup
() 0.01
()
GG
AA
I
I
55
Then 2
2 ( 2) 4(40)( 1)
2(40)
PROBLEM B.23
After a period of use, one of the blades of a shredder has been worn to
the shape shown and is of mass 0.18 kg. Knowing that the mass
moments of inertia of the blade with respect to the AA and BB axes are
2
0.320 g m
and
2
0.680 g m ,
respectively, determine (a) the location
of the centroidal axis GG, (b) the radius of gyration with respect to axis
GG.
PROBLEM B.24
Determine the mass moment of inertia of the 0.9-lb machine component shown
with respect to the axis AA.
PROBLEM B.24 (Continued)
Finally, using Figure 9.28, we have
123
()()()
AA AA AA AA
II I I
(0.038332)(0.8) (0.000399)(0.2) (0.010781)(0.6) (lb s /ft) in
AA
To the instructor:
The following formulas for the mass moment of inertia of thin plates and a half cylindrical shell are
derived at this time for use in the solutions of Problems 9.135 through 9.140.
Thin rectangular plate
2
22
12 2 2
1()
mb h m
12 2 3
Thin triangular plate
1
2
and
3
,area
1
36
z
Ibh
Then
,mass ,area
zz
ItI
Similarly,
2
,mass
1
18
y
Imb
Now
22
,mass ,mass ,mass
1()
18
xyz
III mbh
Thin semicircular plate
2
8
(Continued)
Now
III ma
2
16
1
4
ma
Now
2
,mass ,mass ,mass
1
2
xyz
III ma
22
29
and
2
,mass ,mass
yy
IImz
2
116
49
