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PROBLEM B.14
Determine by direct integration the mass moment of inertia and
the radius of gyration with respect to the x axis of the paraboloid
shown, assuming that it has a uniform density and a mass m.
3
x
PROBLEM B.15
A thin rectangular plate of mass m is welded to a vertical shaft AB as
shown. Knowing that the plate forms an angle
with the y axis,
determine by direct integration the mass moment of inertia of the
plate with respect to (a) the y axis, (b) the z axis.
PROBLEM B.15 (Continued)
12
z
PROBLEM B.16*
A thin steel wire is bent into the shape shown. Denoting the mass per
unit length of the wire by m, determine by direct integration the mass
moment of inertia of the wire with respect to each of the coordinate
axes.
4
y
PROBLEM B.16* (Continued)
Alternative solution:
1/3
22 1/35/3
aa
a
2
z
PROBLEM B.17
Shown is the cross section of an idler roller. Determine its
mass moment of inertia and its radius of gyration with respect
to the axis AA. (The specific weight of bronze is 0.310 lb/in
3
;
of aluminum, 0.100 lb/in
3
; and of neoprene, 0.0452 lb/in
3
.)
22
ft/s (0.310 lb/in ) in. in
432.2 16 8 4
m
32
1.4332 10 lb s /ft
PROBLEM B.17 (Continued)
AA
PROBLEM B.18
Shown is the cross section of a molded flat-belt
pulley. Determine its mass moment of inertia and its
radius of gyration with respect to the axis AA. (The
density of brass is 8650 kg/m
3
and the density of the
fiber-reinforced polycarbonate used is 1250 kg/m
3
.)
3
17.4806 10 kg
PROBLEM B.18 (Continued)
1[(11.4134)(0.005 0.011 ) (2.8863)(0.011 0.017 )
AA
PROBLEM B.19
The machine part shown is formed by machining a conical
surface into a circular cylinder. For
1
2
,bh
determine the
mass moment of inertia and the radius of gyration of the
machine part with respect to the y axis.
50
y
m
y
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
.)
10
2
da
PROBLEM B.21 (Continued)
AA
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.
anem 2
34
AA
l
l
PROBLEM B.22 (Continued)
l
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.
SOLUTION
GG
PROBLEM B.24
Determine the mass moment of inertia of the 0.9-lb machine component shown
with respect to the axis AA.
SOLUTION
PROBLEM B.24 (Continued)
AA
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
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
22
22
() ( )
1()
12 2 2
1()
3
xm x m
IImd
bh
mb h m
mb h
2
2
22
() ( )
11
12 2 3
ym y m
IImd
b
mb m mb
2
2
22
()
11
12 2 3
zzm
II md
h
mh m mh
Thin triangular plate
We have
1
2
mV bht
and
3
,area
1
36
z
Ibh
Then
,mass ,area
3
2
1
36
1
18
zz
ItI
tbh
mh
Similarly,
2
,mass
1
18
y
Imb
Now
22
,mass , mass , mass
1()
18
xyz
III mbh
Thin semicircular plate
We have
2
2
mV at
and
4
,area ,area
8
yz
II a
(Continued)
,mass 2
49
y
