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PROBLEM B.45
A section of sheet steel 2 mm thick is cut and bent into the
machine component shown. Knowing that the density of
steel is 7850 kg/m
3
, determine the mass products of inertia
I
xy
, I
yz
, and I
zx
of the component.
SOLUTION
First compute the mass of each component. We have
12
2
40.195
m 0.082761 m
3
y
PROBLEM B.45 (Continued)
Finally,
zx
PROBLEM B.46
A section of sheet steel 2 mm thick is cut and bent into the machine
component shown. Knowing that the density of steel is 7850 kg/m
3
,
determine the mass products of inertia I
xy
, I
yz
, and I
zx
of the
component.
1111
zx
PROBLEM B.46 (Continued)
1
zx
PROBLEM B.47
The figure shown is formed of 1.5-mm-diameter
aluminum wire. Knowing that the density of aluminum is
2800 kg/m
3
, determine the mass products of inertia Ixy, Iyz,
and Izx of the wire figure.
3
32
36
3
0.89064 10 kg
(2800 kg/m ) (0.0015 m) (0.3 m)
4
1.48440 10 kg
mm
PROBLEM B.47 (Continued)
Now observe that the centroidal products of inertia, ,,and,
x
yyz zx
II I
of each component are zero
because of symmetry.
m, kg ,m
x
,my ,mz 2
,kg mmx y
2
,kg mmy z
2
,kg mmz x
zx z x
zx
PROBLEM B.48
Thin aluminum wire of uniform diameter is used to form the figure shown.
Denoting by m the mass per unit length of the wire, determine the mass
products of inertia Ixy, Iyz, and Izxof the wire figure.
3
sin cos
uv
ma d
PROBLEM B.48 (Continued)
/2 3
2
2
zx
PROBLEM B.49
Brass wire with a weight per unit length w is used to form the figure shown.
Determine the mass products of inertia Ixy, Iyz, and Izx of the wire figure.
g
g
g
PROBLEM B.49 (Continued)
3(1 5 )
w
zx
g
0
PROBLEM B.50
Brass wire with a weight per unit length w is used to form the figure
shown. Determine the mass products of inertia I
xy
, I
yz
, and I
zx
of the wire
figure.
g
2
g
4
g
PROBLEM B.50 (Continued)
4
zx
g
PROBLEM B.51
Complete the derivation of Eqs. (9.47), which express the parallel-axis theorem for mass products of
inertia.
y
z
PROBLEM B.52
For the homogeneous tetrahedron of mass m shown, (a) determine by
direct integration the mass product of inertia I
zx
, (b) deduce I
yz
and I
xy
from
the result obtained in part a.
20
zx
PROBLEM B.52 (Continued)
0
45
bb
20
zx
PROBLEM B.52 (Continued)
Alternative solution for part a:
The equation of the included face of the tetrahedron is
xyz
20
zx
PROBLEM B.53
The homogeneous circular cone shown has a mass m.
Determine the mass moment of inertia of the cone with respect
to the line joining the origin O and Point A.
60 ma 2
OA
PROBLEM B.54
The homogeneous circular cylinder shown has a mass m. Determine the mass
moment of inertia of the cylinder with respect to the line joining the origin O and
Point A that is located on the perimeter of the top surface of the cylinder.
OA
PROBLEM B.55
Shown is the machine element of Problem 9.141. Determine its
mass moment of inertia with respect to the line joining the
origin O and Point A.
32
32
[0.59188 kg (0.04 m)(0.03 m)] [0.39458 kg ( 0.04 m)( 0.02 m)]
(0.71026 0.31566) 10 kg m
0.39460 10 kg m
xy x y
PROBLEM B.55 (Continued)
From the solution to Problem 9.141, we have
32
32
13.98800 10 kg m
x
I
OA
