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Problem 10.105
The cross section of a symmetric
W815
wide-flange I beam has area
AD4:44 in:2
and area moment of inertia about the
x1
axis
Ix1D121 in:4
. Determine the moment
of inertia of the area about the x2axis Ix2.
Problem 10.106
(a) For the L-shaped channel shown, determine the dimension dso that the
origin of the coordinate system, point
O
, is positioned at the centroid of
the area.
(b) Determine the area moment of inertia about the xaxis.
12.4 in:/.4 in:/3C.2 in:/2.4 in:/.4 in:/
1
12.3:5 in:/.3:5 in:/30:5 in:C3:5 in:
22
.3:5 in:/.3:5 in:/ D10:81 in:4;(2)
Problem 10.107
The cross section of the bar shown is symmetric about the xaxis.
(a)
Determine
d
so that the origin of the coordinate system, point
O
, is
positioned at the centroid of the area.
(b) Determine the area moment of inertia about the xaxis.
(c) Determine the area moment of inertia about the yaxis.
Solution
Part (b)
Part (c)
We could follow the usual procedure of determining the moment of inertia about the
y
axis by
using
IyDP.Iy0Cd2A/
where
d
is the shift distance between the
y0
axis for a composite shape and
Problem 10.108
For the uniform solid cone with height
h
and radius
r
, use integration to
determine the mass moment of inertia indicated, expressing your answer in
terms of the mass mof the cone. I´.
Problem 10.109
For the uniform solid cone with height
h
and radius
r
, use integration to
determine the mass moment of inertia indicated, expressing your answer in
terms of the mass mof the cone. Ix.
Problem 10.110
The truncated cone shown has
2000 kg=m3
density. Report your answers for
the problems that follow using kg and mm units.
Use integration to determine the mass moment of inertia about the xaxis.
Problem 10.111
The truncated cone shown has
2000 kg=m3
density. Report your answers for
the problems that follow using kg and mm units.
Determine the mass moment of inertia about the
x
axis using composite
shapes.
Solution
Using the Table of Properties of Solids (inside back cover of book),
Problem 10.112
The truncated cone shown has
2000 kg=m3
density. Report your answers for
the problems that follow using kg and mm units.
Use integration to determine the mass moment of inertia about the ´axis.
Problem 10.113
The truncated cone shown has
2000 kg=m3
density. Report your answers for
the problems that follow using kg and mm units.
Determine the mass moment of inertia about the
´
axis using composite
shapes.
Solution
We use the composite shapes shown, where shape 2 has negative mass.
Using the Table of Properties of Solids (inside back cover of book) and the parallel axis theorem,
I´D3
m1
Problem 10.114
The object shown has
8000 kg=m3
density and has a conical hole. Use integra-
tion to determine the mass moment of inertia about the xaxis.
Problem 10.115
The object shown has
8000 kg=m3
density and has a conical hole. Use integra-
tion to determine the mass moment of inertia about the yaxis.
Problem 10.116
The solid of revolution consists of materials with densities 1and 2.
Fully set up the integrals, including limits of integration, that will yield the
mass moment of inertia about the
x
axis. You are not required to evaluate the
integrals.
Problem 10.117
The solid of revolution consists of materials with densities 1and 2.
Fully set up the integrals, including limits of integration, that will yield the
mass moment of inertia about the
y
axis. You are not required to evaluate the
integrals.
Problem 10.118
A solid of revolution is produced by revolving the area shown 360ıaround the
yaxis. Use integration to determine the mass moment of inertia about the axis
of revolution assuming the solid has uniform density of 2000 kg=m3.
Problem 10.119
A solid of revolution is produced by revolving the area shown 360
ı
around the
y
axis.
Use integration to determine the mass moment of inertia about the axis of revolution
assuming the solid has uniform density of 2000 kg=m3.
Problem 10.120
A beam is constructed of three identical pieces of wood, each piece having
40 mm
by
160 mm
by
500 mm
dimensions, and
2kg
mass that is uniformly
distributed. The cross section of the beam is symmetric about the
x
and
y
axes.
(a)
Determine the area moment of inertia for the cross section about the
x
axis.
(b) Determine the mass moment of inertia for the beam about the xaxis.
12.40 mm/.160 mm/3C21
12.160 mm/.40 mm/3C.100 mm/2.160 mm/.40 mm/(1)
)IxD1:43 108mm4:(2)
Part (b) The mass moment of inertia about the xaxis is
Problem 10.121
A beam is constructed of three identical pieces of wood, each piece having
40 mm
by
160 mm
by
500 mm
dimensions, and
2kg
mass that is uniformly
distributed. The cross section of the beam is symmetric about the
x
and
y
axes.
(a)
Determine the area moment of inertia for the cross section about the
y
axis.
(b) Determine the mass moment of inertia for the beam about the yaxis.
12.160 mm/.40 mm/3C21
12.40 mm/.160 mm/3(1)
)IyD2:82 107mm4:(2)
Part (b) The mass moment of inertia about the yaxis is
Problem 10.122
The object shown is made of thin plate with
0:02 lb=in2
specific weight. Deter-
mine the mass moment of inertia about the xaxis.
Problem 10.123
The object shown is made of thin plate with
0:02 lb=in2
specific weight. Deter-
mine the mass moment of inertia about the yaxis.
Problem 10.124
The object shown is made of thin plate with
0:02 lb=in2
specific weight. Deter-
mine the mass moment of inertia about the ´axis.
