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Problem 10.36
Determine the area moments of inertia Ixand Iy.
12.30 mm/.30 mm/3C.15 mm/2.30 mm/.30 mm/
C1
36.30 mm/.10 mm/3C30 mm C10 mm
321
2.30 mm/.10 mm/
Note that in Eq. (1), for the quarter-circular area (i.e., the last term), the parallel axis shift is not needed
12.30 mm/.30 mm/3C.15 mm/2.30 mm/.30 mm/
C1
36.10 mm/.30 mm/3C30 mm 30 mm
321
2.10 mm/.30 mm/
Problem 10.37
Determine the area moments of inertia Ixand Iy.
Solution
We use the composite shapes shown with the parallel axis theorem.
Note that in Eq. (1), for the triangular area, the parallel axis shift is not needed since the expression
1
, which is obtained from the Table of Properties of Lines and Areas from the inside
36.0:2 in:/.0:2 in:/31
20:2 in:C0:2 in:
32
Problem 10.38
Determine the area moments of inertia Ixand Iy.
12.0:5 in:/.4 in:/3C.2 in:/2.0:5 in:/.4 in:/
C1
12.1 in:/.1 in:/3C.3 in:/2.1 in:/.1 in:/
12.4 in:/.0:5 in:/3C.0:25 in:/2.4 in:/.0:5 in:/
C1
12.1 in:/.1 in:/3C.1 in:/2.1 in:/.1 in:/
Problem 10.39
Determine the area moments of inertia Ixand Iy.
12.100 mm/.60 mm/3C
8.30 mm/41
12.88 mm/.36 mm/3
8.18 mm/4(1)
12.60 mm/.100 mm/3C.50 mm/2.60 mm/.100 mm/
12.36 mm/.88 mm/312 mm C88 mm
22
Problem 10.40
(a) Determine Ix1.
(b)
Use the result of Part (a) with the parallel axis theorem to determine
Ix2
.
Hint: See the common pitfall discussed on p. 586.
12.6 mm/.9 mm/3C.4:5 mm/2.6 mm/.9 mm/C1
12.6 mm/.9 mm/3D1823 mm4
2.6 mm/.9 mm/
With the above result, the location of the centroidal
x0y0
coordinate system is known,
as shown. Using the result for
Ix1
from Part (a), we use the parallel axis theorem
Problem 10.41
A circular hole is to be drilled through the side of a beam as shown. Describe
where the hole should be positioned so that the area moment of inertia about the
centroidal
x0
axis, at the cross section that passes through the hole, is reduced
as little as possible.
Problem 10.42
The cross section of the bar shown is symmetric about either the xor yaxis.
(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.
Hint: If you carried out some of the exercises from Section 7.1, beginning on
p. 445, you may have already solved Part (a) of Probs. 10.46 and 10.47.
12.4 in:/.4 in:/3C.2 in:/2.4 in:/.4 in:/
C1
12.16 in:/.2 in:/3C.1 in:/2.16 in:/.2 in:/
Problem 10.43
The cross section of the bar shown is symmetric about either the xor yaxis.
(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.
Hint: If you carried out some of the exercises from Section 7.1, beginning on p. 445,
you may have already solved Part (a) of Probs. 10.46 and 10.47.
12.15 mm/.20 mm/3C.1 mm/2.15 mm/.20 mm/
1
12.5 mm/.10 mm/3.6 mm/2.5 mm/.10 mm/D8083 mm4
Problem 10.44
The cross section of the bar shown is symmetric about either the xor yaxis.
(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.
Hint: If you carried out some of the exercises from Section 7.1, beginning on p. 445,
you may have already solved Part (a) of Probs. 10.46 and 10.47.
12.2 mm/.8 mm/3C1
12.4 mm/2.2 mm/3)IxD88:0 mm4:(2)
12.8 mm/.2 mm/3C.1 mm/2.8 mm/.2 mm/C1
12.2 mm/.4 mm/3C.2 mm/2.2 mm/.4 mm/
Problem 10.45
The cross section of the bar shown is symmetric about either the xor yaxis.
(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.
Hint: If you carried out some of the exercises from Section 7.1, beginning on p. 445,
you may have already solved Part (a) of Probs. 10.46 and 10.47.
12.2 in:/.4 in:/3C.1:5 in:/2.2 in:/.4 in:/
C1
12.4 in:/.2 in:/3C.1:5 in:/2.4 in:/.2 in:/ )IxD49:3 in:4(2)
Part (c)
Problem 10.46
The cross section of the bar shown is symmetric about either the xor yaxis.
(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.
Hint: If you carried out some of the exercises from Section 7.1, beginning on
p. 445, you may have already solved Part (a) of Probs. 10.46 and 10.47.
Solution
Part (b)
IxD1
12.40 mm/.20 mm/3C.25 mm/2.40 mm/.20 mm/
Part (c)
Problem 10.47
The cross section of the bar shown is symmetric about either the xor yaxis.
(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.
Hint: If you carried out some of the exercises from Section 7.1, beginning on
p. 445, you may have already solved Part (a) of Probs. 10.46 and 10.47.
Solution
Part (b)
Part (c)
Problem 10.48
The uniform slender rod has mass
m
and length
l
. Use integration to show that
the mass moment of inertia about the yaxis is IyDml2=12.
Problem 10.49
The uniform plate has thickness t, radius r, and density .
Use integration with a thin cylindrical shell mass element to determine the mass
moment of inertia about the
x
axis. Express your answer in terms of the mass
m
of the
plate.
Problem 10.50
The uniform plate has thickness t, radius r, and density .
Assuming
tr
, use integration with a mass element parallel to the
y
axis (as
in Example 10.2 on p. 580) to determine the mass moment of inertia about the
y
axis.
Express your answer in terms of the mass mof the plate.
Solution
Using integration with the mass element shown at the right, the mass moment
of inertia about the yaxis is
Problem 10.51
The uniform cylinder has length L, radius R, and density .
Use integration with a thin cylindrical shell mass element to determine the
mass moment of inertia about the
x
axis. Express your answer in terms of the
mass mof the cylinder.
Problem 10.52
The uniform cylinder has length L, radius R, and density .
Use integration with a thin disk mass element to determine the mass moment
of inertia about the
y
axis. Express your answer in terms of the mass
m
of the
cylinder.
ydm (3)
D1
4R2dxR2Cx2R2dx: (4)
L
Z
R4
L
Z
L
L
Problem 10.53
For the uniform solid cone with length
L
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 10.54
For the uniform solid cone with length
L
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´.
