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PROBLEM 9.106* (Continued)
min
PROBLEM 9.107
It is known that for a given area
y
I
48
10
6
mm
4
and
x
y
I
–20
10
6
mm
4
, where the
x
and
y
axes are
rectangular centroidal axes. If the axis corresponding to the maximum product of inertia is obtained by
rotating the
x
axis 67.5
counterclockwise about
C
, use Mohr’s circle to determine (
a
) the moment of inertia
x
I
of the area, (
b
) the principal centroidal moments of inertia.
PROBLEM 9.108
Using Mohr’s circle, show that for any regular polygon (such as a pentagon) (
a
) the moment of inertia
with respect to every axis through the centroid is the same, (
b
) the product of inertia with respect to every
pair of rectangular axes through the centroid is zero.
PROBLEM 9.109
Using Mohr’s circle, prove that the expression 2
x
yxy
I
II
is independent of the orientation of the
x
and
y
axes, where
I
x
,
I
y
, and
I
x
y
represent the moments and product of inertia, respectively, of a given area
with respect to a pair of rectangular axes
x
and
y
through a given Point
O
. Also show that the given
expression is equal to the square of the length of the tangent drawn from the origin of the coordinate
system to Mohr’s circle.
PROBLEM 9.110
Using the invariance property established in the preceding problem, express
the product of inertia
I
xy
of an area
A
with respect to a pair of rectangular axes
through
O
in terms of the moments of inertia
I
x
and
I
y
of
A
and the principal
moments of inertia
I
min
and
I
max
of
A
about
O
. Use the formula obtained to
calculate the product of inertia
I
xy
of the L3
2
1
4
-in. angle cross section
shown in Figure 9.13A, knowing that its maximum moment of inertia is 1.257
in
4
.
PROBLEM 9.111
A thin plate of mass m is cut in the shape of an equilateral triangle of side a.
Determine the mass moment of inertia of the plate with respect to (a) the
centroidal axes AA and BB, (b) the centroidal axis CC that is perpendicular
to the plate.
PROBLEM 9.112
Determine the mass moment of inertia of a ring of mass
m
, cut from a
thin uniform plate, with respect to (
a
) the axis
AA
, (
b
) the centroidal axis
CC
that is perpendicular to the plane of the ring.
PROBLEM 9.113
A thin, semielliptical plate has a mass
m
. Determine the mass moment of
inertia of the plate with respect to (
a
) the centroidal axis
BB
, (
b
) the
centroidal axis
CC
that is perpendicular to the plate.
PROBLEM 9.114
The parabolic spandrel shown was cut from a thin, uniform plate.
Denoting the mass of the spandrel by
m
, determine its mass
moment of inertia with respect to (
a
) the axis
BB
, (
b
) the axis
DD
that is perpendicular to the spandrel. (
Hint:
See Sample
Problem 9.3.)
PROBLEM 9.114 (Continued)
