PROBLEM 9.111
Using Mohr’s circle, prove that the expression
2
x
yx
y
I
II
′′
′
′
−
is
independent of the orientation of the
x
′
and
y
′
axes, where
x
I
′
,
y
I
′
, and
x y
I
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.
SOLUTION
First observe that for a given area
A
and origin
O
of a rectangular
coordinate system, the values of
ave
I
and
R
are the same for all
orientations of the coordinate axes. Shown below is a Mohr’s circle, with
the moments of inertia,
x
I
′
and
,
y
I
′
and the product of inertia.
,
x y
I
having been computed for an arbitrary orientation of the
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 Spring '08
 Jenkins
 Statics, Cartesian Coordinate System, Right angle, xy axes, coordinate axes

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