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MECHANICS. 297
Polar Moment of Inertia.
The polar moment of inertia is a mathematical expression,
used especially when calculating the torsional strength of
beams, shafting, etc. It is very frequently denoted by the letter
J. The polar moment of inertia is the sum of the products of
each elementary area of the sur-
face multiplied by the square of
its distance from the center of
gravity of surface. Suppose (in
Fig. 10) that the area is divided
into circular rings, as #, d, c, d,e,
f, g, h, z, j, k, /, m, n, o,fi, and the
area of each ring multiplied by
the square of its distance from
the center, c ; the sum of all these
products is the polar moment of
inertia. The moment, calculated
this way, will always be a trifle
too small, but the smaller each
ring is taken the more correct the
result will be. If each ring could
be taken infinitely small the result would be correct.
The polar moment of inertia is equal to the square of the
radius of gyration about the geometrical center of the shaft, mul-
tiplied by the area of cross-section of the shaft; therefore, for a
round, solid shaft (as the section shown in Fig. 10), the polar
moment of inertia is always expressed by the formula
:
(Radius)4
X ^ ^ (Diameter)4
X k
2
’
32
For a hollow, round shaft, the polar moment of inertia is
expressed by the formula,
D = Outside diameter. d = Inside diameter.
The fundamental principle for the polar moment of inertia
for any shape of section is that, if two rectangular moments of
inertia are taken, one being the least rectangular moment of
inertia, about an axis passing through the center of gravity, and
the other, the least rectangular moment, about an axis perpendic-
ular to the first one, also through the center of gravity, the sum
of those two rectangular moments is equal to the polar moment.
In Fig. 10, the rectangular moment of inertia about the axis
x y will be
(diameter)4
X
J
64
the axis x> y will also be
(diameter)4
X
64
(diameter)4
X 7r
and the rectangular moment about
; thus the polar moment
will be
32
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