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MECHANICS. 293
The force is applied to more advantage in
Fig. G than in Fig. 5. As a rule, the force should
always be applied so as to act at right angles to
the lever.
Radius of Gyration.
The radius of gyration of a rotating body is the distance
from its center of rotation to its center of gyration.
,, ,. r .. A moment of rotation
Radius of gyration = \ s
—
—
* mass of rotating body
or, for a plane surface:
-p. ,. c .- /moment of inertia
Radius of gyration = \ —
.
* area of surface
The radius of gyration of a round, solid disc, such as a grind-
stone, when rotating on its shaft, is equal to its geometrical
radius multiplied by */ Yz or radius multiplied by 0.7071 very
nearly. The radius of gyration of a round disc, if indefinitely
thin and rotating about one of its diameters, is equal to radius
divided by 2. The radius of gyration of a ring, of uniform cross-
section, rotating about its center, such as a rim of a fly-wheel
when rotating on its shaft, is :
V7?2
_i_
r2
——
—
7? = Outside radius.
r = Inside radius.
The radius of gyration of a hollow circle when rotating
about one of its diameters is
:
Radius of gyration = -v/
~*~ r
R = Outside radius.
r = Inside radius.
Moment of Inertia.
The moment of inertia is a mathematical expression used
in mechanical calculations. It is an expression causing con-
siderable ambiguity, as it is not always used to mean the same
thing.
The least rectangular moment of inertia, as used when
calculating transverse strength of beams, columns, etc., is the
sum of the products of all the elementary areas of cross-sections
when multiplied by the square of their distances from the axis
of rotation. The axis of rotation is considered to pass through
the center of gravity of the section.
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