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2
 
E   (6.6)
2
The form of this expression is analogous to that of translational

kinetic energy E  mv 2 .
2
That is, for rotation arond a fixed axis, this moment of inertia is

analogous to mass m (or inertia), and angular velocity  is analogous
to linear velocity v.

6.2 Calculation of Moments of Inertia

The moment of inertia of a solid body can be calculated by eq.

(6.5). But this equation can be applied directly only in cases when the
body consists of a few point masses. When the body consists of a
continuous distribution of matter, the sum is expressed in terms of an
integral.

Any body can be considered as a system with the material points

of mass dm with moment of inertia dI  r 2 dm.Therefore the moment
of inertia of body may then be expressed as
2
I   r dm. (6.7)
V
(integration over total volume V of body)

Since the density  of the element is its mass per unit volume,

  dm ,
dV
2
I   r   dV . (6.8)
V
If the body is homogeneous, that is, if the density is uniform, then 
may be taken outside the integral:

I    r 2 dV . (6.9)

V
For regularly shaped bodies this integration can often be carried out
quite easily. Three examples are given below.

Example 1. Figure 6.2 shows a slender uniform rod of mass m and
length I We want to compute its moment of inertia about axis OO
through the end of rod and perpendicular to it . Select as an element of
volume dV a short section of length dx and cross-sectional area S.

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