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380 CHAPTER 12 Rotation of a Rigid Body

z
2
I a R ¢m i (12.32)
i i
where R is the radial distance of the mass element m from the axis of rotation. In
i i
the limit m S 0, this approximation becomes exact, and the sum becomes an integral:
i
R 0 2
I R dm (12.33)

In general, the calculation of the moment of inertia requires the evaluation of the
integral (12.33). However, in a few exceptionally simple cases, it is possible to find the
All of mass of hoop is at same
radial distance R 0 from axis. moment of inertia without performing this integration. For example, if the rigid body
is a thin hoop (see Fig. 12.13) or a thin cylindrical shell (see Fig. 12.14) of radius R 0
FIGURE 12.13 A thin hoop rotating rotating about its axis of symmetry, then all of the mass of the body is at the same dis-
about its axis of symmetry. tance from the axis of rotation—the moment of inertia is then simply the total mass
M of the hoop or shell multiplied by its radius R squared,
0
2
I M R 0
All of mass of z If all of the mass is not at the same distance from the axis of rotation, then we must per-
cylindrical shell
is at same radial form the integration (12.33); when summing the individual contributions, we usually
distance R from axis. write the small mass contribution as a mass per unit length times a small length, or as
0
a mass per unit area times a small area, as in the following examples.

R 0
Find the moment of inertia of a uniform thin rod of length l and
EXAMPLE 9
mass M rotating about an axis perpendicular to the rod and
through its center.
SOLUTION: Figure 12.15 shows the rod lying along the x axis; the axis of rotation
is the z axis. The rod extends from x l 2 to x l 2. Consider a small slice
dx of the rod. The amount of mass within this slice is proportional to the length
dx, and so is equal to the mass per unit length times this length:
FIGURE 12.14 A thin cylindrical shell
rotating about its axis of symmetry. M
dm dx
l
2
2
The square of the distance of the slice from the axis of rotation is R x ,so
Eq. (12.33) becomes
2 l>2 2 M M x 3 l>2
I R dm x dx a b`
l>2 l l 3 l>2
(12.34)
3
M 2(l/2) 1 2
M l
l 3 12
A slice of width dx
is located at distance
x from rotation axis.
Rod extends z
from x = –l/2
to x = +l/2.
dx x
O
l

Slice has a fraction dm/M
FIGURE 12.15 A thin rod of total mass equal to its
rotating about its center. fraction dx/l of total length.