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12.5 Kinetic Energy of Rotation; Moment of Inertia 383

1 2 1 2 l 2
Ml Ml M a b (12.39)
3 12 2
which is identically true.
Note that it is a corollary of Eq. (12.38) that the moment of inertia about an axis
passing through the center of mass is always smaller than that about any other paral-
lel axis.
Table 12.3 lists the moments of inertia of a variety of rigid bodies about an axis
through their center of mass; all the bodies are assumed to have uniform density.

The large centrifuge shown in the chapter photo carries the
EXAMPLE 12 Concepts
payload in a chamber in one arm and counterweights at the in
Context
end of the opposite arm.The mass distribution depends on the choice of payload
and the choice of counterweights. Figure 12.18 is a schematic diagram of the mass
distribution attained with a particular choice of payload and counterweights.The
3
payload arm (including the payload) has a mass of 1.8 10 kg uniformly distrib-
3
uted over a length of 8.8 m. The counterweight arm has a mass of 1.1 10 kg
uniformly distributed over a length of 5.5 m, and it carries a counterweight of
3
8.6 10 kg at its end. (a) What is the moment of inertia of the centrifuge for this
mass distribution? (b) What is the rotational kinetic energy when the centrifuge
is rotating at 175 revolutions per minute?

8.8 m
5 5 m
1.8 10 3
kg
1.1 10 3
kg
8.6 10 3
kg
counterweight
FIGURE 12.18 Centrifuge mass distribution.
SOLUTION: (a) The total moment of inertia is the sum of the moments of iner-
3
tia of a rod of mass m 1.8 10 kg, length l 8.8 m rotating about its end; a
1
1
3
second rod of mass m 1.1 10 kg, length l 5.5 m also rotating about its end;
2
2
3
and a mass of m 8.6 10 kg at a radial distance of R 5.5 m. The moments
of inertia of the rods are given by Eq. (12.35), and the moment of inertia of the
2
counterweight is mR . So the total moment of inertia is
2
2
1
1
I m l m l mR 2
3
3
2 2
1 1
2
3
1
1
3
1.8 10 kg (8.8 m) 1.1 10 kg (5.5 m) 2
3 3
3
8.6 10 kg (5.5 m) 2
5
3.2 10 kg m 2
(b) At 175 revolutions per minute, the angular velocity is 18 radians/s
(see Example 5), and the rotational kinetic energy is
1
K I 2
2
5
2
1
3.2 10 kg m (18 radians/s) 2
2
7
5.2 10 J