Exercise

Figure 6.2 shows a cylinder, a disk, and a thin ring, all of the same mass. In each case the axis of rotation is indicated. The radii (or radiuses) of the disk and the ring are the same. In the case of the cylinder, all its particles are close to the axis, so the r _i are small. In comparison, the particles of the disk are spread out to larger distances from the axis. Without any calculation we may conclude that I_A< I_B. Whereas the mass of the disk is uniformly spread out, all the particles of the ring are at the maximum possible distance from the axis. Consequently, I_B < I_C.

Figure 6.3 shows a rigid body rotating with angular velocity ω about an axis that is located at a perpendicular distance h from the CM. Kinetic energy of the body has two terms, K = K _CM+ K _rel. The first term, К _CM, is the kinetic energy associated with the motion of the CM, and K _rel is the kinetic energy of motion relative to the CM. The angular velocity of a rotating body is the same about any point on it. In particular, although the body is rotating about О, the angular velocity about the CM is also ω. Therefore, the total kinetic energy of a body rotating about an axis fixed in direction is

FIGURE 6.3

K = K _CM + K _rel

or

(6.2)

From Eq. 11.5, , and so . The total kinetic energy

becomes . This may be expressed as where

(11.16)

Parallel axis theorem

This relationship, called the parallel axis theorem, relates the moment of inertia I about any axis to the moment of inertia I _CM about a parallel axis through the CM. This result is very helpful since it is often easier to evaluate the moment of inertia about the CM.

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