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382 CHAPTER 12 Rotation of a Rigid Body

TABLE 12.3 SOME MOMENTS OF INERTIA

BODY MOMENT OF INERTIA

Thin hoop about symmetry axis MR 2
R R

R Thin hoop about diameter 1 2 MR 2

R
Disk or cylinder about symmetery axis 1 2 MR 2

R
2
l Cylinder about diameter through center 1 4 MR 12 1 Ml 2

1
Thin rod about perpendicular axis through center 12 Ml 2
l

Thin rod about perpendicular axis through end 1 3 Ml 2
l

R Sphere about diameter 2 5 MR 2

R Thin spherical shell about diameter 2 MR 2
3

It is possible to prove a theorem that relates the moment of inertia I about an
CM
axis through the center of mass to the moment of inertia I about a parallel axis through
some other point. This theorem, called the parallel-axis theorem, asserts that

parallel-axis theorem I I Md 2 (12.38)
CM

where M is the total mass of the body and d the distance between the two axes.We will
not give the proof, but merely check that the theorem is consistent with our results for
the moments of inertia of the rod rotating about an axis through the center
2
2
1
1
[I Ml ; see Eq.(12.34)] and an axis through an end [I Ml ; see Eq.(12.35)].
CM 12 3
In this case, d l/2, and the parallel-axis theorem asserts