8 ROTATIONAL MOTION 8.6 Moment of inertia

x

Figure 76: The perpendicular axis theorem.

̧ ̧

x^2 dx dy

(^)
(^) (8.38)
Iy = M (^) ̧ ̧
dx dy
,
respectively. Here, we have made use of the fact that z = 0 inside the object. It
follows by inspection of the previous three equations that
Iz = Ix + Iy. (8.39)
See Fig. 76.
Let us use the perpendicular axis theorem to find the moment of inertia of a
thin ring about a symmetric axis which lies in the plane of the ring. Adopting the
coordinate system shown in Fig. 77 , it is clear, from symmetry, that Ix = Iy. Now,
we already know that Iz = M b^2 , where M is the mass of the ring, and b is its
radius. Hence, the perpendicular axis theorem tells us that
2 Ix = Iz, (8.40)
or
I =
Iz
x 2

1
M b^2. (8.41)
2
Of course, Iz > Ix, because when the ring spins about the z-axis its elements are,
on average, farther from the axis of rotation than when it spins about the x-axis.
The second useful theorem regarding moments of inertia is called the parallel
axis theorem. The parallel axis theorem—which is quite general—states that if I
is the moment of inertia of a given body about an axis passing through the centre
z
Iz = Ix+ Iy
y