A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.6 Moment of inertia

̧ ̧

= M b 2

ring

Figure 75: The moment of inertia of a ring about a perpendicular symmetric axis.

b is its radius. Each element of the ring shares a common perpendicular distance

from the axis of rotation—i.e., σ = b. Hence, Eq. (8.34) reduces to

I = M b^2. (8.35)

In general, moments of inertia are rather tedious to calculate. Fortunately,

there exist two powerful theorems which enable us to simply relate the moment

of inertia of a given body about a given axis to the moment of inertia of the same

body about another axis. The first of these theorems is called the perpendicular

axis theorem, and only applies to uniform laminar objects. Consider a laminar

object (i.e., a thin, planar object) of uniform density. Suppose, for the sake of

simplicity, that the object lies in the x-y plane. The moment of inertia of the

object about the z-axis is given by

Iz = M

(x^2 + y^2 ) dx dy

̧ ̧

dx dy

,

(8.36)

where we have suppressed the trivial z-integration, and the integral is taken over

the extent of the object in the x-y plane. Incidentally, the above expression fol-

lows from the observation that σ^2 = x^2 +y^2 when the axis of rotation is coincident

with the z-axis. Likewise, the moments of inertia of the object about the x- and

y-^ axes^ take^ the form^

Ix = M

y^2 dx dy

̧ ̧

dx dy

,

(8.37)

axis (^) I
M
b
̧ ̧