A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.6 Moment of inertia

X

i

X

×

×

×

̧ ̧ ̧

Here, the quantity I is termed the moment of inertia of the object, and is written

I =

i=1,N

mi |k ri|^2 =

i=1,N

mi σ 2 , (8.31)

where σi = |k ri| is the perpendicular distance from the ith element to the axis
of rotation. Note that for translational motion we usually write

K =

1

M v^2 , (8.32)

2

where M represents mass and v represents speed. A comparison of Eqs. (8.30)
and (8.32) suggests that moment of inertia plays the same role in rotational
motion that mass plays in translational motion.

For a continuous object, analogous arguments to those employed in Sect. 8.5

yield

I =

∫∫∫

ρ σ^2 dV, (8.33)

where ρ(r) is the mass density of the object, σ = |k r| is the perpendicular

distance from the axis of rotation, and dV is a volume element. Finally, for an
object of constant density, the above expression reduces to

I = M

σ^2 dV

̧ ̧ ̧

dV

.^ (8.34)^

Here, M is the total mass of the object. Note that the integrals are taken over the
whole volume of the object.

The moment of inertia of a uniform object depends not only on the size and

shape of that object but on the location of the axis about which the object is

rotating. In particular, the same object can have different moments of inertia
when rotating about different axes.

Unfortunately, the evaluation of the moment of inertia of a given body about
a given axis invariably involves the performance of a nasty volume integral. In

fact, there is only one trivial moment of inertia calculation—namely, the moment
of inertia of a thin circular ring about a symmetric axis which runs perpendicular

to the plane of the ring. See Fig. 75. Suppose that M is the mass of the ring, and