MOMENT OF INERTIA1
A. Basic Concept: The kinetic energy associated with the rotation of a rigid body
about an axis L can be written in the familiar form as IL2/2. It is thus clear that
the quantity IL should have the similar meaning as the mass, as one compares with
the kinetic energy expression of a particle in linear motion: mV2/2. The quantity
IL is determined by the shape and is viewed as the “rotational mass. It is known as
the moment of inertia with respect to the axis L. This means to define the
“rotational mass,” you will need to know (a) the geometry of the rigid body and
(b) the axis about which the rigid body rotates.
B. Conceptual Difference:
While IL is conveniently viewed as a “mass” quantity, important distinctions must
be made:
 A piece of steel weighs 1 kg is a piece of steel weighs 1 kg, regardless of its
shape. Hence, the IL can not be an intrinsic physical quantity since its numerical
value depends on its geometry.
 Unlike the mass, the “rotational mass,” IL, is not a “stand-alone” concept: you
must specify which axis of rotation it is referring to (such as the axis of rotation
L from above).
C. Technical Details
 If the rigid body is symmetric about the XY plane, then IXZ = IYZ = 0. If the rigid
body is symmetric about the XZ plane, then IXY = IYZ = 0. If the rigid body is
symmetric about the YZ plane, then IXY = IXZ = 0. For rigid body in plane
motion, the coordinate system is normally chosen so that the XY plane is the
plane of motion. We will make one extra assumption in most of the problems
we solve: the rigid body is symmetric about the (XY) plane of motion. So, in
most problems, IXZ = IYZ = 0.
 Parallel Axis Theorem: If the moment of inertia with respect to a given axis, say
L, is given by IL, then the moment of inertia with respect to a second axis
parallel to L, say T, is IT = IL + md2 where m is the mass of the rigid body and d
is the distance between the two axes.
 Radius of Gyration: In many cases, the moment of inertia with respect to a
given axis, say L, is given as IL = mrg2, where rg is called the radius of gyration.
Physically, using the energy concept as in A, the kinetic energy of the rigid
body rotating about the axis L is given by IL2/2 = mrg22/2 = mV2/2. This
means we are putting the entire mass of the rigid body (m) as a point mass
located at a radial distance rg from the axis of rotation L. In practice, when the
radius of gyration is provided, the moment of inertia is calculated as IL = mrg2.
 In the application to kinetics of rigid body, you need to be able to determine
which axis of rotation to be taken to describe the Newton’s Law. There are only
two possibilities for rigid body in plane motion: either taken at the centre of
mass (general plane motion) or at the axis of rotation (fixed axis rotation).
1
Intuitively, “moment” implies “rotation” and “inertia” implies “mass.”