The parallel axis theorem relates a body’s moment of inertia about any axis to its moment about a parallel axis through the centre of mass.
Consider the diagram above, where the green area represents a system of particles, each with mass , rotating about an axis perpendicular to point
. Point
serves as both the origin of the coordinate system and the centre of mass of the system. As a result, point
has the coordinates
, while particle
at point
has the coordinates
.
By definition, the centre of mass with respect to the -coordinate satisfies
. Similarly, for the
-coordinate, we have
. Expanding these two equations and rearranging them yields
where .
Since , we have
Question
Show that the moment of inertia of a system of particles, each with mass
, and located at perpendicular distances
from the axis of rotation, is given by
.
Answer
From eq4, the total rotational kinetic energy of the particles is
where .
The moment of inertia about the axis perpendicular to point is
where is measured from
to
.
Substituting in eq7 and expanding gives
Substituting eq6 in eq8 yields
Since , where
is measured from
to
, and
, eq9 becomes
Eq10 is the mathematical expression of the parallel axis theorem.