8 ROTATIONAL MOTION 8.6 Moment of inertia
̧ ̧ ̧̧ ̧ ̧yringxzFigure 77: The moment of inertia of a ring about a coplanar symmetric axis.of mass of that body, then the moment of inertia IJ of the same body about a
second axis which is parallel to the first is
IJ = I + M d^2 , (8.42)where M is the mass of the body, and d is the perpendicular distance between
the two axes.
In order to prove the parallel axis theorem, let us choose the origin of our
coordinate system to coincide with the centre of mass of the body in question.
Furthermore, let us orientate the axes of our coordinate system such that the z-
axis coincides with the first axis of rotation, whereas the second axis pieces the
x-y plane at x = d, y = 0. From Eq. (8.20), the fact that the centre of mass is
located at the origin implies that
∫∫∫
x dx dy dz =∫∫∫
y dx dy dz =∫∫∫
z dx dy dz = 0, (8.43)where the integrals are taken over the volume of the body. From Eq. (8.34), the
expression for the first moment of inertia is
I = M(x^2 + y^2 ) dx dy dz
̧ ̧ ̧
dx dy dz,(8.44)since x^2 + y^2 is the perpendicular distance of a general point (x, y, z) from the
z-axis. Likewise, the expression for the second moment of inertia takes the form
IJ = M[(x − d)^2 + y^2 ] dx dy dz
̧ ̧ ̧
dx dy dz.(8.45)