Perpendicular Axis Theorem
For a three dimensional object, the moments of inertia about 3 mutually
perpendicular axes are related due to geometry and the definition of the moment
of inertia.
The moment of inertia about an axis is given by
 s dm where s is the radius from
2
the axis of rotation to dm.
[1] Using x, y and z axes, show that
I x  I y  I z  2 r 2 dm
where r is the radius from the origin in spherical coordinates. Start by writing s2
in terms of x2, y2 and z2, as needed.
This can be used to simplify the calculation of the moments of inertia for some
objects. For example, the moment of inertia of a spherical shell can be calculated
this way without needing to set up the rings to integrate over.
[2] Use the perpendicular axis theorem above and symmetry to determine the
moment of inertia of a spherical shell. Write the dm in terms of density and the
spherical radius r.
Another application is the determination of the moment of inertia of a simple
object about an arbitrary axis rotated by some angle from the symmetry axes.
Take a ring with a small thickness. The
moment of inertia about z is Iz = MR2 .
[3] How do the remaining two moments of
inertia compare (Ix and Iy)?
[4] Write dm for a small piece of the ring in
terms of dφ and radius r and keep in mind
that z = 0 for all parts of the ring.
[5] Use these two answers and the perpendicular axis theorem to determine Ix and
Iy :
Having the three moments of inertia, we need one more idea and we can look at
moments of inertia for a different set of axes. Return to your proof of the
perpendicular axis theorem in [1].
[5] What about the proof would change if some other set of axes were used
instead of x, y and z? (Assume the new axes are also mutually perpendicular).
Would the right-hand side be any different? Would the proof still be valid?
[6] From your answer above, what can you say about the sum of the three moments
of inertia about perpendicular axes ?
[7] Now consider our ring with a new set of axes rotated 45° about y. How would
the values for Ix' and Iz' compare? Use symmetry here.
[8] Using the answer above, the value for Iy and the perpendicular axis theorem,
determine Ix' and Iz' .
If you want to get very adventurous, try the last problem for an arbitrary angle of
rotation θ about the y axis.
Adapted from C.E.Mungan AJP 57:181 (1989)