PARALLEL AXIS THEOREM

The Parallel Axis Theorem is used to determine the mass moment of inertia of any rigid body about any axis given the body’s moment of inertia about a parallel axis through the object’s center of mass and the perpendicular distance that separates the two axes. Thus, the moment of inertia about any new axis is given by:

Where:
I_cm is the moment of inertia of the object about an axis passing though its center of mass
M is total mass
r is the perpendicular distance between the two axes

One should be familiar with the moments of inertia about the center of mass before attempting to master the Parallel Axis Theorem. As one can see from the equation above, the Parallel Axis Theorem is a building block that implements the use of the moment of inertia about the center of mass.

Begin by defining the center of mass. One should assume that in a Cartesian coordinate system the perpendicular distance between the axes lies along the x-axis and that the center of mass lies at the origin. Thus, the moment of inertia relative to the z-axis passing through the center of mass is:

The moment of inertia relative to the new axis, perpendicular distance r along the x-axis from the center of mass, is:

By expanding the brackets one gets:

Thus, one should clearly see that the first term is I_cm, the second term becomes Mr², and the final term is zero since the origin is at the center of mass. This proves that the equation shown above is correct.