How do I calculate the moment of inertia of a rotating hollow sphere (rotating about its diameter) about an axis outside the sphere around which it is revolving also (both axes are parallel)?

Answer 1

Use the parallel axis theorem, which says that the moment of inertia of an object around any axis is the sum of the moment of inertia of a point with the object's mass revolving around that axis and the moment of inertia of the object itself around a parallel axis through the object's center of mass:

I' = I + mR²

What does that mean?

Well, say for example that the z-axis runs through the center of your sphere, which has mass m and radius r. You can calculate or look up your sphere's moment of inertia about that axis in terms of m and r. Let's call it I_z.

Now, say you want to find the sphere's moment of inertia around a new axis parallel to the z-axis, call it the z'-axis, and let's say its distance from the z-axis is R. (Do not confuse R with the radius of the sphere r, which can be something else!)

Then, the moment of inertia of a point with the sphere's mass m located at the sphere's center of mass around the z'-axis is:

I_cm = mR².

By the parallel axis theorem, the sphere's moment of inertia around the z'-axis is the sum of the two moments of inertia above:

I_z' = I_z + mR².

I hope this helps!

Aaron