Sometimes it's necessary to find the moment of inertia of a system of rigid bodies about a particular axis. Most machines, for example, have rotating parts. Frequently the first task is to find the center of mass of the system. Let's look at the example that was presented in an earlier module on center of mass, for which we have the center of mass and these data:
Suppose this machine part is rotated about its center of mass (axis is indicated by the dotted line). What is the moment of inertia of the part about this line?
Solution: We have the position of the center of mass from the earlier example - 1.028 m from the left end. If you'd like to review how that was obtained, please click on this link, which will display the example where the center of mass was obtained (in the earlier module).
The parallel axis theorem will have to be used for each portion of the machine part - the disk, the rod, and the sphere. It's important to keep in mind that the parallel axis theorem must be used to move an axis from the center of mass axis for each body to some other axis, in this case, to the center of mass axis for the entire system. The formula to be used is:
where d is the distance between the center of mass of the part and the axis about which rotation is occurring.
I'll use a table to keep the information
organized:
| item | ICM (kg m2) | d (m) | I (kg m2) |
| disk |  |
0.528 | 246 |
| rod |  |
1.46 - 1.028 = 0.432 | 20.6 |
| sphere |  |
1.92 + 0.2 - 1.028 = 1.092 | 318 |
| sum | not applicable | not applicable | 585 |
It is this value of I that would be used
to find the kinetic energy of the system 
if
it were rotated about the center of mass axis of the system at a certain
angular velocity 
.