Understanding Products of Inertia

[AdamX1980X]

We covered this topic in Dynamics yesterday and the text does a less than average job on the topic. I am confused on the argument of symmetry. Does this mean if an item is symmetrical with respect to all planes the Inertia is 0 ( even when it is not bisected by the planes in the axis given. We looked at an example in class dealing with three pipes joined by 90 degree elbows. Two of which laid on the given x and the given y axis. The third however in the z direction laid under the x and y plane and was off of the given z axis. We used the parallel axis theorem which I understand. It's just that pipe in the z direction had a zero in the product of inertia. and I thought that it would have some sort of inertia since it was not centered on the point of rotation. Any help clearing this up would be greatly appreciated. Thanks!

 

[Navin A S]

Symmetry does not necessarily mean that the inertia will be zero. The key to understanding this concept is the parallel axis theorem. This theorem states that when you move an object from one axis to another, the inertia of the object changes due to its changed moment of inertia. In your example, the pipe in the z direction may have been off-center from the given axes, but since it was symmetrical it would still have a moment of inertia. However, since it was not directly on the axes, its inertia would change when it was moved to the given axes, and the parallel axis theorem would allow you to calculate this change.