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Considering the yo-yo like two CDs with a hollow cylinder between them, what is the moment of inertia of that object?

The axis that I must choose can't pass through the CM and be parallel to rotational axis that passes through the center of mass. A force $\vec{T}$ is pulling the object.

I can't use the Steiner theorem here, right? All because that cylinder cannot be ignored.

Another try that I do is calculating the moment of inertia of the three (well, two) objects and sum, but I think that this isn't correct.

enter image description here

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    $\begingroup$ If an object is made of several parts, you can add the moments of inertia of the parts to get the moment of inertia of the whole. Just make sure you're consistently using the same axis of rotation. $\endgroup$ – DarenW Apr 12 '13 at 5:31
  • $\begingroup$ What do you mean by "despised" here? $\endgroup$ – Manishearth Apr 12 '13 at 10:43
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Use Steiner theorem and ignore the cilinder between the two disc.

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