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I want to know in which situation the disk rotates about its center of mass? It seems it’s in the first situation when it’s fixed. But how does it rotates if it’s fixed and what does cause it to rotate?

And if so, how to determine its moment of inertia for pivot $P$? Can I use parallel axis theory when the disk rotates about its center of mass and pivot at same time? And why doesn’t it rotate when it is mounted to the rod by a frictionless bearing so that it is perfectly free to spin? In the second situation why can the disk can be a particle mass point?

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If the disc is fixed, then by definition it is not possible for it to rotate (this is what fixed means). This should be made more clear when the second part of the problem states that it is attached by a "frictionless bearing so that it is perfectly free to rotate" (emphasis mine). If it is free to rotate in the second case due to the bearing, then it must not have been able to do so in the first case.

In the first case, since the disc cannot rotate, then it only contributes to the motion as the standard pendulum bob at the center of mass. In the second case, you have to consider the moment of inertia of the rotation of the now-freely-moving disc.

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  • $\begingroup$ I agree with you. I edited the question so you can see the explanation. But why does the answer to b describe the disk won’t rotate about its center of mass, and see it as a mass point. If the disk doesn’t rotate in both scenarios, how can the moment of inertial contributing to the total motion different? Thank you. $\endgroup$
    – Xiang Li
    Commented Apr 7, 2023 at 8:32
  • $\begingroup$ You do not agree with me: you are reading my answer, as well as the answer key you've edited into the post, completely wrong. The disc does not rotate in the first case; it does rotate in the second case. This is why the moment of inertia does not contribute in the first case but does in the second case. $\endgroup$
    – Kyle Kanos
    Commented Apr 8, 2023 at 13:22

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