Moment of Inertia, difference between Ixc, Ix, and Io?

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SUMMARY

The discussion clarifies the differences between the moments of inertia: Ix, Ixc, and Io. Ix represents the resistance to rotation around a defined X-axis, which may not necessarily pass through the center of mass. Ixc denotes resistance to rotation around an axis parallel to the X-axis that passes through the center of mass. Io refers to the moment of inertia around an axis through the origin, indicating symmetry in all directions around the origin, which coincides with the center of mass.

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  • Understanding of basic mechanics and rotational dynamics
  • Familiarity with the concept of moment of inertia
  • Knowledge of axes of rotation in physics
  • Ability to interpret mathematical notations related to inertia
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Students of physics, mechanical engineers, and anyone studying rotational dynamics will benefit from this discussion on the distinctions between different moments of inertia.

Femme_physics
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So I understand that Ix resistance to rotation around the X axis, Ixc is resistance to rotation around the center of gravity of the shape on its X axis, and Io I was told is also resistance to rotation around the object's center of gravity. So, I'm completely confused as to the difference between Ixc and Io. Except that Io don't seem to define an axis. Can anyone help me understand it?
 
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Femme_physics said:
So I understand that Ix resistance to rotation around the X axis, Ixc is resistance to rotation around the center of gravity of the shape on its X axis, and Io I was told is also resistance to rotation around the object's center of gravity. So, I'm completely confused as to the difference between Ixc and Io. Except that Io don't seem to define an axis. Can anyone help me understand it?

I is always the moment of inertia around some axis.
The index denotes which axis is meant.
Sometimes people get creative how to denote a specific axis.

Ix for instance is around the defined x-axis, which may or may not be through the center of mass.

Ixc would be around an axis parallel to the x-axis, which is through the center of mass.

Io would be the moment of inertia around an axis through the origin. The fact that no distinction is made about which axis seems to imply that the object is symmetric in all directions around the origin, which will be its center of mass as well.
 
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