When it works:
* Spheres: For a solid sphere rotating around an axis passing through its center, the entire mass can be considered concentrated at the center for calculating rotational inertia. This is because the distribution of mass is perfectly symmetrical, and the moment of inertia is simply (2/5)MR², where M is the mass and R is the radius.
* Thin spherical shells: Similar to spheres, the mass of a thin spherical shell can be treated as concentrated at its center for rotational inertia calculations.
When it doesn't work:
* Non-spherical objects: For objects that aren't spherically symmetrical, the mass cannot be considered concentrated at the center. For example, a rod rotating about its center has a moment of inertia of (1/12)ML², where L is the length of the rod.
* Rotation about an axis not passing through the center: Even for spheres, if the rotation axis doesn't pass through the center, the mass cannot be treated as concentrated at the center.
* Objects with irregular mass distribution: Even if the object is spherical, if the mass is not uniformly distributed, the mass cannot be treated as concentrated at the center.
Key takeaway:
The concept of concentrating mass at the center for rotational inertia calculations only applies to a limited number of specific cases, primarily involving perfectly symmetrical objects rotating around their center of mass. For other scenarios, you need to consider the actual distribution of mass and use appropriate formulas to calculate the moment of inertia.
