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While considering the motion of a sphere (rotational motion), do we measure all motion taking the instantaneous axis of rotation as reference( in general,

when axis not mentioned which to be taken). in such cases, the moment of inertia is taken 2/5MR^2 or (2/5MR^2 + MR^2)?



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If the sphere is rotating about its center in any direction and it is not rolling or moving in x, y or z directions, then the moment of inertia = 2/5 * M R²
It does not matter if its rotating about a fixed axis or changing axis.

Similarly, if the sphere is rolling on a planar surface without slipping and without friction, then the moment of inertia is =  2/5 * M R² + M R².
It does not matter if the sphere is rolling in varying directions.  If the sphere is rolling on a planar surface, then v = R ω = velocity of center of mass.

Even if axis is mentioned or not mentioned, when the sphere is not fixed and is symmetrical about any diameter or axis, then the above two formulas are good and valid.

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