Why do we use inertia in rotational motion and not the whole mass of an object ?
What do you mean?
Do you mean rotational inertia? (Moment of inertia.) If so, realize that rotation and translation have their differences.
I remember any introductory text on the subject gives enough arguments and calculations!
Grant R. Fowles' Analytical mechanics is enough I think.
The moment of inertia is a calculation that does consider the whole mass of the object, as well as some information about the distribution of the mass about the rotational axis.
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