rotational dynamics - Why is moment of inertia dependent on $r^2$ and not on $r$? (physical reason)

Moment of inertia is the mass equivalent in rotational dynamics. I know, by mathematical arguments, moment of inertia of a particle is $$ I = \text{mass} \cdot r^2.$$ But what is the physical reason? When I want the other linear equivalents like velocity,acceleration, we only multiply the linear component times the distance from the centre. But, in moment of inertia, it is not distance but distance-square. Why is it so?