It is possible for a free body to make rotational motion with the effect of one only force?
Not "pure" rotational motion. If the single force happens to be through the center of mass then you would get pure linear motion. If the single force is applied on a line not through the center of mass, then you will get a combination of rotation about the center of mass and linear motion. You would need to divide the force into a "couple"- a pair of equal but oppositely directed forces (so they add no linear motion) equally spaced about the center of mass to create the rotation- and a force through the center of mass to create the linear motion. You would need to set the forces so that the change in total momentum and energy is correct for the force applied.
An unbalanced free-body will tend to rotate or otherwise move towards its attractive force. But this is only temporary(until balance is achieved). Sustained peak cycling of this has never been demonstrated to my knowledge.
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