In a 2D plane there is an object, whose center of mass is at the $P$ point. The mass of the object is $m$.
We apply a force $\vec F$ at the point $A$.
If the center of mass is on the line, defined by the $A$ point and the $F$ vector, then the object will not rotate, and the acceleration can be calculated easily with the $\vec a = \vec F/m$ formula.
But where should I start otherwise? How is it possible to calculate the acceleration of the position and rotation of the object?
The only thing I've figured out, is that the object will rotate but not moving, if the center of mass is on the line defined by the $A$ point, and the normal vector of $\vec F$.