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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 $\vec F$.

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    $\begingroup$ The body will always accelerate, following Newton's second law, no matter if center of mass was on the normal to the force or not. $\endgroup$ – Tofi Apr 24 '16 at 15:03
  • $\begingroup$ if the force vector does not go through the center of mass there will be rotation to be considered , it will be combined motion rotational acceleration and linear acceleration $\endgroup$ – anna v Apr 24 '16 at 15:10
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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 $\vec{F⃗}$.

This is not necessarily true. Look at the free body diagram.

Free body diagram.

Decompose $F$ into $x$ and $y$ components. Newton now tells us that:

$$ma_x=\Sigma F_x$$ $$ma_y=\Sigma F_y$$

So if $F$ was the only force then the object will definitely accelerate.

In addition, $F$ exerts torque $\tau$ about the CoG (point $O$):

$$\tau=F|OP|$$

The angular acceleration $\alpha$ is given by:

$$I\alpha=\Sigma \tau$$

So in the absence of other torques the object will definitely start rotating too.

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The laws describing the movement of the body are: $$\sum \vec F=m\vec a_{cm},$$ and $$\sum \vec \tau= \frac{d\vec L}{dt}$$ wehre $\vec F$ is the external force, $\vec \tau$ the external torque, $\vec L$ the angular momentum and $\vec a_{cm}$ is the acceleration of the the center of mass. As you can see from the first formula, the center of mass will always accelerate in the direction of the resulting force. No matter where it is. The second equation will tell you whether the object will rotate or not.

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