Will an object rotate when we apply a force to it? What would happen if the axis of rotation passes through the centre of mass of an object? Will the object rotate when we will apply a force to the object?
Edit: The object is free, is not fixed to an axis of rotation and force is perpendicular to the object.
 A: 
What would happen if axis of rotation pass through centre of mass of
  an object? Will the object rotate when we will apply force to the
  object?

For the sake of future readers, I'll reply to the original question:


*

*Every body has a Centre of Mass, whatever its form. If an object has a regular shape and uniform, homogeneous distribution of mass its CoM coincides with its centre.

*If an object is fixed to a pivot, a fulcrum the axis of rotation will be at the pivot

*If an object is free, not fixed to an artificial axis of rotation any action outside the CoM will make it rotate around it





*

*Suppose now we have an object B (a board, for example, or a door, like in your other question. Its CoM lays at the middle: if we exert a force, an impulse, an impact bang on the CoM, the whole board will move in the same direction.

*If you apply a force on any point except the Com, let's say at one edge, you must specify if the force is rotating with the body. The body will rotate anyway, but if the force always act in the same direction, after a short time it will lose contact with the board.

*Lastly, if a stone, a point mass hits the edge of the board it will move forward and rotate at the same time. Supposing that the projectile has mass 1 and v 20  (p = 20, L = 10, E = 200) and the board has m = 9 and that the collision is elastic, the ball will bounce back at roughly v = -10 m/s , the board will translate at 3.33 m/s and the board will rotate with a frequency $\nu = 2.6 rps$


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A: The question (even after the edit) is not very clear. So I will make some general statements about forces, objects and rotation.
In order to cause a change in the angular momentum of an object (which is one interpretation of "rotate", although it can mean "stop rotating" too), you need to apply a torque.
A torque results from a force applied along a line of response that does not pass through the center of mass.
If you have a pair of forces that cancel each other in magnitude and direction, but that are not along the same line, you will get pure rotation. Any other combination (whose vector sum is non zero) will give rise to acceleration of the center of mass of the object - and further, if the net vector does not pass through the center, will also result in rotation.
I hope that clears up your confusion.
A: I assume that you are essentially asking whether an unhinged rod placed on a smooth surface will start rotating if a force perpendicular to it is applied at one of its ends. 
Yes it will. But why ? Why doesn't the rod as a whole start moving in a purely translational manner in a particular direction when you apply torque to one end ?
You can think of it like this: The center of mass of the rod will always try to resist motion. When you apply a force at a point on the rod perpendicular to it (other than at the center of mass), that point starts moving with a velocity. Now let's say for a moment that the center of mass has a choice whether to move with the same, greater or lower velocity than that point. The center of mass will obviously choose the case where it will have the lowest velocity possible, since it wants to resist motion. Therefore, it will move with a velocity lower than the velocity of that point. Since different points on the rod now have different velocities (and the rod remains rigid), it is said to undergo rotation. 
You asked another question on why the rod rotates only about the COM and not any other point when a force is applied at a point other than its COM in one of your comments. The very definition states that all the translational motion of the system can be described by the motion of the COM. Therefore if you replace the whole rod with its center of mass, you must still be able to explain the whole translational motion of the rod with the help of its COM. The center of mass must show purely translational motion in an isolated, unhinged sytem such as yours; because if the COM itself shows rotational motion, which other point will purely describe the translational motion of the rod?
If the rod starts rotating about any other point (other than the COM) when a force is applied, it means that the center of mass must rotate about that point. However as we have said, the COM is supposed to show only translational motion and not rotational motion in an isolated and unhinged system. Therefore, it poses a contradiction. As a result, the rod must compulsorily rotate about its COM when a force is applied at any other point. 
P.S: In the third paragraph, I have used a slightly philosophical argument to show that the COM must have purely translational motion in an isolated, unhinged system. I hope you understand what I mean over there.
