What happens if a force is applied to the center of a rod? Imagine there is a 1d-rod (representing a simple rigid "body") of length L.
If we apply a force on the end of the rod it will result in a fast rotation around the rod's center, if we apply force near the center of it,
there will still be some amount of rotation (torque) but most of the force will be "consumed" to translate the rod from position A to position B.
But what happens if I apply an arbitrary amount of force from any direction to the center of the rod? In what direction will the rod be translated,
and will there be any rotation after the force is applied?
My guess is there will be no rotation at all (the position vector is zero), but I am not sure with the translation direction.
If the green force is applied to the center of the rod, will the rod move in direction a, b or c?

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 A: You are right in observing that there will be no rotation, since the force is applied directly towards the center of mass. To find the acceleration of the rod, use Newton's second law, $\mathbf{F}=m\mathbf{a}$. The acceleration points along the same direction as the force here, so the rod will move in direction $\boxed{b}$.
It's worth noting that the $\mathbf{a}$ found in this describes the acceleration of the center of mass of the rod. If the force is applied to the end of the rod, the rod will start to rotate, but the center of mass of the rod will accelerate the same regardless of where the force is applied (as long as it is the same force).
A: 
But what happens if I apply an arbitrary amount of force from any
direction to the center of the rod? In what direction will the rod be
translated, and will there be any rotation after the force is applied?

If the line of action of the force is through the center of mass, there will be pure translational motion of (and no rotation about) the center of mass (COM) in the direction of the force.
If the line of action of the force is not through the COM, there will be the same translational motion of the COM but there will also be rotation of the rod about the COM.
You can move a force parallel to its direction to any location as long as you add a couple (two parallel forces that are equal in magnitude, opposite in sense and do not share a line of action) to account for the rotational effect of the force about the COM.
Hope this helps.
