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I have been thinking about this for hours now but I can't seem to find a definitive answer so I hope you can help me. So here is my problem:

If you exert a force perpendicular to a line through the center of gravity at distance L you will produce a torque T, that torque will rotate the body around the center of gravity with a rotational acceleration w_dot = IT where I is the moment of inertia. Because this force is perpendicular to the leverage arm through the cg no force will be applied to the cg so we will have no translation of this point, only rotation. Am I right so far?

If my previous statements were correct I see a problem with my following resoning:

Imagine that the object in space is a long iron beam. If we apply a perpendicular force in one end it will start to rotate about cg. Now imagine that we move the applied force closer and closer to cg, the torque will become smaller but it will still produce only rotational movement as long as the force is perpendicular. As soon as the applied force hits straight on cg we will get zero torque and therefore no rotation but now the cg will translate according to F = ma. Is it really a sudden step between rotational movement and translational movement of the object or what have I missed? Will the cg start to translate even for perpendicular forces applied far from cg? Will objects in space always rotate about cg if torque is applied or can it rotate around other points?

I hope you can help, and I hope it is not something too simple that I have forgot so I don't have too feel stupid :P

Regards

Niclas