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I'm really only working with the case where the object is a bar with all of the mass divided evenly between the ends, but a general solution would be very welcome.

The basic case for the bar is where the force is applied at one of the ends. That end will start to move a little bit away from the force, thus changing the center of mass (and there's no reason why the other end would move up to compensate)--so there must be at least _some_ linear motion. But then the tension between the ends will slow the first end down and start the second end moving towards the first. What is that tension?

If I can find that tension, the problem's solved, because then I can just plug it into the formula for centripetal force with the radius as half of the bar length and the mass as half of the mass of the bar, and solve for speed, and figure that all the rest of the energy must go into linear motion. But how do I find that tension?

This is for a simulation of bars bouncing against one another, so there's the additional complication that it's not a fixed _force_, it's a fixed mass and velocity. But unless you happen to know what happens there off the top of your head, forget it--I think I have an intuitive idea of how it should work (at the middle the bar would act like it had its full mass, and as you moved towards an end it would linearly decrease to acting [for the purposes of figuring collision force] like it had half of its mass).