Richard sent me the following example:
A text book is placed flatly on a door near the doors outer edge. And
someone closes the door shut. Would the book rotate and translate when
it is separated from the door again, but it has all of its motion as it
had when the door was closing?
Martin: If we are thinking about the conservation of angular momentum I think we first have to choose where to measure the angular momentum around. I think I would choose the hinge of the door. I don't think we can choose the centre of the book because, as the door is closing, this will be travelling in a circle which in linear terms means that it is accelerating and is therefore not an inertial frame of reference.
So since we have chosen the hinge to measure the angular momentum about the hinge the, after the door has slammed is the perpendicular distance times the mass of the book. This is the angular momentum that the book had when the door hit the endstop, so the book is not spinning about its own axis at this stage.