Just to make the discussion easier, let's say that an idealized rigid body is composed of component parts which always maintain a fixed distance from each other, and that we can refer to these component parts as ‘atoms’ without getting bogged down with details of how “real” atoms actually jiggle about.
bahamagreen said:
I agree that each point has an instantaneous momentum vector wrt the floor, I was the first to mention cycloid paths (I called them epicycles)
Nicely done.
Just for reference here’s a link I found after a quick Google search which shows shapes of the three types of cycloids.
http://www.daviddarling.info/encyclopedia/C/cycloid.html
A point on the outer perimeter of a wheel rolling on the ground traces a path called a cycloid. A point farther from that edge traces out a path called a prolate cycloid, and a point closer to the center of the wheel traces out a path called curtate cycloid. Also, according to the Wikipedia article on cycloids, the term trochoid is used to refer to any of the three types of cycloids, so I suppose it’s best to make that correction and switch to using that terminology from here on.
Every point of a rigid body which is set in motion in a plane, traces out the path of a trochoid. Think of a wheel of an automobile. Every component part (atom) of the wheel is tracing out the path of a trochoid. As per the OP’s question, there is both translation motion (the car is moving forward) and rotation motion (the wheels are spinning). As the car slows down, each atom of the car slows down in velocity along the trochoid path that it traces. Now here is the important part. As the car comes to a complete stop, all atoms following their respective trochoid paths come to a complete stop at the exact same instant in time!
Now don’t misunderstand me. I’m not saying that the physics here is the same as the physics of the atoms of our book which are tracing out trochoid paths (although I’m not saying it isn’t, either). I’m just pointing out that this is a great example of a solid body composed of component parts which trace out trochoid paths and all tend towards zero velocity at the same time.
bahamagreen said:
To forget about keeping the spin and translation separate seems to miss the problem's point - to know why both types of motion cease at the same time. The question asks why these motions both continue to have magnitude until the book stops.
I don't think I missed the problem's point, I think the problem's point makes false assumptions. But anyway, fair enough. We'll just go along with the books assumptions and speak as if rotational motion and translational motion are separate. So back to launching our book again, but this time let's first consider the physics if there were no friction or other outside influences. When we set the book in motion we give it some combination of translational motion and rotational motion, and that (due to its being a rigid body) sets the component atoms along various but well behaved trochoid paths. We could also set the initial motion as rotation only and no translation, or translation only with no rotation, and those two cases would also be valid because their paths (straight line and circular) are just special cases of the general trochoid.
Due to Newton’s first law, all the component atoms will maintain their trochoid paths indefinitely, due to that fact that at this point we still have no friction. So can we change or remove some or all of the translation motion without affecting the rotation motion, or vice versa?
Absolutely. We can apply a force to some chosen location on the book and the book will react according to Newton's 2nd and 3rd laws. The result will be to change its translation motion relative to its rotation motion.
And what is the fate of our ubiquitous trochoid paths that are traced out by the component atoms? They have all changed. Each and every atom now follows a new trochoid path, reflecting the new ratio of translational motion with respect to rotational motion.
Now we add friction, and ask the question: Can friction change or remove some or all of the translation motion without affecting the rotation motion, or vice versa?
The answer is no, and the reason is that friction acts upon each atom in a manner which is proportional to its instantaneous momentum vector. In other words, it slows each atom down in the direction that the atom traces, and in proportion to its velocity along its trochoid path. If there is no change to the shapes of the trochoid paths of the atoms, then there can be no change in the ratio of rotational motion with respect to the amount of translational motion.