OK, so consider three perpendicular axes through a book:
(1) From the center of the front cover to the center of the back cover.
(2) The long axis, from the top of the book to the bottom of the book
(3) The axis cutting across the book starting from the middle of the spine
I can easily toss the book up in the air and make it flip around axis (1) or axis (2), but if I try to make it rotate around axis (3) it goes into a tumble. (Try it yourself if it’s not clear what I mean.) What is the physical reason for this? Feel free to make your answer as technical/mathematical as you like, although an answer I could easily explain to my non-physics student friend who asked me the question would also be useful.
I just tried it myself and I didn’t have any spectacular tumbles, though I did see something of a wobble with #3. I think it’s because that’s the only case where there’s a substantial difference in the distribution of weight from one end of the axis to the other… (the spine seemed to be slightly denser than the rest of the book.) and so the force of gravity threw it off of spin. Does that notion help at all??
I received a great lecture on this as an undergrad. You can explain it in terms of Euler’s equations for rotation. Basically, the equations for the axes with the largest and smallest moments of inertia are stable, but the intermediate axis is unstable, and the small-angle solution goes exponential on you. In practice, a book or a box that you try to rotate about the intermediate axis will flip over in midair.
I’m not going to try to do more than that online. I think you can find this treated in a good mid-level text like Symon’s Mechanics. I’ll bet someone has a treatment online, too.
The intermediate axis theorem. The book has three principal axes as you describe. Motion about the axes having the most and least rotational inertia are stable. Motion about the third axis – number (3) in your example – is unstable. If you very carefully flip the book so it rotates anout axis (3) and no other, it will stay oriented. Some slight irregularity, though, and it will go unstable. You can see the same effect in tennis rackets and hammers.
A book like you’re holding has three principal moments of inertia, about the three axes you describe above (called the principal axes.) If you number the axes the way you’ve described them above, then for most books (assuming the pages aren’t in landscape mode) we have I[sub]2[/sub] < I[sub]3[/sub] < I[sub]1[/sub], where the I’s are the moments of inertia about the respective axes.
Now, it turns out, if you look at the dynamics of a rotating rigid body (like your book), that it’s perfectly happy to rotate about any one of the axes — as long as you get the axis of rotation perfectly aligned with the principal axes of the book to start with. What you’re observing is what happens when the axis of rotation is slightly off from the principal axis. If the axis of rotation is slightly off from axes 1 or 2, then the axis of rotation will stay in roughly fixed. (In facier language, if you perturb the axis of rotation in these cases, the perturbation is stable.) However, if the axis of rotation is slightly off from axis 3, then that difference will grow in time, and pretty soon your book will be rotating about some axis that’s not even close to axis 3 (hence the tumbling.)
So it’s possible, in theory, to toss up your book so it rotates nicely around axis 3 — but if you start it even slightly off, then the axis of rotation will change pretty quickly. In contrast, rotations about axes 1 and 2 are stable, and so if you start it “slightly off” it won’t get much more “off.”
Why axis 3 is special this way is harder to explain without a few lectures on rigid-body motion; if you look at the equations of motion, it boils down to the fact that there’s one moment of inertia bigger than I[sub]3[/sub] and one less than I[sub]3[/sub]. But a good amount of formalism & mathematics is required to get to that mathematical statement, and this is neither the time nor the place for that.
okay, so I guess my distribution of weight isn’t quite right… or at least, it’s not the only explanation of this behaviour. (Personally, I’m quite sure that the book I was experimenting with DOES have an asymmetrical distribution of weight along this axis, and that’s probably what threw me off.)
