Book stacking - how rigorous is the standard proof?

[bcrowell]

book stacking -- how rigorous is the standard proof?

There is a classic problem in mechanics, which is that you have n identical books, and you want to place them in a stack at the edge of a table so that they stick out as far as possible. Here is a typical, fairly careful statement of the problem with its solution: courses.csail.mit.edu/6.042/fall05/ln8.pdf (see p. 7).

I tried this on the kitchen table with a stack of encyclopedias tonight, hoping to catch the interest of my daughter. I did, and she tried it herself. One thing that she did made me doubt whether the standard solution by induction is really valid. She would make a stack, see it start to tip over, and then put in another book way in back, low down, to shore it up. This violates one of the assumptions that I haven't seen explicitly stated, which is that at any given height, there is only one book.

Is it possible to find a counterexample to the standard bound by using a stack that has more than one book at the same height?

Or, alternatively, is it possible to prove that putting more than one book at the same height is never optimal?

 

[AlephZero]

The CM of the top n books only depends on the positions of those books.

To make a stable stack of n+1 books, you only need to support the top n books at one point, namely directly underneath its CM. Therefore, you only need one book at level n+1.

Real books are not perfectly rigid cuboids with uniform density, and if the surfaces in contact are not all horizontal planes, friction forces come into effect. Practical experiments may be misleading

 

[1994Bhaskar]

The proof in that book is just to show us the application of harmonic progression.Normally in physics we also have to consider the normal from books below, which will form a couple along with the weight force acting downwards.So what your daughter did was just natural to balance the couple and prevent the stack from falling.By the way it was really an interesting application of Harmonic Progression!

 

[bcrowell]

The CM of the top n books only depends on the positions of those books.

To make a stable stack of n+1 books, you only need to support the top n books at one point, namely directly underneath its CM. Therefore, you only need one book at level n+1.

Real books are not perfectly rigid cuboids with uniform density, and if the surfaces in contact are not all horizontal planes, friction forces come into effect. Practical experiments may be misleading

Hi, AlephZero,

Thanks for the reply! I'm afraid this argument doesn't really convince me, though. If you assume one book per level, then the proof by induction is straightforward, because the optimal stack of n+1 must be composed of an optimal stack of n on top of the n+1-th book. This is the standard argument. But if you don't assume one book per level, then you can't necessarily assume that the optimal stack of n+1 must be composed of an optimal stack of n on top of the n+1-th book.

Ben

 

[1994Bhaskar]

hello bcrowell,

Try to understand what I have told you.It doesn't matter whether friction exists,or any other real situation problem.It is normal and weight couple which causes tumbling of books and no other real life force.Though forces like friction will effect but this is the main reason.I repeat that the explanation given in the pdf file mentioned by you was just to demonstrate the use of Harmonic Progression.Just take a look at the attachment image.

Regards,
Bhaskar