online handle: tenifni: [Physics and Cartoons] Stacking Books

The problem: You have a stack of books that are of the same size and mass. You put a book on a table with some overhang, then add another book on top of the first book with some overhang, and so on. What is the minimum number of books needed so that the top book is completely overhanging from the table's edge?

Let's set up a mathematical model to solve this problem. First we assume that the center of mass is the in middle (1/2) of the book. Instead of adding books from bottom to top, it's simpler to look at it as adding books to from top to bottom. Each time we add a new book, the edge of that book must be right on the center of mass of the original stack of books.

As shown, each time we add a new book, the total overhang increases by a certain amount. So how do we figure out that amount and the total overhang if there are n books?

Finding the overhang of n books:

Suppose each book has a horizontal length of l and mass of m. We first consider 2 books:

Then add a third book to the bottom so that its edge is right on the center of mass of those previous 2 books:

Set up a coordinate system and put x = 0 on the center of mass of the first two books.

Now we can write this equation:

Xc*2mg = (l/2-Xc)mg (1)

We can solve this for the center of mass Xc (this will give us the length of the overhang of 3 books). Changing 2 to n, now we have n +1 books in total:

Using equation (1) but replacing 2 with n, we find center of mass of n+1 books:

Plugging some numbers into this equation, we get:

You may have noticed at this point that Xc is the extra overhang added to the original overhang each time we add a new book to the stack.

So total overhang of n books is d = l/2+l/4+...+l/n.

Now back to the question: What is the minimum amount of books needed so that the top book is completely overhanging from the table's edge? This is happens when overhang > l.

From the calculations, we can easily see that when n = 4, the overhang is larger than the length of the book, making the book completely protruding out of the table's edge. Thus the minimum # of books needed is 4.

Since it was hard to find 4 identical books, I used poker cards to experiment (coins also work):

4 cards

The top card is completely overhanging from the edge.

Is there a maximum length of overhang?
If we add all the overhangs together, it turns out to be related to harmonic series.

When k becomes infinitely large, the sum is roughly = 1/2 log(n).

Plot of the sum of the first 100 terms of 1/2 log(n):

This harmonic series is divergent and its sum is infinite. In other words, in theory, you can stack infinitely many books and there is no maximum length of overhang.

Other ways to stack:

Other popular ways are the "3-row inverted" method and "5-diamond" method, which are both proven to be not very stable. But you can stack the books however you want as long as the they don't fall!