## Physics Friday 98: A Classic Problem

Consider a set of identical books (or other rectangular objects), stacked flat one atop another. By offsetting the books, one can have a “leaning” tower that angles to one side.

Is it possible to make such a tower where no part of the top book is over any part of the base book, so that the “lean” is larger than the length of a book? What is the maximum possible horizontal offset between the top and bottom books?

Consider the top book; with (roughly) uniform density, its center of gravity is the geometric center. To keep its center of gravity over the book supporting it, its maximum offset is half the book length; in units of book lengths, we have a maximum offset of h=1/2 for a stack of n=2 books.

Now, that set of two books has a center of gravity that, via symmetry, has horizontal position in the middle of the overlap, and so the maximum offset between the lower book of this pair and a third book supporting the pair is 1/4 of a book length, so we have total offset of the top book of .

If we number the books from the top down, and let  be the displacement of the ith book relative to the one below, so we have  and . Now, consider the top k books, and their displacements relative to the k+1 book. The center of mass of the top k books must be offset from the k+1 book by 1/2. The displacement of book k relative to book k+1 is , that of book k-1 relative to book k+1 is ; book i is deplaced relative to book k+1 by . Then the center of mass of the top k books (relative to book k+1) is the average of these displacements. Reordering the sum of these, we see  appears in the sums for the displacements of books 1 through i, and so the sum of these is
. Thus, we see
,
and so
.
Thus
,
and subtracting this from the above, we see
.
And for a stack of n books, we have total offset
,
where  is the nth harmonic number.

To answer our first question, then, we have h>1 when Hn-1>2. The harmonic numbers pass 2 with H4=25/12, so only five books are needed for this tower. As to the second question, the harmonic series  diverges; so long as we stack enough books, any finite offset can be produced. For large n, we have , where γ is the Euler-Mascheroni constant. Thus, for larger displacements, we have
,
and thus
,
and the number of books needed to achieve a given displacement grows (roughly) exponentially with that displacement.

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