Question

Show that it is possible to stack a pile of identical books so that the top book is as far as you tike to the right of the bottom book. You may use as many books as you need, but for any desired distance it will be some finite number. Hint: Start at the top of the pile to see how to stack them. Each time place the pile already completed on top of another book so that the pile is just at the point of tipping. (In practice, of course, you can't let them overhang quite this much without having the stack topple.) Find the distance from the right-hand end of each book to the right-hand end of the one beneath it. (To find a general formula for this distance, consider the three forces acting on book \(n\), and write the equation for the torque about its right-hand end.) Show that the sum of these distances is a divergent series. (It is said that some students studying in the library tried this experiment one evening and lefe the results, to the consternation of the librarian the next morning. It is suggested that you try it instead with one or more decks of cards. Also see "Leaning Tower of The Physical Reciens," American Journal of Physics, vol. 27, no. 2, p. 12 t.)

Short answer

The harmonic series \( H_n = \sum_{i=1}^n \frac{1}{i} \) is divergent, allowing an arbitrarily large overhang with enough books.

Step-by-step solution

Understand the Problem

The goal is to demonstrate that it is possible to stack books such that the top book can overhang to the right of the bottom book by any desired finite distance using a specific stacking method.

Concept of Overhang

To achieve the overhang, each book in the stack must be placed so that it barely avoids tipping over the edge of the book beneath it. The key idea is to place the top book first, and then successively place each additional book such that the entire stack remains stable.

Calculate Overhang for Book n

Consider the forces acting on each book. For book n, the torque about its right-hand end must be zero for equilibrium. Each successive book adds a smaller overhang distance. If book 1 is placed such that its right edge is just on the edge of book 2, the overhang is given by the formula.

Write the Equation for Torque

The overhang distance for each book due to the torque equilibrium is given by the harmonic series: \( H_n = \sum_{i=1}^n \frac{1}{i} \). The torque equation for book n regarding forces (gravity acting at the center of mass of each book) confirms that the series is divergent.

Divergent Series

The harmonic series \( H_n = \sum_{i=1}^n \frac{1}{i} \) is known to be divergent. This means that as the number of books increases, the total overhang distance can be made arbitrarily large, confirming that any finite distance is achievable with a sufficient number of books.

Key concepts

torque equilibrium

To understand how the books can be stacked to achieve overhang, we need to look at the concept of torque equilibrium. Each book is influenced by gravity, which acts downward at its center of mass. If we want the stack of books to remain stable and not fall over, we must make sure that the sum of the torques around any point is zero.
For book n, we consider the torque about its right-hand end. This means we calculate the forces acting on the book in such a way that they balance each other out. The formula for torque (τ) is given by the force applied (F) times the distance from the pivot point (r), so: \[ \tau = F \times r \]. In this scenario, we place one book on top of another in a manner that balances the torque, preventing the stack from tipping over.

harmonic series

The harmonic series plays a crucial role in calculating the overhang of the books. The harmonic series is an infinite series given by the sum of the reciprocals of the natural numbers: \[ H_n = \sum_{i=1}^n \frac{1}{i} \].
This series helps us understand how the overhangs accumulate as we keep stacking more books. Each term in the series represents the additional overhang distance contributed by each book. So, the top book in the stack may only overhang a small amount relative to the book just below it, but each additional book adds more to the overall overhang.
By using the harmonic series, we can compute the total overhang for any number of books.

divergent series

A series is divergent if its sum approaches infinity as more terms are added. The harmonic series, which is involved in our overhang calculation, is an example of a divergent series. Though each individual term in the harmonic series decreases as we add more books, the series itself grows without bound.
In the context of our stack of books, this divergence indicates that, theoretically, we can achieve an overhang of any desired finite distance simply by using enough books. The sum of the distances contributed by each book’s overhang adds up and does not converge to a finite limit. Therefore, with sufficient books, the overall overhang can always be increased.

overhang calculation

The calculation of overhang in this problem is a practical application of the harmonic series and torque equilibrium. To calculate the overhang, we start with the first (top) book and place each subsequent book so that the entire stack maintains equilibrium.
Let's denote the overhang distance of book n relative to the book below it. The overhang distance contributed by each book due to torque equilibrium is given by: \[ H_n = \sum_{i=1}^n \frac{1}{i} \]. This represents how each book slightly shifts the stack’s center of mass to just before the point of tipping. Through this method, the overhang for each successive book becomes smaller, but the overall effect accumulates.
This method ensures the stack remains stable while gradually increasing the total overhang. To figure out how far to place each book, you use the harmonic series to sum up these small overhang contributions, achieving the desired distance eventually.