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The Leaning Tower of Lire


nathanstack15

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Recently in my BC Calc class, we've been talking about series and in some cases the application of them. The harmonic series is especially applicable to music: in music, strings of the same material, diameter, and tension whose lengths form a harmonic series produce harmonic tones. Another application of the harmonic series is the Leaning Tire of Lire, a theoretical structure. Suppose that an unlimited identical books are stacked on the edge of a table in such a way that the maximize the overhang. In order to maximize overhang but prevent the structure from collapsing, we can apply the formula for calculating center of mass: c= (x1M+ x2M2) / (M1+M2). In order to maximize the overhang, we need to stack the books in a way such that their center of gravity remains at x=0. This prevents the weight of the stack from applying a torque to the stack, which would result in an angular acceleration and the toppling of our structure. If we consider the center of mass of the stack with n+1 books, we get the following: 

lire_stackn_eq.gif

The length of the overhang, therefore, can be modeled by the harmonic series, lire_N_partial.gifTheoretically, the harmonic series will balance with an infinite number of books. It takes 31 books for the overhang to be two books long, 227 books for the overhand to be 3 books long, and over 272 million books for the overhang to be 10 books long. Crazy stuff.

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