How far can a stack of identical books (of mass m and unit length) extend without

Chapter 10, Problem 90

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How far can a stack of identical books (of mass m and unit length) extend without tipping over? The stack will not tip over if the (n + 1)st book is placed at the bottom of the stack with its right edge located at or before the center of mass of the first n books (Figure 6). Let cn be the center of mass of the first n books, measured along the x-axis, where we take the positive x-axis to the left of the origin as in Figure 7. Recall that if an object of mass m1 has center of mass at x1 and a second object of m2 has center of mass x2, then the center of mass of the system has x-coordinate m1x1 + m2x2 m1 + m2 (a) Show that if the (n + 1)st book is placed with its right edge at cn, then its center of mass is located at cn + 1 2 . (b) Consider the first n books as a single object of mass nm with center of mass at cn and the (n + 1)st book as a second object of mass m. Show that if the (n + 1)st book is placed with its right edge at cn, then cn+1 = cn + 1 2(n + 1) . (c) Prove that lim n cn = . Thus, by using enough books, the stack can be extended as far as desired without tipping over.