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\newcommand{\semester}{Spring 2004}
\newcommand{\class}{Ma236--Calculus Laboratory}
\newcommand{\labtitle}{ Center of Mass }

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\begin{center}
{\bf \large \em \labtitle}\footnote{adapted from {\em Calculus: Early
Transcendentals}, 4th ed. Stewart, pg 780}
\end{center}
\vspace{.1in}

Consider the stack of identical books pictured in Figure \ref{stack}.
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\begin{figure}[h]
\center
\includegraphics[width=3in]{stack}
\caption{A precarious stack of books.} \label{stack}
\end{figure}
%
Notice the top book on the stack extends completely beyond the footprint of
the bottom book.  Show that given enough identical books one can create a
stack such that the top book extends as far to the right of the bottom book
as you want.  Let the sequence $\left\{1/2,1/4,1/6,1/8\right\}$ 
be know as the {\bf
overhang sequence}. 

\begin{enumerate}

 \item Begin by computing the center of mass of the stack of books in
 Figure \ref{stack}.  (Hint: compute the center of mass along the $x$-axis
 of each book and average.)   Is the stack stable?  What does the center of
 mass tell about the stability of the stack?

 \item \label{arb} Compute the center of mass for an arbitrary extension of
 the overhang sequence $\left\{1/2,1/4,1/6,1/8,1/10,\ldots,1/(2n)\right\}$.  
 You may
 assume all of the books are of uniform size, density and have length 1.

 \item Compute how far the top book extends beyond the footprint of the
 bottom book for the arbitrary length overhang sequence in Part \ref{arb}.

 \item Explain how this problem is related to the {\em harmonic series}.


\end{enumerate}




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