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I'm currently working on problem 6a of this problem set from MIT Open Course Ware. It's a spin on the book stacking problem. In this scenario, any additional books you stack beyond the first one has half of the weight of the previous book, and you also have a duplicate of the lightest book on the stack.

The solution in the problem set says that each additional book you stack in this case will give you exactly 1/4th of a book length for each additional book you add.

My problem with this conclusion is that the solution assumes "Thus, you can model this problem as stacking two books of equal weight". Isn't this untrue since a stack of books has an uneven weight distribution whereas a single book would have a uniform weight distribution? With a large book stack (> 3), the stack will get lighter the farther you move away from it's center of gravity, which would need to be compensated for when finding the new center of gravity.

Can someone explain how the solution came up with the fraction 1/4 as the constant growth length?

user886596
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1 Answers1

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Suppose we have a stack of $n$ books that is perfectly balanced, with total mass $m$, and that we are going to put a book underneath which also has mass $m$. Since the stack of $n$ books is perfectly balanced, its center of mass will be exactly at the edge. This means that when we add the extra book, it is like having a book of mass $m$ with a point mass of mass $m$ at one end. The center of mass of this configuration will be exactly $3/4$ of the way across the book, and since the center of mass is also the balancing point, this means that we can stick the bottom book out exactly $1/4$ of its length, and the stack will be balanced.

Eric Naslund
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