Balancing books (1 Viewer)

[glittergal96]

Suppose we have an infinite collection of identical planks of wood (which are rectangular prisms that are long in one dimension but short in the other two).

We place the first one on a table so that one of its ends is flush with the edge of the table.

Place the second one on top of it with all faces parallel to those of the first plank but shifted off the table slightly (whilst not being far enough for it to overbalance and fall).

Continue this process.

Q: Is it possible to extend an arbitrary horizontal distance beyond the edge of the table by using enough planks?


(Ignore the title, I changed it to planks for the idea to be clearer).

 

[SilentWaters]

Suppose the table can support an infinite weight.
Then yes, I suppose so.

Balancing is simply dependent on the center of gravity in our book stack, which should be at or behind the table's edge. The greatest horizontal distance beyond the edge occurs when a stack's center of gravity is directly above the edge. For our hypothetical book stack calculations, let us neglect the book touching the table (with its fixed zero overhang), so that the "bottom book" of our stacks is the one directly on top. For some books of mass each, width , we denote the moment of the stack of greatest horizontal overhang with respect to its center of gravity (force x distance) as .

Resolving this into a system composed of the bottom book and the books above it, an equivalent expression can be obtained for the moment. Considering the two centers of gravity, which for the bottom book would simply lie at the midpoint of its width, we note that to maximize overhang the stack of books must have its own COG directly above the edge of the bottom book. Denote this . Taking moments with respect to these COGs, the bottom book contributes to the total moment of the stack of books, while the contribution of the stack is . Hence:



Solving for , we obtain the recurrence relation:



Trivially, the maximum overhang for the bottom book would be . Successive iterations turn out to be:





So that:



The above generates all values for maximum overhang, and we know that by shifting books backward so that COGs recede behind the table's edge, all other values in between are obtained. Noting that the harmonic series is a divergent one*, we can conclude that overhang values tend to infinity with increasing , so that any arbitrary horizontal distance can indeed be attained.

*Divergence can be concluded simply by looking at the graph of to the right of , and noting that the harmonic series constitutes upper bound rectangle sums. Since the area beneath the graph is infinite, the harmonic series must also tend to infinity.

 

[glittergal96]

Yep, good work!

 

[Chlee1998]

Are u guys like part of the physics olympiad or something? ?

 

[glittergal96]

I'm not (I didn't even do physics in the HSC), but I have taught myself a bit by reading.

 

[Chlee1998]

Lol why didn't u join some maths olympiad or something

 

[glittergal96]

Lol why didn't u join some maths olympiad or something

I don't really care about competing, I just learn things because they are cool. Also, I am not good enough for olympiad.