The key idea is this: instead of thinking in terms of building the stack from
the bottom up, let's think in terms of building it from the top down. You'll
see what we mean as we go along.
Assuming that we have only one block, what's the best that we can do? As we've
already seen, we can lay the block halfway on and halfway off the table, so
that it is perfectly balanced. But what makes the block perfectly balanced?
It is that the center of gravity of the block, which is at its midpoint, is
exactly at the edge of the table.
Figure 3: One Block Stacked
Let's develop a coordinate system and use Maple to compute and keep track of
centers of gravity. Imagine a horizontal axis with its origin at the right edge of the table, positive
numbers extending to the right, and divided into units of feet (just like the
blocks). In this system, the center of gravity of the first block is at
coordinate 0.
If you haven't already done so, start up Maple. We'll use it to
remember where the center of gravity of each block lies, beginning
with the first block.
| cg1 := 0; |
Keep in mind: if the center of gravity of a block is zero, it is perfectly
balanced. If it is negative (and thus over the table), the block is still
balanced. If it is positive, the block is unbalanced and will fall.
Joseph L. Zachary
Hamlet Project
Department of Computer Science
University of Utah