After playing with blocks for a while, people often proceed by piling the blocks and gently tapping the top block over till it barely balances, then tapping the second block from the top till it barely balances, and the third, etc. Using center of mass ideas from physics, we can calculate how far we can tap the block until the block will fall.

We make the usual physics assumptions: the blocks are uniform of length 1 and mass 1, perfectly rectangular, and no sudden gusts of wind or earthquakes disturb our delicate balance! 

The top block is easy to place; we can move it halfway over. Note this places the center of mass of this top block exactly at the edge. For convenience define the origin (x=0) to be at the right edge of the original tower of blocks (the "pivot edge").  

Now consider the top two blocks. The center of mass of this system is just the sum of the moments of the two blocks divided by the total mass:
moment top block = 1 * (0)   since the center of mass is exactly at the pivot point x=0
moment second block = 1 * (-1/2)  since its center of mass is half a unit to the left of the pivot edge.  
total moment = 0 + (-1/2) = -1/2
center of mass = total moment / total mass = (-1/2) / 2 = -1/4
Thus, we can move this set of two blocks over 1/4 until they balance right at the pivot edge.

 
Now consider the top three blocks.  As before, the center of mass of the top two blocks is at the pivot edge so its moment is zero.  
moment third block = 1 * (-1/2)  since its center of mass is half a unit to the left of the pivot edge.  
total moment = 0-1/2 = -1/2
center of mass = total moment / total mass = (-1/2) / 3 = -1/6
Thus, we can move this set of three blocks over 1/6 until they balance right at the pivot edge.

Similarly with the top four blocks; the center of mass is at -1/8 so we can move the four blocks over 1/8 to the pivot edge:

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April 2008  Return to Robert Styer Home Page