Balanced Blocks

How far over the edge of a table can a stack of balanced blocks hang?

Figure 1

Before we continue, let's get a clear understanding of the question. Refer to Figure 1. We have three blocks on top of the table (in blue). These three blocks create an overhang (in green). Suppose we have an infinite amount of blocks with no forces on the set-up other than gravity. Each of the blocks are identical with a length "L". Our question today involves finding the maximum possible overhang without the blocks falling off the table.

Figure 2

An important concept in this scenario is center of mass. An object's center of mass is where we can imagine the force of gravity acting on it. For the blocks in this set-up, it is at the geometric center, as shown by the red dot in Figure 2.

To answer our main question, we'll look at a simpler situation first. Imagine just one block on top of a table. How long is the maximum possible overhang while the block is still balanced on the table? We can find the answer by thinking about the center of mass of the block. When the center of mass of a block is hanging off the object under it, gravity makes the block rotate and it will fall. On the other hand, if the center of mass of the block is on the surface below it (not hanging off), it will not fall. We can now rephrase the question. How long is the maximum possible overhang when the center of mass of the block is behind or on the edge of the table? This makes our question easy. Figure 3 shows that to get the maximum possible overhang, we move the block such that the center of mass of the block is directly over the edge of the table. If it was moved to the right, it would fall off. If it was moved left, the overhang would be decreased. Notice that the resulting overhang for the 1-block scenario is "L/2".

Figure 3

Let's take the next step forward. Suppose we have two blocks. What is the new maximum possible overhang? Remember, for maximum possible overhang, a block or set of blocks have to be moved such that their center of mass is positioned on the edge of the surface on which it rests. In the 2-block situation, there are two things to consider. First, the top block has to balance on the bottom block. This means that the center of mass of the top block will be placed on the right edge of the bottom block as shown in Figure 4. This is the exact same situation when there is 1 block and a table. In this case, the "1 block" is the top block and the "table" is the bottom block. The other thing to consider is that the bottom block, along with the top block on it, has to stay balanced on the table. To satisfy this while keeping a maximum overhang, we need to move the center of mass of the 2-block system on the edge of the table. To find the center of mass of the system, we can average the center of masses of each block (purple and pink line in Figure 4) to find the center of mass of the 2-block system (red dot). This is positioned right over the edge of the table. Notice that the top block creates the same overhang as the 1-block system. The bottom block adds an extra overhang of "L/4". The total created overhang is "L/2 + L/4 = 3L/4".

Figure 4

Figure 5

Now let's add a third block. You might have noticed that some of the positionings stay the same when we increase the numberla of blocks. In the 3-block scenario, there are important similarities with the 2-block case that will make our calculations easier (Figure 4 vs Figure 5). In the 3-block case, the top and middle block are positioned the same as the 2-block case, only with the bottom block acting as the table from the 2-block case. The only thing that changes from the 2-block to 3-block case is that an extra block is added under the 2-block case with some extra overhang. This extra overhang from the added block is determined by the position of the average center of mass of all the blocks. In this situation, the center of mass of all 3 blocks is located "L/6" away from the right edge of the bottom block, creating the overhang. Refer to Figure 5 for the 3-block case's positioning. Note that the total overhang is calculated as "L/2 + L/4 + L/6 = 11L/12".

You may notice that a pattern is beginning to emerge. If we skip the calculations, we get that the 4-block overhang is calculated as "L/2 + L/4 + L/6 + L/8 = 25L/24". If we use 5 blocks, the overhang is "L/2 + L/4 + L/6 + L/8 + L/10 =137L/120". We can deduce a formula for the total overhang as shown in Figure 6. The rest of Figure 6 may seem complicated if you don't know calculus, but the important takeaway is that if you continue adding numbers in the pattern of "L/2 + L/4 + L/6 + L/8 + ... ", the sum becomes infinity. We have our answer. If we have an infinite amount of blocks, the maximum possible overhang will be infinite. The formula for overhang allows us to make some interesting calculations. For example, suppose our blocks have a length of 1 foot and a height of 3 inches. If we wanted to reach an overhang of just 4 feet, we would need at least 12,367 blocks. If we multiply this by the height of each block, we get a total height of about 3092 feet. That's about 376 feet taller than the tallest building. Even crazier, if we stacked enough blocks such that it would reach the height of Mount Everest, the maximum possible overhang would still only be about 6.12 feet!

Figure 6