# I am aware of gravity, yes.

July 30, 2013 7:03 AM Subscribe

Why do blocks fall?

I was stacking some of my son's blocks yesterday, sturdy square blocks, and after a certain height, it just seemed like the ability to balance them became more and more difficult, no matter where I placed them. Seriously, why?

I was stacking some of my son's blocks yesterday, sturdy square blocks, and after a certain height, it just seemed like the ability to balance them became more and more difficult, no matter where I placed them. Seriously, why?

Two reasons I can think of: first, are the faces of the blocks perfectly flat? Do they lie perfectly flush with one another? Or can you "rock" them back and forth a little bit? If they can rock at all, then when you have a large number of block stacked, they'll be able to rock a lot.

Second, you're never going to get one block perfectly aligned with the one below it. As you add more and more blocks, each one is going to be slightly misaligned with the next. The cumulative effects of this misalignment grow as the stack gets taller (it's essentially a random walk), and there's always going to be some number of blocks for which the cumulative misalignment makes the tower lean a little too much.

posted by Johnny Assay at 7:12 AM on July 30, 2013

Second, you're never going to get one block perfectly aligned with the one below it. As you add more and more blocks, each one is going to be slightly misaligned with the next. The cumulative effects of this misalignment grow as the stack gets taller (it's essentially a random walk), and there's always going to be some number of blocks for which the cumulative misalignment makes the tower lean a little too much.

posted by Johnny Assay at 7:12 AM on July 30, 2013

If you were gluing the blocks together, it would be much more stable as you went up because you just have one joint to worry about.

The problem is that since you are not gluing them, not only must each block be balanced on the one just below it, each new block must maintain the balance of every joint below it.

posted by seanmpuckett at 7:20 AM on July 30, 2013 [1 favorite]

The problem is that since you are not gluing them, not only must each block be balanced on the one just below it, each new block must maintain the balance of every joint below it.

posted by seanmpuckett at 7:20 AM on July 30, 2013 [1 favorite]

Also, any slight irregularity or movement near the bottom is amplified at the top. If the surface you stack them shifts just a bit (imagine a slightly flexible wooden floor, or carpeting) the taller the stack, the greater lateral movement this will cause at the top. Similarly, if the surface isn't perfectly flat and level, the whole tower will be forced to lean over which clearly is untenable beyond a certain height.

posted by exogenous at 7:41 AM on July 30, 2013 [1 favorite]

posted by exogenous at 7:41 AM on July 30, 2013 [1 favorite]

Even if you are perfectly placing your blocks, you have to deal with torque and moment of inertia.

As you are moving/placing blocks further up you are farther and farther away from the center of mass of the whole tower.

Since you are farther from the center of mass, you are exerting more torque on the tower itself. That is, small slips or pushes as you slide around that block on top have a greater effect on destabilizing the tower as a whole.

Then, the blocks themselves have a low moment of inertia. Put more simply, they are easily rotated. If you were stacking plywood, for example, this would have a higher moment of inertia and you could make much taller stacks than with your blocks before it felt fragile.

posted by vacapinta at 8:08 AM on July 30, 2013 [3 favorites]

As you are moving/placing blocks further up you are farther and farther away from the center of mass of the whole tower.

Since you are farther from the center of mass, you are exerting more torque on the tower itself. That is, small slips or pushes as you slide around that block on top have a greater effect on destabilizing the tower as a whole.

Then, the blocks themselves have a low moment of inertia. Put more simply, they are easily rotated. If you were stacking plywood, for example, this would have a higher moment of inertia and you could make much taller stacks than with your blocks before it felt fragile.

posted by vacapinta at 8:08 AM on July 30, 2013 [3 favorites]

If there's a little piece of crud (dust, cookie crumb, etc) on any part of even one block, that's going to make that block sit at a slight angle to the surface you set it on.

In theory, the probability is that pieces of crud would distribute themselves around such that it didn't cause significant asymmetry in the block stack as a whole (i.e. not creating the leaning tower of Pisa) but it would definitely introduce some wobble into things.

posted by aimedwander at 8:14 AM on July 30, 2013

In theory, the probability is that pieces of crud would distribute themselves around such that it didn't cause significant asymmetry in the block stack as a whole (i.e. not creating the leaning tower of Pisa) but it would definitely introduce some wobble into things.

posted by aimedwander at 8:14 AM on July 30, 2013

To go a little more structural engineering-y, blocks are a material with good compressive strength, but no tensile (pulling) strength. Since they are probably pretty light for their size, there isn't a whole lot of weight "gluing" them together. Since there is very little (relatively) force holding the blocks together, and as the tower gets higher and higher there are more and more forces trying to make the tower sway, the blocks are unable to use their own weight to hold together.

Another way to think about it is to imagine a stick made of the same material as the blocks, and the same length of the height of the tower you are creating. If it seems hard to imagine that stick standing up on its own, it would be even more difficult to get a similar stack of blocks to stand on its own.

Another way to visualize it is to think of the concept of angle of repose. The force of gravity is unalterably *straight* down. Build a pretty good stack of blocks to the point that it is getting difficult to keep up. Then put a weight on the end of a string. This will act as a reference for what direction gravity is pulling. Compare it to your tower. You'll probably see that there is some empty space directly below some of the upper blocks. That means that gravity is trying to pull those blocks off the tower, in a sense. Gravity is pulling straight down, but the tower isn't perfectly, directly underneath itself, so it's pulling at an angle compared to the base of the tower. As the center of mass goes up in the structure, it becomes easier and easier for it to move out from the center axis of the tower and find itself no longer directly supported. This shifts the demands put on the material from directly compressive to tensile, and the tower fails.

posted by gjc at 8:17 AM on July 30, 2013 [1 favorite]

Another way to think about it is to imagine a stick made of the same material as the blocks, and the same length of the height of the tower you are creating. If it seems hard to imagine that stick standing up on its own, it would be even more difficult to get a similar stack of blocks to stand on its own.

Another way to visualize it is to think of the concept of angle of repose. The force of gravity is unalterably *straight* down. Build a pretty good stack of blocks to the point that it is getting difficult to keep up. Then put a weight on the end of a string. This will act as a reference for what direction gravity is pulling. Compare it to your tower. You'll probably see that there is some empty space directly below some of the upper blocks. That means that gravity is trying to pull those blocks off the tower, in a sense. Gravity is pulling straight down, but the tower isn't perfectly, directly underneath itself, so it's pulling at an angle compared to the base of the tower. As the center of mass goes up in the structure, it becomes easier and easier for it to move out from the center axis of the tower and find itself no longer directly supported. This shifts the demands put on the material from directly compressive to tensile, and the tower fails.

posted by gjc at 8:17 AM on July 30, 2013 [1 favorite]

Tall thin columns like the one you describe can fail due to buckling.

posted by Rob Rockets at 8:50 AM on July 30, 2013

posted by Rob Rockets at 8:50 AM on July 30, 2013

Imagine your tower of blocks is one big cuboid, then (assuming the blocks are all identical and perfect rigid cubes etc), the centre of mass will be halfway up the tower, in the middle of it. Now picture a vertical line travelling from the centre of mass to the floor. Our slightly abstract tower will keep standing up as long as this line goes through its 'footprint', the area of it touching the floor. For example, if you give the tower a little nudge, it will return to the standing position as long as the line doesn't move outside the footprint (page 14 of this sheet has a good picture).

As you build your tower higher, its centre of mass also gets higher, but the size of the footprint stays the same, so its tipping angle, or the nudge required for it to fall gets smaller, making it less stable. In the case of a tower made from real blocks, the problem is made even worse as the centre of mass is going to move away from the central axis of the tower, since they aren't being stacked perfectly. Eventually, the line from the centre of mass will go outside the footprint, either due to the placement of a brick or a vibration/draught knocking it, and then it will fall over.

The fact that your tower is made of bricks that can slip off each other rather than it being a single object makes it a bit more complicated to work out when it will fall (if a nudge isn't big enough to topple the tower, but it causes a brick to move then the tower is less stable as its centre of mass has moved further out over its footprint) but the basic idea is still the same, a big base is good, a high centre of mass is bad.

posted by Ned G at 9:46 AM on July 30, 2013 [1 favorite]

As you build your tower higher, its centre of mass also gets higher, but the size of the footprint stays the same, so its tipping angle, or the nudge required for it to fall gets smaller, making it less stable. In the case of a tower made from real blocks, the problem is made even worse as the centre of mass is going to move away from the central axis of the tower, since they aren't being stacked perfectly. Eventually, the line from the centre of mass will go outside the footprint, either due to the placement of a brick or a vibration/draught knocking it, and then it will fall over.

The fact that your tower is made of bricks that can slip off each other rather than it being a single object makes it a bit more complicated to work out when it will fall (if a nudge isn't big enough to topple the tower, but it causes a brick to move then the tower is less stable as its centre of mass has moved further out over its footprint) but the basic idea is still the same, a big base is good, a high centre of mass is bad.

posted by Ned G at 9:46 AM on July 30, 2013 [1 favorite]

Or to simplify things a bit, think of a lever. The farther you are from the fulcrum, the less force it takes to push it. So the farther your tower of blocks gets from the floor, the larger effect small imbalances are going to have.

posted by valkyryn at 9:55 AM on July 30, 2013 [1 favorite]

posted by valkyryn at 9:55 AM on July 30, 2013 [1 favorite]

*after a certain height...*

Easy. The blocks aren't perfectly square. They have manufacturing tolerances ("make all blocks .5mm...plus or minus .08mm"). As you stack more blocks up, the tolerances add up to, adding sway...which eventually leads to collapse.

This, is one of the reasons real buildings use some kind of malleable paste (like concrete) between bricks like that. To fill in for the tolerances.

posted by hal_c_on at 1:07 PM on July 30, 2013

I missed a quantum mechanics lecture (I wasn't very good at universitying) where the lecturer proved that even an absolutely perfect pencil balanced on its tip would be expected to fall over within a few seconds, so even if you had flawless blocks you'd run into trouble eventually.

posted by lucidium at 3:09 PM on July 30, 2013

posted by lucidium at 3:09 PM on July 30, 2013

The argument that appeals to me most is that the effect of any deviation from perfect flatness (the extent to which the two sides of the block which are in contact with other blocks in the stack are not perfectly parallel) in a given individual block tends to accumulate as blocks pile up above it.

For example, suppose you had a very large number of perfect blocks and one block which was not perfectly flat.

If you put the imperfect block near the bottom of your stack and stacked the other blocks above it, no matter what you did the blocks above would lean in a certain direction more with each individual block added, and eventually the line drawn from the center of gravity of the blocks above would fall outside the footprint of the imperfect block and the stack would topple.

In a set of real blocks, however,

The arrow drawn from the center of gravity of the blocks above would therefore move in a random walk around the footprint of any given block in the stack as each new block was added, and as soon as that random walk reached the edge of the footprint of any block in the stack, the whole thing would topple.

In a standard random walk, where each step is of equal magnitude, a random walk will eventually go beyond any fixed distance from its starting point, and my argument would have shown that the stack must eventually topple.

In this case, however, the magnitude of each succeeding step in the random walk is 1/n, where n is the total number of blocks above a given block, because the mass of that nth block is only 1/n of the total mass of the blocks above that given block.

But the total distance the random walk has covered in this case after the nth block (as opposed to the distance from the origin) is 1+1/2+1/3+ ... +1/n (the harmonic series) which grows without limit just as n itself does (though much more slowly), and so the distance from the starting point of the random walk in this case will also grow without limit (I haven't actually proved this last assertion, but I hope I've made it more likely)-- and so the stack must topple.

posted by jamjam at 6:15 PM on July 30, 2013

For example, suppose you had a very large number of perfect blocks and one block which was not perfectly flat.

If you put the imperfect block near the bottom of your stack and stacked the other blocks above it, no matter what you did the blocks above would lean in a certain direction more with each individual block added, and eventually the line drawn from the center of gravity of the blocks above would fall outside the footprint of the imperfect block and the stack would topple.

In a set of real blocks, however,

*all*the blocks are imperfect, and the deviations would add together and cancel each other out in a random fashion with the addition of each new block to the stack.The arrow drawn from the center of gravity of the blocks above would therefore move in a random walk around the footprint of any given block in the stack as each new block was added, and as soon as that random walk reached the edge of the footprint of any block in the stack, the whole thing would topple.

In a standard random walk, where each step is of equal magnitude, a random walk will eventually go beyond any fixed distance from its starting point, and my argument would have shown that the stack must eventually topple.

In this case, however, the magnitude of each succeeding step in the random walk is 1/n, where n is the total number of blocks above a given block, because the mass of that nth block is only 1/n of the total mass of the blocks above that given block.

But the total distance the random walk has covered in this case after the nth block (as opposed to the distance from the origin) is 1+1/2+1/3+ ... +1/n (the harmonic series) which grows without limit just as n itself does (though much more slowly), and so the distance from the starting point of the random walk in this case will also grow without limit (I haven't actually proved this last assertion, but I hope I've made it more likely)-- and so the stack must topple.

posted by jamjam at 6:15 PM on July 30, 2013

This thread is closed to new comments.

posted by phunniemee at 7:06 AM on July 30, 2013