The Tallest Tree
Often on Fizzix Phriday, it seems that I talk about things that are crazy far away in space, or so small you can't see them, or weird esoteric things, which, while super cool, aren't something you can go out and see yourself. So today, I'm going to talk about something that's probably near you, unless you life in a desert or grassland: trees. Now, I've been known to be upset and angry about how big trees can get, because the General Sherman tree (a Giant Sequoia) is just too big to be imaginable. But it's not the tallest tree out there. The tallest trees are the redwoods, which are also mind-bogglingly huge. I'd highly recommend seeing them at some point in your life. However, I'm not here to talk about how tall the California Redwoods are, I'm here to figure out how tall they are allowed to be. And that allowed means we're bringing in some physics.
What really limits the height of a tree? Well, there are essentially two factors: the energy it takes to draw water to the leaves at the top of the tree from the ground, and whether the trunk of the tree can support its own weight. The first one is basically a function of photosynthesis and energy produced by leaves, and is really a biology and chemistry thing. This also comes into play sooner than the maximum height limit given by the physics of the tree itself, so a tree will always stop sooner than the point it would collapse. According to this article, this point is somewhere between 400 and 426 feet (122-130 meters). However, because this is Fizzix Phriday, we're going to do some physics to figure this out. And, as is often the case in physics, we're going to begin by assuming that the other height restriction doesn't matter, or can be somehow overcome. We're looking for how tall it is possible to make a tree, even if the tree has to defy biology to get to that height.
So, at what point will the tree collapse under its own weight? Well, that will depend on the compressability of the wood the tree is made of. At the point where the base of the trunk can no longer support the full weight of the tree, the tree will collapse. The best part of this is that the width of the tree doesn't matter at all here, only the height. Why is that? Well, there's a really neat principle, which, as it was first noticed by the artist Leonardo Da Vinci, is called Da Vinci's Rule. It states that the cross-sectional surface area of a tree at any point along the tree vertically is the same. What this means is that, if you take a slice of the tree at the base, and then another slice way up where there are tons of branches, and then you add together the tiny area slices of all the branches, the area you get will be the same as the area of the slice of the trunk at the base. What this means even further is that we can model our tree as simply a tall cylinder made of wood, and it's totally ok, and not a cheap spherical-cow-in-a-vacuum cheat.
But I said tree width didn't matter, and we'll see how that falls out. That's because of the math of the compressive force on the base of the tree. If we want to find the weight of the tree, which is really just a measurement of the downward force on the very base of the tree, we have to multiply the mass of the tree times g, the acceleration due to gravity. The mass of the tree is simply the density of the wood itself times the volume of the tree. Since the tree can be modeled as a cylinder, the volume is the height of the tree times the area of the base of the tree. To get the compressive stress on the base of the tree, we divide by the area that the force is distributed over, as the stress is in units of force over area (think things like psi—pounds (force) per square inch (area)). Oh, wait, that area is the area of the base of the tree. We divide out the area, and since the area of a circle is pi times the radius of the circle squared, the radius drops right out of our equation.
When you think about it, it also logically makes sense. If the tree is a cylinder, it can be thought of as a bunch of identical really tall pieces, the base of which each is only experiencing the compressive stress of the weight of the tiny vertical slice above them. You could keep making the size of these slices thinner and thinner, so the width of the tree doesn't matter, only the height and the density of the wood (because the weight of that slice will depend on the density). You can think of a thick tree as being composed of large numbers of basically infinitely thin trees, each of which has to deal with the compressive force of their own weight. In that way, how large or small you make the tree, only the height matters.
So, that's the compressive stress on the base of our tree due to its height. Since we have that, we can simply solve for what height the compressive stress is equal to what the type of wood the tree is made of can take, and that's the limit to the height of that kind of tree. Redwood has a density of 450 kg/m3, and a compressive strength of 8,273,709 Pa (Pascals, N/m2, which is force (Newtons) over area (square meters)). That means the theoretical maximum height of our redwood before it collapsed under the pressure of its own weight would be about one and a half kilometers. Now you can see why we build so many things from wood.
When it comes to compression, on a scale of one to strong, wood is very.