 So today we're going to talk about a concept, a relatively complex concept, a geometrical concept known as the area moment of inertia, or, as it's often called as well, the second moment of area. So when we were introduced to moment, we considered a force. For example, the force on a seesaw applied at some distance from some axis point. So if I consider two weights applying a force, and they're each a distance from the axis point in this picture here, we're going to recognize that there's some effect with a magnitude that depended on this perpendicular distance, d. And we call this the moment arm. And our example in this case was something that we actually called moment, which is also sometimes in physics class called torque. And in this case, the moment was equal to the force, the force being equal to the weight of the objects here, times the distance. And we typically consider the total sum of all these moments. So the forces due to weight one and the forces due to weight two were summed together to give a total moment around the axis point, some moment m. And if we were in equilibrium or looking for equilibrium, we would find conditions that resulted in that moment being zero. This dependence on distance is called a first moment. And it's a first moment of whatever it's applied to. So in this case, this would be perhaps a first moment of force. Although generally we kind of just leave out the terms of force when we're using the term moment because that was the first sort of definition that we use for a moment. Sometimes this is also called weighted by. In other words, our moment weighted by the force, okay, referring to sort of our initial concept of applying this to weights on a seesaw. It turns out in physics, and in engineering, and in other disciplines, including statistics, that other effects might depend on other exponents of the distance. For example, we can have the zero with moment, the first moment, which we just saw the second, the third, the fourth, and so on. And we might represent them this way. The zero with moment would be equal to the force times the distance to the zero with power. Well, as it turns out, the distance to the zero with power is just a value of one. So that would end up being the force itself, not particularly interesting and no real reason to describe the zero with moment that way. However, the first moment is the one we just discussed, the force times the distance to the first power or simply the force times the distance. And then there are other moments. Let's put little brackets around these. The second moment might be considered the force times the distance squared, or the third moment is the force times the distance cubed, or the fourth moment, etc., is the force times the distance times whatever exponent we choose to use. Similarly, it could also just depend on other weighting choices or other weighting choices. So instead of talking about the moment of force, we might talk about the moment of mass, m d one, m d squared, m d cubed, for example. That would be the third moment of mass or of area, a d one, a d squared, a d cubed. That would be the third moment of area or the second moment of area. Notice our title talked about the second moment of area, which is the one we're interested in discussing here. The second moment of mass represents the resistance of a body to rotational moment around an axis. So if I have a mass and I'm trying to make it spin, perhaps it's attached to an axis or otherwise related to some particular axis at a distance r, there is a value called the second moment of mass, and it's also called the moment of inertia and abbreviated i. i equals m r squared. This is also sometimes called rotational inertia. And by the way, you can think about this axis as going sort of into the piece of paper or into the page and that the mass is being attempted to be spun around that, and it has a resistance to it and that resistance is this moment of inertia. There's a similar resistance associated with area, but area has sort of a particular orientation. If you take an area, it's facing a particular direction. We'll sort of assume that this area is facing toward the right and that it's moving back into the page in my third dimension there. And we can take the axis of rotation, or if we consider the axis of rotation, to be in the same plane as whatever that area is. So here's the area a. It's facing off to the right, but we're going to maybe rotate it around a plane that's going into the page, or an axis going into the page, or we can rotate it around an axis that's vertically. But that axis is in the same sort of location or in the same plane as the area. So I can take this area and flip it around this axis, or maybe I take this area and rotate it around a vertical axis. If I try to do either of those, there's a resistance that's associated with that. And this resistance is similarly called the moment of inertia, which can be confusing unless you're paying close attention to your context. So to distinguish between the two of them, instead of calling this one moment of inertia, often it will be called the area moment of inertia to distinguish for them the fact that we're actually using the area, or better yet, the second moment of area. And notice based on what we said before, if there's a second moment, it must be associated with some distance squared, scaled by or weighted by the area. The big question is, however, what this distance actually is, because that distance depends on what axis you're choosing to rotate by. If I decide to call this, for example, my x-axis and call this vertical one my y-axis, I could choose to spin myself around either of those axes. Well, in the first case, if I want to rotate around the x-axis, the distance to this point a is actually a distance y, measured along the y-axis. To spin around x, I have to measure along y. Similarly, to spin around y, I have to measure along x. So looking at this area system a little more directly, here's my x-axis, here's my y-axis, and here's my area A. I actually have two different ways that I can define the second moment of area. One of them is abbreviated again with the i, the same moment of inertia i, but I will use two x's here to represent that we're doing a rotation through something that's parallel to the x-axis. And this may be the x-axis itself, or more typically it is something that's parallel to the x-axis, but through the middle of the area. And we define that as the area times the distance y squared, where the distance is how far we are from the x-axis. Notice if the x-axis runs through the middle, then that value goes to zero, and it doesn't really resist the rotation. We can also define i, y, y, which is how much this is tended to resist a spin around the y-axis. And in that case, the distance we use would be the distance x, how far we are from the y-axis. So our moment of inertia around a vertical axis is going to be scaled by the area times the distance x from that vertical axis. So this is for a little piece or a little part of area, but a total second moment of area can be found by summing up, or if we use the calculus terms, integrating the product of area, here's the area, pieces. So if we take the area and we chop it up into little pieces, we can integrate the product of those area pieces times the distance from whatever rotational axis we choose. And unless otherwise noted, we choose to have that axis go through the centroid or the middle of the cross-sectional area. Let's consider, for example, a very typical construction form of an i-beam. And we're going to think about this i-beam as being a series of little chunks of area, and we'll make them all equal for the time being, although if you actually do a calculation you don't have to do that. We're going to run the x-axis through the middle, there's a symmetry to this i-beam, we'll run the x-axis through the middle of that beam, and we'll define the distance for each of these pieces with the letter y. And we'll say y i to represent a series of each pieces. For this, my pp is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We can label them all with a number. We'll let i represent the number of which piece we're talking about. And if I want to find my moment of inertia or my second moment of area around this neutral x-axis, I add up, that's the sigma notation meaning the sum of a whole series of things, each area times its distance squared. So each of these pieces has an area, a i, and it has a distance from the x-axis, y i. Note that all the contributions here are positive because a i is positive value, our area is always a positive area. And even if we go in the opposite direction and have a negative value for our y, it's going to be squared and therefore made positive. So each piece has a positive contribution to this geometric value called the area moment of inertia or the second moment of area. Let me take that same picture, replicate it here. And now this time we want to see how it rotates around its center, but this time the vertical center. You could either think about this as turning and rotating the entire form and laying it down another way or we could just consider it rotating around a vertical axis. Now in this case each of our little pieces here still has area a i, but now its distance from the rotational axis is in the x direction, so it has a distance x i. And we would define the second moment of area as being around a y-axis and we would do that by summing all these area pieces times that x-distance squared. Notice the units for both of these, units of area are going to be in a length times a length or length squared. And this value x i is also a length which is in turn squared. So now we have length times length squared or length to the fourth. So our units for these type of geometric constructs for the second moment of area are always going to be length to the fourth. For example, centimeters to the fourth, inches to the fourth, or perhaps meters or feet to the fourth, but usually it's much smaller scale for the types of things we calculate. So how does this relate to what we're doing? If we look at this picture here you can see this is a cross section of two different beams. The first one on the left is an i-beam, you can see the i-shape, and the one on the right is a half beam, sort of a channel beam laid on its side. And as we've learned before, if there is a moment, a type of bending moment applied to a beam, that can be considered as a series of distributed stresses where there is a lot of stress along the top and the bottom of the beam, and then that stress reduces to zero through the middle of the beam. True on both of these beams here, if we're bending in that particular direction. Well, if the moment is going to be resisted by the material, the amount of resistance is going to be greater in the area where there is more material. Material further from the center has a significantly larger effect. It's a squared effect because we said this distance, this distance y here goes a times y squared. So the things that are further away are much more effective in resisting the material. And there's a couple reasons for this. The first reason is because, well, it's further away. And like we said, the moment sort of depends on that distance. And the second reason is because the stress that's being applied there is also further away. So its distance from the center gets applied effectively twice. And so this particular value, this second moment of area is useful in representing how much a particular cross section in a beam resists the tendency to spin or resists internal bending moments that are applied. As it turns out, typical geometries for beams and rods, etc. have already been calculated. And there are numerous formulas that you can use to find cross-sectional areas and cross-sectional moments of inertia for typical beam forms. For example, simple rectangles or box hollow rectangle sections, circular sections, whether hollow or not. And then the sort of typical eye beam or even a triangular section all have typical formulas that can be used based on the sizes you might choose.