Hi and welcome to module 31 of two-dimensional dynamics. Today, we're gonna determine the Mass Moments of Inertia and products of inertia for bodies in 2-D planar motion. We derived their expressions last time and then I wanna explain the concepts of mass moment of inertia and product of inertia to give you a feel for what they are. And so this is where we left off last time. We had the angular momentum about any point for a body in 2-D planar motion, it ended up being this. And if we picked a particular point for P as either the mass center or a point is zero velocity, it reduced to this. Where these Ixz and Iyz are called products of inertia and Izz down here is called the mass moment of inertia. And so how can you determine first of all, these products of inertia and these mass moments of inertia for different bodies? And one way you can do it is you can do the integral mathematically, cuz you can see that you have an integral that you can do over the body. And so as an example, I just showed a calculation of the mass moment of inertia for a homogeneous solid cylinder about the z-axis. And so this ends up being a triple integral, you can go through the math. This is one standard type shape that you can calculate. This, as I say is the calculation for mass moments of inertia. You can do the same sort of thing for other bodies, other shapes and also for the products of inertia. But actually, since most of our engineering systems, either a standard body shape or a combination or a compilation of standard body shapes. We have tables, which you can find on the Internet or a text book reference for different types of shapes. And so as an example, here's the values for a rectangular solid cylinder and a thin right triangle. So this is a rectangular solid. We've got our x, y, z-axis welded at the mass center and oriented as shown. In this particular case, we only have mass moments of inertia and these are their values. Here's a solid cylinder. Again, the coordinate axis is welded at the mass center, oriented as shown and we have the Ixx and Iyy mass moments of inertia are the same and the Izz mass moment of inertia is different. And then finally, I've got a thin right triangular plate. And here, I included this example, because we have an Ixx mass moment of inertia, a Iyy mass moment of inertia about the y-axis and Izz mass moment of inertia about the z-axis. And this is a case where we actually have a product of inertia, as well. And I would encourage you to go in and find a text reference or on the internet, look at and get a feel for how you would find these different mass moments of inertia and products of various shapes. So now I wanna explain the concept of the mass moment of inertia. It is the integral about the z-axis. It is the integral over the body of (x squared + y squared)dm. And so the z-axis is in the center of the rotation and how far we go in the x and the y-axis to any little piece of mass and then integrated over the entire body gives us the mass moment of inertia. So we can say that this is how much mass is located? How far from the axis of rotation? And let's look at a quick demo here. Here I have an x, y, z-axis. I'm gonna rotate this body about the z-axis. If we would be able to calculate the mass moment of inertia for this body and then as a feel for what the mass moment of inertia is if I moved those masses further out away from the z-axis. So they are further out in the x and y directions, then I would have a larger mass moment of inertia. So that gives you a physical feel. [SOUND] How about the Products of Inertia? Well, this is the mathematical expressions for the product of inertia. Products of Inertia measure a lack of symmetry. And so if the products, you can have the Products of Inertia = 0. And so if there's a symmetry with a z-axis or if there's symmetry in the xy plane, then we get the products of inertia as being zero. And so here's an example of a cone. This cone shape, the z-axis is an axis of symmetry here. Here's my z-axis or the xy plane is also a plane of symmetry. So here's my xy plane. With my xy plane, if I do a mirror image above and below, it's the same. So, it's symmetric. In both of those cases, you would find that the product of inertia would be = 0. So I wanna show you one more example, a physical example. So we'll come over here. Here's a wheel, it's in planar motion and so it's spinning. This would have Products of Inertia that are equal to 0, because it's symmetric about the z-axis. And as well, it's symmetric about the xy plane. So a good example of a case, where we would only have a mass moment of inertia and no products of inertia. Thank you. Just because a body does not belong to either of these two examples, it doesn't mean it cannot have zero product of inertia. You can't always see when the products of inertia, you can't see the symmetry when the Products of Inertia = 0. But these are common cases that are worthy of note and give you some physical field for what the products of inertia are and it's common to choose coordinates. You can see, if I chose coordinates that were not welded in these shapes, then I would not have the symmetry and the products of Inertia for those coordinate axes would include a product of inertia. And so it's often common for us to choose coordinates such that the products of inertia do equal zero and we have only mass moment of inertia. Given all that, I'd like to give you an exercise to actually calculate the mass moment of inertia for an uniform hemispherical solid about the lateral axis x sub c through it's mass center. You can do it through integration. You can try to find this standard shape in a reference, but I'd like you to find the mass moment inertia for this body spinning about the x sub c axis. And I've got the solution in the handouts, so I can see how you did.