 Moment of inertia. If we go back to the inertia concept that we've looked at before, we first saw this with linear motion. And inertia was a resistance to the change in motion. And in linear motion, that was directly related to the mass. The more mass it was, the harder it was to get that mass to change its motion. If I've got rotational motion, it still depends on the mass, but it also depends on how that mass is distributed. And by distributed, I mean, how is it arranged compared to the axis of rotation? Now, we're going to start with just a single particle. If I have a single particle moving in circular motion, there's an inertia with keeping that particle moving in that circular motion. And it's the mass times the radius squared. So the m is our mass, and r is the radius of the circular motion, or it could be expressed as the distance from the axis. Now, the units for this is going to be kilograms for my mass and meters squared for the radius squared, giving me kilogram meters squared. Now, conceptually, the moment of inertia, a higher inertia means I've got a harder object to rotate. A lower inertia means it's easier to get that object rotating. Now, if I go back to this first equation I have for a single particle, then I can also see that just like my regular inertia, more mass means more inertia. But now, I've got the distance from the axis, and the further away from the axis it is, the harder it is to get it rotating, meaning I've got a higher moment of inertia. Now, this was all looking at a single particle. But if I have a group of particles, I have to include a contribution from each one of them. So if I have a mass 1 out at a distance of r1 and a mass 2 at a distance of r2 and a mass 3 at a distance of r3, et cetera, I would calculate the individual moment inertia for each one of those masses and then add them together to look at my total moment of inertia. Now, because I'm adding these together, I could also use the math notation that I'm doing the sum of the individual inertia is using the individual masses and distances. If I've got a continuous distribution, that means I don't have a group of single, easily separated little particles, but I've got a solid object. I can divide that solid object into individual little masses where they're all right up against each other. Now, if I can use calculus, I can take those individual little masses and use calculus to find the total inertia over the entire solid. Now, the notation there is rather than looking at the summation, I'm looking at the integral. And I'm actually integrating over the mass. Practically speaking, trying to integrate over the mass is not quite as easy as integrating over some sort of geometric parameters. So instead, we represent this mass as being a combination of the density, row, and the volume. So I can integrate over the density times the radius squared, integrated over the volume. Now, if we've got some common shapes, we don't have to go through that messy calculus-based integral because they've already done that messy integral for some of the common shapes. So for example, if I've got a solid sphere, it's got an axis along its diameter. Each little piece of this is one of my little DMs that I'm integrating over. And doing the particular integration over that surface, we end up finding that the moment of inertia for that entire solid object is 2 fifths the total mass times the radius of the sphere squared. Now, this is just one example. What really happens is we've got tables listing a whole bunch of the different common shapes for various different axes of rotation. And it's going to give us the formula for that particular shape. So you could look those values up from a table. So that's our introduction to the moment of inertia. We still have to look at some examples for how to calculate it for individual particles or for larger distributions.