Hey, guys. So here's another pretty straightforward moment of inertia question. The only difference here is that we're going to have density to deal with. So let's do this real quick. Since we have a planet that is nearly spherical, nearly spherical means that it's a sphere. Okay. So you can basically ignore the word nearly. It means we're going to approximate it as a sphere with nearly continuous mass distribution. Again, you can ignore the word nearly and assume that it has continuous mass distribution. Continuous mass distribution, we'll talk more about this later, but it basically means that the sphere has the mass evenly distributed throughout the sphere. So sometimes you see drawn like something like this. This is a solid sphere and this is opposed to a hollow sphere, which is a sphere that has nothing inside. It's not continuously or evenly distributed. All the mass is concentrated on the edges. This is not what we have here. This is what we have here. The reason why that's important is because you're going to get a different I equation, a different moment of inertia equation depending on what kind of sphere you have. And the moment of inertia equation for this guy here is:
2 5 m r 2The question didn't give you this, but you would look this up in your book or in a test. The professor would have to give you this somehow, unless your professor requires you to memorize this, then you'd have to do that. But most of them don't. Alright? So that's the equation you're supposed to use. I give you the radius right here. Radius is \(8 \times 10^7\) meters and I give you the density. Density, you can use d, but the official variable, if you will, is rho. Right? It's \(10,000 \, \text{kg/m}^3\). I want to remind you that if you have a volume, the density of a volume is mass over volume, and you could have seen this from the units, kilograms per cubic meter. Right? So it's a volume. It's a 3-dimensional object and that's it. That's all we're given. I also give you here the equation for the volume of a sphere. Volume right here of a sphere. Now if you were looking for I and if you start putting stuff in here, you would realize you don't have m, but you have r. Okay. So we don't have m. We gotta figure this out. And if you look around, you realize, well, I have another piece of information that has some connection to m. So maybe I can use this to solve for m and that's exactly what we're supposed to do. So I want to find m. I have 10,000, but I don't have v. But once again, I have another piece of information here that allows me to find v. V equals, the volume of a sphere is \(4/3 \pi r^3\) and I know r. I know r, so I can get v. I'm going to know v, so I'm going to be able to get m. I'm going to know m, so I'm going to be able to get I. Okay. That's how it's going to flow. Alright. So what I'm going to do is right here, I'm going to solve for m. In other words, I'm going to move v over here. So m equals 10,000v and v is according to this equation right here, \(4/3 \pi r^3\). So I'm going to get this whole thing, which is m and I'm going to stake it in here. Okay. So
2 5 10,000 4 3 π r 3 r 2These two terms combined \( r^5 \). So it's going to be \( 8 \times 10^7 \) to the 5th. Okay, so you should expect a pretty big number and I get \(5.49 \times 10^{43}\). Now, how exactly you arrive at this number doesn't really matter. It's I, so it's kilograms meter squared. You could have, you know, gotten a number here, plugged in here. It really doesn't matter as long as you arrive here. It's a bunch of multiplication. Cool. So that's it for this one. Let me know if you have any questions.