Inertia of planet of known density

by Patrick Ford
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Hey, guys. So here's another pretty straightforward moment of inertia. Question three Only difference here is that we're gonna have density to deal with. So let's do this real quick because 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 it's fear has the mass evenly distributed throughout the sphere. So you sometimes you see drawn like something like this. This'd 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 gonna get a different I e. 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 to over five m r Square. The question didn't give you this, but you would look this up in your book or on a test. He would have to give you this somehow. Unless you're Professor requires to memorize this thes, and then you have to do that. But most of them don't. All right, so that's the equation you're supposed to use. I give you the ratings right here. Radius is eight times 10 to the 7 m, and I get to the density density you can use D. But the official variable, if you will, is row right. It's 10,000, um, kilograms per cubic meter. I want to remind you that if you have a volume, the density of the volume is mass over volume. And you could have seen this from the units. Kilograms, um, cubic meter cube. Right. So it's a volume. It's a three dimensional objects. And and that's it. That's all we're given. I also give you hear the equation for volume of a sphere volume right here of a sphere. Now, if you were looking for I and if you start plugging stuff in here you would realize you don't have em, but you have are okay, so we don't have em. We gotta figure this out. And if you look around, you realize, Well, I have another piece of information that has some connection to em. So maybe I can use this to solve for M, and that's exactly what we're supposed to do. So I wanna find em. I have 10,000, but I don't have the. But once again, I have another piece of information here that allows me to find V V equals the volume of the spheres. Four thirds pi r cube. And I know our I know are So I can get the I'm gonna novi. So I'm gonna be able to get em. I'm gonna know em. So I'm gonna be able to get I Okay, That's how it's gonna That's how it's gonna flow. All right, So what I'm gonna do is right here. I'm gonna solve for em. In other words, I'm gonna move you over here, so M equals 10,000. The and V is according to this equation right here. Um, four thirds pi r cubed. So I'm gonna get this whole thing, which is M. And I'm gonna stick it in here. Okay, so to over five times 10,000, four thirds pi r cute. This is just m And then I also have the R Square here. Tons of ours. Okay. And if you multiply this whole crap you're gonna end up with Let's see, this is 18. Um, I'm sorry. This is eight. So this is gonna be 80, divided by 15. So it's 80,000 pie. I got that right. Yep. Divided by 15 times are to the fifth. These two guys combined are to the fifth. So it's going to be eight times 10 to the seven. This thing to the fifth. Okay, So you should expect Ah, pretty big number. And I got 3.15 times. 10 to the 45. Now, how exactly you arrive at this number? Um, doesn't really matter its eyes. So it's kilograms meters square. You could have, you know, gotten a number here plugged in here. It really doesn't matter as long as you arrived here. It's a bunch of multiplication. Cool. So that's it for this one. Let me know if you have any questions.