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Moment of inertia of Earth

Patrick Ford
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Hey, guys. So, as you know, the Earth is rotating. Therefore, it has a moment of inertia. And if we make some assumptions about the shape of the Earth, we can actually calculate the moment of inertia. The earth. Let's check it out. So it says here the earth has a mass and radius given by these big numbers. And then I also tell you that the radio distance between the earth and the sun is this What I mean by radio distance is that, um if the sun is here, um, the earth spins around the sun in this distance here, little are is 15 times to the 11. Squeeze it in their cool. That's what I mean by that on. Then I gave you the mass of the Earth and the radius of the Earth as well. I want to know the moment of inertia of the earth as it spins around itself. And as it spins around the sun. As you know, the Earth has two motions on beacon. Calculate a moment of inertia about or relative to those two motions or for those two emotions, remember, Moment of inertia depends on the axis of rotation. That's why these numbers will be different. So if you want to know the moment of inertia of the earth around self, you would have to treat the earth as a zey as an object with a significant size you can't treat it is a tiny object. Eso What we do here is we're gonna treat the earth as a solid sphere. Okay, It's a solid sphere. So the earth is a big ball that spins around itself. Now, technically, it's at an angle like that, but it doesn't really matter. You could just do this. Okay, so it's spinning around itself, and your book would show you that solid spheres have a moment of inertia given by this equation right here. So when I tell you, solid sphere, I'm indirectly telling you Hey, use this equation for I, Okay, so for part a, we're going to do Party's over here. We're gonna say I equals to over five m r squared, and all we gotta do is plug in the numbers here. So M is the mass of the earth, which is 5.97 times 10 to the 24th, and R is the radius of the earth, which is this and not the radio distance. It's the earth going around itself. So it's the radius of the actual object to the sphere. Um, 6.37 times 10 to the six square. Okay, if you look at this number, I got a 24 and then I got a six squared. So you should imagine that this is gonna be, ah, gigantic number. And it is. I multiplied everything. I get 9.7 times 10 to the um, kilograms meters square. The earth has a lot of inertia. And what that means is that it would be incredibly hard to make the earth stops spinning. Okay, now, if you were to Google this number, you would see that that's actually a little bit off. The actual moment of inertia is a little bit off, and that's because the Earth is not a perfectly a perfect sphere. It's got different layers. It's not even sphere. Um, so but this number is a pretty good approximation for part B. We want to find out what is the moment of inertia of the earth as it spins around the sun. Now, in this case, relative to the sun. The earth is tiny, so we're gonna treat it as a point mass, which is crazy. The earth is huge thing and you're gonna just treat it as a little point massive, negligible radius. And that's because relative to the sun, the earth is negligible in size. Okay, so I'm gonna put the earth here as a tiny M Earth. Um, and it's going around the sun and the distance here. The radial distance, which is little, are big r is radius of an object and little artist. Distance to the center is 1.5 times 10 to the m. In this case, we're going to use instead of 2/5 m r. We're gonna use em are square because the earth is being treated as a point mass. Here M is the mass of the object itself, right? It's the object that spinning It's not son s. So it's gonna be 5.97 times 10 to the 24th same thing, But our is going to be the distance to the center, which is 1.5. So 1.5 times 10 to the 11th square. I got a 24 I got 11 square. This is gonna be again a gigantic number 1.34 times 10 to the 47. This number is, um, like, a billion times bigger than the other number. Right? So, as hard as it would be to stop the Earth from stopping to get the earth to stop spinning, Um, it would be way harder, right? It would be 10 to the 10 times harder to make the earth stops going around the sun. Um, and that's it. So that's a finished ones. Very typical classic problem. Hopefully, it makes sense. You should try this out on your own and let's keep going.