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Center of Mass Distance

Patrick Ford
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Hey guys. So we saw from the universal law of gravitation that we could calculate the force between two objects. Now, if these objects are relatively small, like we have here in this diagram, we call those objects point masses because we can treat them as points. So in this case, if these point masses have mass one and mass to and they're separated by some distance are the universal law of gravitation says that we could calculate the gravitational force between these two things, and that's given by this equation up here g m one m two over our square. But let's say we have a different situation in which we have something huge like a planet. And then we're calculating the force on a person that's orbiting some distance above that. So the universal law of gravitation still says that we need G times, the product of both of the masses. But in this situation, we're working with large objects instead of using em ones and M twos. The big object. It's a capital M, and you'll see the small object has a lower case M. So it's still the product of the two masses. Little big M and little em over R squared. This is really the same equation. It's the same thing. I'm not telling you anything. I'm not giving a different equation. It's just for a separate, um, situation. Okay, so what is this little our distance between the centers of Mass for point masses? It was just this distance between the two objects. But in this situation, in this diagram here, it's a little bit more complicated because now, if we want to get from the center of mass of the earth to the center of mass of the astronaut, because this little are is the center of mass distance between the two objects. First we have to go from the core or the center of the earth, out to the surface, and then we have to go some extra distance here above that surface. So, really, the whole center of mass length is this. Little are here, and that's made up of two lengths. The first length from the core out to the surface is called the radius. If you think about the earth is just a giant sphere in the court of the surface is capital are, which is the radius of the earth. And this extra distance here above the surface is called the height. So that little H is called the height. And we can see from this diagram that this little are is really just the sum of both of the radius and the height. So for large objects, the little R is gonna be capital R plus H. Now I want to point out that I have two different kinds of variables. Here are two different kinds of letters. I haven't uppercase letter or capital letter and a lower case letter. So in physics, a capital letter is always used to represent Constance. So, for instance, the radius of the earth a constant never changes. Whereas this lower case letter H here represents a variable because you could go any distance away from the surface of the Earth. But the radius always stays the same. So that's basically it, guys. So let's go ahead and start working out this example. So this example we have the height above the earth. So we're asked for at what height above the Earth is the gravitational force on a satellite equal 2000 Newtons. Let's go ahead and draw a little diagram here and figure out what's going on. So we have the earth representatives, that sphere, and I'm just gonna go ahead and represent the satellite is dot Don't want to show you my bad drawing skills. Okay, so we have, um Let's see, we've got the force of gravity. So we got the force. The gravitational force is equal to 1000 Newtons, and we have the mass of the satellite is equal to, ah 1000 kg. And we're because we're working with the gravitational force here. I want to go ahead and write out that equation. So I've got f g equals G times the big M little em over r squared. Remember, Because we're working with a planet and a small mass, we're gonna use that. And really, what am I looking for while I'm actually looking for the height above the surface. So I'm looking for a church, but I'm gonna use this gravitational force in order to solve for that. Okay, So what you might be tempted to dio is you might be tempted to replace this formula with our plus H capital R plus H squared and then use that to solve for little H, but I wanna warn you against doing that because you're actually gonna make the math a little bit more complicated than it needs to be. So instead of doing this here, I've got a pro tip for you guys. If you ever have a problem that asks you to solve for capital are or H first, go ahead and solve for little. Our first using Newton's law of gravity and then using this equation, you consult for whatever you want, so don't do this instead, What you're gonna do is we're gonna solve for f g equals G m m over little r squared. Go ahead and sulfur little r and then we can use this equation r equals big R plus h in order to figure out the variable that we're looking for And this problem We're looking for this h variable here. So let's go ahead and solve for that gravitational force and see if we can find the little our distance. Okay, so let me go ahead and write out all of my knowns here. Um right, So we've got the in this diagram. We've got the center of the earth. And then if I wanted to find a little our distance. That's gonna be two things. I've got the radius of the earth and that I've got a height above the center so that height is really what I'm looking for. And I've got this Little are here That is that distance we're gonna be solving for that first. Okay, so I've got that the, um the mass of the satellite is equal to 1000. What about Capital M? Because we need to know what capital M is. Well, I've got I've got G, which is the gravitational constant. I've got capital M, which is actually given over here. The mass of the earth. Remember? That's a capital letter. So gets a constant. I have the mass of the satellite, and I also have the gravitational force between them, so I could go ahead and use on soft for little are let me go ahead and write all that stuff out. So I've got that capital M is equal to 5 97 times 10 of the 24th, and I know G is and then yeah, that's basically awesome. So if I go ahead and rearrange this equation right here, so this this gravitation equation. I could come over this expression r squared. If I move that to the other side and then move the F G down is equal to G times capital, M lower case M over the force of whoops, the force of gravity. So because this is a square, I could take the square root of both sides, and I'm gonna get that r equals the square root of GM little em over the force of gravity. I'm gonna go ahead and start moving this over here so I can actually just go ahead and start plugging in values for this. So I've got 6.67 times 10 of the mice, 11 that I've got the mass of the earth 5 times 10 to the 24th. And then I've got the mass of the satellite, which is 1000. And then I've got the force of gravity, which is also 1000. If you go ahead and plug all of this stuff into your calculator, you should notice that we should get two times 10 to the seventh meters. So we're done, right? Well, no, because remember, this number here on Lee represents the full center of mass distance, not the H, which is what we're really looking for. So our last step is we're basically just gonna have to solve using the r equals R plus H equation. So if I wanted to figure out what H is going and use this equation and figure out that H is equal to little ar minus big are which is the radius of the earth. But what is that value? What is that capital are we haven't been given a value for that yet. Well, if I look here in my gravitational constants that capital r is just represents the radius of the earth which is given right here as this number. So I've got that from my final answer. I've got h is equal to little are, which is two times 10 to the seventh minus big are, which is 6.37 times 10 to the six, and that's gonna be in meters. So if you go ahead and work this out for the finance so you get 1 times 10 to the seventh, and that's gonna be in meters, so that's about 13,600 kilometers above the surface. So that is the answer for this. Let me know if you guys have any questions with this