Hey, guys. So we saw from the universal law of gravitation that we could calculate the force between 2 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 1 and mass 2 and they are separated by some distance r, 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, F=Gm1m2r2. 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 m ones and m twos, the big object, it's a capital M, and then you'll see the small object has a lowercase m. So it's still the product of the 2 masses, big M and little m over r squared. This is really the same equation. It's the same thing. I'm not telling you anything new. It's just for a separate, situation.
Okay, so what is this little r 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 r 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 r here and that's made up of 2 lengths. The first length from the core out to the surface is called the radius. If you think about the earth just a giant sphere, then the core to the surface is capital R, 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 r 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+H.
Now, I want to point out that I have 2 different kinds of variables here or 2 different kinds of letters. I have an uppercase letter or a capital letter and a lowercase letter. So in physics, a capital letter is always used to represent constants. So for instance, the radius of the earth, a constant, never changes. Whereas this lowercase letter h here represents a variable because you can go any distance away from the surface of the earth, but the radius always stays the same. So that's basically it, guys. Let's go ahead and start working out this example. So in this example, we have the height above the earth. If we're asked at what height above the earth is the gravitational force on a satellite equal to 1000 newtons, let's go ahead and draw a little diagram here and figure out what's going on. So we have the earth represented as that sphere, and I'm just gonna go ahead and represent the satellite as a dot. Don't wanna show you my bad drawing skills.
Okay. So we have, 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 1000 kilograms. And because we're working with the gravitational force here, I wanna go ahead and write out that equation. So I've got Fg=GMmr2. Remember, because we're working with a planet and a small mass, we're going to use that. And really, what am I looking for? Well, I'm actually looking for the height above the surface, so I'm looking for H. But I'm gonna use this gravitational force to solve for that.
Okay, so what you might be tempted to do is replace this formula with r +h, capital R +H squared, and then use that to solve for little h. But I want to warn you against doing that because you're actually going to 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 R or H, first go ahead and solve for little r first using Newton's law of gravity, and then using this equation, you can solve for whatever you want. So don't do this. Instead, what you're gonna do is gonna solve for Fg=GmM1r2. Go ahead and solve for little r, and then we can use this equation, r equals big R+h, in order to figure out the variable that we're looking for. In 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 r distance.
Okay. So let me go ahead and write out all of my knowns here. Right. So we've got the Well, in this diagram, we've got the center of the earth. And then if I wanted to find the little r distance, that's going to be 2 things. I've got the radius of the earth and then I've got a height above the center. So that height is really what I'm looking for. And I've got this little r here, that is that distance. We're going to be solving for that first. Okay. So I've got that 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 it's a constant. I have the mass of the satellite, and I also have the gravitational force between them. So I can go ahead and use and solve for little r. Let me go ahead and write all that stuff out. So I've got that capital M is equal to 5.97 times 10 to the 24th. And, I know what G is, and then Yeah. That's basically it. Awesome. So if I go ahead and rearrange this equation right here, so this, this gravitational equation, I can come up with this expression. R squared, if I move that to the other side and then move the Fg down, is equal to G times capital M, lowercase m over the force of gravity. So because this is a square, I can take the square root of both sides, and I'm gonna get that r equals the square root of Gmlittlem 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 to the minus 11. Then I've got the mass of the earth, 5.97 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 this stuff into your calculator, you should notice that, well, you should get 2 times 10 to the 7th meters. So we're done, right? Well, no, because remember this number here only 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 +H equation. So if I wanted to figure out what H is, I can go ahead and use this equation and figure out that H is equal to little r minus big R, which is the radius of the earth. But what is that value? What is that capital R? We haven't been given a value for that yet. Well, if I look here at 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 for my final answer, I've got H is equal to little r, which is 2 times 10 to the 7th, minus big R, which is 6.37 times 10 to the 6th, and that's gonna be in meters. So if you go ahead and work this out for the final answer, you get 1.36 times 10 to the 7th, 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.