Newton's law of gravitation
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Gravitation (part 2)
Welcome back. So I was trying to rush and finish a problem in the last two minutes of the video, and I realize that's just bad teaching, because I end up rushing. So this is the problem we were going to work on, and you'll see a lot of these. They just want you to become familiar with the variables in Newton's law of gravitation. So I said that there's two planets, one is Earth. Now I have time to draw things, so that's Earth. And then there's Small Earth. And Small Earth-- well, maybe I'll just call it the small planet, so we don't get confused. It's green, showing that there's probably life on that planet. Let's say it has 1/2 the radius, and 1/2 the mass. So if you think about it, it's probably a lot denser than Earth. That's a good problem to think about. How much denser is it, right? Because if you have 1/2 the radius, your volume is much less than 1/2. I don't want to go into that now, but that's something for you to think about. But my question is what fraction, if I'm standing on the surface of this-- so the same person, so Sal, if I'm on Earth, what fraction is the pull when I'm on this small green planet? So what is the pull on me on Earth? Well, it's just going to be-- my weight on Earth, the force on Earth, is going to be equal to the gravitational constant times my mass, mass of me. So m sub m times the mass of Earth divided by what? We learned in the last video, divided by the distance between me and the center of the mass of Earth. Really, my center of mass and the center of mass of Earth. But this is between the surface of the Earth, and I'd like to think that I'm not short, but it's negligible between my center of mass and the surface, so we'll just consider the radius of the Earth. So we divide it by the radius of the Earth squared. Using these same variables, what's going to be the force on this other planet? So the force on the other planet, this green planet-- I'll do it in green-- and we're calling it the small planet, it equals what? It equals the gravitational constant again. And my mass doesn't change when I go from one planet to another, right? Its mass now is what? We would write it m sub s here, right? This is the small planet. And we wrote right here that it's 1/2 the mass of Earth, so I'll just write that. So it's 1/2 the mass of Earth. And what's its radius? What's the radius now? I could just write the radius of the small planet squared, but I'll say, well, we know. It's 1/2 the radius of Earth, so let's put that in there. So 1/2 radius of Earth. We have to square it. Let's see what this simplifies to. This equals-- so we can take this 1/2 here-- 1/2G mass of me times mass of Earth over-- what's 1/2 squared? It's 1/4. Over 1/4 radius of Earth squared. And what's 1/2 divided by 1/4? 1/4 goes into 1/2 two times, right? Or another way you can think about it is if you have a fraction in the denominator, when you put it in the numerator, you flip it and it becomes 4. So 4 times 1/2 is 2. Either way, it's just math. So the force on the small planet is going to be equal to 1/2 divided by 1/4 is 2 times G, mass of me, times mass of Earth, divided by the radius of Earth squared. And if we look up here, this is the same thing as this, right? It's identical. So we know that the force that applied to me when I'm on the surface of the small planet is actually two times the force applied on Earth, when I go to Earth. And that's something interesting to think about, because you might have said initially, wow, you know, the mass of the object matters a lot in gravity. The more massive the object, the more it's going to pull on me. But what we see here is that actually, no. When I'm on the surface of this smaller planet, it's pulling even harder on me. And why is that? Well, because I'm actually closer to its center of mass. And as we talked about earlier in this video, this object is probably a lot denser. You could say it's only 1/2 the mass, but it's much less than 1/2 of the volume, right? Because the volume is the cube of the radius and all of that. I don't want to confuse you, but this is just something to think about. So not only does the mass matter, but the radius matters a lot. And the radius is actually the square, so it actually matters even more. So that's something that's pretty interesting to think about. And these are actually very common problems when they just want to tell you, oh, you go to a planet that is two times the mass of another planet, et cetera, et cetera, what is the difference in force between the two? And one thing I want you to realize, actually, before I finish this video since I do have some extra time, when we think about gravity, especially with planets and all of that, you always feel like, oh, it's Earth pulling on me. Let's say that this is the Earth, and the Earth is huge, and this is a tiny spaceship right here. It's traveling. You always think that Earth is pulling on the spaceship, right? The gravitational force of Earth. But it actually turns out, when we looked at the formula, the formula is symmetric. It's not really saying one is pulling on the other. They're actually saying this is the force between the two objects. They're attracted to each other. So if the Earth is pulling on me with the force of 500 Newtons, it actually turns out that I am pulling on the Earth with an equal and opposite force of 5 Newtons. We're pulling towards each other. It just feels like the Earth is, at least from my point of view, that the Earth is pulling to me. And we're actually both being pulled towards the combined center of mass. So in this situation, let's say the Earth is pulling on the spaceship with the force of-- I don't know. I'm making up numbers now, but let's say it's 1 million Newtons. It actually turns out that the spaceship will be pulling on the Earth with the same force of 1 million Newtons. And they're both going to be moved to the combined system's center of mass. And the combined system's center of mass since the Earth is so much more massive is going to be very close to Earth's center of mass. It's probably going to be very close to Earth's center of mass. It's going to be like right there, right? So in this situation, Earth won't be doing a lot of moving, but it will be pulled in the direction of the spaceship, and the spaceship will try to go to Earth's center of mass, but at some point, probably the atmosphere, or the rock that it runs into, it won't be able to go much further and it might crash right around there. Anyway, I wanted just to give you the sense that it's not necessarily one object just pulling on the other. They're pulling towards each other to their combined center of masses. It would make a lot more sense if they had just two people floating in space, they actually would have some gravity towards each other. It's almost a little romantic. They would float to each other. And actually, you could figure it out. I don't have the time to do it, but you could use this formula and use the constant, and you could figure out, well, what is the gravitational attraction between two people? And what you'll see is that between two people floating in space, there are other forms of attraction that are probably stronger than their gravitational attraction, anyway. I'll let you ponder that and I will see you in the next video.