D1.1 Gravitational fields and force [IB Physics SL/HL]

追加: 2026-05-05

[00:00:00]Obey gravity. It's the law. We're going to be talking about gravitational fields and force. So, first I think it helps to learn about gravitational field lines.

[00:00:10]So, these are going to be defined as uh these are these, you know, artificial lines that we kind of invent here that we draw them in the direction that a test mass would go. So, it seems a bit weird to do this, but it really helps us to know the direction of things going by. So, you know, we're we're always going to say that uh an object if it's got mass, then it will have these gravitational field lines, and they will go radially inward. So, they're always going to go from the outside to the inside. And that's because if I dropped a little mass, let's say I had a little mass right here. I put it here to the right. Where would it go? Well, this is the Earth. It would fall inwards, right?

[00:00:46]It would go, you know, this way. And then, of course, if I put it over here, it would go this way. And of course, if it was from the top, it would go down.

[00:00:53]It would go up. and so on and so on. So that's why we say in this exam tip that I have down below, I say that we draw them going radially inwards towards the center. Now, uh usually we're going to be assuming that the planet or the moon or whatever it is, it has all the masses located in the center. So we're going to assume it's kind of like a point mass.

[00:01:14]In other words, if we remove the surface here, you know, everything would just go right towards the center. We're also going to be assuming things like, you know, uniform density and composition.

[00:01:22]So, we're going to try to make our lives a little bit easier. So, we have Newton's universal law of gravitation. I like this. And I'm attracted to you. And according to Newton's laws of uh universal gravitation, you're attracted to me, too. I think we should talk about it. So, this universal law of gravitation, it's just this. It's universal because we assume it holds true everywhere in the universe. It's not entirely true anymore because we know something more about dark energy, but for the most part, this is going to this is going to be fine. So, it's going to be the force of attraction between two masses and that are separated by distance. Remember, like the center of their masses. That means if I have one mass and another mass, they're going to attract each other. Doesn't matter what.

[00:02:04]Um, so we have an equation for it that you don't have to memorize. You can look it up. And it goes like this. f= g m1 m2 over r^2.

[00:02:18]Now this is in your data booklet so you don't have to memorize it which is good.

[00:02:21]Um there's another common way to write it. I just want you to be used to it in case you see it. Sometimes you like to write you know like one big mass let's say like the mass of the planet will be big m and then the mass like a little m will be like the moon or the satellite or something like that. So sometimes you'll see it as GMM / R 2 like that but same same. So let's define all our variables here. So F is the force which is in Newtons. Uh M1 and M2 well those are the masses. So those are in kilogram. And then we have this thing called un uh the Newton's universal gravitational constant capital G. That has a number at 6.67 * 10 -1 Newton m^ 2 per kilogram squared. Now finally we have r which is the distance between the center of those two objects. So that'll be measured in meters. So let's do an example here. We have a situation where we have a center of two different planets and maybe I'll have them you know mass m and mass m like this.

[00:03:15]Distance between them is r and I'm going to say well the gravitational force between the two planets is f which means the big capital m one here it feels um a gravitational force caused by m going to the right cuz it's attractive. And this one over on the right feels that same gravitational force on the left because it's attractive. Now, what's the force between these two though if the separation increases to 4R? In other words, I take these two masses and I I pull them further apart. So, what does your intuition say? I hope your intuition tells you, well, it should be less force because they're further apart. Now, these types of questions like this, I call them ratios questions because, well, you have a situation where yeah, you're not given all the details necessarily. You're just given letters. for example, like just variables and then you're told like something quadruples or triples or doubles or whatever. And I solve these all in the same way. So I'm going to show you first of all two ways to solve this. First of all, I'm going to call this like the brute force method. This will be uh kind of the slower way to do it. And after that, I'll show you like a quicker way. But uh the brute force method is just to use this uh well I need an equation that governs the behavior. First of all, I'm just going to write myself an equation. F equals and remember because this is Newton's law of gravitation, I'm going to say it's m GMM over R2 or use a capital R here. And in fact, that'll be my first equation. I call that sort of old equation. Now, I need a new one. So, I'm going to write one maybe F2. And I'll call that well GMM because the masses haven't changed. But the difference is the distance is 4R. Now, I'm going to make a mistake on purpose. So, watch carefully right here. What I've done right here, it seems really reasonable, doesn't it? This is a mistake that a lot of students make you. It's not 4 r 2, it's 4 r all that squared. In other words, you have to do 4^ 2 r 2. So that's really really important not to forget that. So if we do this here, let's see, I keep going. So g m over let's see 4^ 2 is 4 * 4, which is 16 * r 2. Okay, so that'll be my sort of new equation. And then the way I solve all ratios questions like this, I do new over old. So that means I'll just divide these. I'll say f_sub_2 equals. Now this is a little bit long to write and it's annoying, but I hope you'll get the idea here. That's why I called it brute force. Uh gm mm over r2.

[00:05:35]Now what happens when you divide a fraction by a fraction? Well, you can multiply by the reciprocal. In other words, you take this bottom one and you flip it and put it on top. So I'm going to keep my top one first of all. So the top one is still GMM over uh 16R 2. But then I'm multiply that by let's see R 2 first. So it's going to be R 2 all that over GMM like this. And then you'll see a miracle occurs. The GMMs cancel out. The R squares cancel out. And I end up with just F2 over F equals let's see 1 / 16 is all that's left. Therefore, I can state without a doubt that the new force then is just going to be, let's see, I multiply the F to the top over here. I multiply both sides by F. I end up with F over 16. In other words, the force is 116th what it was before. So, this is my final answer. Now, that was the brute force way. But there's actually a quicker way to do it. So, I'll say maybe this. I'll say quicker. And what you do is you just use the fact that uh these things are proportional. So, we don't actually look at all the details. We just look at what's changed. So, if the masses didn't change, then the only thing that changed is the radius or the distance between them. Then, I'm just going to only care about the piece that's changed. So, in other words, I'm going to say the force is proportional to 1 / r 2. I don't care about the gms.

[00:06:58]See, since they were the same, they're all going to cancel out. I just look at this. All right. So, I can keep going then. So I can say my uh new one, let's say I call it F2, is going to be proportional to 1 over, but it's going to be 4 R2. But I don't care about the Rs cuz they're going to be used up and they're going to cancel each other out.

[00:07:14]I just care about the number that changed. So it's a four that was squared. So that means my force F_sub_2 is proportional to 1 / 16. And because that then I can say, ah, that means I know then that F_sub_2 is just equal to the old force divided by 16. That was like a quicker way to get there. I mean, whatever works for you, right? Whatever gets you there. But the key is just to remember that you can always use um this Newton's universal law of gravitation.

[00:07:42]Whenever you have the force between two objects, you know, mass m and m that are separated by distance r,