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Universal law of gravity example

Let's explore how the force between masses change when the distances and the masses change using the universal law of gravity. Created by Mahesh Shenoy.

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Video transcript

- [Instructor] We know that every object in this universe attracts every other object due to their masses, and we call this force, Gravity. Like say for example, the sun and the earth pull on each other due to that force of gravity. What we want to explore in this video, is how would that force change, if we changed the distances or the masses of these objects. For example, what would happen to this force if the earth were to go twice as far away? Or what would happen if the mass of that sun were to double? Or maybe the mass of that earth were to triple? How would that force change? That's what we will explore in this video. Now, before we begin, we've already seen how to calculate this force of gravity between any two objects. It's given by what is called the Universal Law. And it looks like this. Where F is the force of gravity, G is a universal constant, It's a very, very tiny number, 6.67 times 10 to the minus 11, and m one m two are the masses of the two bodies. So in our example, it would be the mass of the sun. And m two might be the mass of the earth. And d represents the distance between the centers of the two bodies. So from the center of the sun to the center of the earth. Of course this diagram is not to scale, And if you feel that you need more clarity on this, then we've talked a lot about this in a previous video called Intro to Gravity, so you can just go back over there and watch that, and then come over here. Anyways, now let's see what happens to force of gravity, as you start messing with these values. So first let's say the distance between them doubles. What happens to that force? Well, if the distance it's between them doubles, then the earth would now be twice as far from the sun as before, which means this distance is now going to be two d, because before it is d, doubles means now becomes two d. And so, what now happens to this force? What is this force equal to? How do we calculate that? Well, let's call it force F dash. It's a new force, right? Let's call it as F dash. We can use the universal law. So we could say, look, F dash is gonna be equal to, G, times m one m two, well, their masses have not changed, right? Their masses are still the same, only the distance has changed. So it'll still be m one times m two, divided by the new distance between them is two d. So, it's going to be two d the whole squared. Do you understand that? Because the universal law says, you'll divide by the distance squared. So the new distance is two d, so two d whole squared. And so if we simplify this, we get G times m one times m two, divided by two squared is four, and d squared. Okay. Now what I want to know is, what happens to the force of gravity? Has it doubled? Has it become a half? What happened to it? Which means I need to compare F dash and F. Now, one of the ways to do this is you can just divide F dash and F, that's one way, but another way over here is we can look at this equation over here for F dash. This is F dash, and we can say, "Hey, look, look at this part. G m one m two divided by d square, that's F. That's our initial force, Isn't it? Look, that's just F" Which means from this, we could say that F dash has become one fourth, of F. And that's our answer. So when the distance doubles, the force becomes one fourth, it reduces all right, that kind of makes sense, right? The farther you go away, as the distance increases, the force of gravity would reduce, but when the distance doubles, the forces doesn't become half, it becomes one fourth. And guess what? With practice, such problems can also be solved directly in our head, without having to write any steps. Let me just show you what would that look like? So if we didn't want to write any steps, what we could do is we could say, "look, the only thing that has changed is that the distance has doubled." So we'll go to our original formula, and we'll say that instead of d there'll be a two d, and that whole thing will be squared. And because of the square, that two squared becomes four. And that's how a four comes in the denominator. And as a result, our original force has been divided by four. And that's why the force now becomes one fourth. Okay? But of course, initially when you're doing it, we can do it with the steps, but with practice, we'll be able to do it in our head as well. Okay. Now, can you try one? What do you think will happen if the distance were to triple? Just pause the video and give it a shot. All right. Well, if the distance becomes tripled, this number would be three, and as a result over here, we get a three d squared, and so, three squared will become nine. And as a result, our new force would be one ninth, of the initial force. That means the force would reduce and become one ninth. And again, we can do this directly, and you can just see there's a three, there's, a square of three will become nine in the denominator, and so it becomes a one ninth of the initial force. Okay. Let's try one more. In this case, we have distance between them halves, and one of the masses doubles. Okay. Again, let's draw this. So the distance between them halves, that means they come closer to each other, and now the distance becomes d by two. And one of the masses doubles, then are you in which mass? So we can assume. So let's say the mass of the sun remains the same. Let's assume that the mass of the earth doubles. So now the mass of the earth becomes two to m two. And again, we need to find out what happens to this force. So what is this new force now? How, will it change? Again, can you pause the video and see if you can give this a shot? Go ahead. Give this a try. All right. So we'll do the same thing as before. The new force F dash is going to be G times mass of the sun remains m one, but the mass of the earth has now become two m two. That's the new mass of the earth. And so instead of m two, you put two m two, divide by distance has become d by two, the whole squared. So this is d by two, the whole squared. And now we just have to simplify this. Let me pull that two out. So you get two G m one, times m two, divided by d by two whole squared becomes d squared divided by four. Now let's be a little careful because there's a fraction in the denominator. Here what I'd like to do is I'd like to remember that dividing by something, is the same thing as multiplying by it's reciprocal. And so we can now say, "F dash is going to be two G m one m two, into the reciprocal of this four over d squared." What does that give us? Well two times four is eight. So I end up with, eight G m one m two by d squared. And again, we can say the G m one m two by d squared, is the same thing as the original force F. And as a result notice, with our new force becomes eight times F. And so it has increased eight times. It kind of makes sense, right? They have come closer and the mass has also doubled. So we would expect the force to increase a lot. And again, if we were to do this in our head, what we would do, well, let me again just black this thing out. So what we would do is we could say, look, the mass has doubled, and as a result there is a two in the numerator, and the distance has become half. There won't be a half in the denominator because that half has to be squared, d squared. So that half squared becomes one fourth in the denominator. That's basically what we did with the steps. And two divide by one fourth, that becomes eight. And that's how the force becomes eight times as much as before. Okay, one last, just for fun and practice. Here's the big one. The distance between them quadruples, one mass doubles, another mass quadruples, quadruple means becoming four times as much. Now again, can you pause and see if you can try this whole thing yourself, make a drawing and see what happens. All right. So here's what the drawing would look like. The distance between them has become now 40, because it has been quadruple four times. Let's assume that the mass of the sun doubled, and let's say the mass of the earth quadrupled. It doesn't matter, even if this was to become four times, and this was to become two times the answer would still remain the same. All right. Now I'm pretty sure you can do the steps all by yourself. So let me just go ahead and do this directly. So I would look at this equation and I say, "Hey, m one has doubled and because of that, there's a two in the numerator." M two has become four times, and so there's another four being multiplied in the numerator, divided by what happens to d? Well, d has now becomes four times, but they won't be a four because d has to be squared, so that four squared will become 16. Okay? Four times two is eight, and eight goes one times, eight goes two times, that means there is just a two in the denominator, and so the force becomes half. So that's our answer. The force becomes half. And so if you want to figure out how the force of gravity changes, when the masses or the distances change, then we redraw, rewrite the new force and substitute the new values. And once you get that, we compare this new equation with the old one, and see what relationship we get between the new force and the old force.