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- 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 the Sun were to double. Or maybe the mass of the 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 time 10 to the minus 11. And "m1" "m2" are the masses of the two bodies, so in our example it would be the mass of the Sun, and "m2" 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 the force of gravity as we start messing with these values. So first, let's say the distance between them doubles. What happens to that force. Well, if the distance between them doubles than the Earth would now be twice as far from the Sun as before. Which means this distance is going to be "2d". Because before it is "d". Doubles means now it becomes "2d". And so, what now happens to this force, what is this force equal to? How do we calculate that? Well let's call that Force "F'", It's a new force right. Let's call it "F'". We can use the universal law, so we can say, look "F'" is going to be equal to "G" times "m1" "m2", well their masses have not changed right? There masses are still the same. Only the distance has changed. So it will still be "m1" times "m2", divided by the new distance between them, is "2d", so. It's going t o be "2d", the whole square. You understand that? Because the universal law says, you are divided by the distance squared. So the new distance is "2d", so "2d" whole squared. And so if we simplify this, we get "G" times "m1" time "m2" divided by 2 square is 4. And "d" square. Okay, now what I want to know is what happens to the force of gravity. Has it doubled, has it become half, what happened to it? Which means I have to compare "F'" and "F". Now one of the ways to do this is you can just divide "F'" and "F". That's one way. But another over here, is we can look at this equation over here for "F'", this is "F'". And we can say, "Hey look, look at this part." "G", "m1", "m2" 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 can say that "F'" has become 1/4 of "F". And, that's our answer. So when the distance doubles, the force becomes 1/4. 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 force doesn't become half, it becomes 1/4. 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 that would look like. So if you 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 go to our original formula and you'll say, that instead of "d" there'll be a "2d" and that whole thing would be squared. And because of the square, that two square becomes four. And that's how 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 1/4. Okay? But of course initially when you are 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 would happen if the distance were to triple? Just pause the video and give it a shot. All right, well if the distance becomes triple, this number would be three and as a result over here we get a "3d" square, and so, three square would become nine, and as a result our new force would become 1/9 of the initial force. That means the force would reduce and become 1/9th. And again we can do this directly. You can just say there's a three, the square of three would become nine in the denominator, and so it becomes 1/9th 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/2". And one of the masses doubles. They don't give you which mass, so we can assume. So let's say the mass of the Sun remains the same. This leaves that the mass of the Earth doubles. So now the mass of the Earth becomes "2m2". 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'" is going to be "G" times, mass of the Sun remains "m1", but the mass of the Earth has now become "2m2" that's the new mass of the Earth. And so instead of "m2" we put "2m2" divided by distance has become "d/2" the whole square. So this is "d/2" the whole square. And now we just have to simplify this, let me pull that two out, so we get "2" "G" "m1" times "m2" divided by "d/2" whole square becomes "d"square divided by four . Now let's be a little careful because there's a fraction in the denominator. Here what I like to do is, I like to remember dividing by something is the same thing as multiplying by it's reciprocal. And so we can now say "F'" is going to be "2Gm1m2" into the reciprocal of this. 4/d square. What does that give us. Well two times four is eight, so I end up with, "8Gm1m2" by "d" square. And again we can say that "Gm1m2" by "d" square is the same thing as the original force "F" and as a result notice, our new force becomes eight times "F", and so it has increased eight times. It kind of makes sense right, they'll 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 the half has to be squared, "d" square. So than half square becomes 1/4 in the denominator. That's basically what we did with the steps. And two divided by 1/4, 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 "4d" because it has been quadrupled, 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 will look at this equation and I say hey, "m1" has doubled, and because of that there's a two in the numerator. "m2" 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 become four times, but there won't be a four because "d" has to be square, 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's 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.