Newton's law of gravitation | Physics | Khan Academy

The mass of the Earth is about 6 * 10 ^ 24 kg. But you know what? I always wondered, how did we figure this out? How on Earth do you figure out the mass of a planet? Well, we did that by using Newton's universal law of gravity, and in this video, we're going to see exactly how.

So what is gravity? Gravity is a force of attraction between any two masses in the universe. For example, Earth has mass, and this apple also has a mass. Clearly, the drawings are not to scale, okay? But since both of these have masses, they will attract each other, and this is the force of gravity.

Now remember, the Earth is not the only one that's pulling on the apple. The apple also pulls back on the Earth—Newton's third law. It puts an equal and opposite force back on Earth. So this is the force of gravity.

But there's nothing special about Earth and the apple. Any two masses will attract each other. This means, for example, you and the moon are also attracting each other because you both have masses. You and this apple are attracting each other. You and the Earth are attracting each other. Every two masses, any two masses in the universe will attract each other because of their mass—and that is the force of gravity.

But this immediately brings up a question: why don't we feel this force of attraction, say, between each other? Or why doesn't all the furniture in our house just stick to each other? Shouldn't we feel that? Why do we only feel the attraction to Earth? That's a really good question!

To answer that, we need to investigate this force of gravity a little bit more. So let's do that, and let's just stick to the Earth and the apple for now. The first question we could ask ourselves is: what will happen to this force of gravity if the mass of the Earth or the mass of the apple were to increase? What do you think would happen? Well, since the force of gravity is an attractive force between masses, we can expect that if the masses increase, the force of gravity should also increase.

So we can predict that the force of gravity should be directly related to both the masses of the interacting bodies. Okay, the next question we could have is: what will happen to this force of gravity if these masses went farther away from each other?

Interesting! Well, here's how I like to think about it. There are so many massive stars and galaxies very far away, right? Why don't we feel their force of gravity? I mean, they are very massive. Maybe it's because they're so far away that their force of gravity is very weak. From this, we can probably predict that the farther we go, the weaker the force of gravity gets. And if you go too far away, then gravity becomes so weak that it'll hardly affect you.

So this means that the force of gravity should be inversely related to the distance between the objects. Now, it turns out Newton had similar lines of reasoning. By looking at the observations of how the moon goes around, and looking at the data of how the planets go around the sun, he was actually able to come up with an expression for this force of gravity.

So what is that expression, you ask? Well, for that, let's label these things. Let's say the mass of the planet, mass of the Earth is M1, and the mass of the second object is M2. And let's say the distance between their centers—okay, not the surfaces, but the distance between the centers—is R. Then the force of gravity between those two objects is given by this expression.

We'll come back to what G is in a second, but just by looking at this, we can see, hey, the force of gravity is directly related to the masses, exactly as we thought. We also see that the force of gravity is inversely related to the distance between them—again, exactly as we thought—but there's a square! Again, we'll come back to that.

But there's a third interesting thing: see, whether you're calculating the force of gravity on the apple or on the Earth, in both cases you have to input the value of both the masses, which means you'll get the same value whether you consider the force of gravity on the apple or the force of gravity on the Earth. That's also nice because we already saw that that has to be the case from Newton's third law.

So this is awesome; everything works out. And of course, this not only holds true for Earth and the apple; it will hold true for any two masses in the universe, which is why this is called the universal law of gravitation. This expression is called so.

Okay, and now, what is G? Well, G is a constant; it's a universal constant. Its value happens to be— we found it out experimentally. Its value happens to be about 6.67 * 10 ^ (-11) units. But what are its units? It would be a good idea to pause and see if you can figure out the units yourself. You just have to rearrange this equation to do that. So can you try?

Okay, let's do that. So we rearrange it in such a way we isolate G on one side. And so what we get on the other side—well, after multiplying and dividing by stuff, you'll get r² here, divided by M1 M2. So now the unit of G should be the same as the unit of this number. So this would be—well, there's a Newton; there's a radius or distance—sorry, not the radius, distance. So it's meters squared divided by kilograms, so kilogram * kilogram. Kilogram squared, so this is our universal gravitational constant value.

And look at that number; it is so tiny: 10^{-11}. So if I put in a few kilograms—if you have, like, consider the force of attraction between, I don't know, maybe you and an apple—then you put a few kilograms over here, and the value would be very small—very, very small: 10^{-11}. Very tiny! And so that force, for all practical purposes, would be negligible.

So that force is there, but it's so tiny we don't see it. However, the mass of the Earth is incredibly huge. We saw that in the beginning; it has a mass of 10^{24}. So because of this enormous value, the force of gravity between the Earth and all the other things on Earth dominates because Earth is very massive. Of course, we're also very close to Earth; we're almost on top of the Earth right now!

Let's look at the denominator over here. We see that there's an R squared. Since the force of gravity is inversely related to the square of the distance, we also call this the inverse square law. But what does that mean? Well, this means if you were to double the distance between the center of the Earth and, say, this apple over here, what would happen to the force? Well, it wouldn't become half. It'll reduce, but it won't be half!

How much will it be? Well, let's see. If I were to double this over here, I will now get 2R, the whole square. But 2R, the whole square—well, there's a square over here as well, so I get 4R squared, which means, look, the force of gravity has become 1/4 of what we had before because there's a four in the denominator. So when I double the distance, the force of gravity becomes 1/4—not half.

Similarly, what would happen if I were to triple the distance? Well, now I have a 3R, which now becomes 9R squared, which means, look, I have a 9 in the denominator, so the force of gravity becomes 1/9. So when I go three times as far away, the force of gravity becomes not 1/3 but 1/9.

We can now nicely show this by drawing a graph of the force of gravity versus the distance. So again, at R, let's say the force is F, or F of G. If I go twice the distance away, the force will not become half; it'll become 1/4. So it'll be somewhere over here. If I go thrice, it'll become 1/9.

So notice the force of gravity will decrease as you increase the distance. But how does it decrease? If you were to draw a graph, you can kind of imagine the graph looks somewhat like this. This is the graph of the inverse square law.

Okay, we now know how to calculate the force of gravity between any two objects, but a lot of times we are interested in calculating the force of gravity between Earth and other objects very close to Earth, like when we drop an apple or the force between Earth and me—things like that. So in such cases, what will this simplify to?

Alright, so that's the last part. So, oops, let's consider close to Earth. For that, let's say the mass of the Earth is just capital M, big M for big Earth, and the mass of the other objects which are close to Earth—we'll just call it small m. The force of gravity would now become what it would be? Well, the same thing: G times mass of the Earth times mass of the small object, whichever object we are considering close to Earth, divided by the distance from the center of the planet—the center of Earth—all the way to that particular object.

But if that object is very close to Earth, that distance is pretty much going to be the radius, isn't it? Think about it. For example, if you're considering, let's say, a skyscraper over here. Now, if you consider the force of gravity on the ground or the force of gravity between the Earth and you on the top over here, it's going to be pretty much the same because remember this is not an up-to-scale drawing, okay? If I were to try and draw to scale, it would look somewhat like this.

And look, it is so tiny that the distance from the center to the ground or the center to the top is pretty much just the radius. In fact, this is also not to scale. At this scale, we wouldn't even be able to see the skyscraper. So for all practical purposes, the distance will stay the same.

This means, long story short, if you're close to Earth, we can pretty much approximate this distance between the center of the planet and the center of whatever the object is. We can say it's pretty much the radius of the Earth. And so if we plug that in, we will just get R squared over here.

And now what's interesting is that since we're calculating this for the Earth, these numbers are fixed. G is a universal constant, which we already know. M is the mass of the Earth, which we saw we already know it, and we'll see how we calculate it in a second now. And R is the radius of the Earth. We also know that value; it is approximately about 6,400 kilometers.

So since these values are fixed for Earth, when we are close to them, instead of substituting every single time, we can just bunch it all together, substitute once, and just remember it. Therefore, we can now simplify this and we can say the force of gravity on objects close to Earth is equal to—we can bunch this all together, and we can substitute it—but we'll give it a symbol; we'll call it small g.

It's not the same thing as capital G, okay? So the force of gravity would be M into small g, M into small g. So this becomes our simplified expression for the force near Earth.

Now notice this is not a new expression for the force of gravity close to Earth; it's the same as the universal law of gravity; we just simplified it. But now what is the value of g? Well, we can substitute the values of capital G, mass of the Earth, and the radius of the Earth and all of that. If you do it, I'm pretty sure you can do that; if you do it, we'll get the answer to be about 9.81.

But what exactly is it? What are the units over here? Well, look, if this is from Newton's second law, we know that if this is a force and this is mass, this must be acceleration. So this g, this 9.81, is actually the acceleration that objects experience close to Earth. That is the acceleration an apple would feel if you drop it and if you ignored all the other forces—if you only had the force of gravity.

So let's say in vacuum. What's interesting is that every object—notice this number is independent of the mass of the object. It only depends on the mass of the planet, mass of the Earth, the radius of the Earth, and the value of capital G. That means every object close to Earth, if you drop it, it will accelerate at 9.81 m/s². And so this is called acceleration due to gravity.

Now this is the acceleration due to gravity of Earth. Different planets will have different accelerations due to gravity because they will have different masses and they will have different radii.

Alright, so now we can answer our original question: how did we figure out the mass of the Earth? Well, guess what? We already knew the value of small g, the acceleration due to gravity of Earth, because we calculated it experimentally. And you can do that just by dropping something and figuring out from what height it dropped and how long it took to fall down.

You can figure out this value experimentally, and we had already done that. We can also do it using a pendulum. Okay, so we knew the value of this; we knew the value of the radius of the Earth that was also calculated a long time back. It was very interesting stuff, using shadows and things like that; it was very interesting stuff.

And then finally, once we figured out the value of capital G, we could just equate them and figure out the mass of the Earth. That's how we actually figured out the mass of the Earth. Isn't that interesting? Isn't it amazing?

A question for you now to think about is: how would you figure out the mass of the sun, for example? I mean, you could go to the sun, drop an apple, and figure out the acceleration due to gravity, but we can't do that, right? So how do you do that? How do you figure out the masses of the sun, the mass of the other planets, the mass of all the other moons? For example, how would you figure that out? Something to ponder upon.