What's up, guys? So we took a look at the gravitational forces between objects, whether it was point masses or a planet and something else. But in a lot of problems you're going to see, you're going to need to calculate the acceleration due to gravity. So we're going to cover that in this video. Now, that acceleration due to gravity, we can figure out from Newton's law of gravity. And the letters that you'll see for that sometimes are \( a_g \), but most of the times you'll see little \( g \) for that. We're going to cover that. And it's got 2 different forms depending on how far you are away from a planet. If you're any distance away from the planet, it's going to be:
GM r 2But if you're near the surface or on the surface, it's going to be:
GM R 2Notice the derivation. So how do we get the acceleration from a force? How do we get \( a \) from \( f \)? We always use \( f = ma \). So in this case, we have a force that's acting on this person. We know that's the gravitational force. So we write that out. The gravitational force is just \( m \), except instead of \( a \), you'll just see this become \( g \). So now we know what the gravitational force is. It's:
gmm ⁄ r 2 = mgSo if we just cancel this \( m \)'s out that appear on both sides, we get to this expression right here. And by the way, this works for any distance away. So let's look at a special case when you're on the surface of this planet. So we're going to start off with this equation right here. We know that little \( g \) is just \( G \) times \( M \) over little \( r \) squared. But now when you're on the surface here, your little \( r \) distance is equal to big \( R \) + \( h \). But now when you're on the surface, this \( h \) distance, which remember is your height above the surface, is so small compared to the big radius of the earth that we're just going to cancel that out and approximate it to 0. So we can only do this because your height is just so small, we're just going to cancel it out. So what happens is your little \( r \) basically becomes your big \( R \). So now what happens is that this equation becomes \( G \) surface and now we're just going to write this as:
GM R 2That's basically the derivation. So there are a couple of key points to remember about these equations. So whenever you're specifically asked or given a height, you're always going to use this little \( g \) equation right here because it depends on the height. Whereas you're always going to use \( g \) surface, this guy right here, whenever you're standing on the surface. And in problems, you'll see the words like surface gravity or free fall due to gravity or something like that at the surface. And also, you'll note that in these equations, they only depend on big \( M \) and not little \( m \). So it only depends on the mass of that thing in the middle, the big planet or whatever. Finally, we know that \( g \) surface, because it depends on a whole bunch of capital letters, big letters, that it's a local constant, which means it's just a constant that depends or, that's going to be the same anywhere you are on the surface. Whereas \( g \) is a variable and it's going to decrease as \( r \) is increasing. So as you get farther away, your acceleration is going to decrease. We can actually see that because as your \( r \) increases, we know that the force of gravity gets weaker, which means that your acceleration is going to get weaker as well. Now finally, the last point is that your weight, what we define as the weight at any distance away from a planet, is just going to be the force of gravity. We know that's \( gmm \) over \( r \) squared, but now we can actually write this as \( mg \), with the little \( g \). Now we can actually write it in terms of little \( g \). And on the surface, this becomes \( mg \) at the surface. And we've actually dealt with this a lot before. And that's basically it, guys. So let's go ahead and take a look at an example. So we've got, we're trying to figure out what the acceleration due to gravity is on Mount Everest, and we're specifically given what the height is. So we know we're going to use little \( g \) for that because we're given a height. So we're just going to start out with that equation \( G \) \( M \) over little \( r \) squared. Now I know that little \( r \) is just equal to \( R \) + \( h \). So now I just have to make sure I have everything in that problem. I know that these are just constants and I have what the big radius of the earth is. I can just use my table right here and I have what the height is. So I'm just going to go ahead and plug everything in. I've got:
6.67 × 10 - 11 , 5.97 × 10 24 , 6.37 × 10 6 + 8.85 + 8850Then you square that. Just make sure you're adding this before you square it. And, yeah. So if you do that, right, you should get \( 9.79 \) meters per second squared. So now let's take a look at the surface gravity of the earth because we're asked to compare the surface gravity of the earth with this guy. Right? So we have \( g \) surface, so that's the surface gravity, and that's just going to be \( g \) \( m \) over big \( R \) squared. Now hopefully from forces and kinematics and stuff, you remember that this number right here, that \( g \) is just \( 9.8 \). But if you forgot for some reason, you could actually get back to it from this equation. So if you just go ahead and plug in all the numbers that we had for this except for this 8850, you're actually going to get that this is \( 9.81 \) meters per second squared. So that's where this number actually comes from. It comes from this equation. And so we can see that this is \( 9.81 \) meters per second squared and this number is really close. And that's because your height, even though you're standing on Mount Everest, which is the tallest mountain in the world, that thing, that height is so small compared to the huge radius of the earth that it doesn't really have a big effect on this. So what happens is \( h \) is really small even though it's technically the tallest mountain on our planet. Alright, guys, let me know if you have any questions.
