Hey, guys. So, we always use Newton's law of gravity to figure out the gravitational force between any two objects of mass. Well, some problems will involve an object that is in the interior of the Earth or a planet or something like that, and we're gonna need a new force of gravity to solve those problems. Now, just remember, the force of gravity is always gonna act as if all of the mass beneath us or beneath you is concentrated at the center. So when we worked with the earth, we were either on or outside or at some height of the Earth. We always just basically assume that all the mass was concentrated at a point at the center. And to calculate the force, we just needed the 2 masses and the distance, and that was either big R or big r+h, depending on where we were on the surface or at some height.

Now what happens is you're gonna grab a shovel and you're gonna dig down towards the center of the earth, and you wanna figure out what the gravitational force is here. So there are two differences. Right? So now what happens is that the center of mass distance is this little r, so that's the difference. That's one difference. And the other difference is that all of the mass beneath you is actually all of the mass that is concentrated within this spherical shell that I have in the dotted line. So it's all of the mass that is closer toward the center of the Earth. Now we can still sort of the force of gravity still act as if all of it is concentrated at the center, but now instead of being the total mass of the Earth over here, this is actually the inside mass of the shell. That's what we're gonna use for our equations. So in other words, it's almost as if we were standing on the surface of a smaller Earth that was less massive, and all of this stuff that's outside or above you is not going to matter. All of this stuff is not gonna contribute any force. So, just as over here, the force of gravity depended on the total mass of the earth, the gravitational force on you here is gonna depend on the mass that's inside of the shell.

Now, there's a much more common and powerful formula that you're gonna need, that's related to this, and that's gonna be Gmmr^{3} × r And the most important thing that you need to know is that we're always gonna use this equation for gravitation inside Earth whenever the center of mass distance is less than big R. In other words, whenever this is less than this. So in other words, we're inside of the Earth. Okay? So where does this equation actually come from? Because some of you might need to know that. I'm just gonna go ahead and quickly show you. This equation actually comes from this. If we wanted to find the inside gravitational force, that depends on the inside mass divided by r squared. The problem is in these problems, you have no idea what this inside mass could be, so we're gonna need some relationships to sort of get rid of that variable in that equation. So one thing we can do is we can say that if this sphere here is a smaller subset, or a smaller piece of the total amount of earth, then one of the things we can do is actually relate the density of these two things. Remember, the density is given by the Greek letter rho. And so we can say is that this sphere is a smaller piece of the larger one, then the density of the inside sphere is gonna be equal to the density of the overall Earth as a whole. Now, remember that this density here is just the mass over volume. So, in other words, the mass of the inside divided by the volume of the inside. And that's gonna be the total mass of the earth divided by the total volume of the earth. So now we actually have an expression for this inside mass over here, and now we can just say that we can move this over, this volume inside over, and then mass inside is gonna be v inside divided by v Earth times the mass of the Earth. Okay? Let me just make more maroon over here. So the volume of the inside is gonna be the volume of a sphere. Right? So we have a spherical shell like this, and the volume of a sphere is 4/3×πr^{3} because that's the radius of the inner shell. The volume of the earth is 4/3×πR^{3}, and now we have the mass of the earth. So now what happens is the 4/3×πs cancel, and this is actually gonna be equal to the inside mass. So now what happens is we have an expression for this, and if we plug this expression inside for this mass inside, you guys can work it out for yourselves. You cancel a whole bunch of stuff. We're gonna get back to this expression right here. Cool.

Now, here's what one really important thing you should know. It's kind of a conceptual point. This derivation in this equation only works if the density of the earth is approximated to be constant. So that's how we were able to set these two densities equal to each other, just assumed it was constant. Alright? So now we actually have a complete idea of what the force of gravity looks like with distance. Right? So what happens is when you are inside of the Earth, which is over here, so you're inside, the force of gravity actually grows because it is proportional to r, and that's in the numerator. So, in other words, F is proportional to r, and it increases. That's one really important point. Then what happens is you hit this boundary over here where r is equal to big R. In other words, we're actually on the surface, so this is the on the surface. And then what happens is that the force of gravity is going to decrease when you're outside. And we know that. That's just F is proportional to 1 over r squared. So it's gonna decrease as you get away from the surface. So that's the total picture of how the force of gravity changes with distance. Now, one really important, one interesting sort of quirk of that is that at the Earth's center, in other words, when r is equal to 0, so that's actually gonna be right over here. The force of gravity is 0, so you'd actually be weightless at this point. And that's because all of the mass sort of spherically surrounding you is gonna cancel itself out. Okay? So that's it for this one. Let me go ahead and show you an example of how this works. So we have a person that has a surface weight of 780 Newtons. It's gonna drill down, and we're supposed to be figuring out how far from the center of the Earth. So what variable is that? That's the letter r. Is their weight, so we're going to be in, 80% of the surface weight. So, in other words, the weight inside is gonna be 0.80 of the weight at the surface. So, in other words, we need to figure out what distance little r this actually happens at. So, in other words, this is gonna be the weight inside, and this is gonna be the weight at the surface, and you're gonna have a little person right here. Cool.

So, let's see. We actually have what this number is, so we know we're gonna be using the f inside equation. So, by the way, your weight inside is equal to the force of gravity inside, because weight and force of gravity equal each other. That's gonna be Gmmr^{3} r And so that is gonna be a number, and that's gonna be 0.80 of what the surface value is. So, in other words, 0.8 times 780. So it's actually gonna be 624. So, in other words, we need to figure out what distance of little r is the weight and the interior are gonna be 624 Newtons. So if we're gonna rearrange for that equation, we just have r is equal to We have to bring this over to the other side and this has to go down to the other side as well. So that's gonna be like that. So we're gonna have r is equal to 624 times r cubed divided by G m times little m. So let's look through our variables. I know what the radius of the earth is. That's just a constant. And I know G and the mass of the earth. The only problem is I actually don't know what the mass of the person is. I know what their weight is, but that's not the same thing as the mass. So I need to go over here and figure out what the person's mass is equal to. Now, one of the things I do have is their weight. Remember that's 780 Newtons. So we can just use the familiar W is equal to mg, and that's gonna be at the surface. And so we just have 780 Newtons is equal to m, and then G at the surface is equal to 9.8. So if you bring this over to the other side and divide it, we're just gonna get that the mass is equal to 79.6 kilograms. So now, we can stick that back into this equation. So, that means that this r distance over here is going to be 624. The radius of the Earth is 6.37 times 10 to the 6th. Gonna cube that. Now we have to divide it by G, which we have. Do you remember G is just that constant? The mass of the Earth is 5.97 times 10 to the 24th. And now this m here is 79.6. So if you work this all out in your calculator, you're gonna get an r distance. That's 5.09 times 10 to the 6th meters. So this is not quite our answer yet, because we actually have to express this as a multiple of the Earth's radius. So all we have to do here is just say that r divided by big R is equal to 5.09 times 10 to the 6th, divided by the radius of the earth, 6.37 times 10 to the 6th. If you work this out, this actually ends up being 0.8 or 80%. And that is no coincidence, because remember that we are looking for the weight, which is 80% of the surface value. So that means that the r is gonna be 80% of the total radius. Because remember that your weight inside is proportional to r. So that's actually why we get the same number here, so it's no coincidence. Alright, guys. Let me know if you guys have any questions with this.