Alright, guys. Let's check out this problem. It is a classic one in gravitation, so we're going to work it out together. So we've got 2 spheres, and we're going to position them along a line like this. I've got 2 spheres like this, I know what the masses are and I know the distance between them. So, I'm going to call the left mass the 10-kilogram mass, mass a, and the 25-kilogram one, mass b. And I know what the distance between them is. I'm told that the distance between them is 5 meters, that's little r. So the whole point of this problem is that I'm going to position a mass somewhere along the line between them, and I know what the I'm just going to call that mass, mass c. And I don't actually know what that mass c is, but I want to figure out where I have to place it so that the gravitational forces, the net gravitational force, is equal to 0. So it means I need to find out what this distance is, and I'm going to call that ra. So that's my target variable, really. That's what I'm trying to solve for. So I've got ra as my target variable and I want the net gravitational force to equal 0. Well, we know that Newton's law or Newton's gravitational law says that there are forces between any two objects. So there is a force from this guy, which I'm going to call FAC, and there's a force from that guy over there, and that's FBC. And I know how to express that as the equation. And so we want these things to basically cancel out, which means that the magnitude of FAC needs to be the magnitude of FBC, but they have to be equal and opposite in direction. So I'm actually going to write out the gravitational forces for those. We're going to start off with Newton's law of gravity. So I've got gmamc/ra2. And then over here, you've got gmbmc/rb2. Where that rb is just the distance right here between ma or mc and mb, so that's rb. Alright? Cool. So we can take a look at this equation. We can actually cancel some terms out. I noticed that g pops up on both sides and then also the mc also pops up on both sides. So it's actually good. We didn't need to know the mass of that center thing in there. And then the ra is my target variable. Now I know what mass a and mass b both are, so all I need to do is just figure out what this rb is. That's my only unknown variable. I'm not told what that distance is. So I actually have to go here and figure it out. So what is the distance between this thing and mb? Well, we don't know what the individual distances are between ra and rb. We don't know what those things individually are, but we do know that the whole entire distance between the two spheres is 5. So we can come up with another equation for this. We know that ra + rb is equal to 5. So now what happens is, in this equation, where we've come up with 2 unknowns, we've got this other equation that we're going to use to help solve it. So if I can figure out what rb is, then I can plug it back into that equation. rb is just, if I move this guy over, just going to be 5 - ra. So now I'm just going to substitute this equation back into that guy right there, and now we're only going to have one variable. So now we've got mass a divided by ra2 equals mass b divided by (5 - ra)2. Notice how we've gone from 2 unknown variables to now only 1, and that's the one that I need to find out. So because I need to figure out what that variable is, I want to start getting everything over to one side. So what I'm going to do is I'm going to take this expression right here and move it up to the other side on the left and then the mb is going to come down and trade places with it. So when I do that, I get (5 - ra)2 / ra2 equals mb / ma. Now I actually know what these two masses are, right? This mass b was equal to 25, so 25. And I've got this mass a was equal to 10. So how do we get rid of this whole, like, squared thing? Right? We've got this, like, expression here that's squared. You might be tempted to, like, foil it out and start like basically multiplying everything out, but that's actually going to be way too complicated. Notice how both of these things on the top and bottom are both squared. So we can actually take the square root of both sides. So don't go don't don't go ahead and foil it because it's actually going to be, like, way more complicated. So instead, what we can do is basically square roots_of_i sides of this thing. And when we do that, we get that the squares cancel, 5 - ra / ra. And then this is just a number. Right?
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Newton's Law of Gravity
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