Hey, guys. So now that we've gotten a good feel for the universal law of gravitation in one dimension, it's time to look at how it works in 2D. But we're going to see very similar stuff that we've done before. So, we're working with net forces in one dimension; then, if we have multiple forces on an object, for instance, like this force right here that was, I don't know, 3 newtons, and we had another force that was acting on it like negative 4 newtons, then we can figure out what the net gravitational force is just by subtracting or adding these forces together. So I'd get a net force of negative one newton that points in that direction. It's simple addition, right? Well, in this case, let's check it out. So we have a sort of triangle of masses, so all these things are little m's. Newton's laws say they're all pulling on each other. So this bottom left one feels a force that points in that direction and it also feels a force on it that points in that direction. So let me just make up some numbers here. I'm just making this up. So let's say this was like 3 newtons and now this was like 4 newtons. How would I figure out the net gravitational force? So these questions will ask you what the net gravitational force is, but you can't just add these things together because they point in different directions. And so to solve for net forces in non-linear arrangements, you have to use vector addition. And remember that the force of gravity is a force and forces are vectors, So we can always take a vector like this and break it up into components. Alright. So the point is, if you have this vector here that points in some off angle, you can always decompose it into its x and y components just using your sine and cosine stuff from trig before. So we'd have to know some angle theta and stuff like that, right? So now what we can do is now we can take these components, add them up, and then figure out what the net gravitational force is like that. Okay. To sort of refresh ourselves with all of this stuff, let's go ahead and just do a quick example.

We're supposed to calculate what the magnitude and the direction of this net force is on the bottom mass, the m1 in the figure. So the first thing is we just have to label the forces that are acting on this. So I've got a force that points in this direction between these two masses and then we've got a force that force that points to the right from those two masses. Now this is between m1 and m2, so I'm going to call this f12. And then this is f13, so I'm going to call this f13. Okay. And so how is what is the net gravitational force gonna be? Well, I can't just add these things together so I have to use vector addition. If I add these 2 vectors tip to tail, what I'm gonna get is I'm going to get a net gravitational force that points in that direction. Right? So if these things pointed this way, the net gravitational force has to be between them. So this is the net gravitational force, and I'm to figure out the magnitude of that. So how do I figure out the magnitude? Well, remember that if we're figuring out the magnitude of a force and we have these sort of components like the x and y components, we do that using Pythagoras' theorem. So we've got f13 is the x component, so we've got to square that. And then we've got f12, that's the y component, and then we've got to square that. So this is like the hypotenuse of this triangle, right? So if we're trying to figure out what this distance is right here, then we have to figure out what sort of like the hypotenuse of this triangle that it makes, right? Okay. So now we actually have to figure out what these two forces are; what is f13 and what is f12? Well, I can figure out f13 just by using Newton's law of gravitation for point masses. So I've got big G times mass 1, mass 3 divided by the distance between r1 and 3 or m13 squared. So that distance between m1 and m3 is equal to 0.1. And then I have both of those masses, so I can go ahead and just figure that out. We've got 6.67 × 10 - 11 and the product of the 2 masses, 25 and 20 divided by 0.1 2 . So what you get for that is you get 3.34 × 10 - 6 . So I've got 10 - 6 there. Alright. And if I do the same exact thing for f1 and 2, I've got g m1 m2 divided by the distance between 12 squared, which is at 0.15. So just got the same setup here, 6.67 × 10 - 11 , then I've got 25, and now 30, and now divided by 0.15 2 . So what I get is 2.23 × 10 - 6 . And both of these things are in newtons. Alright. So now that I actually have the components of these and the actual forces, now I can go ahead and just figure out the net gravitational forces. So the magnitude of f net is going to be the square roots and then I've got just basically plugging these values. I've got 3.34 × 10 - 6 + 2.23 × 10 - 6 . If you go ahead and plug that in, you should get a net force of 4.02 × 10 - 6 and that's in newtons. How about the direction? How do I figure out what the direction is? Remember that that direction is going to be this angle theta, right? And theta, how do we relate theta? From the tangent. So we relate the tangent theta is equal to the y component divided by the x component. So in other words, it is the f12 divided by f13. So to go ahead and figure out this angle here, I've just got to take the inverse tangent. So that's going to be the inverse tangent of f12, which is going to be 2.23 × 10 - 6 ÷ 3.34 × 10 - 6 . If you do that in your calculator, what you should get is 33.7 degrees. Alright, guys. So that is the magnitude and the direction of this net gravitational force and, let's keep going on. Let me know if you have any questions.