Gravitational Forces in 2D

by Patrick Ford
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Hey, guys. So now that we've got a good feel for the universal law of gravitation in one dimension, it's time to look at how it works in two D. But we're going to see this very similar stuff that we've done before. So we're working with Net forces in one dimension. Then if we had net force, if we have multiple forces on an object, for instance, like this force right here, that was, I don't know, three Newton's and we had another force. It was acting on it like negative four Newtons, that we could figure out what the Net gravitational forces just by subtracting or adding these things together. So I get a net force of negative one Newton that points in that direction. It's a 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 a little EMS, Newton's losses, they all pulling on each other. So this bottom left one fields of force that points in that direction, and it also feels of 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 three Newton's. And now this was like four Newtons. How would I figure out the gravitation, 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 think they point in different directions. And so for to solve for net forces in nonlinear arrangements, you have to use Vector Edition and because 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. All right, 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 you have to know some angle theta and stuff like that, Right? So now we can do is now we can take these components, add them up, and then figure out what the Net gravitational forces like that. Okay, sort of like refresh ourselves all of this stuff. Let's go ahead and just do a quick example. So we're supposed to calculate with the magnitude and the direction of his net force is on the bottom mass the M one in the figure. So the first thing is, we just have to label the forces they're acting on this. So I've got a force that points in this direction between these two masses. And then we got a voice, that force that points to the right from those two masses. Now, this is between m one and M two, so I'm gonna call this F 12 and then this is F one and three. So I'm gonna call this F 13 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 Edition. If I add these two vectors tip to tail. What I'm gonna get is I'm gonna get a net gravitational force that points in that direction. Right? So 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 we're figuring that the magnitude of a force and we have the sort of components like the X and Y components. We do that using Pythagorean serum. So we've got F 13 is the ex components. We've got a square that and then we've got f 12 That's the white component. And then we've got a square that so this is like the high pot news of this triangle. Right? So we're trying to figure out what this distance is right here. Um, that we have to figure out what sort of like the have partners of this triangle that it makes right? Okay, So now we actually have to figure out what these two forces are. What is F 13 and what is F one to? Well, I can figure out f 13 just by using Newton's Law of gravitation for point masses. So I've got big G times mass one mass three divided by the distance between our one and three, or and one and three squared. So that distance between and one and then three is equal to 0.1, and then I have both of those masses so I could go ahead and just figure that out. We've got 6.67 times 10 to the minus 11 than the product of the two masses 25 20 divided by 0.1 squared. So when you get that is, you get to their sorry 3. 34 times 10 to the minus six. So I got 10 of minus six there. Alright? And if I do the same exact thing for F one and two, I've got G m one m two divided by the distance between one and two squared, which is at 0.15. So just got the same same set up here 6.67 times, 10 to the minus 11. Then I've got 25 now, 30 now divided by 0.15 squared. So what I get is 2. 23 times 10 to the minus six. And both of these things are in students. All right, so now that I actually have the components of these in the actual forces now, I could go ahead and just figure out the net gravitational forces So the magnitude of F net is going to be square roots. And then I've got just basically plugging these values. It got 3 34 times, 10 to the minus six squared, plus 2.23 times 10 to the minus six squared. If you go ahead and plug that in, you should get a net force of 4.2 times 10 to the minus six. And that's the Newton's. How about the direction? How do I figure out what the direction is? Remember that that direction is gonna be this angle theta right and theta. How do we relate data from the tangent? So we really the tangent data is equal to the y component divided by the ex components. So in other words, it is the, uh uh f 12 divided by F 13 So we'll go ahead and figure out this angle here. I've just got to take the inverse tangent, so that's gonna be the inverse tangents of the F one to which is going to be that 2.23 times 10 to the minus six, divided by 3.34 times 10 to the minus six. If you do that on your calculator, what you should get is 33.7 degrees. All right, guys, 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.