Using Symmetry in 2D Gravitation

by Patrick Ford
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Hey, guys, So often problems that you'll see in two D gravitation. We'll have a lot of big numbers and a lot of steps to follow. So this video, I want to give you a shortcut in order to minimize the amount of work that you have to dio. So let's check it out. So let's say I was trying to figure out what the gravitational forces on this top mass right here. So I wanna find F net. Now, How do I do that? Well, first I have to go ahead and label my forces. So there's two masses on this triangle and they're both exerting gravitational forces on this top mass right here. So we've got a gravitational force from this guy that points in that direction and then f g over here. Okay, so now that I've labeled the forces, I have to calculate them. But notice how if these masses are the same, the little EMS are the same, and I haven't given numbers for those. I've just I've just said that they were both little EMS, and the distances between them are the same. So this is the distance between them, then these things have to have the same gravitational force. These have to have the same f g. Remember that F g is just big G times the mass of both objects divided by their distance square. So if you have the same EMS and the same ours, gravitational force is the same. So now that we've calculated with the gravitational force ours, If I actually had numbers for this, I'd have to decompose the vectors into their its components Now, just to just to sort of refresh that the way I would do that is, if you have to factor that points off in this direction, then you could always split off into its components. So I have an X component, which is F X, and this is a white components F y. If this is a force or vector like F right and we do that using TRIG, right, If long as I have the angle theta relative to the X axis, I could get its components. So the next step for this is now that I have the forces Aiken split off into its components. So if this is the angle relative to the X axis, that means it's gonna have a X components here, and that X component is gonna be related to the coastline of data. Whereas the why component down here that points in this direction is gonna be related to the sign of data. All right, But I have the same exact forces. And if these things have the same angles and the same forces, then you're gonna end up with the same exact components in the X and Y directions. So the tip is if you end up with two components in the in the other direction that are equal, but opposite you can always cancel them if they're opposite. So this is another way where you can use symmetry to reduce the amount of work that you've done. So let me show you. So I have the X and Y components. But this force over here on the left is gonna break up into the same exact components F x and F y, except the white components are gonna add together here, whereas the X components are equal, but they point in opposite directions. So we don't even have to do this in our calculators. We won't have to worry about it because the Net force is gonna end up canceling both of those out. And so instead, what's gonna happen is that the f Y components are gonna add together and your net force is gonna be pointing down here. And that net force, which I wanted is just gonna be equal to times f Why? So this is one way that symmetry can reduce the amount of work that you have to dio. So I wanna attach some numbers to this stuff. So let's go ahead and work out this quick example. So go ahead and pause the video after looking this figure and see if you can figure out what the net gravitational force is on this little M from these two forces to these two masses over here on the rights. Okay, cool. So let's get into it. The first thing we have to do is we have to label our forces and calculate them. So I've got a force that points off in this direction. And I'm told that that F G is equal to five Newtons and I have another force that points off in this direction so that F G is equal to five Newtons and I'm told it feels a five Newton force from each em on the right. And that makes sense because we have the same exact are and same m for both of them. So we know that those forces are equal. So now we just have to split it up into its components. So I've got this data angle here that's relative to the X axis. So that means that this component here is gonna be FX and then I've got f y. That points off over here, So I've got f y. But this angle right here relative to the X axis is going to be the same as this angle right here because I had the same distances involved are, and we can assume that these masses right here are sort of like connected by that vertical line. So that means that the angles for both of them are the same. Which means that the why components are gonna be pointing in opposite directions, but they're going to be canceling each other out so that I can cancel out my f y with this F y right here. And the FX components are actually going to add together. So that means that the net gravitational force is going to be pointing off in this direction. So what are the X components for? Both of these things will remember that the X components if I look at my vector edition is just f times cosine theta. So I actually can put I could put numbers to that because I know what the force is. I know that these five Newtons and as for the angle I'm given that the angle is 53. degrees. So that means that each FX component is three Newton's. And as for the Net force these things we're gonna add together and they're gonna be perfectly equal to each other in terms of the components. So that means that the F Net is going to be two times FX. So that means that the net gravitational force is two times three, which is just six Newton's. Alright, guys, let me know if you have any questions with stuff