Hey, guys. So often problems that you'll see in 2D gravitation will have a lot of big numbers and a lot of steps to follow. In this video, I want to give you a shortcut in order to minimize the amount of work that you have to do. So, let's check it out. Let's say I was trying to figure out what the gravitational force is on this top mass right here. I want to find fnet. Now, how do I do that? Well, first, I have to go ahead and label my forces. There are 2 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 fg 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 m's are the same, and I haven't given numbers for those. I've just said that they're both little m's, 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 fg. Remember that fg is just big G times the mass of both objects divided by their distance squared. So if you have the same m's and the same r's, the gravitational force is the same.
Now that we've calculated what the gravitational forces are if I actually had numbers for this, I'd have to decompose the vectors into its components. Just to refresh that, the way I would do that is if you have this vector that points off in this direction, then you can always split it off into its components. So I have an x-component which is fx and this is a y-component fy, if this is a force or a vector like f. Right? And we do that using trig. Right? As long as I have the angle theta relative to the x-axis, I can get its components. So the next step for this is, now that I have the forces, I can split it off into its components. So if this is the angle relative to the x-axis, then that means it's going to have an x-component here, and that x-component is going to be related to the cosine of theta, whereas the y-component down here that points in this direction is going to be related to the sine of theta.
But I have the same exact forces, and if these things have the same angles and the same forces, then you're going to end up with the same exact components in the x- and the y-directions. So the tip is, if you end up with 2 components in the y in either direction that are equal but opposite, you can always cancel them out if they're opposite. So this is another way where you can use symmetry to reduce the amount of work that you have to do. Let me show you. So I have these x- and y-components, but this force over here on the left is going to break up into the same exact components, fx and fy, except the y-components are going to 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 going to end up canceling both of those out. Instead, what's going to happen is that the fy components are going to add together and your net force is going to be pointing down here. That net force, which I wanted, is just going to be equal to 2 times fy.
This is one way that symmetry can reduce the amount of work that you have to do. So I want to attach some numbers to this stuff. Let's go ahead and work out this quick example. Go ahead and pause the video after looking at 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 right. Okay, cool. 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 fg is equal to 5 newtons. And I have another force that points off in this direction, so that fg is equal to 5 newtons. And I'm told it feels a 5-newton force from each m on the right, and that makes sense because we have the same exact r and the 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 theta angle here that's relative to the x-axis. So that means that this component here is going to be fx, and then I've got fy that points off over here.
But this angle right here relative to the x-axis is going to be the same as this angle right here because I have the same distances involved r, 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 y-components are going to be pointing in opposite directions, but they're going to be canceling each other out. So that I can cancel out my fy with this fy right here, and the fx components are actually going to add together. That means that the net gravitational force is going to be pointing off in this direction. What are the x-components? So I actually can put, I could put numbers to that because I know what the force is. I know that it’s 5 newtons. As for the angle, I'm given that the angle is 53.1 degrees. So that means that each fx component is 3 newtons. And as for the net force, these things are gonna add together, and they're gonna be perfectly equal to each other in terms of the components. So that means that the fnet is going to be 2 times fx. So that means that the net gravitational force is 2 times 3, which is just 6 newtons.
Alright, guys. Let me know if you have any questions with this stuff.
