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15. Rotational Equilibrium

Review: Center of Mass


Intro to Center of Mass

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Hey, guys. So in this video, I'm going to talk about center of Mass, which is a pretty straightforward concept in physics. Now, the ideal center of mass is that when you have a bunch of objects in a system, you can simplify that system and represented as a single object. For example, if you have thousands or even millions of planets and stars in the galaxy spread all over the place, you could treat the entire system of millions of things as one object. Okay, let's check it out. So, as I said, sometimes it's useful to simplify a system of objects, a collection of objects by replacing all of them with a single equivalent object. So, for example, instead of having a bunch of things moving this way, I can simplify this and just say that there is a single object that goes that way. Okay, now this single object who have mass m equals the sum, the summation of all the individual masses, right? So since you're combining, you add up all the masses and the system will be located right at the systems. The single object will be located, this systems center off mass, and here's a really simple example of that. Let's say I wanna combine the system made up of two masses, 2 kg and 2 kg into a single object. So first of all, the total combined mass will be 4 kg. Now, where would it go? Well, I hope you're thinking that if I wanted to simplify instance one thing the center of this whole combination of things is actually right down the middle. Okay, so I would have something like this where this gap here is 5 m, Okay. And this would be a 4 kg objects. Now, the reason it's down the middle is because this is a balanced system. The left and the right have exactly the same masses. But if, for example, I had something like a two here in a 10 year, the center of Mass would be much closer to the 10. Would be somewhere over here. Okay, so that's the idea. Now, the middle thing Onley works. If they are the same, you're not gonna get that. So you're gonna need to use an equation. You're gonna need to use the center of mass equation to figure out where the center of this combination of masses is and you're going to use, um, the exposition. So where along the horizontal the center of mass is is going to use this equation some of x. Sorry, some of em times X divided by some of em. And I'll explain what that means. I'll give you an example where you have, let's say, three objects. Um, some of them Max looks like this m one x one plus m two x two plus groups m three x three. That's what it means to do the summation of the max you're gonna have m one x one m two x two and three extra. And the some of em's is just m one plus m two plus M three. Now, this is in the case like this, where both objects on the X axes you draw a line between them and the center of mass will be somewhere along that line. But if you have objects in a two dimensional plane, something like this, and notice that I'm intentionally drawing the ball is different sizes. Okay, if you have a system made up of four objects, okay, the center of mass will be somewhere in the middle of the system. In this case, the left side balances with the rights, so it's gonna be somewhere down the middle. But the bottom is much heavier than the top. So you're not gonna have the center of Mass be along a line here. It's going to be further down somewhere here. Okay, a little bit closer. This is where the center mass is. So I hope you see how this is two dimensions because I have X and y eso. In that case, you're going to use not only the y equation, but you're also going to use right in addition to the Y equation you're going to also use and X and y, And the equation is the same. Some of em instead of x some of m y divided by some of em. Okay, so this looks like this m one y one m two y two m, three y three m one plus m two plus M three. Now X and y is the X and y position of these objects. Let's do two quick examples, um, to see what this looks like in practice, so I have to mass is placed along the X axis, so Here's the X axes. One mass is mass A at 0 m. So let's say zeroes over here. So I got at 0 m, I got a which is 10 kg and at 4 m I have be, which is 20 kg and I wanna know what is the center of mass. This is X X is only so I'm gonna say that the center of mass is the ex center of mass and the equation is this So it's m one x 12 x two. So what I'm gonna do is write the masses here, 10 x 20 x 10 plus 20. Okay, Now all I gotta do is plug in the numbers. What is the exposition off the 10. The 10 is at zero and the 20 is that four. Okay, so I put a four here and that's it. Very straightforward. This cancels I have 80/30 and the exposition of this thing is to 67 meters. Okay, Now the middle between zero and four is too. You should have been expecting that the actual center of mass is somewhere to the right of the middle because the system is heavier on the right side, and that's what it is. Ex center of mass. That's what we get. 2.67 So you're able to if you think about it a little bit, sometimes you be ableto sort of. Guess where the answer will be. Okay, I wanna make a quick point, and then we're gonna jump into the next example. Um, there are two terminal terminal two terms that are similar. Um, one is center of gravity in the other one center of mass. Now, in this video, we're only talking about center of mass. We're not talking about center of gravity, and we're not really gonna do problems with center of gravity. But I do want to clarify that they're two different terms and they mean different things now without getting into its center of gravity, is I will just tell you that they actually are the same thing if the gravitational field is constant. So this is a conceptual point that I want you to remember. If the gravitational field is constant, then center of mass instead of gravity or the same thing, we can use them interchangeably. So what does it mean for the gravitational field to be constant? well, let me draw something real quick. You don't have to draw this. This is just conceptual. But let's say here's the earth and you're here and your sisters here or whatever. Um, you're very close to each other and I wanted you form a system. The gravity where you are is, let's say, 9.8, and it points straight down. Your sister's right next to you. So the gravity that she feels is also 9.8, and it's also straight down. If you guys are close enough together, the gravity will be almost identical. Um, in fact, will be so close that we can consider them to be the same. So because the two objects feel the same gravity, the gravitational fields is the same for both. Um, the center of mass is the same as the center of gravity. That's the idea. Okay, now, if to people on Earth are really far apart, they will feel different. Gravity's because even if it's 9.81 is being pulled this way, the other ones being pulled this way. Okay, so it's conceptual point for you to know. Ah, lot of professors don't even get into this so if you didn't really hear the distinction between two don't even worry about it. It's not that big of a deal. Okay? The last point I wanna make here is that unless otherwise state stated, we're just going to assume that this here is the case. Okay, we're going to assume that the gravitational field is constant. So what does that mean? Well, that means that these two things mean the same Okay, are the same thing. So if gravitational fields constant, these two words mean the same. We're going to assume that it's constant. So we're going to assume that these terms are the same thing. We can use them interchangeably. Okay. Again. Not gonna dio problems with center of gravity. But I do want to touch up on the conceptual point there. All right, So let me quickly do example, too. And we'll be done with this concept. That's all we're doing. Um, so three masses were placed on the X y plane, so I'm gonna draw a little X Y plane like this. Why? X mass a, um, is placed at 0000 is right here. Remember, guys, uh, coordinate systems are X comma y. So this means X equals zero and y equals zero. This is objects A, which has a massive 10 being. Is that 03? So this is X and Y X equals zero is on this line here and three. I'm sorry. X equals zero. Is this year okay? And three going up. Oops. Sorry about that. Three going up will be somewhere here. So this is zero on the water in the X axis and three. Okay, So zero on the X axis means you haven't moved left or right here in the middle. And then three in the Y axis means you go up three be a kilograms, and then see, is that 40? So the X values four. I go for this way, but I stay on the on the x axis. I don't go up or down. Forward. Common zero Looks like this. And this is seen, which has a massive 6 kg. Here's a diagram. I wanna know what is the X and y coordinates of the center of mass of the system you might be thinking of the center of mass of the system is somewhere here. Okay? You can actually think about this in terms of X and Y. Let's try to look at this. So I in the y axis, I have this guy on the Y axis and these two guys in the Y axis. Okay, don't make all these scribbles, but notice how this one is eight. And these two here combined, uh, to be 16. So the y axis is heavier towards the bottom. So I'm going to predict that this thing will be, um, somewhere like around one. Okay. Not three. Not the middle, but closer to zero. Now, on the X X is, um you have these two here in this here, so the center of mass will be somewhere in the middle. The left. I have 10 and eight. That's 18. And to the right, I have six. Okay, um, to the right, I have six. So this this thing is much heavier towards the left. So instead of being in the middle between zero and four, which would be to I'm going to guess it's gonna be to the left of two. I'm gonna guess that's gonna be around one again. I'm just doing rough estimate estimates so that we can later see if it makes sense with what we expected, Okay? We're just gonna plug it in and we're done. Ex senator Mass is going to be. The masses are 10. So I'm gonna do this. Eight and six, divided by 10 plus eight plus six andan the Y center of mass is the same thing. 10. Hey and six divided by 10 plus eight plus six. The only difference is that here, I'm gonna add X values. And on the other one, I'm gonna add y values. Okay, So what's the X value of the 10? Just look right here. It's zero of the eight is zero. And of the six is four. What about the Y value? The Y value of the A zero of B is three and of c is zero as well. So when you do this, when you do this, you get six times 4 24 divided by 24. Okay, so it's a coincidence that the numbers happened to divide so neatly 1 m, and then here, this cancels in this cancels, um, I get eight times 3 24 as well. Again, a coincidence that this happens to be the same as that divided by 24 1 m. So I actually, um, sort of my rough estimate. What happened to be dead on? Um, but usually you just know that it's roughly a number. It's 11 So you could say that the center of mass of this system is at position. One comma, 1 m. Okay, that's the final answer. All right, that's it for center of mass. Um, let me know if you have any questions.