Hey, guys. So in this video, I'm going to talk about the relationship between an object's center of mass and whether the object will balance itself on a surface, whether the object will stay balanced or tilt, at the edge of a surface. Let's check it out. So, first of all, remember that an object's weight, mg, always acts on the object's center of gravity. It's called center of gravity because that's where gravity acts. Okay? Now, for most of you, most of the time, center of gravity means the same thing as center of mass. If your professor has made a big deal about the difference between the two, then you need to know the difference between them. I'm not going to talk about it in this video. For a vast majority of you guys and for a vast majority of physics problems, all you need to know is that the two things are really the same. So, I'm gonna call this center of gravity or center of mass. In fact, some of you will never really see a problem where they are different. Okay? So, remember also that if an object has what's called uniform mass distribution, this means that mass is evenly distributed in an object. For example, if you have a bar, this means that you have the same amount of mass in every piece of the bar as opposed to this is a uniform mass distribution as opposed to if you have a bar that has way more mass here than in other parts. This is not a uniform mass distribution. Guess what? A vast majority of physics problems will be like this. I'm sorry, like this will be uniform mass distribution. All right? So that's good news. If you have uniform mass distribution, the object's center of mass will be in its geometric center. What geometric center means is it's going to be in the middle, okay, middle. So it's just going to be dead in the center right there. And what that means is that mg will act here. mg always acts on the center of gravity, and the center of gravity is almost always in the middle. It is in the middle if you have uniform mass distribution. Okay. If you have an object sticking out of a surface like this, it will tilt if its center of mass is located beyond the support's edge. So that's two situations here. I got the same bar on two desks, but this one is located here. The center of mass is within the table, right. In this case, it's right in the middle. And then here it is, it is beyond the table. What that means is that here, the object will not tilt. You can try this at home, but the acceleration will be 0. Right? So there's and this is at equilibrium, it won't tilt. Here, the object will tilt. There will be an acceleration that is not 0 and this is not equilibrium. So, if you want an object to tilt, if you want an object not to tilt, you want this situation here, and this is static equilibrium. So, some questions will ask, what's the farthest you can place this object so that it doesn't tilt? And we're gonna solve these problems using the center of mass equation, which I'll show you here, which is actually gonna be much simpler. These are not torque problems, though they show up in the middle of a bunch of torque equilibrium questions. Okay? So, the equation here is that, let's say if you have two objects, m1 here and then m2 here, and you want to find the center of mass between them. The x position of the center of mass will be given by the sum of mx divided by the sum of m. And what this means for two objects, just to be very clear, it's something like m1x1 + m2x2 divided by m1 + m2. If you had three objects, you keep going. M1x1, M2x2, M3x3. M are the masses, and x is the x position of that object. Alright. So, let's check out this example here. So here we have a 20-kilogram plank that is 10 meters long. So massive plank, 20, length of plank, 10. It's supported by two small blocks right here, 1, 2. 1 is at its left edge. So this is considered to be all the way at the left, even though it's a little, it's wide here. You can just think of it being right here at the very left. And the other one is 3 meters from its right edge. So, the right edge of the plank is here. This is 3 meters away. The entire thing is 10 meters. So, if this is 3, this distance has to be 7. A 60-kilogram person walks on the plank. So, this guy right here, I'm gonna call it big M equals 60. And I wanna know, what is the farthest a person can get to the right of the rightmost support before the plank tips. So, I want to know, what is this distance here. Okay, what is this distance here? Alright, and the idea is this is not really a torque, this is not really an equilibrium question we're gonna solve with torque. Instead, it's an equilibrium question we're gonna solve with the center of mass equation. And the idea is, if this person as this person changes position, the center of mass of the system will change. The system here is made up of Planck plus person. You can imagine if the guy is somewhere over here, don't draw this because I'm going to delete it. If the guy is somewhere here, the center of mass of the tube will be somewhere like here. Right? If this thing was really long, and the guy was, whoops, if this thing was really long and the guy was here, you would imagine that the center of mass between the two would be somewhere here, which means it would definitely tip because it's past the rightmost support point. It's past the edge. Okay? So, what you want to find the right, the rightmost he can go, the farthest he can go, is you wanna know what position does he have to have so that the center, the center of mass of the system or the combination of the two, will end up here. This is the farthest that the center of mass can be before this thing tips. So basically, you wanna set the system's center of mass to be at this point, right, which is 7 meters from the left. Okay. So the idea is, if the center of mass can be as far as 7, what must x be? This distance here, we're going to call this x. What must x be to achieve that? Right? So that's what we're going to do. And what we're going to do to solve this is we're going to expand the xcm equation. I have two objects. So it's going to be, m1x1 + m2x2 divided by m1 + m2. And this equals 7. And the tricky part here is going to be not the masses, but the xs. Alright? The distances. The first mass is 20. It's the mass of the plank. The x of the plank is where the plank is. Now the plank is an extended body. So where the plank is is really the plank's center of mass, which because the plank has uniform mass distribution, it doesn't say this in the question, but we can assume it. Because the plane, because it has uniform mass distribution. I'm gonna assume this happens in the middle, mg, little mg. The guy has big mg over here. This happens at a distance of 5 meters, right down the middle. So I'm gonna put a 5 here. What about the guy? Well, the guy's position is over here, which is 7+x. I hope you see this is x and this whole thing here is 7. Right? This whole thing this whole thing here is 7. So this is going to be oops. Sorry. So this entire distance from the left is 7+x. So that's what we're going to do here. M2 is the guy 67+x divided by the two masses which are 2-60, and this equals 7. This is a setup. If you got here, you're 99% done. We just got to get x out of here by using algebra. So, we're going to multiply these 2, 100. I'm going to distribute the 60. 60 times 7 is 420 plus 60 x. This is 80. If I multiply 7 times 80, I get 560. Okay. 7 let me put 7 times 80 here, and that's going to be 5, 60. I forgot that this is 60 x, of course. So I'm gonna send these two guys to the other side. So I'm gonna get 60 x equals 460. I'm sorry, 560 minus these two, which is 520. And the answer here is or the result here is 40. So I have x equals 40 divided by 60 and 40 divided by 60 is four over six or two over three, which is 0.67 meters. This means that x is 0.67 meters. It's how much farther he can go beyond that point. That's not much, right? So, even though this bar is 10 meters long and it's supported here, the guy can only walk a little bit more, and that's because he's much heavier than the bar. So, this should make some sense if you can somehow picture a 10-meter-long or a 30-foot-long bar. You can only walk a few steps beyond its 7-meter point or 70% length of the bar before the bar starts tipping if you are much heavier than the bar. All right. So that's it. That's how you would find this, and I hope it makes sense. Let me know if you have any questions and let's keep going.
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Center of Mass & Simple Balance
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