Center of mass

I will now do a presentation on the center of mass. And the center mass, hopefully, is something that will be a little bit intuitive to you, and it actually has some very neat applications. So in very simple terms, the center of mass is a point. Let me draw an object. Let's say that this is my object. Let's say it's a ruler. This ruler, it exists, so it has some mass. And my question to you is what is the center mass? And you say, Sal, well, in order to know figure out the center mass, you have to tell me what the center of mass is. And what I tell you is the center mass is a point, and it actually doesn't have to even be a point in the object. I'll do an example soon where it won't be. But it's a point. And at that point, for dealing with this object as a whole or the mass of the object as a whole, we can pretend that the entire mass exists at that point. And what do I mean by saying that? Well, let's say that the center of mass is here. And I'll tell you why I picked this point. Because that is pretty close to where the center of mass will be. If the center of mass is there, and let's say the mass of this entire ruler is, I don't know, 10 kilograms. This ruler, if a force is applied at the center of mass, let's say 10 Newtons, so the mass of the whole ruler is 10 kilograms. If a force is applied at the center of mass, this ruler will accelerate the same exact way as would a point mass. Let's say that we just had a little dot, but that little dot had the same mass, 10 kilograms, and we were to push on that dot with 10 Newtons. In either case, in the case of the ruler, we would accelerate upwards at what? Force divided by mass, so we would accelerate upwards at 1 meter per second squared. And in this case of this point mass, we would accelerate that point. When I say point mass, I'm just saying something really, really small, but it has a mass of 10 kilograms, so it's much smaller, but it has the same mass as this ruler. This would also accelerate upwards with a magnitude of 1 meters per second squared. So why is this useful to us? Well, sometimes we have some really crazy objects and we want to figure out exactly what it does. If we know its center of mass first, we can know how that object will behave without having to worry about the shape of that object. And I'll give you a really easy way of realizing where the center of mass is. If the object has a uniform distribution-- when I say that, it means, for simple purposes, if it's made out of the same thing and that thing that it's made out of, its density, doesn't really change throughout the object, the center of mass will be the object's geometric center. So in this case, this ruler's almost a one-dimensional object. We just went halfway. The distance from here to here and the distance from here to here are the same. This is the center of mass. If we had a two-dimensional object, let's say we had this triangle and we want to figure out its center of mass, it'll be the center in two dimensions. So it'll be something like that. Now, if I had another situation, let's say I have this square. I don't know if that's big enough for you to see. I need to draw it a little thicker. Let's say I have this square, but let's say that half of this square is made from lead. And let's say the other half of the square is made from something lighter than lead. It's made of styrofoam. That is lighter than lead. So in this situation, the center of mass isn't going to be the geographic center. I don't know how much denser lead is than styrofoam, but the center of mass is going to be someplace closer to the right because this object does not have a uniform density. It'll actually depend on how much denser the lead is than the styrofoam, which I don't know. But hopefully, that gives you a little intuition of what the center of mass is. And now I'll tell you something a little more interesting. Every problem we have done so far, we actually made the simplifying assumption that the force acts on the center of mass. So if I have an object, let's say the object that looks like a horse. Let's say that object. If this is the object's center of mass, I don't know where the horse's center of mass normally is, but let's say a horse's center of mass is here. If I apply a force directly on that center of mass, then the object will move in the direction of that force with the appropriate acceleration. We could divide the force by the mass of the entire horse and we would figure out the acceleration in that direction. But now I will throw in a twist. And actually, every problem we did, all of these Newton's Law's problems, we assumed that the force acted at the center of mass. But something more interesting happens if the force acts away from the center of mass. Let me actually take that ruler example. I don't know why I even drew the horse. If I have this ruler again and this is the center of mass, as we said, any force that we act on the center of mass, the whole object will move in the direction of the force. It'll be shifted by the force, essentially. Now, this is what's interesting. If that's the center of mass and if I were to apply a force someplace else away from the center of mass, let' say I apply a force here, I want you to think about for a second what will probably happen to the object. Well, it turns out that the object will rotate. And so think about if we're on the space shuttle or we're in deep space or something, and if I have a ruler, and if I just push at one end of the ruler, what's going to happen? Am I just going to push the whole ruler or is the whole ruler is going to rotate? And hopefully, your intuition is correct. The whole ruler will rotate around the center of mass. And in general, if you were to throw a monkey wrench at someone, and I don't recommend that you do, but if you did, and while the monkey wrench is spinning in the air, it's spinning around its center of mass. Same for a knife. If you're a knife catcher, that's something you should think about, that the object, when it's free, when it's not fixed to any point, it rotates around its center of mass, and that's very interesting. So you can actually throw random objects, and that point at which it rotates around, that's the object's center of mass. That's an experiment that you should do in an open field around no one else. Now, with all of this, and I'll actually in the next video tell you what this is. When you have a force that causes rotational motion as opposed to a shifting motion, that's torque, but we'll do that in the next video. But now I'll show you just a cool example of how the center of mass is relevant in everyday applications, like high jumping. So in general, let's say that this is a bar. This is a side view of a bar, and this is the thing holding the bar. And a guy wants to jump over the bar. His center of mass is-- most people's center of mass is around their gut. I think evolutionarily that's why our gut is there, because it's close to our center of mass. So there's two ways to jump. You could just jump straight over the bar, like a hurdle jump, in which case your center of mass would have to cross over the bar. And we could figure out this mass, and we can figure out how much energy and how much force is required to propel a mass that high because we know projectile motion and we know all of Newton's laws. But what you see a lot in the Olympics is people doing a very strange type of jump, where, when they're going over the bar, they look something like this. Their backs are arched over the bar. Not a good picture. But what happens when someone arches their back over the bar like this? I hope you get the point. This is the bar right here. Well, it's interesting. If you took the average of this person's density and figured out his geometric center and all of that, the center of mass in this situation, if someone jumps like that, actually travels below the bar. Because the person arches their back so much, if you took the average of the total mass of where the person is, their center of mass actually goes below the bar. And because of that, you can clear a bar without having your center of mass go as high as the bar and so you need less force to do it. Or another way to say it, with the same force, you could clear a higher bar. , Hopefully, I didn't confuse you, but that's exactly why these high jumpers arch their back, so that their center of mass is actually below the bar and they don't have to exert as much force. Anyway, hopefully you found that to be a vaguely useful introduction to the center of mass, and I'll see you in the next video on torque.