Guys, some professors are going to give you a system of objects, multiple objects, and they're going to ask you to calculate the center of mass. In this video, I'm going to introduce what the center of mass is and the equation that we use to calculate it. It's very straightforward. So let's go ahead and take a look here. The idea here is that sometimes it's useful to simplify a group or a system of objects by replacing them with a single equivalent object. The idea here is that a lot of times in problems, instead of having to deal with lots of small little masses and having your equations be really long and sort of tedious, you can actually replace it or simplify your problem by just considering a single object. There's a couple of interesting things that happen. So let's take a look at our example here. We want to take these two objects. Right? So we have 10 and 10, these two masses, m_{1} and m_{2}, and we'll replace this system here with a single equivalent object. So how do we do that? Well, it turns out that when you combine objects, the mass of the combined object is just the sum of all the masses of the individual objects that make it up. It's just going to be the sum of all the little m's. So really, what we want to do here is that we want to replace this system of objects with a single m, and that's just going to be 10 plus 10, and so that's just going to equal 20 kilograms. We want to replace this system here with a single 20-kilogram object. The problem here is I don't know where exactly that 20-kilogram object is going to be located along this number line, and that's what I want to figure out. These two objects here are at x equals 0 and x equals 4. I want to figure out where along this number line I'm going to replace these two objects with a single 20-kilogram object, and that's where the center of mass comes in. So the mass is the sum of all the masses of the objects, but the center of mass is going to be the average position of all of the objects in the system. It's going to basically tell you where the location of that single equivalent object is going to be. So let's go ahead and take a look at some very quick examples here. So very quick conceptual examples. So if you have a single object, the center of mass is pretty easy to find. A single object, the center of mass is just going to be basically right in the center of that object. If you have two objects, it's going to be a little bit more complicated, a little bit more interesting. So last, imagine we had these two blocks of mass m like this. The center of mass is going to be somewhere in between them. But if they are equal mass, the center of mass is actually going to be directly in between them. The center of mass for two equivalent or equal mass objects is going to be directly in between them. That's where the center of all of the mass that's concentrated in the system is. When you have two objects of unequal mass, it's actually a little bit even more complicated. So now let's say you have a block of m_, and then another block of 2 m. Now what happens is that the center of mass is not actually going to be directly in between them. The center of mass is actually going to be skewed towards the heavier one because that's where more of the mass center is located in the system.

Alright. So how do we actually calculate the center of mass? That actually brings me to the equation and I'll just give it to you. The center of mass here is going to be the sum of all the masses times the positions of all the objects in the system divided by the sums of all the masses. So basically, the way that it works is if you have m_{1}, you're just going to multiply m_{1} times its position, m_{1} x_{1}, and then m_{2} times x_{2}, and then m_{3} times x_{3}, and then so on and so forth, however many masses you have in the system, divided by the sum of all the masses. So m_{1}, m_{2}, m_{3}, and so on and so forth. Alright? So this is the center of mass equation. Let's go ahead and take a look at our examples and finish them off. So we know we want to calculate these, with the center of mass here because we want to figure out where we can put the single equivalent 20-kilogram block. So to do that, we're going to use the center of mass equation. So this is going to be the sum of all masses times the positions divided by the sum of all masses. There's only two here. We have m_{1} x_{1} plus m_{2} x_{2} divided by m_{1} plus m_{2}. So we have the masses. Right? They're both just 10 and 10. So we're going to set this up the way we'd normally so the way we kinda settle up a momentum equation. We have the masses. So we're just going to plug in those masses, and then we have to figure out what goes inside the parentheses here. So 10, plus 10 on the bottom. So what goes inside this parenthesis? Well, this is just going to be the position of this first 10-kilogram box, and that's actually x equals 0. So there's a 0 that goes here and it goes away. And then the x equals 4 is for the second, 10-kilogram mass. That's what goes inside this parenthesis here, and then we're done. There's only two masses. So this is going to be 40 divided by 20. So your center of mass is going to be 2 meters, and this should make some sense. If you have these two boxes, x equals 0 and 4, the center of mass, because they are equal mass, is actually just going to be directly in between them. So your center of mass is actually going to be right over here. And that's actually exactly what we said for the situation right here. If they're equal mass, the center of mass is directly in between them. Let's take a look at our other example. It's the same setup except now we have a 10 and a 30-kilogram block here. So just before we start, where do you think the center of mass is going to be? Do you think it's going to be at x equals 2, or do you think it's going to be to the left or right? Well, hopefully, you guys realize that this object is heavier and so the center of mass is going to be somewhere over here. Let's go ahead and check this out. So these total mass of this object is just going to be 40 kilograms. How do we calculate the center of mass? Again, we just use the equation. Again, there's only two objects, so so I'm just going to use m_{1} x_{1} plus m_{2} x_{2} divided by m_{1} plus m_{2}. So here I have 10 times something plus 30 times something divided by 10 +30. So what goes inside here? Well, remember that this 10-kilogram block is going to be at x equals 0. So I'm just going to plug in 0 there. And this 30-kilogram block is at x equals 4, so I plug in a 4 here. What you end up getting here is you end up getting a 120 divided by 40 and you're going to get an x_{center of mass} position to be 3 meters. So if we look at our number line, it turns out we were right. This is x equals 2, but this is not the center of mass because the heavier one sort of skews the center of mass towards the right. So this is actually your center of mass right here at x equals 3. So the center of mass is not necessarily in the middle of objects. As we've said before, if you have different masses, it's usually going to be closer towards the heavier objects. Alright, guys. So that's it for this one. Let me know if you have any questions.