11. Momentum & Impulse

How to Identify the Type of Collision

# Intro to Center of Mass

Patrick Ford

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Hey guys, so some problems going to give you a system of objects, multiple objects, and they're gonna ask you to calculate the center of mass. So in this video I'm going to introduce you what the center of mass is and the equation that we used to calculate it. It's very straightforward. So let's go ahead and take a look here. So 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 objects. So 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 to 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 things, interesting things that happen. So let's take a look at our example here. So we want to do is we want to take these two objects, Right? So we have 10 and 10. These two masses M. One, I'll call them M. One and M. Two. And want to replace this system here with a single equivalent objects. 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 the sum of all the little Ems. 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 gonna be 10 plus 10. And so that's just gonna equal 20 kg. So, we want to replace this system here with a single 20 kg object. The problem here is, I don't know where exactly that 20 kg object is going to be located along this number line. And that's what I want to figure out. So these two objects here are X equals zero and X equals four. I want to figure out where along this number line, I'm going to replace these two objects with a single 20 kg 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 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, a 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 going to be basically right in the center of that object. If you have two objects, it's gonna be a little bit more complicated. A little bit more interesting. So that imagine we have these two blocks of mass. M like this, the center of mass is going to be somewhere between them. But if they are equal mass, the center of mass is actually gonna 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 is 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 em and then another block of two 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 gonna be skewed towards the heavier one because that's where more of the mass is central is located in the system. All right. 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 gonna 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 it works is if you have M. One, you're just gonna multiply M one times its position and one X one and then M two times X two and then M three times X three 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 one, M two M three and so on and so forth. All right, so this is the center of mass equation. Let's go ahead and take a look at our examples and finish them off. So, you know, we want to calculate the uh the center of mass here because we want to figure out where we can put the single equivalent 20 kg block. So to do that, we're gonna use the center of mass equation. So this is gonna be the sum of all masses times the positions divided by the simple masses. There's only two here, we have M one X one plus M two X two divided by M one plus M two. So we have the masses, right? They're both just 10 and 10. So we're going to set this up the way we would normally. So we we kind of set up a momentum equation. We have the masses. So we're just gonna 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 these parentheses? Well, this is just gonna be the position of this 1st 10 kg box and that's actually X equals zero, so there's zero that goes here and it goes away and then the X equals four is for the 2nd 10 kg mass. That's what goes inside this parentheses here and then we're done, there's only two masses. So it's gonna be 40 divided by 20. So your center of mass is going to be two m. And this should make some sense if you have these two boxes, X equals zero and four. 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. Now, we have a 10 and a 30 kg box 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 two? 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 the total mass of this object is just gonna be 40 kg. How do we calculate the center of mass? Again? We just use the equation again, there's only two objects. So I'm just gonna use em one X one plus M two X two divided by M one plus M two. So here I have 10 times something plus 30 times something divided by 10 plus 30. So what goes inside here will remember that this 10 kg block is gonna be at X equals zero. So we're just gonna plug in zero there and this 30 kg block is at X equals four. So I plug in a ford here. What you end up getting here is, you know, beginning 120 divided by 40 and you're gonna get an X center of mass position to be three m. So, if we look at our number line turns out we were right, this is X equals two, but this is not the center of mass because the heavier one sort of skews the center of mass towards towards the rights. So this is actually your center of mass right here at X equals three. 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. All right, guys, So that's it for this one. Let me know if you have any questions.

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