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Non-Uniform Mass Distributions (Find Center of Mass)

Patrick Ford
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Hey, guys. So in most problems we've seen so far involving a extended object, so not a point mass, a tiny point mass, but a long extended object or extended body. We've had uniformed mass distribution, which means that the center of mass of the object is in the middle. The object is evenly distributed in the center of Mass is assumed to be in the middle. Now that's not always going to be the case. And in some problems you have what's called non uniformed mass distribution. Andi, I want to show you what this looks like in some examples. Let's check it out. So unless otherwise stated, you can safely assume that a rigid body and extended body will have uniformed mass distribution. Okay, uniformed mass distribution again means that mass is evenly distributed. This is good news, because then you can assume that the weight acts on the objects center. Okay, but if you don't have uniformed mass distribution, it means that you cannot assume the location of the objects center of Mass. Therefore, you can't assume where MG happens. Okay, now there's 22 types of problems. I mean, there's more than two but two of the basic types of problems will be Either It will be giving you a center of mass and ask you to calculate something else or the other way around. It will give you some other information and ask you to calculate the center of mass. Alright, so I want to show you this example here. So you get a new idea of how this works. I have an 80 kg man that is 2 m tall and he lies horizontally as it's shown here on a 2 m long board off negligible mass. So the mass of the man is 80 both him in the border, 2 m long. So we're gonna assume that his head matches the right end and his feet are exactly touching the left end. Um, he has a mass. Uh, but the board doesn't have any mass there to scales that are placed under him. So they're here, and the readings of the scale at the ends. The reading the scales are left 3 20 right. Um, left 3. 20 and right for 80. The reading of the scale is the normal force. So that means that there's a normal force up here. Okay, so normal. One is 3. 20. Normal two is 4 80. Those are two forces. There's one more force the board is massless of. The board doesn't have an MG, but the guy has an MG. Now, if you didn't know where to draw his center of mass because that's what we're looking for, you could have just put in the middle. Or even better if you knew that the human body has its center of mass closer to the head than the feats. If you knew that, you could just draw out there. If you didn't know that, you could tell by the numbers here by the by the normal forces the fact that this, uh this is a greater force, um, is telling you that the center of mass is closer to that right scale. All right, so I'm gonna do it a little bit off to theme the little closer to the head here. So I wanna know how far from his head is the center of mass. So I wanna know this distance X here. Now, in all of these problems where you're looking for a distance these air tricky You wanna write every other distance as the variable that you're looking for. So here, if I'm looking for this piece and I call this piece X, I need to try to rewrite this other piece as a function of X. So in terms of X, So if the whole thing is to and then this piece is X, then this piece is two minus x. Okay. Now, to solve this problem, this is a a complete equilibrium. Questions so I can use I'm gonna use some of our forces equals zero, and I'm gonna use some of all topics equals zero. Let's start with the easy one. Some of all forces equals zero. I'm gonna go through this one very quickly because all this equation does it's gonna be pretty useless. Um, it's just gonna tell us that everything ends up, so check it out. I have n one plus into going up equals M g going down. And if you plug in these numbers 3 20 plus 4 um, M g is 80 times 10. This just tells us that 800 equals 800 which is cute, but not really useful. So I'm gonna quickly jump into some of all torques equals zero. Now, remember, we can do this for any points, but it might make sense to pick a point where it makes sense to pick a point with a force happen. So either one to or three. And in this particular problem, it really doesn't matter which point you pick, you'll be able to find the answer. Um, either way, Okay, So I'm just gonna go ahead and pick one here, um, sort of at random, and that's what we're gonna get. So some of all talks about 10.1 equals zero. This means that this is the axis of rotation and I have n one happening here, and the one will not produce a torque because it happens in the axis of rotation. Mg over here will produce a torque. It will produce a torque in this direction. Torque of M. G. Um, that's because imagine the access is here and you are pushing down. So this thing is trying to spend this way. This is clockwise, which is positive and then normal. Chew right here is, um causing it torque in this direction again. You got the project here held here. Then you push this way you go this way, which is in the direction of the unit circle, which is counter clockwise. Eso it is. I'm sorry. This is negative. Clockwise, and this is positive. Hopefully you caught. That's those are the only two talks I have. They have to cancel. So that means I can, Right? That's the torque of M G equals the torque of N two. Torque is force, so M g r. And then the sign of data and force and to our sign of data. So let's draw our our vectors. For these two forces, this is the axis of rotation. The R four mg is this one or M G. And they are for normal two. Is this one notice that both of them make 90 degrees with their respective forces. So this would be sign of 90 inside of 90. They both become simply one. Andi, Even if it wasn't one, they would cancel each other anyway. Okay? And then the distances. What's the distance to M. G? It's this distance here, which we called, um, to minus X right. This is the distance we're talking about right here. And so I'm gonna put here tu minus X in the distance toe and choose the entire length of this thing, which is to so see how there's an X here. That's my answer. Okay, so we're gonna get that in just a little bit. So now we're just gonna plug it in and get the X out of there. So massive, the guy is 80 gravity 10 times two minus X equals and to I know into its 4 80 for 80 times too. So we're gonna This is 800. I'm gonna distribute 800 times. Two is 1600 minus 800 X equals this, which is 9 60. Okay, so I gotta move some stuff around. This guy's negative. I'm gonna move it over here. But then I'm gonna move this guy over here so that the access by itself. So I'm gonna have 1600 minus 9. 60. Um, equals 800 x. So 1600 minus 9. 60 is 6 40. Divided by 800 6 40. Divided by 800. That's my ex. I didn't calculate this in advance, but we're gonna do this real quick. Um, 64 80 both have eight as a factor. So this is eight times eight on this is eight times 10. So I do this eight divided by 10.8. That's my X. Okay, So if the exes 0.8, the whole thing is, too. Which means this will be 1.2. You can see how the middle would've been one. Right? You can see how this is sort of 80% of your height is where your center of mass is in this problem, right? This isn't necessarily fully accurate, and it also changes from person to person. But that's it. That's where your center of masses and this is how you can calculate a center of mass. Um, if you're given two measurements for supporting forces, okay? A different version of this question, by the way, could have had tensions, instead holding an object and then trying to figure out center mass that object based on the difference between the two tensions. All right, um, there's some shortcut you could have used here to solve this problem. For example, you might have seen you could have used, like some ratios between 4. 80 and 3 20 to figure out your how far you have to be in terms of center of mass So if you do, for example, for 80 divided by, that's this one here divided by the whole thing, right, This is 0.6, which means it's 60% of your your 60% of the way now 600.6 times 2 m would give you 1.2, and that would give you the distance from your feet up. Right? So you couldn't use that a za quick shortcut. I'm showing that just because some of you might have noticed or wondered about that. But what I really recommend you do just to play it safe is just keep it standard to use the Tory equation to solve it. All right, so that's it for this one. Hope it makes sense. Let me know if you have any questions and let's keep going.