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Hint #1:Sum forces to find the weightHint #2:Sum moments to find the location of the center of massHint #3:Parallel Axis Theorum...

Question

Hint #1:Sum forces to find the weightHint #2:Sum moments to find the location of the center of massHint #3:Parallel Axis Theorum

Hint #1: Sum forces to find the weight Hint #2: Sum moments to find the location of the center of mass Hint #3: Parallel Axis Theorum



Answers

For the following exercises, compute the center of mass $(\overline{x}, \overline{y}) .$ Use symmetry to help locate the center of mass whenever possible.
$$ \begin{array}{l}{ \rho=3 \text { in the triangle with vertices }(0,0), \quad(a, 0)} \\ {\text { and }(0, b)}\end{array} $$

Let us find a send off. Most off the given dado morning in cagey located at X equal to Did he meet death? Yeah. Do you goto three kg located at X equal to minus one? We did the former force and rough Mosses exporting Would do you have more next? Warrantless? Um do we extrude? You were in vain. You have one less m two send it does sketches given system on the number line. You want you going to do 10 kg? You said three Yeah, until you go to three k g He said minus one Let us a place given value in the in this form Gloves and Ralph Moss ex money could do then in country unless, you know minus one Do it anyway then. Less thing The golden 30 minus treaty with waiter Dean So the center of Mosses its body could do the sound You're better, Dean Approximately I knew approximately 20. Sound your weight. 13 decoder 2.8 Study on this On your bettered Any goto who 0.8 So the central mosque is located at this hour You're a waiter

Okay, So for this problem, we're told that we have three masses and there are all off the corners of an equilateral triangle when you want to find the center of mass. So let's draw this out so mass mass mass? We want to get the center of mass. Um, so we want to get probably. We want to get in the horizontal and vertical direction. By symmetry, The, uh, the center of mass horizontally is just gonna be right in this centre. Oops. I meant to draw a street. Why? Let's see if I could do a better job. It's go ahead. So here's our sort of line of symmetry. So we know the center of mass is gonna lie on here on now. We need to get the, uh we need to get the vertical center of mass position. So go ahead and call this. Why Center of mass Since why is used generally from vertical, the vertical dimension, and then in general again, that's gonna be the some of the position of the mass times and the mass of that particular position divided by the total mass. So and this also depends on the origin, so stop once she was in origin. You and general wanted choose an origin that makes your cop calculation is easy as possible. So, um, you want to choose your origin to be where most of the masses? Because that means that the terms will not We'll have a zero position so they won't contribute here. Um, just so you don't have to type this watch into your calculator. So I'm gonna let my origin my y equals zero line. Be right here. So now I can evaluate this. So the these two masses all right here. So they're not gonna contribute. Um, but this third mass with mass m is going to be at this position now, uh, we need to get its height. H um, in terms of the dimensions given in the triangle. And so the triangle has a side L So we need to do a little trade to figure out what the height is. So in a, um, equilateral triangle can It was probably better ways to do this. I'm thinking of just breaking this up into a little right triangle. So this is, um so this is like in this triangle right here, doing a different color. So this has like, Oh, this is the height that we want cause that's gonna be our position. And this is 60 degrees. So we can use right triangle trade to say that the sign of 60 is h. Or how so h is gonna be l signed 60. And then the sign of 60 degrees is, um, square to 3/2. So I'll squirt a three divided by two is our height, so we can take that and put that in here for the white position. So it's gonna be all squared a 3/2. And now for the denominator, we want to just divided by the total mass to the total mass of the object is three m double check that. That's the givens. Well, I guess it's not really matter cause the masses you're gonna cancel. Um, so let's go ahead and simplify this. So those go like that and then then you have l divided by, um Let's see. Let's absorb this sort of three into here, So it's gonna be l divided by the square root of three and then my not bring this one a half on the bottom. So this is the final answer

Right? Or ask this old for, um, ex and and Why Anderson's arrest? Given that R. P. Is equal to two grams over son Amir swag and were asked to find the area between why you could x squared and why is he called nine X? And this is over, girl three. Right? So it's start out by graphing our equations forgiven. Why's he got to X squared? That's just a problem with the origin of the center that gold upwards like this. And then we have wide 69 next, when Why is equal to Darryl accessible to zero as well? And when executed l why is also equal to be Oh, so we have an intersection Where our exit Why intercepts at Joe? Well, let's see when acquired And nine intersect the night next intersect the X squared minus night x Did you cut it out? That's ex times X minus night addict. All right, so we have an intersection at nine. Well, why's equal to nine times nine? That gives me that gives me equal one. So the Intersect that nine comma 81. Okay, I'm just gonna draw like yeah, just just like somewhere that I don't need to know somewhere really far. So, nine, it suits like that. Scotus nine. But we don't need to worry about that because we're only looking at the interval from Joe today. So we're looking at something like right here. You get okay. Since we know that our top the graph is why is equal to Nynex and are bottom is why's he could to x squared to knowing this It's all or I'm white using equations one Well, im why is equal to okay. Times interval from zero street of x Times nine x to the top graft minus our bottom graft. Which is X word the ex. What actually is equal to the inner girl of nine X squared minus x cute The X. All right, Integrating this, we get P Times nine x cubed over three. Excuse me. Three x cubed. Why does X four over four and then evaluated at three. And so, looking in three, we get P times 325253 which is 27 times three, which is 81 minus 81 over four. This is minus evaluation at zero, which is myself. So any one times four give me 3 24 minus 81 over four times Peak. That gives me 3 24 months to anyone. Let it before. Well, let's just leave it at a fraction to Brittany. Four point is maybe one. Yeah, me, too. Three p over four. This is why. All right, Know solving for and excusing equation three. Um, X is equal to one over GOP integrated at Joe. It's a drink. Um, top graph the Nynex squared minus our bottom graph, which is X squared. You are, aren't you d X? But that's just equal to 1/2 p integrated at Nynex No. Nine to party, which is 81 x squared minus fixing for four. The ex. That gives me 1/2 p and integration of this, which is 81 over three x cubed minus x 50 bober five Evaluated at three. Enzo. Okay, so you get 1/2. You need to fire three. 27 times 81. Give me 2187 Divided by three is me 2187 3 7 to 9. Minus 3 to 4 or five between the 243 over five. All right, that's equal to 1/2. Seven too far. Nine times five is 36465 over five minutes to 43 over five, minus two for three. That gives me 34 You know, my peas. Okay, on half p. Let's do like this right shoe when we get 1701 p over. There's rmX. All right now, looking for our area, Which is easy. Question in a girlfriend. Jonesthree over top graph, which is nine x minus X squared. The X that gives me i n X squared over two minus X cubed over three, integrated at three. And so, for about a way to that green zone that gives me nine and 32 birth rate. There's nine, which is 243 over two when it's 3 to 3. 27 over the tree, don't you 43 over to minus 27 over three? No. Oh, my three alternates. What? PS Well, forgive me. It's like ew! You to part two, which is nine times nine is 81. Sorry about that. This is actually 81 because that's three to part two, which is lying in a nine times nine is 81. So we get 81 over two when it's 27 over three, which is 81 over two. And when? This night. That gives me anyone over to minus 18. Over too. So anyone minus 18 is equal to 63 over two times, Pete. Okay, this is a mess. Okay, So, given that our mass is equal to 63 over to pee, um, why is equal to 243 pee over three, um, ex to put 43 over four. You're an X is equal to 1701 over five. Piece. We get a better set of Bess. Is he could sue XY. Um, Why CME, which is equal to and why over em, comma and X over which is equal to and why would just two for three p over four times M, which is to over 63 peaks. And then X, which is 1 71 p over five times. And we're just too over 63 fee. Okay, This simplifies dude too. You're going to Mmm. See what else we can do? That's two for three over ones two or three over 63 times. Deal, which is 1 26 when we get 340 to over 315 That gives us two for three over 1 26 is one times 13 over a fortune and 1 71 times two divided by five, some 63. Give me 10 for over 100. Oh, so this is the center mass and our area and our moments.

Welcome to new problems. This time we have objects or you can call the masses. And these masses are on right? Trying also the you know, one of them is that 75 kg and a watchful did not 75 kg, but 75 g. So we can call that what we can start with the with the law one. So this one is M one has a muscle 100 g and then we have to to right here and them to us and myself. 250 g and then and three has, um ourselves 75 g and all these are connected. So, you know, think of a right triangle like that. These are three Mosses. Our goal in this particular problem is to figure out what the sent off muss eyes going to be. That so golden the problem. You know what's what's going to be the center of mass and it's a system. So we're looking for the center off mass for this system, and we have a lot. These three points have a radius. So this is are one. And, um, this is the second one is our to and then the third one is are three. So we have three radius is so the position off. The central months would have started right there. So the position off the scent off muss is, um m one are one plus him to aren't too. I was are three, and then m one him to o. M three. So them is the mouse, and then our is the position. So these are the positions were dealing with where they have the masses and in terms off these positions, you can see that we have specific specific radius is so you know way. Want to clear this up? I want to clear this up a little bit. So this one right here, this is our one. It's the same us. Um, if we take our measurements from from this point, this would be are one. And then the second one for the radius is radius would be hard to, And then the third radius would be our three. So in terms off positions, this is what we're saying. We're taking this as the origin. So this is why this is your ex direction. So are one is legacy four centimeters R two, which is that the origin is going to be zero centimeters and then are three, which is on this side. It's gonna be, um, the recent theaters. So those of the positions you're dealing with in terms off the are so can always compute. That sent off radius 100 grooms minus four centimeters plus 75 rooms, three centimeters. And this is Jay because it's going in the positive y direction. And this is I because it's going in the negative X direction. Remember, I heart is the unit vector, and that's the unit vector in the X. And then Jay Hart is the unit vector in the no y direction. Oh, So then we defied that by 100 rooms Waas 1 50 rooms plus 75 rooms. This is negative. 400 rooms was sending me to and then plus 2 25 rooms setting me toe. Then this is Jay, huh? Then this is 3 25 runs. And so that center of radius is gonna be the same us. Um uh, this is negative. Plus 2 25. So negative 1.2 should be centimeters and then plus 0 +1697 images. Uh, J and so the scent of moss, the scent of muss. Um, assuming assuming 150 g is the origin his ex senators, It was too negative. 1.23 slim images. And then why senators is 0169 centimeters. So those are the two positions that you're looking at? Um and you know, I hope you enjoy the problem, Phil Few to send any questions or comments and have a wonderful day about. So just to go over the problem again, you know, we were supposed to get the scent center of mass for this problem, and we assumed that's the on to was the origin. And so the formula for the sent off mass is the product of the mosque and the radius. Um, that's the numerator and was summing them up. And then we sum up the masters together and plug in the numbers, assuming the 1 50 g is the radius always the origin. So we are used that Azzaro benchmark so are two is going to be zero are ones that before centimeters and then our threes three centimeters will bunch up those together. And so we get our sent off mass in the X direction is negative, 1.23 centimeters and then obviously the scent of mass in the Y direction 0.69 centimeters. No. So I hope you enjoyed the problem. Feel free to send in questions or comments and have a wonderful day.


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