for-the-following-exercises-compute-the-center-of-mass-overlinex-overliney-use-symmetry-to-help-lo-5

Question

For the following exercises, compute the center of mass $(\overline{x}, \overline{y}) .$ Use symmetry to help locate the center of mass whenever possible.
$\rho=7$ in the square $0 \leq x \leq 1, \quad 0 \leq y \leq 1$

Mauricio Araiza Canizales · Accepted Answer

in the following problem. We want to compute the center of mass which is given by these terms right here with information that the density function. Ricky recycle to seven on the interval from Syria to one for X and on the interval. For what? From why sickle to Syria, to a white school to one. So let&#x27;s first for a picture of how this looks like. So we have over here Xs equal to one and why every six equal to one. So we&#x27;re looking at these points right here. So this is a rick check. Sorry. A square with Britta&#x27;s is at one comma one zero comma one zero comma zero no one comma zero. So for the center of master will actually need to use integral Sze However, we can skip that process by looking at this shape right here We know that this is square right here has symmetry are prone to lines The line X is equal to 1/2 and the line why sickle to 1/2 Because this is square has symmetry around these two lines and because the density is a constant function, we can ensure that the center of a mess it&#x27;s gonna be looking at this point right here in the middle, in the intersection of this to excess of symmetry, which is gonna be at the point 1/2 coma, 1/2 using symmetry, we actually skip the process of using integral to find the center of mess.

Amy Jiang · Answer

Okay, so we have that y is equal to co sign a Becks. And why people to negative post in a box. And this is between ex being negative fire before or part of a tutu. However, so are grass looks something like this. So this is some pie over too. I&#x27;m gonna have to fire with Chip. So then it is something like this. That&#x27;s co sign of exit. This is negative. Close side of ex. Okay, Were asked to you something to help locate sense of mass whatever possible. Well, if you recall since co sign of X or actually, yeah. If we reflect coastline of X across the X axis, it just becomes negative posts on our backs. So our top portion and are bottom portion are so much. So based on this, we could know that if our action, if we know that they are sick, the magic across both the x and y axis, then our center of mass is just a origin. And in this case, origin is Darryl comments. So

Vikash Ranjan · Answer

in the problem. Thanks. The drag A ll has, uh, uniformed dance A D That is we not? Is that the density is constant. Okay, the sent it off. Marce off the triangle is you. And why? Expert Why? But this is equally true. Do you know? Plus a plus, Joe off on three on DDE. You know, plus to that Plus be a phone three. Therefore we have a bar on what about us, your country and be upon us. Sudan, sir given problem.

Amy Jiang · Answer

Okay, so we have wise equal to two X and two X squared Y is equal to zero. And let&#x27;s see, explica todo to one. Okay. Were asked to draw out the region and in computer center of Mess, and we could use somebody to help locate the region. Okay, so let&#x27;s draw out our graph first. Well, two X squared is a problem. We have something like this, and then it cuts up between Joe and one. So it&#x27;s returned. One is over here. So we&#x27;re looking at this region over here. Okay, Well, looking at this, it seems like we don&#x27;t have any cemetery, so because we don&#x27;t have any cemetery we had to use, um, any properties or any shortcuts of calculating it, So we&#x27;re gonna do it directly. So we&#x27;re gonna take export to equal interval from Gerald to one of X times. Why, that&#x27;s two x cubed over the integral of just Joe&#x27;s one from of exports eggs. The X does that Give me that. Gives me. Mmm mmm. Oh, it is it to over here to actually Cassidy Stew&#x27;s. Okay, So I had just the interval from Joe&#x27;s one of Excuse that you explore over four over X three over three, evaluated at one, and it&#x27;s usually one over four over one over three, which is equal to three over four. Now let&#x27;s find White Boy. It&#x27;s equal to 1/2 of the interval, just one of two X squared times X ray just to be ex. All right, which is execute the X over in ago. Joe&#x27;s one of two X squared DX. What does that give me? That gives way. Oh, wait, no, I&#x27;m squaring it right, I&#x27;m staked. So I get actually to exploit the power of Jews. That&#x27;s four x for X over Indigo from Joe&#x27;s one. Just why would just two x squared x Okay, so we get well, our 1/2 can&#x27;t so that this because it too. And then I choose chances that would get X over five over five extra para fireball in five over X to part three over three about looting at one and tell excuse me. Three over five So our center mass becomes really over four comma, three over five

Mauricio Araiza Canizales · Answer

in the following problem. We want to find the center of mass of the region in between the functions. Why is equal to two X square and waste equal to zero for X in between zero and one. So let&#x27;s first draw sketch for this graph. So over here we&#x27;re going to draw the functional bicycle to Texas Square, which we know is the problem and the journal for myself. Gravel. It&#x27;s gonna be this one right here. And let&#x27;s just a point to substitute X just to pinpoint. So we&#x27;ll have the point X is equal to zero in the point X equal to one. So let&#x27;s find what is the white co ordinate for this point. So the Y sickles zero. We substitute that over here. Well, find when zero. I&#x27;m gonna use Rhett to determine at this point. And now at X equal to one y sickle to two. So we&#x27;re here with a 0.2. Call me one. Sorry. One Coma too. The 0.0 commence here. Now the line y sickle to zero is just the X axis. Is this one right here? So the region aren&#x27;t that we&#x27;re trying to find its center of mess is this one right here? So we have drawn the function in the graph. Let&#x27;s find out the center of mess. So we know that the center of mass it&#x27;s located. But this coordinates and we know that this special cases of X and Y are located at the moment with respect to the Y Axis awaited. But the total mass of the system and the way coordinate it&#x27;s located at the moment would respect to the ex access be raided by the mass of the system? So let&#x27;s now calculate these three quantities we&#x27;re here. So the start of by calculating the moment would respect to the wax. This is equal to the integral from A to B of the density of the density of the function. Tim&#x27;s X temps, the upper function Buying is lower function times DX, in this case are bounce. We&#x27;re gonna being from Syria to one Temps road Temps X. So FX is gonna be to excess square And gee, if X is gonna be zero, So we&#x27;re just gonna leave that a serious now we have the integral from Syria to one of Roe terms to excuse the ex we can now perform this integral, which is released to compute, which is just gonna be extra fourth divided by four which in this case of was ballots are invalid from Syria to one. And if we succeed that these will have to roam times one over four, which is just roll over too. In their white words went to calculate the mass of the system. Just gonna be the integral from a TV of room tempts the upper function minus the lower one. The bounce are going to stay the same So from Syria to one times, Rowe says We know that ji, if X is equal to zero, would just substitute over here to exit square the ex. We can now perform this integral which is just X cube divided by three invalid from Syria to one. So the mass of the system is just your row terms one be better by three, which is just 2/3 of Rome. Finally, let&#x27;s calculate the moment with respect to the X axis. Now this is equal to 1/2 times the integral from a TV of roe Times Square fffx minus the square of GI effects the ex We can now sit sit through their bounce. And since we know that GI effects zero, we&#x27;ll just have to excess square square. This is equal to 1/2 Tim&#x27;s room times for extra the fourth. Yes, sorry. Forgotten DX over here. So the moment would respect to the X axis. It&#x27;s not just went too weak to Thames Row times this integral right here, which is just X to the fifth power divided by five validated from Syria to one which is just to row. He read it by five. Now we have calculated all of these three quantities. We can find out the location of the center of mess, which is going to be We know that X according it&#x27;s gonna be at the moment of inertia. Would respect to the Y axis derided by M and why is gonna be the moment of inertia with respect to the X axis? You bet it by m. So the first term is gonna be a road divided by two divided by to Rome, divided by three the white coordinate It&#x27;s going to be to row invited by five divided by two row, divided by three. The center of mass is just then we&#x27;ll have three of Rome divided by four row coma. You wrote temps three debated by five, divided by to row. So in this case, the X coordinate is gonna be three divided by four. The white coordinates gonna be three divided by five. And this is a solution to this.

Mauricio Araiza Canizales · Answer

in the following question. We want to find the center of mass located in the intersection between the functions. Why? Sickle to cosign effects on bicycle to native Kocian effects for X in between negative pirate chewed to pirate tube. We&#x27;re getting that The density in between this reed in between these two functions is equal to two. So it&#x27;s first pro on a proxy in a sketch of her intersection. Are the region are, which is the intersection of these dysfunctions? Looks like So we&#x27;re here. We&#x27;ll have the point. Negative one y school too negative one and one and then point pie Sequel to two Hi, Divided by two sorry and negative piety Right about you. So we know that cosign effects starts from one and then a pirate to zero. Then it repeats itself and it&#x27;s the same around the negative. That was Fricks. So this is the line. Why sickle to co sign effects with green. And when I draw the line, why sickle to native coast and effects, which is the same. However, it&#x27;s just reflected across the Y axis, so this will be the graph of negative ghost and effects. So this is why sickle to negative coast and effects. And this will be the region. Are Did you notice something similar in our well are has symmetry. Sorry around the X axis. And why does bar has symmetry around the X axis? Well, we know that this shape right here is the same once this one And because of that, we can see that it has similar around the XX is however eat old soon has symmetry around the y axis. And we know that because this shape right here and this one are the same. So because our has symmetry sorry, because this two functions have symmetry around the x and Y axes. We can actually compute the center of mass without actually using any integral. And in this case, is gonna be the intersection of this juice lines of symmetry which in this case is just one on this one, and they intersect at the 10.0 comma cereal. So because our because these two functions have symmetry around the x and Y axes, the center of mass is located at the intersection of these two alliance which

Problem

For the following exercises, compute the center o…

Problem 270 Easy Difficulty

For the following exercises, compute the center of mass $(\overline{x}, \overline{y}) .$ Use symmetry to help locate the center of mass whenever possible.
$\rho=7$ in the square $0 \leq x \leq 1, \quad 0 \leq y \leq 1$

Answer

$\left(\frac{1}{2}, \frac{1}{2}\right)$

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Gilbert Strang,

Calculus

Grace H.

Anna Marie V.

Kristen K.

Samuel H.

Integration Review - Intro

Definite vs Indefinite

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$\left(\frac{1}{2}, \frac{1}{2}\right)$

Calculus

Grace H.

Anna Marie V.

Kristen K.

Samuel H.

Integration Review - Intro

Definite vs Indefinite

Gilbert Strang,Calculus

Grace H.

Anna Marie V.

Kristen K.

Samuel H.

Gilbert Strang,

Calculus