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Problem 32 Hard Difficulty

A thin wire has the shape of the first-quadrant part of the circle with center the origin and radius $a$ . If the density function is $\rho(x, y)=k x y,$ find the mass and center of mass of the wire.

Answer

$\left(\frac{2 a}{3}, \frac{2 a}{3}\right)$

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Video Transcript

So this question were given that we have a thin wire and it has this shape of Sandy's surf semi circle of radius R. So I'm were given that the Ben City Oh, the density function is equal to K X one, and they were asked to determine the mass in the center mess. All right, so the first thing we're gonna do we're gonna parameter rise this this current. So since we're given that it's a certain part of her circle, we know that our X and part of the Circle Freddy's Bar of Radius eight we know that X is going to be a co sign t. And why is gonna be a society? That's how we correct tries a circle and it goes from zero to privatise tears between zero and five or galicians your own apartment. All right. Now, to determine the mass, the mass is just divine into wrote brought off the density function. I know we want to write everything in terms of keep so first followed. Yes. So, first of all, we know that we can write row in terms of TV because X is in terms of tea and wines in terms of tea. So that's simple. I know you're not. We want to write the yes, in terms of TG or GT. So we know the formula, sir, The yes is the square root of the X by people, I swear what people are like that sorry DX by DT swear plus divided squared. So now the derivative of EKO sank A with respect to t is negative. A sign, T and interpretive of plywood perspective teas. Just a closer no, only square those we get a sward which we can follow as a common factor. And what we're left with is, um, co sine squared plus science Berg, which is one. So we get that, I guess, is just a The square is a square root on this were cancelled left. So that's DS. So now we're gonna put this back Our mass function our mass. So m is square root from zero to buy over to k a co sign t a sign T e d t. Um, what we can do is we're gonna unplug the constants which are a que Pincay. But now we're gonna do something that might know you clear it first. So we're gonna divide by two and multiply so just this like a sneaky one. So I pulled out 1/2 so I'm gonna multiply. What's inciting to grow by two. All right, now what do we know? Where? Trigger identities. We know that to close on 90 scientists just to sign up to. And now what's Indians? But we can stop the internal that we have. We can solve using you substitution. Zor gonna get you equal to t So do you is just to DT. And if we divide by 21 of both sides, we get the details. Justin, you divide effective. So now we determined our DT, which is just do you divide it by two. And now this integral there's really and just one last thing explosions. The tea goes from zero to high over to are you is gonna go from zero because we're gonna multiply zero by two. This is zero. And then, instead of ending it to ah, over to you is gonna end up time because two times So we're using this relation right here. So two times 5/2 is just so that's why our limits right now we can pull, but half to be outside. You know, we're left with big cube times k divided by four and the integral of sine of you d years. Just negative co sign for and are integral are limits are zero. All right, so now we put high and zero in Do co sign. So we get, uh, a cute K divided by four times negative. Negative one plus ones of this whole thing right here. It's two and two and the four simplify. So I left with a cube times Katie about it, too. So that's ordinance. All right. And now we need to determine the center of mass of center of massive pecs we're gonna name it. Accepts e is just one divided by the mass and the integral of X raw. Devious. So we know that I am Is a cube k divided by two. So do one divided by that, and then we're gonna plug in now. Everything else become great in terms of, so we just park that in, and then we're gonna get we're gonna pull out the constants which Peter the four times K. And here we can simplify cubes. And either the power forward just left with a case canceled or just left with chili. And now we have course. France were signed TVT Well, this one again. We can use U substitution. Zoff said you equal to pro sign TV that you is negative. Sign tgt, Which means that scientist DT is negative to use of this whole thing right here is just negative. So now we get this into girls right over here. All right. Now, what's the integral all, um, of negative? You swear to you that's just negative, You cube you. But are you escrow sine of t? So that's how we get this integral right here. And then Now we just poured in the limits, which is part of her two and zero. So we get to a times zero lost 1/3 or to be divided by three. That's our X coordinate. Now we need to find our bike work again. This is exactly the same steps as before, except instead of having X, we have one. So we're gonna follow exactly the same steps as before. We're gonna pull up Constance and so on. But the only difference now is that the integral right inside here is a bit different in the case above, we had co signed squared scientist here we have sine squared closely. So instead of taking you to be co sign, we use u substitution, we let you be equals B with you equal sign. So the you just close on TV. All right, so now we go listen to over here. All right. Now, what's the integral of your square to you? Well, that's just you, Cube. You, but we know that you assigned t so this is just signed Cube t divided by All right. So the integral a few square to use you, Cube, divide three. But are you? It's signed. So now we get this in general, right here and that. Sorry. Now we get this function right here. This question and what we do is we know, plug in the limit. All right, So sign of pile or two is just one. So we get to a times 1/3 minus sign of 00 But workers on zero, which is just to a divided by. So now we determined both coordinates for the center of mess, which is to a derived by three on 2 to 8, divided by three. So our center master coordinate Is this right here? This spirit to a divided by three common to