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Problem 31 Hard Difficulty

A thin wire is bent into the shape of a semicircle $x^{2}+y^{2}=4, x \geqslant 0 .$ If the linear density is a constant $k$ find the mass and center of mass of the wire.

Answer

Mass of the wire is 2$\pi k$
Centre of mass of the wire is $\left(\frac{4}{\pi}, 0\right)$

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Video Transcript

open this question. We're told that we're giving a wire. It's bent into a semi circle of Brady's, too, and that this wire has constant when you're dancing. All right, so we're supposed to find the mass on the center of this. So, um, the first thing we're gonna have to do is parameter rise. Uh, this curve eso the semi circle. So we know that for a mature ization of a circle is just or co scientific for X and R 74. Why? But our is to the radius is too. So are para mature. Ization is X equals a coast to coast 90 and wise, it was to say, and t ranges from negative flyover to depart to since it's a seven. All right, Now, uh, just we're gonna use line into crows were going to write everything in terms of t. So yes is square root of alive I d squared, plus the expected square. So the derivative drivers to sign to you and respected teas to close 90. Then we're gonna square that and the derivative of X, which is to co sign tea with respective. He is negative. Two sites and the world. We're gonna square that and take it square root We can pull out a war is a common factor and then we're left with co sign square T plus sine squared. All right, now close time. Square foot sign. Square block. So we're left with the square root affording TV, which is the find of minutes, the masses just the line, integral. Ah, but a long sea off the density, uh, with DS and we know that the density is constant. So it's just k and yes, we determined it to be to Aditi. So now we can pull up to okay. On the outside on the integral of DT is just teen and TV has the lower limit of negative by over two and the upper limit off. All right, so now if we plug in these values of t But we care is two k times y over two minus negative by over two. So our masters just to pikit. All right, great. Now we need to find the center of mass. So where is the center of mass located again? We know this Thean tha group, This is the line integral to find the center of Mass. Ah, And then we already determined what the Big M is with the masses. It's just to buy K and then exes to co sign T on The density is constant K and yes, Stoudt. Now, when we pull out the constants on the outside, we have four K in the numerator to play candor, denominator in case cancel the tour and the floor canceling You're just left with, too, so divided by playing on the outside. And now we have the integral from negative pi over to defy over to co sign TV. Well, the integral society is just scientific, and the limits are negative. Pi over to an appointment. Well, right now we're plugging these values were t we get two divided by pi time signed by over two minus sign of native pirate, too. But that's just two divided by pi times one minus negative one. So that's two divided by apply times two or for dividing. And then now we're going to do the same thing except our interview work, trying to find a like ordinate of the center of mass. So that's one divided by AM into growth. Negative flyover to defy over to of why which is to 70 times the density function turns the yes, and then now we're gonna do exactly the same steps as before. We pull out the constants. We cancel the K more in the two week and simplifies was just two divided by pi and the integral of science. He's just negative coastline tst Granges from negative, however, to supply with. So if we plug in negative co side of our virtue is zero minus negative co side of negative fiver do well, that's minus minus. It's plus and co sign of negative by over +20 Sorry, guys have zero. So finally we determined the X and the Y coordinates of our center of mass. It's just high, divided by four comma zero right on her.

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