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Problem 33 Hard Difficulty

(a) Write the formulas similar to Equations 4 for the center of mass $(\overline{x}, \overline{y}, \overline{z})$ of a thin wire in the shape of a space curve $C$ if the wire has density function $\rho(x, y, z)$ . (b) Find the center of mass of a wire in the shape of the helix $x=2 \sin t, y=2 \cos t, z=3 t, 0 \leqslant t \leqslant 2 \pi,$ if the density is a constant $k$ .

Answer

a)
$$\overline{x}=\frac{1}{m} \int_{C} x \rho(x, y) d s$$
and
$$\overline{y}=\frac{1}{m} \int_{C} y \rho(x, y) d s$$
and
$$\overline{z}=\frac{1}{m} \int_{C} z \rho(x, y) d s$$
b)
The center of mass of the helix is $(0,0,3 \pi)$

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Video Transcript

All right, So now if we want to, um, determine the center of mass, uh, in the x y c plane. So in three D space, But it's very similar to that planes, except that now we're gonna add an extra component, which is just so the X component is one divided by em, which is the mass of the integral of X times. Roll backs wise ease. And our density function depends on the three position directors. Yes. Live Pratt Lie bar. Also, the same thing is going to be one divided by em. Integral. Why roll X y z? Yes. And now the extra component is gonna be a Z bar, just one divided by am of the lining to grow along. See off Z now row the density function of X y z Dia's all right and then in the second part party were given the curb X is equal to a society. Why is equal to to co sign T and Z is equal to three t So, like if you think about this, if you recall, this is a helix and T goes from 0 to 2 pi and were given to the density function schools and were asked to determine the center of mess. So first of all, uh, as usual, we first find RDS because we want to write everything in terms of teeth. Now he s is just the derivative Wrexham of respect Pity squared plus the derivative of why would respect a T Square plus the derivative of Zeeland respectively square. Then we take the square root of bed and multiply it by detail. So we get the square root of two co scientist squared plus negative two scientists Square Plus Three squared et And then if we pull out of Flora's a common factor, we have four science Where was signed? Square plus nine are just there sport Times one plus nine which is 13. So that the skirt of 13 So now we determined Dia's not to find a mass. The mass is just the lining to rely on the courtesy off row of X y z Yes, And now we're gonna write this in terms of tea. So you know that Rose just k uh no. Sorry, Rose K and yes is a square root of 13 e t now. Okay. And we're rid of 13 are constants we can pull them to the outside and the integral DT is just tea on. The limits of integration is from 0 to 2 pi. All right, so now we have to pile on. Is there are just two pies or soup? I case where? 30. That's the mess. Now we're gonna use this mass on the equations and port A so that we can determine the integral that Sorry. The coordinates of the center of best. So now again, we're just gonna plug in everything. So we're gonna plug in. Uh, so this right here is the mess, um, to sign T this is the X. Okay, raw. And this right here? Yes. And if you notice everything is written in terms of teeth so we can pull out the constants of the two under to cancel the K indicate the cancel the square of their distributive working cancel. So we're just left with one divided, but are now the integral of sine of t is negative co sign and the limits of integration off 2.0. So now if we put this backhand so negative of co sign, uh, two ponds, just negative one and co sign up. Zero is one. So we get negative one plus one. That's our exporting. Now, if we're gonna look for the white Courtney, well, we have to do is the following. Uh, it's just again. One divided by and the integral of Why co signing a sorry. Why grow? Yes. Okay. And then now we can put the constants we simplify. Just left with one divided by pi in general co sign of tea and science T And that and the limits of integration are from 0 to 2 pi that plug that in sign of two pi zero sign of 00 So in the end, just get serious. That's the white corn yet the center of this. And now the sea Cornett of the center of mass Again, it's one divided by and the integral off Ze ro. Uh, yes. And if we plug everything in, we pull out the constants. Case cancels river for teens. Cancel. We're just left with three divided by two pi and what remains inside his TV, You know, the integral of TV tea is just tea square, divided by two on the limits of integration of from 0 to 2 pi free plug that Andrea three divided by two pi four place square, divided by two. And then this is just for price. Great about like users to buy, Swear to and to cancel square and pride. Cancel. So we're left with three. So finally, the coordinates center of massive 003 point.