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Moment of inertia about a line A wedge like the one in Exercise 22 has $a=4, b=6,$ and $c=3 .$ Make a quick sketch to check for yourself that the square of the dist...

Question

Moment of inertia about a line A wedge like the one in Exercise 22 has $a=4, b=6,$ and $c=3 .$ Make a quick sketch to check for yourself that the square of the distance from a typical point $(x, y, z)$ of the wedge to the line $L: z=0, y=6$ is $r^{2}=(y-6)^{2}+z^{2}$ Then calculate the moment of inertia of the wedge about $L$

Moment of inertia about a line A wedge like the one in Exercise 22 has $a=4, b=6,$ and $c=3 .$ Make a quick sketch to check for yourself that the square of the distance from a typical point $(x, y, z)$ of the wedge to the line $L: z=0, y=6$ is $r^{2}=(y-6)^{2}+z^{2}$ Then calculate the moment of inertia of the wedge about $L$



Answers

Set up, but do not evaluate, integral expressions for (a) the mass, (b) the center of mass, and (c) the moment of inertia about the z-axis.

The solid of Exercise 21; $ \rho (x, y, z) = \sqrt{x^2 + y^2} $

So what we're looking to do here is toe find the moment of inertia of this circle of given with the equation X squared plus y squared is equal a squared on We know that the density of the circle, the spin hoop is delta. So the first thing that I'm gonna do is to dress the draw a circle, um, to show what we're working with here to find the density of, uh and so we know that based off of the form of the equation, we have a circle, and we also know that has a radius of a so 01 these access in the X Y plane. Um, we're dealing with a circle of radius a Ah, And in order to find the moment of inertia here, um, which is what we're looking for. We use the point mass moment of inertia formula, but we're going to adjust it so that we're doing it for each point along the circle. So we're adding up the point mass a moment of inertia in order to find the point mass woman of inert are in order to find the moment of inertia for the entire circle. Um, and So this formula that we're gonna be using here is based off of, um, the point mass, which is and square for Sorry. Um, m times R squared, where m is mass and r is radius eso in order to find the mass looking at what's given, we know that the circle has a constant density of Goethe. Um and so we confined the mass of the whole hoop the whole circle by integrating the density about integrating the density along the curves. We're gonna do Delta DS along the curve, c, um, and thus were like adding up all the densities at each point on the curve booth. And so that will give us our max, which is equal toe m. Ah, And then in order to find the radius, we look at the equation of the circle that we were given above. And we know that the general equation of circle is X squared. That's why squared is equal to R squared, which in this case is equal to a squares are radius is a and already squared is a squared. So now we have both our, um our, uh, radius squared and our mass, um and so Now we have the two components, um, of the moment of inertia formula so we can multiply them together as above gonna use a new color here for that. And so we have a squared times integral over the curb of Delta G s. Um And so, in order to find the moment, I'm inertia along the circle. We simplify this integral, and we know that Delta is a constant here because we're given that a problem so we can just move that out front. And so we have a squared delta into Girl of C DS on now, in order to further simplifies integral To get a final answer, we have to simplify, um, interval over. So over the curb of DS, which is just the length of the curve. And we know in our case that this cur is a circle. In order to find the length of a circle, we use the circumference formula, which happens to be two pi r on. Earlier, we found that our radius R R is just a So are circumference is two pi day. Um and so in order to find. And so now we found that integral C. D. S is equal to too high A. And so the solution to our problem for the moment of inertia is Isay. Because we're finding it over. The Z axis is equal to a square delta times two pi A which is equal to two pi a cubed delta. And now we're done. We found the moment of inertia of this thin hoop, uh, over the that is that lies along the circle given above.

Problem. It is required to find the moment of inertia about ex acts off a sim plate haunted boys a problem X equal to Y minus y square and the line X plus y equals zero If the dentist e equal Tow X plus why we restored by sketching the graph So we have two equation. 1st 1 is X plus y equals to zero, which is something like that, and the thick and one ese x equal toe. Why minus why square, which is this curve cutting boy eggs at one. And if we and substitute by or sold the two questions off the car, we can get this point and why equal tow to So now he's asking about this region and we restored our solution. Boy. Completing M So am I will be equal toe integration from zeal to two and integration in wider veteran from negative y to y minus y square for our dynasty, which is X plus y. The ex knew I, which will be equal toe integration from zero to toe for Why are four over too minus two Y or three loss two way over, too. Do you want, which will be equal toe eight over 15. Now we can calculate I Iove x nursey about X x, which is digression from 0 to 2 and the fame integration in my direction. From why to why minus y square for y square and or dynasty, which is exit lous y the ex the wine. So it will be equal toe integration from 0 to 2 or wipe or six overto minus to wipe or five. Unless to Roy or four. Do you, Roy, which will be equal toe 64 over one or five. Thank you.

Okay, so here we have two equations. First, let's sketch. Thea, that's first sketch. Three gin. We have exactly what's weird here, which is a parappa, and we have X close. Why equals to two white? Basically another problem. So X equals two to light minus y squared. Okay, so every wise from 0 to 1 and we're gonna sketch this to probably here. This is Axl's Y squared. And for another one we have they're here. That's because 21 minus y squared. So, um, this intersection that 0 to 1 up. So the intersection, it's 11 and its projection on the Y axis 01 it's over the region. So now we can send this in to grow. We have wise from 0 to 1. Have you all here and for access from the lower down. This y squared the upper bonus two Armanis was squared And was it density that this it is? Depends on why. So this was what This one And we voted to this integral developing. Evaluating with respect to X, we're gonna have everything regarding why. So this is the museum to 12 y squared minus two. What before and the wire here equals 24 over picked a kn and the first moments him with respect. Why, it is the same setting. But we insert X here. So, for for the definition of the first moment, soon again, we have the x t y. About this, we're gonna have to time served one. Why? Cubed times are minus what? The fifth d Y, and the answer is 1/6.

Okay. In this problem, it's required to find the center of mass and moment off inertia about the boy eggs I y off a scene. Rectangular plate got from the first quadrant by a line X equal to six and another line. Why equal to one for a dentist? E equal tow X plus y plus one so we can start boys Kitsch X equal to six is a line cutting X X at six and the second line. Why equal to one even other line cutting y x at one. So force the quadrants. Who is asking about this region? Now we can start calculating mess so mass will be equal to integration from Ciro 21 and second Integration from 0 to 6 for our dynasty, which is X lost Boy plus one the ex Do you want equal to integration from 0 to 1 six white clothes 24. Do you? Why? Which is equal to 27 and uh, M X equal to integration from 0 to 1 and second integration from 0 to 6 for why And then it's t x plus Why plus one e x. And this is the y. Do you, Roy equal to integration from zero 21 poor boy. But deploy six. Why? Plus 24 Do you, Roy? Equal to 14 And for him off eggs. Mm, boy. Sorry. It is same integration from 0 to 1 and another integration from 0 to 6 for eggs. Except Lewis Roy. Just one. The ex knew Roy equal to integration from 0 to 1 onto employees. 18 boy plus 90 knew Roy equal to 99. So now, easily, we can kill plate export as and Malfoy over M equal to live in over three and white board as am affects over M equal to 14 over 27. Also, it is required to complete EIroy. So why? Why will be equal to integration from 0 to 1 and integration from 0 to 6 x square, X plus y plus one the ex. Do you, Roy? Which equal to integration? From 0 to 1, 216 I'll deploy. Why? Over three loss 11 over seeks, do you? Why? Which will be equal toe? 432 Thank you.


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