Meet students taking the same courses as you are!Join a Numerade study group on Discord



Numerade Educator



Problem 37 Hard Difficulty

If a wire with linear density $ \rho(x, y) $ lies along a plane curve $ C $, its $ \textbf{moments of inertia} $ about the $ x $- and $ y $-axes are defined as
$ I_x = \int_C y^2 \rho(x, y) ds $ $ I_y = \int_C x^2 \rho(x, y) ds $

Find the moments of inertia for the wire in Example 3.


$\boldsymbol{I}_{x}=k\left(\frac{1}{2} \pi-\frac{4}{3}\right), \boldsymbol{I}_{y}=k\left(\frac{1}{2} \pi-\frac{2}{3}\right)$


You must be signed in to discuss.

Video Transcript

So here the problem to find a new physical quality coal, moments of inertia and the formulas. So here we have. Ah, wait. Who and for example, st thiss our information for example three. So, in order to complete the inter grow if Neto first paramour tries this life which sorry This semi circle which was given by we'LL use a center paramilitarism ation co sign t sign and, uh and he goes from zero to pi poses opera semi circle So he s should be Ah, swearing off Science will prosecute Answer So that part just want years is the same stT So this one we can compute high ex Pecos into grow tea Found zero to Opie. Why Square Ralph? Ex Weiss. Okay. Want minus Why Sci fi? Yes, it's the same stT. So we have a federal. Okay, we have sai square of t minus. I q t I mean open on the other page. So you have case I hey, integral from zero to pie sai skirt T minus. I cube of tea. So I see. So the first part I want to write it us co in terms of co sign so ah one minus co sign Tootie over two and this one You probably want to write us Sai Square writers one minus coz I square so mine in society plus cose I scurti society Katie so we can write out into crow So the entire the room chief So we have one over to tea home minus sign Tootie over four plus co sign t And here this parts of simple use A fusion co signed He wants you Then nephew of science will be the scientist if he open to you So share you cubed over three Always attentive sign from Yeah I'LL take the room to be a saree in front of music pop on their website here That's right. So we plug Imp, I wish you were my one over to pie. Ah, this one problem pie side two by zero co sign pi sniff If one cause I'm pies there before so negative for Cuba's name before with a ninety seven plus one over three and minus. If we probably zero or your zero zero cause on there is who won and then the anniversary here. So we have one over two pi minus one plus one over three minus one Pross One or three. That should be okay. Pie over to minus hello for over three. And similarly the other integral We have another one to compute. They're ordering The crow is zero to pie X chorus Koh sai Security and row ex wives always should be Okay one minus scientist It's okay. One minus society. The deity. So we have cose. I score t minus cose I scurti sci fi so No, no, not there Quick. We have to You see toppling Go formal. So one plus one over two plus one over to co sign Tootie minus thiss the only hand right out and tight derivative One over to tea. This one is for this one is one over four. Sign Tootie, This one thing in terms of use substitution this will be close and he will be you and natural scientist it you'll be to you so you cuba over Siri So plus co sign Cuba t over three and you probably in pi We get this side two pies syrup, coz I paice negative one. So it's minus one over three and you're probably zero zero zero and one over through here. So we have K one over two pi minus two over three. So that's just as the moment of inertia with respect, the Y axis and with three with respect to X axis, so is this's I X, and this should be I y.