A circular wire hoop of constant density $\delta$ lies along the circle $x^{2}+y^{2}=a^{2}$ in the $x y$ -plane. Find the hoop's moment of inertia about the z-axis.

$I_{z}=2 \pi \delta a^{3}$

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Okay, folks. So now we're gonna talk about problem number 37 and for this problem, we're gonna be looking for the moment of inertia of a circular wire hoop about the Z axis and let me just go ahead and first of a graft a circle out for you. We have a circle, basically Ah, that that looks like this. He's got a radius of a Raisa parts of number, and it's got a constant density delta and we're gonna be looking for the moment of inertia. Now, the moment of inertia is a It is a three by three matrix. But because we're because our object is just a two dimensional object and we're only looking for the component of this matrix, along is the axis. So we can way have a very simple expression for for the moment of inertia. And that expression is just an integral of delta multiplied by r squared minus are three squared Where r three is This notation are three just means Z is the third component of the of the position vector The first component being X and the second quantities wife anyway multiplied by Yes, And because Delta is a constant I'm gonna pull the constant out. So no harsh squared minus are three squared Let me remind you something are square to just x squared plus y squared Plus these word. And now if you subtract off the Z squared you just uses half x squared plus y squared. But there's something very special about this expression because we had When you have a circle, every point on that circle X squared plus y squared gives you a constant value. And as you can probably guess, that constant values just eight. So we have a squared, Yes, but because of the fact that a is a constant, we can pull that constant out again. So we end up with DS Inter Girl Integral DS. But what is integral DS? Well, interval ds is just the circumference of the circle and the circumference is just you have you know, I don't know what the circumference of a circle is asked us to pay a So when you Ah, when you multiply the 22 days together you get two pi delta A to the power three and that's it for this video. Thank you for watching. Um but I

University of California, Berkeley