00:01
Here we're going to be working with the moment of inertia tensor, which is a way to handle situations where you're not rotating about what is called a principal axis.
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So we are going to work out the elements of this tensor.
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For example, that's a solid object.
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So let's take a look at how the terms are defined.
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So you could do the moment of inertia tensor for discrete points, point masses, or for a continuous solid.
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And really all you're doing is turning a sum into an integral.
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But what you're basically doing is figuring out for each mass how far it is away from your axis.
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So, for example, ixx, what you're thinking about is your x -axis, and then your distance is measured by the sum of the y -coordinate squared plus the z -coordinates squared.
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The off -diagonal elements are a little bit harder to explain, but they're easy to calculate.
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You simply take the product of the two coordinates multiplied by the mass.
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And those are negative.
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We won't derive these elements.
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You can derive them by considering angular momentum in different directions.
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In any event, what we are going to be doing is finding the moment of inertia tensor for a square plate with the origin at its corner.
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And it's in the first quadrant, just make life simple square plate.
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Origin equals corner.
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Then we will use that moment of inertia tensor and a calculation of angular momentum and kinetic energy.
02:29
So really kind of some matrix manipulations.
02:34
All right, so let's take the first element ixx.
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So we'll do the on -diagonal elements first.
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So i -xx.
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And here we're going to use a density equal to sigma.
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And instead of a volume, we'll have a d area.
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Sigma times d area.
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And here we're going to have y squared.
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And iyyy is going to be sigma times x squared, d area.
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And we can see from symmetry, these are going to be equal to each other.
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So that will just get us down to one particular integral.
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And izz is sigma times x squared plus y squared times d area.
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And that's just going to be equal to ixx plus iyy.
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And i'll point out that due to the fact that z points out of the plane, we're using z equals zero.
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Now what this means is that i, xy, which is equal to iyx, is the only off -diagonal element that we have to worry about.
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We're going to have i -z equal to i -z x, equal i -z, equal i -z, equal to i -z, equal to i -z -y -z, all those are going to be equal to zero, because there's no, none of that material that extends up into the x -axis.
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And this is the idea that it's a very thin plate.
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So we don't have to worry about all that.
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So all we have to figure out is ixy, and that is equal to minus sigma xy d a.
05:08
Okay, so there's really only two integrals that we have to work out.
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And that makes life a little bit easier, maybe.
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So i'll put a star.
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We just have these two integrals.
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We call this one, one, and the other one is the off diagonal.
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So goody gumdrops, life is easy.
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But let's see how we would do this integral for ixx.
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So fortunately we're in cartesian coordinates, and we have dx, dy, y squared, y, goes from 0 to a, the same with x going from 0 to a.
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So that's 0 to a d x.
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And then we have 8 cubed over 3.
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And that is simply 8 cubed over 3.
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And there's a sigma in front times a.
06:40
Okay, and there's a sigma in front.
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Whoops, i keep forgetting my sigma.
06:44
That is not good.
06:46
That's going to give us the correct units.
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And we probably don't want sigma floating around.
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So the mass of this plate is equal to sigma a squared.
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So when we're done, we can replace sigma by just taking mass over a squared.
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So we have one element, and now we know all the diagonal elements.
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Okay.
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And we'll write it when we get to that point.
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So let's see, i -x -y is the second.
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Integral that we have to do.
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And we'll go ahead and replace sigma with m over a squared.
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And then we have x.
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Now we can actually just separate our integrals.
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We can always do that.
08:00
These are separable.
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So each of those is a squared over two.
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And let's see, this is equal to minus m a squared over four.
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So our final moment of inertia tent looks like, and we're going to bring out the m .a.
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Squared.
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And maybe with a 12 in front.
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And let's see.
08:57
Then we have four minus three, zero, minus three, four, zero, zero, zero, and eight.
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And i like integers.
09:29
I just like integers.
09:31
It's not necessary, but getting integers.
09:34
Inside of that is going to be a little bit helpful.
09:38
So looking at the structure of this, what i notice is, of course, z sticks out by itself, and that's going to make life a little bit easier.
09:51
I -z -z axis is a principal axis with characteristic moment of inertia.
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I -z equals...
10:12
2 thirds m a squared, coming from that 812 sma squared.
10:27
And we'll have to work out the stuff going on in the x, y, plane.
10:34
Okay, but let's see how we would use this moment of inertia tensor.
10:40
So we're going to do some sample calculations before we go on and find the principal axes in the x, y, plane, which will work out to be intuitively what you would expect.
10:51
But we're going to do some sample calculations.
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So the first one we're going to do is find the angular momentum for the plate rotated about its diagonal.
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And that goes for the origin.
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So, yeah, just kind of pointing out where that axis is.
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Draw that axis on the picture here.
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We have a good color.
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Let's use red.
12:05
Okay, so we are going to define that axis, and we're going to use it.
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But the definition of angular momentum is it's a vector, which is given by the moment of inertia tensor times omega vector.
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Now, what is the omega vector? the omega vector turns out to be omega -tensor.
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The axis, which we'll just define as n -hat.
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And in this case, n -hat is equal to x -hat plus y -hat over the square root of 2.
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So we're really just defining how that occurs in space.
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And we want to represent that as maybe a column vector so that we can do a multiplication...