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Second Moment of Inertia using Green's Theorem

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Green's Theorem: double intergral ( curl vector(F) dA ) = closed integral ( vector(F) dot vector(ds) ) What I need is double integral ( x^2 + y^2 dA ), right? Knowing that vector(F) = <F1, F2>.. and applying green's theorem to this problem we get: dF2/dy - dF1/x = x^2 + y^2. I just chose F1 = 0. So F2 = x^2*y + y^3/3. My areas are polygonal, so I am just going to parameterize the boundaries into segments and from there it should go alright. So my question is: Is my work correct?

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You're not going to get too far using two dimensional vectors with the curl operator. Unless, of course, you want to get an answer that has only a Z component.

Or were you just being brief?

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OK. And just to double check, we're talking about plane deformation and not rotation?

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What rotation/deformation.. If u're talking about curl operator.. Well it's a differential operator. curl( F1, F2 ) = dF2/dx - dF1/dy

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Ah, no. Your topic is about the second moment of inertia (aka second moment of area) but the math looks like it's for the moment of inertia. In either case, can I assume the axis of rotation is k?

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Ah alright, got u.. It's for rotation.. and axis of rotation is indeed k.

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Yes..
But in later stages in code, there's gonna be some finite element stuff, so let's build on the simple stuff first, and then I'll see how to deal with the changing stuff.

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You're doing fine.

And it *is* called the moment of inertia (or second moment of area), not second moment of inertia.

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Quote:
Original post by arithma
Green's Theorem:

double intergral ( curl vector(F) dA ) = closed integral ( vector(F) dot vector(ds) )

What I need is double integral ( x^2 + y^2 dA ), right?
Knowing that vector(F) = <F1, F2>.. and applying green's theorem to this problem we get: dF2/dy - dF1/x = x^2 + y^2. I just chose F1 = 0. So F2 = x^2*y + y^3/3.

My areas are polygonal, so I am just going to parameterize the boundaries into segments and from there it should go alright.

So my question is: Is my work correct?

It's almost correct. Curl(F) dot <0,0,1> gives you [(∂F2/∂x) - (∂F1/∂y)]. Letting that equal x2 + y2, and letting F1 = 0 (or in general any constant) gives you F2 = x3/3 + y2x. Note that this is only true if you integrate the contour with respect to the parallel component 'dr'. You can also integrate with respect to the perpendicular component nds = -perpdot(dr), and you'll end up with the divergence theorem (more info about its relationship to Green's theorem). This gives you [(∂F1/∂x) + (∂F2/∂y)], which when equal to x2 + y2 gives you F2 = x2y + y3/3 (with F1 being a constant again). I'm partial (no pun intended) toward the divergence theorem due to my studies in electrostatics, but how you evaluate the integrals is up to you.

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