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Math Help - center of mass with varying density

  1. #1
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    center of mass with varying density

    Hi, I'm having incredible trouble getting a reasonable answer to this problem:
    Find the center of mass of a thin plate covering the region bounded below by the parabola y=x^2 and above by the line y=x if the plate's density at the point (x,y) is d(x)=12x. I have an idea of what to do for the x coordinate, but I'm confused as to how to do the y coordinate. Thanks!
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    Quote Originally Posted by musicman314 View Post
    Hi, I'm having incredible trouble getting a reasonable answer to this problem:
    Find the center of mass of a thin plate covering the region bounded below by the parabola y=x^2 and above by the line y=x if the plate's density at the point (x,y) is d(x)=12x. I have an idea of what to do for the x coordinate, but I'm confused as to how to do the y coordinate. Thanks!
    We can also write d(x,y) = 12x. Does that make it look more doable?

    I think you'll want to integrate with respect to y first, because then the innermost integrand will just be a constant (with respect to y). The limits on your integral should be

    outer: 0 to 1

    inner: x^2 to x

    EDIT: I was thinking in a probability distribution mode. I'll have to see how well it applies to center of mass. Anyway the integral limits should be fine.
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    A quick question, is that a double integral that you were describing? I probably could do that, but since this is for a calc 1 class, the professor might not appreciate that too much...
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    Quote Originally Posted by musicman314 View Post
    A quick question, is that a double integral that you were describing? I probably could do that, but since this is for a calc 1 class, the professor might not appreciate that too much...
    All right, yes, sorry that was a double integral I was describing, and turns out I was telling you how to calculate the mass, not the center of mass.

    So of course you're going to calculate the x-coordinate and the y-coordinate separately... I think this thread could help you, I'm currently looking it over.

    Edit: Oy, that link I gave you is for a homogeneous plate. Sorry again.
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    Senior Member AllanCuz's Avatar
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    Quote Originally Posted by musicman314 View Post
    Hi, I'm having incredible trouble getting a reasonable answer to this problem:
    Find the center of mass of a thin plate covering the region bounded below by the parabola y=x^2 and above by the line y=x if the plate's density at the point (x,y) is d(x)=12x. I have an idea of what to do for the x coordinate, but I'm confused as to how to do the y coordinate. Thanks!
    C.O.M = (\bar x, \bar y)

     \bar x = \frac{ \iint x \rho dxdy } { \iint \rho \space dxdy }

     \bar y = \frac{ \iint y \rho dxdy } { \iint \rho dxdy }

    But since your density only depends on X, those double integrals can be reduced to single integrals. This can also be done by parametrization of x and y.
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    MHF Contributor undefined's Avatar
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    Quote Originally Posted by AllanCuz View Post
    C.O.M = (\bar x, \bar y)

     \bar x = \frac{ \iint x \rho dxdy } { \iint \rho \space dxdy }

     \bar y = \frac{ \iint y \rho dxdy } { \iint \rho dxdy }

    But since your density only depends on X, those double integrals can be reduced to single integrals. This can also be done by parametrization of x and y.
    OK, I'm pretty sure this method is valid, avoiding double integrals.

    First we need mass M.

    M = \int_a^b \rho(x) (g(x)-f(x)) dx

    Where in this case g(x) = x, f(x) = x^2, \rho(x)=12x, a = 0 and b = 1.

    Then we write

     \bar x = \frac{1}{M} \int_0^1 x\rho(x)(g(x)-f(x)) dx

     \bar y = \frac{1}{M} \int_0^1 \rho(x)\left(\frac{1}{2}\right)(g^2(x)-f^2(x)) dx

    I got these equations by combining this reference (which seems to have a typo) and this reference, with modifications.
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  7. #7
    Senior Member AllanCuz's Avatar
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    Quote Originally Posted by undefined View Post
    OK, I'm pretty sure this method is valid, avoiding double integrals.

    First we need mass M.

    M = \int_a^b \rho(x) (g(x)-f(x)) dx

    Where in this case g(x) = x, f(x) = x^2, \rho(x)=12x, a = 0 and b = 1.

    Then we write

     \bar x = \frac{1}{M} \int_0^1 x\rho(x)(g(x)-f(x)) dx

     \bar y = \frac{1}{M} \int_0^1 \rho(x)\left(\frac{1}{2}\right)(g^2(x)-f^2(x)) dx

    I got these equations by combining this reference (which seems to have a typo) and this reference, with modifications.
    This is correct. You would get to this point had you evaluated the double integrals for dy only (leaving dx as a single integral).
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  8. #8
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    Quote Originally Posted by AllanCuz View Post
    This is correct. You would get to this point had you evaluated the double integrals for dy only (leaving dx as a single integral).
    Yeah, assigning this problem in a non-multivariable calculus class is like a lot of introductory physics classes -- give people the equations without saying where they came from.
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  9. #9
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    thanks much guys! I have no idea why we're even doing this for a calc 1 class, since my friends started with stuff like this in calc 2. At least when I get outta high school, I'll have a teacher for my math class. (UW-Extension: not my best choice)
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