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Thread: dense ball

  1. #1
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    dense ball

    A solid spherical ball of radius $\displaystyle R$ is created in a way such that the density at the point $\displaystyle (x,y,z)$ is proportional to the point's distance from the origin.
    What is the mass of the ball?
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  2. #2
    fgn
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    I hope you didn't see my first solution. This time I have actually done some thinking.
    The density $\displaystyle \delta(r) = kr $ for some $\displaystyle 0 < k $

    The surface of a sphere with radius $\displaystyle r $ is $\displaystyle r = 4\pi r^2$
    A sphercial shell $\displaystyle dV = 4\pi r^2\,dr $
    $\displaystyle dm = \delta\,dV $
    $\displaystyle dm = 4k\pi r^3\,dr $
    Thus,
    $\displaystyle m = 4k\pi\int_0^R r^3\,dr = \pi kR^4 $
    Last edited by fgn; Dec 8th 2006 at 02:36 AM.
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  3. #3
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    Quote Originally Posted by putnam120 View Post
    A solid spherical ball of radius $\displaystyle R$ is created in a way such that the density at the point $\displaystyle (x,y,z)$ is proportional to the point's distance from the origin.
    What is the mass of the ball?
    The rule is,
    $\displaystyle m=\int \int_S \int \rho(x,y,z) dV$
    In this case,
    $\displaystyle \rho(x,y,z)=\kappa \sqrt{x^2+y^2+z^2}=\kappa \rho$

    Use spherical coordinates,
    $\displaystyle \int_0^{2\pi} \int_0^{\pi} \int_0^R \kappa \rho^3 \sin \phi d\rho\, d\phi\, d\theta$
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