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Mass of a sphere where density varies

  1. Feb 26, 2013 #1
    Consider a sphere of radius r where its density at any point is f(d) with d being the distance of the point from the origin and f(d) being an algebraic function and thus integrable. What is the function (ideally expressed as one integral & using constants such as Pi) for the mass of the sphere? PS please also supply the proof.
     
  2. jcsd
  3. Feb 26, 2013 #2

    mathman

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    4π∫f(r)r2dr

    Proof is obvious. I'll leave it to you.
     
  4. Feb 26, 2013 #3
    Is below valid proof?
    Consider sphere made up of infinite number of spherical shells. Assume that a spherical shell is at distance x from centre with infinitesimal thickness dx..
    Volume of spherical shell = Its surface area * thickness = 4Pi(x^2)dx
    Mass of spherical shell = mass of spherical shell * density of spherical shell
    = 4Pi(x^2)dx * f(x) = 4Pi(x^2)f(x)dx
    Mass of sphere = Integral from r to 0 of 4Pi(x^2)f(x)dx
     
    Last edited: Feb 26, 2013
  5. Feb 26, 2013 #4
    My answer IMO is the same as yours, mathman.
    Reason I asked this is that Brian Cox & Jeff Forshaw's book titled 'The quantum universe: everything that can happen does happen, they state (on page 235) that where g(a) represents the fraction of a star's mass lying in a sphere of radius a is:
    4Pi(R^3)p * Integral from a to 0 of (x^2)f(x)dx
    where R is the radius of the star & p is the average density of the star.
    I think it should be:
    Integral from a to 0 of 4Pi(x^2)f(x)dx / (4Pi(R^3)/3)p)
    = Integral from a to 0 of (x^2)f(x)dx / ((R^3)/3)p)

    PS apologies for the use of Pi, ^, brackets & the Integral ( as I don't know how to create the appropriate symbols).
     
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