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Finding center of mass of surface of sphere contained within cone.

  1. Dec 7, 2013 #1

    s3a

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    1. The problem statement, all variables and given/known data
    Problem (also attached as TheProblem.jpg):
    Find the center of mass of the surface of the sphere x^2 + y^2 + z^2 = a^2 contained within the cone z tanγ = sqrt(x^2 + y^2), 0 < γ < π/2 a constant, if the density is proportional to the distance from the z axis.

    Hint: R_cm = ∫∫_S δR dS / ∫∫_S δ dS, where δ is the density.

    Solution:
    The solution is attached as TheSolution.jpg.

    2. Relevant equations
    Spherical coordinates:
    x = p sinϕ cosθ, y = p sinϕ sinθ, z = pcosθ
    ∫∫_E∫ f(x,y,z) dV = ∫∫_E∫ f(p sinϕ cosθ, p sinϕ sinθ, p cosθ) p^2 sinϕ dp dθ dϕ

    3. The attempt at a solution
    The first thing I'm stuck on is knowing what δ and dS are. How do I determine what those are (without “cheating” and looking at the solution)?

    Any input would be greatly appreciated!
     

    Attached Files:

  2. jcsd
  3. Dec 7, 2013 #2

    LCKurtz

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    Science Advisor
    Homework Helper
    Gold Member

    That isn't a relevant equation. Surface integrals are double integrals, not triple integrals. The radius of the sphere given as ##a##.

    Surely your text gives ##dS## for spherical coordinates. As for ##\delta##, what is the point on the ##z## axis nearest to ##(x,y,z)##? What is the distance from that point to ##(x,y,z)##? You could answer that in rectangular coordinates and change it to spherical, or write it directly from the figure in spherical coordinates as the solution does.
     
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